text stringlengths 55 9.15k | synonym_substitution stringlengths 55 9.22k | butter_fingers stringlengths 55 9.15k | random_deletion stringlengths 47 4.48k | change_char_case stringlengths 55 9.15k | whitespace_perturbation stringlengths 60 7k | underscore_trick stringlengths 55 9.15k |
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}^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$, then $S\subset \overline{S}'$.
ii. If $S'=S$, then $\eta'\in \Xi\left(X\left({}^{\vee}{G}^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$.
We are going to be mostly interested in the case of micro-packets attached to Langlands parameters coming from Arthur parameters. ... | } ^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$, then $ S\subset \overline{S}'$.
ii. If $ S'=S$, then $ \eta'\in \Xi\left(X\left({}^{\vee}{G}^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$.
We are going to be mostly interested in the case of micro - packet attach to Langlands parameters coming from Arthur param... | }^{\Gamla}\right),{}^{\vee}{G}\right)_{S}^{mic}$, thek $S\subset \overliuw{S}'$.
ii. Ih $S'=S$, thsn $\eta'\in \Xi\left(X\left({}^{\vee}{G}^{\Gamma}\right),{}^{\vex}{G}\ritht)_{S}^{muc}$.
We are going to be mustly intvrested ib tht case of micro-pariets atbcched bo Laugoands parametets coming frmm Arthur parakegexs. ... | }^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$, then $S\subset \overline{S}'$. ii. If $S'=S$, \Xi\left(X\left({}^{\vee}{G}^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$. are going be mostly interested attached Langlands parameters coming Arthur parameters. These of micro-packets, are going to be Adams-Barbasch-Vogan... | }^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$, thEn $S\subset \oVerliNe{S}'$.
Ii. IF $S'=s$, theN $\eta'\In \Xi\left(X\left({}^{\vEE}{G}^{\GaMma}\right),{}^{\vee}{G}\right)_{S}^{mic}$.
WE are gOiNG to bE MoStly iNterestED iN THe cAsE oF miCrO-PaCkets AttAched to langlands pAraMeTers coming frOM ARthur paramEteRs. ... | }^{\Gamma}\right),{}^{\vee }{G}\right )_{S} ^{m ic} $, the n $S \subset \overl i ne{S }'$.
ii. If $S'=S$, t hen $ \e t a'\i n \ Xi\le ft(X\le f t( { } ^{\ ve e} {G} ^{ \ Ga mma}\ rig ht),{}^ {\vee}{G}\ rig ht )_{S}^{mic}$ .
We are goi ngto be mostly in terest ed in the c ase of m icro-p a cketsattachedto Langla ... | }^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$, then_$S\subset \overline{S}'$.
ii._If $S'=S$, then $\eta'\in_\Xi\left(X\left({}^{\vee}{G}^{\Gamma}\right),{}^{\vee}{G}\right)_{S}^{mic}$.
We are_going_to be_mostly_interested in the_case of micro-packets_attached to Langlands parameters_coming from Arthur_parameters._... |
to be weak, accordingly, the correlation length, or the characteristic size of the pair is $l_{corr}>>n^{-1/3}_{bp}$. The concentration of bipolarons (i.e. Cooper pairs) $n_{bp}$ in the BCS is very large and for $T=0$ coincides with the concentration of charge carriers in metals.
In the above considered theory EPI is... | to be weak, accordingly, the correlation length, or the characteristic size of the couple is $ l_{corr}>>n^{-1/3}_{bp}$. The assiduity of bipolarons (i.e. Cooper copulate) $ n_{bp}$ in the BCS is very large and for $ T=0 $ coincides with the assiduity of charge carriers in metals.
In the above consider theory EPI is... | to be weak, accordingly, the correlation leutth, or the cgaracterkstic size of the pair is $l_{clre}>>n^{-1/3}_{bp}$. Uke concentration of cipolarond (i.e. Cooper kairs) $n_{bp}$ in the UDS is vcxy ladne anb hor $T=0$ coincides with the woncentration mf ckarge carriers in metals.
In the above considrrfd theory EPI ys... | to be weak, accordingly, the correlation length, characteristic of the is $l_{corr}>>n^{-1/3}_{bp}$. The pairs) in the BCS very large and $T=0$ coincides with the concentration of carriers in metals. In the above considered theory EPI is considered to be therefore the correlation length is $l_{corr}<<n^{-1/3}_{bp}$. At... | to be weak, accordingly, the corRelation leNgth, oR thE chArActeRistIc size of the paiR Is $l_{cOrr}>>n^{-1/3}_{bp}$. The concentration Of bipOlARons (I.E. COoper Pairs) $n_{bP}$ In THE BCs iS vEry LaRGe And foR $T=0$ cOincideS with the coNceNtRation of charGE cArriers in mEtaLs.
In the above ConSidereD tHeoRY EPI iS... | to be weak, accordingly,the correl ation le ngt h, orthecharacteristic size of the pair is $l_{co rr}>> n^ { -1/3 } _{ bp}$. The co n ce n t rat io nofbi p ol arons (i .e. Coo per pairs) $n _{ bp}$ in theB CS is very l arg e and for $T =0$ coinc id esw ith t heconce ntrati o n of c harge car ri e rs inm etals.... | to_be weak,_accordingly, the correlation length,_or the_characteristic_size of_the_pair is $l_{corr}>>n^{-1/3}_{bp}$._The concentration of_bipolarons (i.e. Cooper pairs)_$n_{bp}$ in the_BCS_is very large and for $T=0$ coincides with the concentration of charge carriers in_metals.
In_the above_considered_theory_EPI is... |
B,C}\sum_{j_3=A,B,C}$.
In order to speed up the calculations the parallelization of the code is used. Additionally, due to the presence of the Boltzman factor in Eq. (\[eq:M\_cf\_Center\]), only the $k_{max} < W$ lowest energy levels and eigenfunctions need to be calculated in order to obtain a very good approximation... | B, C}\sum_{j_3 = A, B, C}$.
In order to speed up the calculations the parallelization of the code is used. Additionally, due to the bearing of the Boltzman agent in Eq. (\[eq: M\_cf\_Center\ ]), only the $ k_{max } < W$ lowest department of energy levels and eigenfunctions want to be calculated in order to receive... | B,C}\skm_{j_3=A,B,C}$.
In order to speed mp the calculations the paralmelizatiun of the code is used. Additmonaoly, dye to the presence of ghe Boltzlan factir ii Eq. (\[eq:M\_cf\_Center\]), only the $k_{max} < A$ locewt energy leveks and eiganfunctions nead tl be calculated in order to obtain w very bold approximatijn... | B,C}\sum_{j_3=A,B,C}$. In order to speed up the parallelization the code used. Additionally, due Boltzman in Eq. (\[eq:M\_cf\_Center\]), the $k_{max} < lowest energy levels and eigenfunctions need be calculated in order to obtain a very good approximation of $M(B,T)$ curves. $W$ is equal to $25$, $25^2$, $25^3$, $25^4$... | B,C}\sum_{j_3=A,B,C}$.
In order to speed up The calculaTions The ParAlLeliZatiOn of the code is uSEd. AdDitionally, due to the presEnce oF tHE BolTZmAn facTor in Eq. (\[EQ:M\_CF\_cenTeR\]), oNly ThE $K_{mAx} < W$ loWesT energy Levels and eIgeNfUnctions need TO bE calculateD in Order to obtaiN a vEry gooD aPprOXimatIon... | B,C}\sum_{j_3=A,B,C}$.
In order tospeed up th ecalc ulat ions the paral l eliz ation of the code is u sed.Ad d itio n al ly, d ue to t h ep r ese nc eofth e B oltzm anfactorin Eq. (\[ eq: M\ _cf\_Center\ ] ), only the$k_ {max} < W$ l owe st ene rg y l e velsand eige nfunct i ons ne ed to beca l culate d in ord ... | B,C}\sum_{j_3=A,B,C}$.
In order_to speed_up the calculations the_parallelization of_the_code is_used._Additionally, due to_the presence of_the Boltzman factor in_Eq. (\[eq:M\_cf\_Center\]), only the_$k_{max}_< W$ lowest energy levels and eigenfunctions need to be calculated in order to_obtain_a very_good_approximation... |
right)^2 -\frac{\partial^2 \zeta}{\partial x^2}\frac{\partial^2 \zeta}{\partial y^2},$$ where the linear operator $\mathcal{D}^4$ is defined by $$\mathcal{D}^4 \equiv S_{11} \frac{\partial^4}{\partial y^4} + (2 S_{12} + S_{66}) \frac{\partial^4}{\partial x^2 \partial y^2} + S_{22} \frac{\partial^4}{\partial x^4}
-2 S_... | right)^2 -\frac{\partial^2 \zeta}{\partial x^2}\frac{\partial^2 \zeta}{\partial y^2},$$ where the linear operator $ \mathcal{D}^4 $ is defined by $ $ \mathcal{D}^4 \equiv S_{11 } \frac{\partial^4}{\partial y^4 } + (2 S_{12 } + S_{66 }) \frac{\partial^4}{\partial x^2 \partial y^2 } + S_{22 } \frac{\partial^4}{\partial... | rigjt)^2 -\frac{\partial^2 \zeta}{\partixl x^2}\frac{\partial^2 \zeta}{\pactial y^2},$$ where tfe linear operator $\mathcal{D}^4$ ms dwfinee by $$\mathcal{D}^4 \equiv S_{11} \wrac{\partiwl^4}{\partiao y^4} + (2 S_{12} + S_{66}) \frac{\'zrtial^4}{\partial w^2 \parciel y^2} + S_{22} \frac{\pattial^4}{\partial x^4}
-2 S_... | right)^2 -\frac{\partial^2 \zeta}{\partial x^2}\frac{\partial^2 \zeta}{\partial y^2},$$ where operator is defined $$\mathcal{D}^4 \equiv S_{11} + \frac{\partial^4}{\partial x^2 \partial + S_{22} \frac{\partial^4}{\partial -2 S_{16}\frac{\partial^4}{\partial x \partial y^3} - S_{26} \frac{\partial^4}{\partial x^3 \parti... | right)^2 -\frac{\partial^2 \zeta}{\partiAl x^2}\frac{\parTial^2 \zEta}{\ParTiAl y^2},$$ wHere The linear operaTOr $\maThcal{D}^4$ is defined by $$\mathcAl{D}^4 \eqUiV s_{11} \fraC{\PaRtial^4}{\Partial Y^4} + (2 s_{12} + S_{66}) \FRAc{\pArTiAl^4}{\pArTIaL x^2 \parTiaL y^2} + S_{22} \frac{\Partial^4}{\parTiaL x^4}
-2 s_... | right)^2 -\frac{\partial^2 \zeta}{\p artia l x ^2} \f rac{ \par tial^2 \zeta}{ \ part ial y^2},$$ where thelinea ro pera t or $\ma thcal{D } ^4 $ isde fi ned b y $ $\mat hca l{D}^4\equiv S_{ 11} \ frac{\partia l ^4 }{\partial y^ 4} + (2 S_{1 2}+ S_{6 6} ) \ f rac{\ par tial^ 4}{\pa r tial x ^2 \parti al y^2} + S... | right)^2 -\frac{\partial^2_\zeta}{\partial x^2}\frac{\partial^2_\zeta}{\partial y^2},$$ where the_linear operator_$\mathcal{D}^4$_is defined_by_$$\mathcal{D}^4 \equiv S_{11}_\frac{\partial^4}{\partial y^4} +_(2 S_{12} + S_{66})_\frac{\partial^4}{\partial x^2 \partial_y^2}_+ S_{22} \frac{\partial^4}{\partial x^4}
-2 S_... |
L65
---
abstract: 'We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M){\rightarrow}J^r(D,N)$ is generically (for $f:M{\rightarrow}N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a... | L65
---
abstract:' We prove a generalization of Thom ’s transversality theorem. It gives conditions under which the k function $ f_*|_Y: Y\subseteq J^r(D, M){\rightarrow}J^r(D, N)$ is generically (for $ f: M{\rightarrow}N$) transverse to a submanifold $ Z\subseteq J^r(D, N)$. We apply this to study transversality ... | L65
---
wbstract: 'We prove a genevalization of Thom’s traisversamity theurem. It gives conditions undxr wyich uke jet map $f_*|_Y:Y\subsetdq J^r(D,M){\rihhtarrow}H^r(D,N)$ us genericemly (for $f:M{\riggbarroc}N$) transverse to a submanixold $Z\subseteq J^f(D,U)$. We apply this to study transversalyty prolegties of a... | L65 --- abstract: 'We prove a generalization transversality It gives under which the generically $f:M{\rightarrow}N$) transverse to submanifold $Z\subseteq J^r(D,N)$. apply this to study transversality properties a restriction of a fixed map $g:M{\rightarrow}P$ to the preimage $(j^sf)^{-1}(A)$ of a $A\subseteq J^s(M,N)... | L65
---
abstract: 'We prove a generaliZation of ThOm’s trAnsVerSaLity TheoRem. It gives condITionS under which the jet map $f_*|_Y:y\subsEtEQ J^r(D,m){\RiGhtarRow}J^r(D,N)$ IS gENEriCaLlY (foR $f:m{\RiGhtarRow}n$) transvErse to a subManIfOld $Z\subseteq j^R(D,n)$. We apply thIs tO study transvErsAlity pRoPerTIes of A... | L65
---
abstract: 'We pr ove a gene raliz ati onof Tho m’stransversality theo rem. It gives conditio ns un de r whi c hthe j et map$ f_ * | _Y: Y\ su bse te q J ^r(D, M){ \righta rrow}J^r(D ,N) $is generical l y(for $f:M{ \ri ghtarrow}N$) tr ansver se to a sub man ifold $Z\su b seteqJ^r(D,N)$ .W e appl y this ... | L65
_---
abstract: 'We_prove a generalization of_Thom’s transversality_theorem._It gives_conditions_under which the_jet map $f_*|_Y:Y\subseteq_J^r(D,M){\rightarrow}J^r(D,N)$ is generically (for_$f:M{\rightarrow}N$) transverse to_a_submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a... |
.61 &0.02 &59.66\
G19.01-0.03 & 18 25 45.153 & 0.023 & -12 22 57.650 & 0.023 & 0.20 &0.02 &59.82\
G19.01-0.03 & 18 25 45.106 & 0.011 & -12 22 29.798 & 0.011 & 0.46 &0.02 &59.82\
G19.01-0.03 & 18 25 45.078 & 0.028 & -12 22 30.322 & 0.027 & 0.18 &0.02 &59.82\
G19.01-0.03 & 18 25 44.812 & 0.015 & -12 23 01.354 & 0.014 & 0... | .61 & 0.02 & 59.66\
G19.01 - 0.03 & 18 25 45.153 & 0.023 & -12 22 57.650 & 0.023 & 0.20 & 0.02 & 59.82\
G19.01 - 0.03 & 18 25 45.106 & 0.011 & -12 22 29.798 & 0.011 & 0.46 & 0.02 & 59.82\
G19.01 - 0.03 & 18 25 45.078 & 0.028 & -12 22 30.322 & 0.027 & 0.18 & 0.02 & 59.82\
G19.01 - 0.03 & 18 25 44.812 & 0.015 & -... | .61 &0.02 &59.66\
G19.01-0.03 & 18 25 45.153 & 0.023 & -12 22 57.650 & 0.023 & 0.20 &0.02 &59.82\
G19.01-0.03 & 18 25 45.106 & 0.011 & -12 22 29.798 & 0.011 & 0.46 &0.02 &59.82\
G19.01-0.03 & 18 25 45.078 & 0.028 & -12 22 30.322 & 0.027 & 0.18 &0.02 &59.82\
G19.01-0.03 & 18 25 44.812 & 0.015 & -12 23 01.354 & 0.014 & 0... | .61 &0.02 &59.66\ G19.01-0.03 & 18 25 0.023 -12 22 & 0.023 & 18 45.106 & 0.011 -12 22 29.798 0.011 & 0.46 &0.02 &59.82\ G19.01-0.03 18 25 45.078 & 0.028 & -12 22 30.322 & 0.027 & 0.18 &59.82\ G19.01-0.03 & 18 25 44.812 & 0.015 & -12 23 01.354 & & &0.02 G19.01-0.03 18 25 44.672 & 0.025 & -12 22 29.417 & 0.024 & 0.20 &0.... | .61 &0.02 &59.66\
G19.01-0.03 & 18 25 45.153 & 0.023 & -12 22 57.650 & 0.023 & 0.20 &0.02 &59.82\
G19.01-0.03 & 18 25 45.106 & 0.011 & -12 22 29.798 & 0.011 & 0.46 &0.02 &59.82\
G19.01-0.03 & 18 25 45.078 & 0.028 & -12 22 30.322 & 0.027 & 0.18 &0.02 &59.82\
G19.01-0.03 & 18 25 44.812 & 0.015 & -12 23 01.354 & 0.014 & 0... | .61 &0.02 &59.66\
G19.01-0 .03 & 18 2 5 45. 153 &0. 023& -1 2 22 57.650 &0 .023 & 0.20 &0.02 &59.82\G19.0 1- 0 .03& 1 8 2545.106& 0 . 0 11&-1 2 2 22 9. 798 & 0. 011 & 0 .46 &0.02&59 .8 2\
G19.01-0. 0 3& 18 25 45 .07 8 & 0.028 &-12 22 30 .3 22& 0.02 7 & 0.18 &0.02 &59.82 \
G19.01- 0. 0 3 & 18 25 44.8 1 2 & 0.... | .61 &0.02_&59.66\
G19.01-0.03 &_18 25 45.153 &_0.023 &_-12_22 57.650_&_0.023 & 0.20_&0.02 &59.82\
G19.01-0.03 &_18 25 45.106 &_0.011 & -12_22_29.798 & 0.011 & 0.46 &0.02 &59.82\
G19.01-0.03 & 18 25 45.078 & 0.028 &_-12_22 30.322_&_0.027_& 0.18 &0.02 &59.82\
G19.01-0.03 &_18 25 44.812 & 0.015_& -12_23 01.354 & 0.014 & 0... |
mathcal{E}^4k_{\bot}^2}{2\omega_2^2\mathcal{N}}|\mathbf{k}|^2\Big[&+(4k_{\parallel}^2+k_{\bot}^2)\omega_2^4+4\mathcal{E}k_{\parallel}k_{\bot}^2\omega_2^3+\big(4k_{\parallel}^4+2k_{\bot}^2k_{\parallel}^2+(\mathcal{E}^2-2)k_{\bot}^4\big)\omega_2^2
\notag \\
&+4\mathcal{E}k_{\parallel}k_{\bot}^2|\mathbf{k}|^2\omega_2+k_{\... | mathcal{E}^4k_{\bot}^2}{2\omega_2 ^ 2\mathcal{N}}|\mathbf{k}|^2\Big[&+(4k_{\parallel}^2+k_{\bot}^2)\omega_2 ^ 4 + 4\mathcal{E}k_{\parallel}k_{\bot}^2\omega_2 ^ 3+\big(4k_{\parallel}^4 + 2k_{\bot}^2k_{\parallel}^2+(\mathcal{E}^2 - 2)k_{\bot}^4\big)\omega_2 ^ 2
\notag \\
& +4\mathcal{E}k_{\parallel}k_{\bot}^2|\mathbf... | matjcal{E}^4k_{\bot}^2}{2\omega_2^2\mathcal{N}}|\mauhbf{k}|^2\Big[&+(4k_{\parallel}^2+k_{\bot}^2)\omeja_2^4+4\mathczl{E}k_{\paraulel}k_{\bot}^2\omega_2^3+\big(4k_{\parallel}^4+2k_{\bov}^2k_{\paealleo}^2+(\mathcal{E}^2-2)k_{\bot}^4\big)\omega_2^2
\nutag \\
&+4\mathbal{E}k_{\paraolel}j_{\vot}^2|\mathbf{k}|^2\omega_2+k_{\... | mathcal{E}^4k_{\bot}^2}{2\omega_2^2\mathcal{N}}|\mathbf{k}|^2\Big[&+(4k_{\parallel}^2+k_{\bot}^2)\omega_2^4+4\mathcal{E}k_{\parallel}k_{\bot}^2\omega_2^3+\big(4k_{\parallel}^4+2k_{\bot}^2k_{\parallel}^2+(\mathcal{E}^2-2)k_{\bot}^4\big)\omega_2^2 \notag \\ &+4\mathcal{E}k_{\parallel}k_{\bot}^2|\mathbf{k}|^2\omega_2+k_{\... | mathcal{E}^4k_{\bot}^2}{2\omega_2^2\mathcal{N}}|\Mathbf{k}|^2\Big[&+(4K_{\paraLleL}^2+k_{\bOt}^2)\OmegA_2^4+4\matHcal{E}k_{\parallel}K_{\Bot}^2\oMega_2^3+\big(4k_{\parallel}^4+2k_{\bot}^2k_{\pArallEl}^2+(\MAthcAL{E}^2-2)K_{\bot}^4\bIg)\omega_2^2
\NOtAG \\
&+4\MatHcAl{e}k_{\pArALlEl}k_{\boT}^2|\maThbf{k}|^2\omEga_2+k_{\... | mathcal{E}^4k_{\bot}^2}{2\ omega_2^2\ mathc al{ N}} |\ math bf{k }|^2\Big[&+(4k _ {\pa rallel}^2+k_{\bot}^2)\ omega _2 ^ 4+4\ m at hcal{ E}k_{\p a ra l l el} k_ {\ bot }^ 2 \o mega_ 2^3 +\big(4 k_{\parall el} ^4 +2k_{\bot}^2 k _{ \parallel} ^2+ (\mathcal{E} ^2- 2)k_{\ bo t}^ 4 \big) \om ega_2 ^2
\no t ag \\&+4\mathc al... | mathcal{E}^4k_{\bot}^2}{2\omega_2^2\mathcal{N}}|\mathbf{k}|^2\Big[&+(4k_{\parallel}^2+k_{\bot}^2)\omega_2^4+4\mathcal{E}k_{\parallel}k_{\bot}^2\omega_2^3+\big(4k_{\parallel}^4+2k_{\bot}^2k_{\parallel}^2+(\mathcal{E}^2-2)k_{\bot}^4\big)\omega_2^2
\notag \\
&+4\mathcal{E}k_{\parallel}k_{\bot}^2|\mathbf{k}|^2\omega_2+k_{\... |
hardware complexity.
Massive MIMO with low-resolution ADCs has attracted much attention over the past few years. Great efforts have been made to understand the effects of low-resolution ADCs on the performance of MIMO and massive MIMO systems. Specifically, by assuming full knowledge of channel state information (CSI... | hardware complexity.
Massive MIMO with low - resolution ADCs has attracted much attention over the past few days. big efforts have been made to understand the impression of low - resolution ADCs on the operation of MIMO and massive MIMO systems. Specifically, by assuming full cognition of distribution channel state ... | hagdware complexity.
Massive MIMO with low-rgsilutioi ADCs gas attrxcted much attention over thx pawt feq years. Great efforts fave been made to undtrstand the effecva of low-resolhbion CDRs on the perfotmance of MIKO and massive MKML systems. Specifically, by assuming sull knpwpedge of channgl stsee ihformation (CSI... | hardware complexity. Massive MIMO with low-resolution ADCs much over the few years. Great understand effects of low-resolution on the performance MIMO and massive MIMO systems. Specifically, assuming full knowledge of channel state information (CSI), the capacity at both finite infinite signal-to-noise ratio (SNR) was ... | hardware complexity.
Massive mIMO with loW-resoLutIon aDcs haS attRacted much atteNTion Over the past few years. GreAt effOrTS havE BeEn madE to undeRStAND thE eFfEctS oF LoW-resoLutIon ADCs On the perfoRmaNcE of MIMO and maSSiVe MIMO systEms. specifically, By aSsuminG fUll KNowleDge Of chaNnel stATe infoRmation (CSi... | hardware complexity.
Mas sive MIMOwithlow -re so luti on A DCs has attrac t ed m uch attention over the past f e w ye a rs . Gre at effo r ts h ave b ee n m ad e t o und ers tand th e effectsoflo w-resolution AD Cs on theper formance ofMIM O andma ssi v e MIM O s ystem s. Spe c ifical ly, by as su m ing fu l l kn... | hardware_complexity.
Massive MIMO_with low-resolution ADCs has_attracted much_attention_over the_past_few years. Great_efforts have been_made to understand the_effects of low-resolution_ADCs_on the performance of MIMO and massive MIMO systems. Specifically, by assuming full knowledge_of_channel state_information_(CSI... |
\in \RR^2$ and $\tb \in \RR_H^3$ as column vectors, then it follows from that $$\label{x-t-x}
x = \tfrac13 H (t_1 - t_3, t_2-t_3)^{{\mathsf {tr}}}= \tfrac13 H (2 t_1 + t_2, t_1+2t_2)^{{\mathsf {tr}}}$$ upon using the fact that $t_1 + t_2 + t_3 =0$. Computing the Jacobian of the change of variables shows that $d x =... | \in \RR^2 $ and $ \tb \in \RR_H^3 $ as column vectors, then it follows from that $ $ \label{x - t - x }
x = \tfrac13 H (t_1 - t_3, t_2 - t_3)^{{\mathsf { tr}}}= \tfrac13 H (2 t_1 + t_2, t_1 + 2t_2)^{{\mathsf { tr}}}$$ upon using the fact that $ t_1 + t_2 + t_3 = 0$. calculate the Jacobian of the variety of vari... | \in \RR^2$ and $\tb \in \RR_H^3$ as colmmn vectors, then it folnows fdom that $$\label{x-t-x}
x = \tfrac13 H (t_1 - t_3, t_2-t_3)^{{\larhsf {ug}}}= \tfrac13 H (2 t_1 + t_2, t_1+2t_2)^{{\mxthsf {tr}}}$$ lpon usint tht fact that $t_1 + t_2 + t_3 =0$. Compmcing fme Jaeouian of the chakge of varidbles shows thdt $d x =... | \in \RR^2$ and $\tb \in \RR_H^3$ as then follows from $$\label{x-t-x} x = t_2-t_3)^{{\mathsf \tfrac13 H (2 + t_2, t_1+2t_2)^{{\mathsf upon using the fact that $t_1 t_2 + t_3 =0$. Computing the Jacobian of the change of variables shows $d x = \frac{2 \sqrt{3}} {3} d t_1 dt_2$. A function $f$ is [*periodic*]{} respect th... | \in \RR^2$ and $\tb \in \RR_H^3$ as column vecTors, then it FolloWs fRom ThAt $$\laBel{x-T-x}
x = \tfrac13 H (t_1 - t_3, t_2-t_3)^{{\mAThsf {Tr}}}= \tfrac13 H (2 t_1 + t_2, t_1+2t_2)^{{\mathsf {tr}}}$$ upOn usiNg THe faCT tHat $t_1 + t_2 + T_3 =0$. ComputINg THE JaCoBiAn oF tHE cHange Of vAriableS shows that $D x =... | \in \RR^2$ and $\tb \in \ RR_H^3$ as colu mnvec to rs,then it follows fr o m th at $$\label{x-t-x}
x = \tfr ac 1 3 H (t _1 -t_3, t_ 2 -t _ 3 )^{ {\ ma ths f{ tr }}}= \t frac13 H (2 t_1+ t _2 , t_1+2t_2)^ { {\ mathsf {tr }}} $$ upon usin g t he fac ttha t $t_1 +t_2 + t_3 = 0 $. Com puting th eJ acobia n of the ... | \in_\RR^2$ and_$\tb \in \RR_H^3$ as_column vectors,_then_it follows_from_that $$\label{x-t-x}
x =_\tfrac13 H_(t_1 - t_3, t_2-t_3)^{{\mathsf_{tr}}}= \tfrac13__H (2 t_1 + t_2, t_1+2t_2)^{{\mathsf {tr}}}$$ upon using the fact that $t_1 +_t_2_+ t_3_=0$._Computing_the Jacobian of the change_of variables shows that $d_x =... |
the reverse deterministic flow (i.e. ${\partial}_t \varphi^u = - ({\partial}^2_{xx} \varphi^u + g(\varphi^u)) )$ during the time interval $[t_1,t_2]$ then, similarly to Proposition \[Lyapunov\], we have $$\frac{1}{2}\int_{t_1}^{t_2} \int_0^1 |{\partial}_t \varphi^u - ({\partial}^2_{xx} \varphi^u + g(\varphi^u))|^{2} =... | the reverse deterministic flow (i.e. $ { \partial}_t \varphi^u = - ({ \partial}^2_{xx } \varphi^u + g(\varphi^u) )) $ during the time interval $ [ t_1,t_2]$ then, similarly to Proposition \[Lyapunov\ ], we get $ $ \frac{1}{2}\int_{t_1}^{t_2 } \int_0 ^ 1 |{\partial}_t \varphi^u - ({ \partial}^2_{xx } \varphi^u + g(\varp... | thf reverse deterministic nlow (i.e. ${\partial}_t \varphi^n = - ({\parfial}^2_{xx} \vxrphi^u + g(\varphi^u)) )$ during the tume ibterval $[t_1,t_2]$ then, similafly to Prlpositiob \[Lyepunov\], we have $$\fczc{1}{2}\int_{t_1}^{t_2} \int_0^1 |{\pzvtial}_c \tarphi^u - ({\partiak}^2_{xx} \varphi^g + g(\varphi^u))|^{2} =... | the reverse deterministic flow (i.e. ${\partial}_t \varphi^u ({\partial}^2_{xx} + g(\varphi^u)) during the time Proposition we have $$\frac{1}{2}\int_{t_1}^{t_2} |{\partial}_t \varphi^u - \varphi^u + g(\varphi^u))|^{2} = 2 \int_{t_1}^{t_2} S(\varphi^u(t,\cdot))}{dt} = 2 (S(\varphi^u(t_2)) - S(\varphi^u(t_1)))$$ from wh... | the reverse deterministic flOw (i.e. ${\partiaL}_t \varPhi^U = - ({\paRtIal}^2_{xX} \varPhi^u + g(\varphi^u)) )$ duRIng tHe time interval $[t_1,t_2]$ then, siMilarLy TO ProPOsItion \[lyapunoV\], We HAVe $$\fRaC{1}{2}\iNt_{t_1}^{T_2} \iNT_0^1 |{\pArtiaL}_t \vArphi^u - ({\pArtial}^2_{xx} \vaRphI^u + G(\varphi^u))|^{2} =... | the reverse deterministic flow (i.e . ${\ par tia l} _t \ varp hi^u = - ({\pa r tial }^2_{xx} \varphi^u + g (\var ph i ^u)) )$ duri ng thet im e int er va l $ [t _ 1, t_2]$ th en, sim ilarly toPro po sition \[Lya p un ov\], we h ave $$\frac{1}{ 2}\ int_{t _1 }^{ t _2} \ int _0^1|{\par t ial}_t \varphi^ u- ({\pa ... | the_reverse deterministic_flow (i.e. ${\partial}_t \varphi^u_= -_({\partial}^2_{xx}_\varphi^u +_g(\varphi^u))_)$ during the_time interval $[t_1,t_2]$_then, similarly to Proposition_\[Lyapunov\], we have_$$\frac{1}{2}\int_{t_1}^{t_2}_\int_0^1 |{\partial}_t \varphi^u - ({\partial}^2_{xx} \varphi^u + g(\varphi^u))|^{2} =... |
{g}^*})$ is compact it is proper, by Lemma \[vanishing\_proposition\] (2), we have that $H^2_{\mathrm{diff}}(G\ltimes \mathbb{S}({\mathfrak{g}^*});E)=0$. The corollary follows from Theorem \[Theorem\_THREE\].
Let $P$ be the Poisson homotopy bundle of $S$, and let $G$ be its structure group. By hypothesis, $P$ is smoot... | { g}^*})$ is compact it is proper, by Lemma \[vanishing\_proposition\ ] (2), we have that $ H^2_{\mathrm{diff}}(G\ltimes \mathbb{S}({\mathfrak{g}^*});E)=0$. The corollary follows from Theorem \[Theorem\_THREE\ ].
Let $ P$ be the Poisson homotopy package of $ S$, and lease $ G$ be its structure group. By hypothesis, ... | {g}^*})$ id compact it is proper, bn Lemma \[vanishiny\_proposmtion\] (2), se have ghat $H^2_{\mathrm{diff}}(G\ltimes \mathub{S}({\mqthfrqk{g}^*});E)=0$. The corollary foluows from Theorem \[Theieem\_THREE\].
Lev $P$ be tmz Poiadon konotopy bundle pf $S$, and lat $G$ be its stsuztbre group. By hypothesis, $P$ is smoot... | {g}^*})$ is compact it is proper, by (2), have that \mathbb{S}({\mathfrak{g}^*});E)=0$. The corollary $P$ the Poisson homotopy of $S$, and $G$ be its structure group. By $P$ is smooth, $1$-connected and has vanishing second de Rham cohomology. Let $\mathcal{G}:=P\times_{G}P$ the gauge groupoid of $P$. The $s$-fibers of... | {g}^*})$ is compact it is proper, by LemMa \[vanishinG\_propOsiTioN\] (2), wE havE thaT $H^2_{\mathrm{diff}}(G\lTImes \Mathbb{S}({\mathfrak{g}^*});E)=0$. The coRollaRy FOlloWS fRom ThEorem \[ThEOrEM\_tHReE\].
leT $P$ bE tHE POissoN hoMotopy bUndle of $S$, anD leT $G$ Be its structuRE gRoup. By hypoTheSis, $P$ is smoot... | {g}^*})$ is compact it isproper, by Lemm a \ [va ni shin g\_p roposition\] ( 2 ), w e have that $H^2_{\mat hrm{d if f }}(G \ lt imes\mathbb { S} ( { \ma th fr ak{ g} ^ *} );E)= 0$. The co rollary fo llo ws from Theore m \ [Theorem\_ THR EE\].
Let $ P$be the P ois s on ho mot opy b undleo f $S$, and let$G $ be i... | {g}^*})$ is_compact it_is proper, by Lemma_\[vanishing\_proposition\] (2),_we_have that_$H^2_{\mathrm{diff}}(G\ltimes_\mathbb{S}({\mathfrak{g}^*});E)=0$. The corollary_follows from Theorem_\[Theorem\_THREE\].
Let $P$ be the_Poisson homotopy bundle_of_$S$, and let $G$ be its structure group. By hypothesis, $P$ is smoot... |
finite words are closed under sum. The cost is $O(n_1\cdot n_2)$ for ${{\sf Last}}$- and ${\mathsf{Sum}}$-automata, and $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$ for ${\mathsf{Sup}}$-automata.
It is easy to see that the synchronized product of two ${{\sf Last}}$-automata (resp. ${\mathsf{Sum}}$-automata) defines the sum ... | finite words are closed under sum. The cost is $ O(n_1\cdot n_2)$ for $ { { \sf Last}}$- and $ { \mathsf{Sum}}$-automata, and $ O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$ for $ { \mathsf{Sup}}$-automata.
It is comfortable to examine that the synchronized product of two $ { { \sf Last}}$-automata (resp. $ { \mathsf{Sum}}$-... | fijite words are closed unaer sum. The cosj us $O(n_1\cvot n_2)$ fkr ${{\sf Lart}}$- and ${\mathsf{Sum}}$-automata, and $O(b_1\cdot m_1 \cdot n_2 \cdot m_2)$ for ${\mxthsf{Sup}}$-altomata.
It is tasy to see that vge syncmxonizsf prmvuct of two ${{\sf Kast}}$-automada (resp. ${\mathsf{Vuo}}$-abtomata) defines the sum ... | finite words are closed under sum. The $O(n_1\cdot for ${{\sf and ${\mathsf{Sum}}$-automata, and m_2)$ ${\mathsf{Sup}}$-automata. It is to see that synchronized product of two ${{\sf Last}}$-automata ${\mathsf{Sum}}$-automata) defines the sum of their languages if the weight of a joint is defined as the sum of the weig... | finite words are closed under Sum. The cost Is $O(n_1\cDot N_2)$ foR ${{\sF LasT}}$- and ${\Mathsf{Sum}}$-automATa, anD $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$ for ${\mAthsf{suP}}$-AutoMAtA.
It is Easy to sEE tHAT thE sYnChrOnIZeD prodUct Of two ${{\sf last}}$-automaTa (rEsP. ${\mathsf{Sum}}$-auTOmAta) defines The Sum ... | finite words are closed u nder sum.The c ost is $ O(n_ 1\cd ot n_2)$ for $ { {\sf Last}}$- and ${\maths f{Sum }} $ -aut o ma ta, a nd $O(n _ 1\ c d otm_ 1\cd ot n_ 2 \cd otm_2)$ f or ${\math sf{ Su p}}$-automat a .
It is eas y t o see that t hesynchr on ize d prod uct of t wo ${{ \ sf Las t}}$-auto ma t a (res p ... | finite_words are_closed under sum. The_cost is_$O(n_1\cdot_n_2)$ for_${{\sf_Last}}$- and ${\mathsf{Sum}}$-automata,_and $O(n_1\cdot m_1_\cdot n_2 \cdot m_2)$_for ${\mathsf{Sup}}$-automata.
It is_easy_to see that the synchronized product of two ${{\sf Last}}$-automata (resp. ${\mathsf{Sum}}$-automata) defines the_sum_... |
uler}$$ where $y'(t)\equiv \frac{d}{dt}y(t)$. The function (\[defi0\]) describes a real growth process.
As far as there is no universal pricing model, the individual evaluations by the market participants differ and the price deviates from the fundamental average. These different evaluations which are changing in time... | uler}$$ where $ y'(t)\equiv \frac{d}{dt}y(t)$. The function (\[defi0\ ]) describes a real growth procedure.
equally far as there is no universal pricing exemplar, the individual evaluation by the market participants disagree and the price deviates from the fundamental average. These unlike evaluation which are chang... | uleg}$$ where $y'(t)\equiv \frac{d}{dt}y(u)$. The function (\[deyu0\]) desccibes a real gruwth process.
As far as there ms ni unicersal pricing model, tfe indivifual evaouatmons by the markxf partignpanta difyec and the price deviates xrom the fundakevtcl average. These different evaluatiogs whicn wre changing ig tikq... | uler}$$ where $y'(t)\equiv \frac{d}{dt}y(t)$. The function (\[defi0\]) real process. As as there is individual by the market differ and the deviates from the fundamental average. These evaluations which are changing in time lead to some kind of spontaneous oscillations. far as each market has another scale it is useful... | uler}$$ where $y'(t)\equiv \frac{d}{dt}y(t)$. the functioN (\[defi0\]) DesCriBeS a reAl grOwth process.
As fAR as tHere is no universal priciNg modEl, THe inDIvIdual EvaluatIOnS BY thE mArKet PaRTiCipanTs dIffer anD the price dEviAtEs from the funDAmEntal averaGe. THese differenT evAluatiOnS whICh are ChaNging In time... | uler}$$ where $y'(t)\equiv \frac{d}{ dt}y( t)$ . T he fun ctio n (\[defi0\])d escr ibes a real growth pro cess.
A s fa r a s the re is n o u n i ver sa lpri ci n gmodel , t he indi vidual eva lua ti ons by the m a rk et partici pan ts differ an d t he pri ce de v iates fr om th e fund a mental average. T h ese... | uler}$$ where_$y'(t)\equiv \frac{d}{dt}y(t)$._The function (\[defi0\]) describes_a real_growth_process.
As far_as_there is no_universal pricing model,_the individual evaluations by_the market participants_differ_and the price deviates from the fundamental average. These different evaluations which are changing_in_time... |
, i.e. subsets that are the union of some minimizing orbits. More precisely, we would like to know if we can say something about the regularity of such subsets (we will be more precise very soon. It’s a kind of differentiability) and particularly if there is a link between the dynamic of the flow restricted to such a s... | , i.e. subsets that are the union of some minimizing orbits. More precisely, we would wish to acknowledge if we can say something about the regularity of such subsets (we will be more accurate very soon. It ’s a kind of differentiability) and particularly if there be a link between the dynamic of the stream restricted ... | , i.e. subsets that are the unlon of some minimizing mrbits. More prdcisely, we would like to knox if we cqn say something about the reguparity od surh subsets (we will be movz predlse vzrb soon. It’s a kikd of diffesentiability) atd pcrticularly if there is a link betweqn the cyjamic of the fjow geftridneb to such a s... | , i.e. subsets that are the union minimizing More precisely, would like to something the regularity of subsets (we will more precise very soon. It’s a of differentiability) and particularly if there is a link between the dynamic of flow restricted to such a set and the regularity of the set. The result this concerns ti... | , i.e. subsets that are the union oF some minimIzing OrbIts. moRe prEcisEly, we would like TO knoW if we can say something abOut thE rEGulaRItY of suCh subseTS (wE WIll Be MoRe pReCIsE very SooN. It’s a kiNd of differEntIaBility) and parTIcUlarly if thEre Is a link betweEn tHe dynaMiC of THe floW reStricTed to sUCh a s... | , i.e. subsets that are th e union of some mi nim iz ingorbi ts. More preci s ely, we would like to know if w ec an s a ysomet hing ab o ut t here gu lar it y o f suc h s ubsets(we will b e m or e precise ve r ysoon. It’s akind of diff ere ntiabi li ty) and p art icula rly if thereis a link b e tweent he dyna m ... | , i.e._subsets that_are the union of_some minimizing_orbits._More precisely,_we_would like to_know if we_can say something about_the regularity of_such_subsets (we will be more precise very soon. It’s a kind of differentiability) and_particularly_if there_is_a_link between the dynamic of_the flow restricted to such_a s... |
each $h \in \mathcal{C}(\mathopen{[}0,R\mathclose{]})$, $Q_\varphi(h) = Q_\varphi(h(\odot)) \in \mathbb{R}$ is defined as the unique solution of the equation $$\int_0^R \varphi^{-1}\left( -h(r) + Q_\varphi(h)\right)\,\mathrm{d}r = 0.$$ It is worth noticing that, compared with the Neumann case, this formulation is slig... | each $ h \in \mathcal{C}(\mathopen{[}0,R\mathclose{]})$, $ Q_\varphi(h) = Q_\varphi(h(\odot) ) \in \mathbb{R}$ is defined as the unique solution of the equation $ $ \int_0^R \varphi^{-1}\left (-h(r) + Q_\varphi(h)\right)\,\mathrm{d}r = 0.$$ It is deserving comment that, compared with the Neumann case, this conceptualiz... | eafh $h \in \mathcal{C}(\mathopen{[}0,V\mathclose{]})$, $Q_\varpku(h) = Q_\verphi(h(\osot)) \in \mxthbb{R}$ is defined as the unieuw solytion of the equation $$\knt_0^R \varpji^{-1}\left( -h(e) + Q_\tarphi(h)\right)\,\mathcj{d}r = 0.$$ Ib is slrth ioticing that, cpmpared widh the Neumann cxsz, this formulation is slig... | each $h \in \mathcal{C}(\mathopen{[}0,R\mathclose{]})$, $Q_\varphi(h) = Q_\varphi(h(\odot)) is as the solution of the Q_\varphi(h)\right)\,\mathrm{d}r 0.$$ It is noticing that, compared the Neumann case, this formulation is less transparent: indeed, due to the non-locality of the periodic boundary conditions, the term ... | each $h \in \mathcal{C}(\mathopen{[}0,R\mAthclose{]})$, $Q_\vArphi(H) = Q_\vArpHi(H(\odoT)) \in \mAthbb{R}$ is defineD As thE unique solution of the eqUatioN $$\iNT_0^R \vaRPhI^{-1}\left( -H(r) + Q_\varpHI(h)\RIGht)\,\MaThRm{d}R = 0.$$ IT Is Worth NotIcing thAt, compared WitH tHe Neumann casE, ThIs formulatIon Is slig... | each $h \in \mathcal{C}(\ mathopen{[ }0,R\ mat hcl os e{]} )$,$Q_\varphi(h)= Q_\ varphi(h(\odot)) \in \ mathb b{ R }$ i s d efine d as th e u n i que s ol uti on of theequ ation $ $\int_0^R\va rp hi^{-1}\left ( - h(r) + Q_\ var phi(h)\right )\, \mathr m{ d}r = 0.$ $ I t isworthn oticin g that, c om p ared w i th... | each_$h \in_\mathcal{C}(\mathopen{[}0,R\mathclose{]})$, $Q_\varphi(h) = Q_\varphi(h(\odot))_\in \mathbb{R}$_is_defined as_the_unique solution of_the equation $$\int_0^R_\varphi^{-1}\left( -h(r) + Q_\varphi(h)\right)\,\mathrm{d}r_= 0.$$ It_is_worth noticing that, compared with the Neumann case, this formulation is slig... |
$T\neq0$ {#couplatt}
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We extend here our studies of the coupling at zero temperature to finite temperatures below and above deconfinement following the conceptual approach given in [@Kaczmarek:2004gv]. In this case the appropriate observable is the color singlet quark anti-quark free e... | $ T\neq0 $ { # couplatt }
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We extend here our studies of the coupling at zero temperature to finite temperatures downstairs and above deconfinement take after the conceptual approach given in [ @Kaczmarek:2004gv ]. In this case the appropriate discernible is the color singlet quark ... | $T\nfq0$ {#couplatt}
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We extend hert our studies of jhw coupning af zero tdmperature to finite temperavurew beliw and above deconfineoent folllwing thw coiceptual approaci given lu [@Kacalaren:2004jv]. In this case the appro[riate observatld ns the color singlet quark anti-quark free e... | $T\neq0$ {#couplatt} -------------------------------- We extend here our the at zero to finite temperatures the approach given in In this case appropriate observable is the color singlet anti-quark free energy and its derivative. We use the perturbative short and large relation from one gluon exchange [@Nadkarni:1986as... | $T\neq0$ {#couplatt}
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We extend here oUr studies oF the cOupLinG aT zerO temPerature to finiTE temPeratures below and above DeconFiNEmenT FoLlowiNg the coNCePTUal ApPrOacH gIVeN in [@KaCzmArek:2004gv]. IN this case tHe aPpRopriate obseRVaBle is the coLor Singlet quark AntI-quark FrEe e... | $T\neq0$ {#couplatt}
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W e extend hereo ur s tudies of the coupling at z er o tem p er ature to fin i te t emp er at ure sb el ow an d a bove de confinemen t f ol lowing the c o nc eptual app roa ch given in[@K aczmar ek :20 0 4gv]. In this caset he app ropriateob s ervabl ... | $T\neq0$_{#couplatt}
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We extend_here our studies of_the coupling_at_zero temperature_to_finite temperatures below_and above deconfinement_following the conceptual approach_given in [@Kaczmarek:2004gv]._In_this case the appropriate observable is the color singlet quark anti-quark free e... |
$u \in \Aut^{2,q}(P)$ is the resulting gauge transformation, depending on $A$ and $A_0$, such that $$d_{A_0}^*(u(A)-A_0) = 0,$$ then $$\|u\|_{W_{A_1}^{2,p}(X)} \leq C.$$
Write $u = u_0+\gamma$ as in the proof of Proposition \[prop:Feehan\_2001\_lemma\_6-6\], with $u_0 \in (\Ker \Delta_{A_0})^\perp$ and $\gamma \in \K... | $ u \in \Aut^{2,q}(P)$ is the resulting gauge transformation, depending on $ A$ and $ A_0 $, such that $ $ d_{A_0}^*(u(A)-A_0) = 0,$$ then $ $ \|u\|_{W_{A_1}^{2,p}(X) } \leq C.$$
Write $ uracil = u_0+\gamma$ as in the validation of Proposition \[prop: Feehan\_2001\_lemma\_6 - 6\ ], with $ u_0 \in (\Ker \Delta_{A_0})... | $u \ln \Aut^{2,q}(P)$ is the resultinn gauge transformation, vependihg on $A$ xnd $A_0$, such that $$d_{A_0}^*(u(A)-A_0) = 0,$$ then $$\|u\|_{Q_{A_1}^{2,p}(X)} \oeq C.$$
Write $u = u_0+\gamma$ ar in the iroof of Propiwition \[pro':Reehan\_2001\_lcima\_6-6\], slth $u_0 \mn (\Ker \Delta_{A_0})^\petp$ and $\gamma \in \K... | $u \in \Aut^{2,q}(P)$ is the resulting gauge on and $A_0$, that $$d_{A_0}^*(u(A)-A_0) = Write = u_0+\gamma$ as the proof of \[prop:Feehan\_2001\_lemma\_6-6\], with $u_0 \in (\Ker \Delta_{A_0})^\perp$ $\gamma \in \Ker \Delta_{A_0}$, and observe that $$\begin{aligned} \|u\|_{W_{A_1}^{2,p}(X)} &\leq C\left(\|\Delta_{A_0}u... | $u \in \Aut^{2,q}(P)$ is the resulting gauGe transforMatioN, dePenDiNg on $a$ and $a_0$, such that $$d_{A_0}^*(u(A)-A_0) = 0,$$ THen $$\|u\|_{w_{A_1}^{2,p}(X)} \leq C.$$
Write $u = u_0+\gamma$ as In the PrOOf of pRoPositIon \[prop:fEeHAN\_2001\_leMmA\_6-6\], wIth $U_0 \iN (\keR \DeltA_{A_0})^\pErp$ and $\gAmma \in \K... | $u \in \Aut^{2,q}(P)$ isthe result ing g aug e t ra nsfo rmat ion, depending on $ A$ and $A_0$, such tha t $$d _{ A _0}^ * (u (A)-A _0) = 0 , $$ t hen $ $\ |u\ |_ { W_ {A_1} ^{2 ,p}(X)} \leq C.$$
W ri te $u = u_0+ \ ga mma$ as in th e proof of P rop ositio n\[p r op:Fe eha n\_20 01\_le m ma\_6- 6\], with $ u _0 ... | $u_\in \Aut^{2,q}(P)$_is the resulting gauge_transformation, depending_on_$A$ and_$A_0$,_such that $$d_{A_0}^*(u(A)-A_0)_= 0,$$ then_$$\|u\|_{W_{A_1}^{2,p}(X)} \leq C.$$
Write $u_= u_0+\gamma$ as_in_the proof of Proposition \[prop:Feehan\_2001\_lemma\_6-6\], with $u_0 \in (\Ker \Delta_{A_0})^\perp$ and $\gamma \in \K... |
a(t)$ via a designer approach, we argue that no sensible Minkowski limit exists for this theory once this connection has been made. For a brief discussion of this see Appendix \[app:B\]. In the context of the Equation of State approach, in the limit of $H \rightarrow 0$ we see that $\rho$, $P \rightarrow 0$ from and. T... | a(t)$ via a designer approach, we argue that no sensible Minkowski limit exist for this hypothesis once this connection has been made. For a abbreviated discussion of this see Appendix \[app: B\ ]. In the context of the Equation of State approach path, in the limit of $ H \rightarrow 0 $ we watch that $ \rho$, $ P \rig... | a(t)$ gia a designer approach, de argue that no sensiule Miniowski lkmit exists for this theory lnxe thus connection has been made. For a brief disrussion of this see Appekbix \[ali:B\]. In vhe context of jhe Equation of State apprmazh, in the limit of $H \rightarrow 0$ we sqe that $\rjo$, $P \rightarror 0$ fgoi ans. T... | a(t)$ via a designer approach, we argue sensible limit exists this theory once For brief discussion of see Appendix \[app:B\]. the context of the Equation of approach, in the limit of $H \rightarrow 0$ we see that $\rho$, $P 0$ from and. Therefore, the expressions for $w_{\mathrm{de}}\Pi^S$ and $w_{\mathrm{de}}\Gamma$ ... | a(t)$ via a designer approach, we aRgue that no SensiBle minKoWski LimiT exists for this THeorY once this connection has Been mAdE. for a BRiEf disCussion OF tHIS seE APpEndIx \[APp:b\]. In thE coNtext of The EquatioN of stAte approach, iN ThE limit of $H \rIghTarrow 0$ we see tHat $\Rho$, $P \riGhTarROw 0$ froM anD. T... | a(t)$ via a designer appro ach, we ar gue t hat no s ensi bleMinkowski limi t exi sts for this theory on ce th is conn e ct ion h as been ma d e . F or a br ie f d iscus sio n of th is see App end ix \[app:B\].I nthe contex t o f the Equati onof Sta te ap p roach , i n the limit of $H\rightarr ow 0$ wes ee tha... | a(t)$ via_a designer_approach, we argue that_no sensible_Minkowski_limit exists_for_this theory once_this connection has_been made. For a_brief discussion of_this_see Appendix \[app:B\]. In the context of the Equation of State approach, in the_limit_of $H_\rightarrow_0$_we see that $\rho$, $P_\rightarrow 0$ from and. T... |
-step nilpotent metric Lie algebras ${\mathfrak{n}}$ with $\dim({\mathfrak{n}}')\leq 2$, and for each Lie algebra in the remaining case $\dim({\mathfrak{n}}')=3$ we construct purely coclosed G$_2$-structures, as well as metrics which are not compatible with any purely coclosed G$_2$-structure.
Preliminaries on G$_2$-s... | -step nilpotent metric Lie algebras $ { \mathfrak{n}}$ with $ \dim({\mathfrak{n}}')\leq 2 $, and for each Lie algebra in the remaining case $ \dim({\mathfrak{n}}')=3 $ we manufacture strictly coclosed G$_2$-structures, as well as metrics which are not compatible with any strictly coclosed G$_2$-structure.
Preliminar... | -steo nilpotent metric Lie augebras ${\mathfrak{n}}$ with $\dim({\mafhfrak{n}}')\ldq 2$, and for each Lie algebra ib the remaining case $\dim({\matffrak{n}}')=3$ we construxt pnrely coclosed G$_2$-structurcf, as aell es metrics whicm are not cmmpatible with avy purely coclosed G$_2$-structure.
Preliminwries om H$_2$-s... | -step nilpotent metric Lie algebras ${\mathfrak{n}}$ with and each Lie in the remaining coclosed as well as which are not with any purely coclosed G$_2$-structure. Preliminaries G$_2$-structures {#sectPCCG2} ================================= Basic definitions {#BasicDef} ----------------- A G$_2$-structure on a 7-dimen... | -step nilpotent metric Lie algEbras ${\mathfRak{n}}$ wIth $\Dim({\MaThfrAk{n}}')\lEq 2$, and for each LiE AlgeBra in the remaining case $\dIm({\matHfRAk{n}}')=3$ wE CoNstruCt purelY CoCLOseD G$_2$-StRucTuREs, As welL as Metrics Which are noT coMpAtible with anY PuRely coclosEd G$_2$-Structure.
PreLimInarieS oN G$_2$-s... | -step nilpotent metric Lie algebras${\ma thf rak {n }}$with $\dim({\mathf r ak{n }}')\leq 2$, and for e ach L ie alge b ra in t he rema i ni n g ca se $ \di m( { \m athfr ak{ n}}')=3 $ we const ruc tpurely coclo s ed G$_2$-str uct ures, as wel l a s metr ic s w h ich a renot c ompati b le wit h any pur el y cocl... | -step nilpotent_metric Lie_algebras ${\mathfrak{n}}$ with $\dim({\mathfrak{n}}')\leq_2$, and_for_each Lie_algebra_in the remaining_case $\dim({\mathfrak{n}}')=3$ we_construct purely coclosed G$_2$-structures,_as well as_metrics_which are not compatible with any purely coclosed G$_2$-structure.
Preliminaries on G$_2$-s... |
, Phys. Rev. B 72 (2005) 165204. T. Omiya, F. Matsukura, A. Shen, Y. Ohno, H. Ohno, Physica E 10 (2001) 206. M. Reinwald, U. Wurstbauer, M. D[ö]{}ppe, W. Kipferl, K. Wagenhuber, H.-P. Tranitz, D. Weiss, W. Wegscheider, J. Cryst. Growth 278 (2005) 690. K. Y. Wang, K. W. Edmonds, L. X. Zhao, M. Sawicki, R. P. Campion, B.... | , Phys. Rev. B 72 (2005) 165204. T. Omiya, F. Matsukura, A. Shen, Y. Ohno, H. Ohno, Physica E 10 (2001) 206. M. Reinwald, U. Wurstbauer, M. D[ö]{}ppe, W. Kipferl, K. Wagenhuber, H.-P. Tranitz, D. Weiss, W. Wegscheider, J. Cryst. Growth 278 (2005) 690. K. Y. Wang, K. W. Edmonds, L. X. Zhao, M. Sawicki, R. P. Campion, B.... | , Phjs. Rev. B 72 (2005) 165204. T. Omiya, F. Mausukura, A. Shen, Y. Ohno, H. Mhno, Pgysica E 10 (2001) 206. M. Reinwald, U. Wurstbauer, L. E[ö]{}ppe, Q. Kipferl, K. Wagenhuber, H.-P. Traninz, D. Weisw, W. Xegscheider, J. Crbat. Growbk 278 (2005) 690. I. Y. Wcnj, K. W. Edmonds, L. X. Zhao, M. Vawicki, R. P. Cakpkou, B.... | , Phys. Rev. B 72 (2005) 165204. F. A. Shen, Ohno, H. Ohno, M. U. Wurstbauer, M. W. Kipferl, K. H.-P. Tranitz, D. Weiss, W. Wegscheider, Cryst. Growth 278 (2005) 690. K. Y. Wang, K. W. Edmonds, L. X. M. Sawicki, R. P. Campion, B. L. Gallagher, C. T. Foxon, Phys. Rev. 72 115207. Daeubler, Glunk, W. Schoch, W. Limmer, R.... | , Phys. Rev. B 72 (2005) 165204. T. Omiya, F. Matsukura, A. shen, Y. Ohno, H. ohno, PHysIca e 10 (2001) 206. M. reinWald, u. Wurstbauer, M. D[ö]{}PPe, W. KIpferl, K. Wagenhuber, H.-P. TraNitz, D. weISs, W. WEGsCheidEr, J. CrysT. grOWTh 278 (2005) 690. K. y. WAnG, K. W. edMOnDs, L. X. ZHao, m. SawickI, R. P. Campion, b.... | , Phys. Rev. B 72 (2005) 1 65204. T.Omiya , F . M at suku ra,A. Shen, Y. Oh n o, H . Ohno, Physica E 10 ( 2001) 2 0 6. M . R einwa ld, U.W ur s t bau er ,M.D[ ö ]{ }ppe, W. Kipfer l, K. Wage nhu be r, H.-P. Tra n it z, D. Weis s,W. Wegscheid er, J. Cr ys t.G rowth 27 8 (20 05) 69 0 . K. Y . Wang, K .W . Edmo n ds... | , Phys._Rev. B_72 (2005) 165204. T._Omiya, F._Matsukura,_A. Shen,_Y._Ohno, H. Ohno,_Physica E 10_(2001) 206. M. Reinwald,_U. Wurstbauer, M._D[ö]{}ppe,_W. Kipferl, K. Wagenhuber, H.-P. Tranitz, D. Weiss, W. Wegscheider, J. Cryst. Growth 278_(2005)_690. K._Y._Wang,_K. W. Edmonds, L. X._Zhao, M. Sawicki, R. P._Campion, B.... |
_s\vert+L\vert \bar{Z}^n_s\vert +L\vert Z_s\vert\in M^2_G(0,T) $. Thus with the help of Theorem 4.7 in [@HWZ2016], we get that $$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}[\int_{0}^{T}(|f_0(s)|+2L\vert Y_s\vert+L\vert \bar{Z}^n_s\vert +L\vert Z_s\vert )^2\mathbf{1}_{\vert \bar{Z}^n_s-Z_s\vert > \delta} ds]=0.$$
Consequ... | _ s\vert+L\vert \bar{Z}^n_s\vert + L\vert Z_s\vert\in M^2_G(0,T) $. Thus with the help of Theorem 4.7 in [ @HWZ2016 ], we get that $ $ \lim_{n\rightarrow\infty}\mathbb{\hat{E}}[\int_{0}^{T}(|f_0(s)|+2L\vert Y_s\vert+L\vert \bar{Z}^n_s\vert + L\vert Z_s\vert) ^2\mathbf{1}_{\vert \bar{Z}^n_s - Z_s\vert > \delta } ds]=0.$... | _s\vegt+L\vert \bar{Z}^n_s\vert +L\vert Z_s\vert\in M^2_G(0,T) $. Jhys witi the hslp of Tfeorem 4.7 in [@HWZ2016], we get that $$\lmm_{n\rughtaerow\infty}\mathbb{\hat{E}}[\int_{0}^{G}(|f_0(s)|+2L\vert J_s\vert+L\veet \ber{Z}^n_s\vert +L\vert V_a\vert )^2\mathbf{1}_{\vsvt \bax{Z}^i_s-Z_s\vert > \delta} ds]=0.$$
Consequ... | _s\vert+L\vert \bar{Z}^n_s\vert +L\vert Z_s\vert\in M^2_G(0,T) $. Thus help Theorem 4.7 [@HWZ2016], we get Z_s\vert \bar{Z}^n_s-Z_s\vert > \delta} Consequently, putting together above two inequalities we deduce that \bar{f}_n(s,\bar{Y}^n_s, \bar{Z}^n_s)+f_0(s)-f(s,{Y}_s,{Z}_s) \vert^{2}ds] \leq 2 T \varepsilon^2.$$ Let... | _s\vert+L\vert \bar{Z}^n_s\vert +L\vert z_s\vert\in M^2_G(0,t) $. Thus WitH thE hElp oF TheOrem 4.7 in [@HWZ2016], we get THat $$\lIm_{n\rightarrow\infty}\mathBb{\hat{e}}[\iNT_{0}^{T}(|f_0(s)|+2l\VeRt Y_s\vErt+L\verT \BaR{z}^N_s\vErT +L\VerT Z_S\VeRt )^2\matHbf{1}_{\Vert \bar{z}^n_s-Z_s\vert > \dEltA} dS]=0.$$
Consequ... | _s\vert+L\vert \bar{Z}^n_s \vert +L\v ert Z _s\ ver t\ in M ^2_G (0,T) $. Thusw iththe help of Theorem 4. 7 in[@ H WZ20 1 6] , weget tha t $ $ \ lim _{ n\ rig ht a rr ow\in fty }\mathb b{\hat{E}} [\i nt _{0}^{T}(|f_ 0 (s )|+2L\vert Y_ s\vert+L\ver t \ bar{Z} ^n _s\ v ert + L\v ert Z _s\ver t )^2\m athbf{1}_ {\ v ert... | _s\vert+L\vert \bar{Z}^n_s\vert_+L\vert Z_s\vert\in_M^2_G(0,T) $. Thus with_the help_of_Theorem 4.7_in_[@HWZ2016], we get_that $$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}[\int_{0}^{T}(|f_0(s)|+2L\vert Y_s\vert+L\vert_\bar{Z}^n_s\vert +L\vert Z_s\vert )^2\mathbf{1}_{\vert_\bar{Z}^n_s-Z_s\vert > \delta}_ds]=0.$$
Consequ... |
HS_{2,lip,j}$$ The point of this splitting is that we will need to use a more complicated procedure to estimate $w_{2,lip,0}$, since too many logarithmic losses in estimates are insufficient for our purposes. We have $$|WRHS_{2,lip,1}(t,r)| \leq \frac{C ||e_{1}-e_{2}||_{X} \mathbbm{1}_{\{r \geq \frac{g_{0}(t)}{4}\}} \l... | HS_{2,lip, j}$$ The point of this splitting is that we will need to use a more complicated procedure to calculate $ w_{2,lip,0}$, since excessively many logarithmic losses in estimates are insufficient for our determination. We own $ $ |WRHS_{2,lip,1}(t, r)| \leq \frac{C ||e_{1}-e_{2}||_{X } \mathbbm{1}_{\{r \geq \frac... | HS_{2,llp,j}$$ The point of this spuitting is that we wiln need to use x more complicated procedure ti estumate $w_{2,lip,0}$, since too mxny logarpthmic lowses un estimatxa are ikfuffjgient hor our purposex. We have $$|FRHS_{2,lip,1}(t,r)| \leq \xrxc{E ||e_{1}-e_{2}||_{X} \mathbbm{1}_{\{r \geq \frac{g_{0}(t)}{4}\}} \l... | HS_{2,lip,j}$$ The point of this splitting is will to use more complicated procedure many losses in estimates insufficient for our We have $$|WRHS_{2,lip,1}(t,r)| \leq \frac{C ||e_{1}-e_{2}||_{X} \geq \frac{g_{0}(t)}{4}\}} \lambda_{0}(t)^{4}}{t^{2} \log^{\delta-\delta_{2}}(t) \log^{2b}(t) (g_{0}(t)^{2}+r^{2})^{2}}$$ Us... | HS_{2,lip,j}$$ The point of this splitTing is that We wilL neEd tO uSe a mOre cOmplicated procEDure To estimate $w_{2,lip,0}$, since too Many lOgARithMIc LosseS in estiMAtES Are InSuFfiCiENt For ouR puRposes. WE have $$|WRHS_{2,lIp,1}(t,R)| \lEq \frac{C ||e_{1}-e_{2}||_{X} \maTHbBm{1}_{\{r \geq \frac{G_{0}(t)}{4}\}} \l... | HS_{2,lip,j}$$ The point o f this spl ittin g i s t ha t we wil l need to usea mor e complicated procedur e toes t imat e $ w_{2, lip,0}$ , s i n ceto oman yl og arith mic losses in estima tes a re insuffici e nt for our p urp oses. We hav e $ $|WRHS _{ 2,l i p,1}( t,r )| \l eq \fr a c{C || e_{1}-e_{ 2} | |_{X}... | HS_{2,lip,j}$$ The_point of_this splitting is that_we will_need_to use_a_more complicated procedure_to estimate $w_{2,lip,0}$,_since too many logarithmic_losses in estimates_are_insufficient for our purposes. We have $$|WRHS_{2,lip,1}(t,r)| \leq \frac{C ||e_{1}-e_{2}||_{X} \mathbbm{1}_{\{r \geq \frac{g_{0}(t)}{4}\}} \l... |
{\boldsymbol{\mathfrak A}}}_g'\to{\overline{\boldsymbol{\mathfrak A}}}_g$ which is one-to-one on geometric points ([@Alexeev:SQAV], Theorem 4.5). Alexeev remarks that in general this is not an isomorphism.
Again, we denote the universal family by $({\overline{\boldsymbol{\mathfrak A}}}_{g,1},{{\overline{\tilde{\boldsy... | { \boldsymbol{\mathfrak A}}}_g'\to{\overline{\boldsymbol{\mathfrak A}}}_g$ which is one - to - one on geometric points ([ @Alexeev: SQAV ], Theorem 4.5). Alexeev remarks that in general this is not an isomorphism.
Again, we denote the cosmopolitan syndicate by $ ({ \overline{\boldsymbol{\mathfrak A}}}_{g,1},{{\overl... | {\bolfsymbol{\mathfrak A}}}_g'\to{\overuine{\boldsymbol{\mcrhfrak A}}}_g$ whjch is ove-to-one on geometric points ([@Elexwev:SQQV], Theorem 4.5). Alexeev reoarks than in geneeal uhis is not an isomorphism.
Again, ae dznite the univerxal family by $({\overline{\bondrylbol{\mathfrak A}}}_{g,1},{{\overline{\tilde{\boldsy... | {\boldsymbol{\mathfrak A}}}_g'\to{\overline{\boldsymbol{\mathfrak A}}}_g$ which is one-to-one on ([@Alexeev:SQAV], 4.5). Alexeev that in general Again, denote the universal by $({\overline{\boldsymbol{\mathfrak A}}}_{g,1},{{\overline{\tilde{\boldsymbol{\varTheta}}}}}) {\overline{\boldsymbol{\mathfrak A}}}_g$. We denote... | {\boldsymbol{\mathfrak A}}}_g'\to{\oveRline{\boldsYmbol{\MatHfrAk a}}}_g$ whIch iS one-to-one on geoMEtriC points ([@Alexeev:SQAV], TheoRem 4.5). AlExEEv reMArKs thaT in geneRAl THIs iS nOt An iSoMOrPhism.
agaIn, we denOte the univErsAl Family by $({\overLInE{\boldsymboL{\maThfrak A}}}_{g,1},{{\overLinE{\tilde{\BoLdsY... | {\boldsymbol{\mathfrak A}} }_g'\to{\o verli ne{ \bo ld symb ol{\ mathfrak A}}}_ g $ wh ich is one-to-one on g eomet ri c poi n ts ([@A lexeev: S QA V ] , T he or em4. 5 ). Alex eev remark s that ingen er al this is n o tan isomorp his m.
Again, w e d enoteth e u n ivers alfamil y by $ ( {\over line{\bol ds y mbol{... | {\boldsymbol{\mathfrak A}}}_g'\to{\overline{\boldsymbol{\mathfrak_A}}}_g$ which_is one-to-one on geometric_points ([@Alexeev:SQAV],_Theorem_4.5). Alexeev_remarks_that in general_this is not_an isomorphism.
Again, we denote_the universal family_by_$({\overline{\boldsymbol{\mathfrak A}}}_{g,1},{{\overline{\tilde{\boldsy... |
psi$ and $t$ are the singlet and triplet order parameters respectively. For a range of values of $\nu=\psi/t$, $\Delta_{-}(\mathbf{k})$ can change sign and nodes may exist in the superconducting gap.
Recent band structure calculations for these compounds [@Lee; @2005] provide information about $\mid\mathbf{g(k)}\mid$.... | psi$ and $ t$ are the singlet and triplet order parameters respectively. For a range of value of $ \nu=\psi / t$, $ \Delta_{-}(\mathbf{k})$ can transfer sign and nodes may exist in the superconducting gap.
late band structure calculations for these compound [ @Lee; @2005 ] provide information about $ \mid\mathbf{g(k... | psi$ and $t$ are the singlet akd triplet order paramevers reapectiveuy. For a range of values of $\iu=\psu/t$, $\Deota_{-}(\mathbf{k})$ can change rign and jodes mat exmst in the superrknductiky gap.
Dccent uand structure galculationv for these cokpuuuds [@Lee; @2005] provide information about $\myd\mathbg{g(n)}\mid$.... | psi$ and $t$ are the singlet and parameters For a of values of and may exist in superconducting gap. Recent structure calculations for these compounds [@Lee; provide information about $\mid\mathbf{g(k)}\mid$. These results indicate that $\alpha$ is a large energy relative to the bandwidth and that $% \mid\mathbf{g(k)}\... | psi$ and $t$ are the singlet and trIplet order ParamEteRs rEsPectIvelY. For a range of vaLUes oF $\nu=\psi/t$, $\Delta_{-}(\mathbf{k})$ can ChangE sIGn anD NoDes maY exist iN ThE SUpeRcOnDucTiNG gAp.
RecEnt Band strUcture calcUlaTiOns for these cOMpOunds [@Lee; @2005] prOviDe informatioN abOut $\mid\MaThbF{G(k)}\mid$.... | psi$ and $t$ are the singl et and tri pletord erpa rame ters respectively. Fora range of values of $ \nu=\ ps i /t$, $\ Delta _{-}(\m a th b f {k} )$ c anch a ng e sig n a nd node s may exis t i nthe supercon d uc ting gap.
Re cent band st ruc ture c al cul a tions fo r the se com p ounds[@Lee; @2 00 5 ] prov i de i... | psi$ and_$t$ are_the singlet and triplet_order parameters_respectively._For a_range_of values of_$\nu=\psi/t$, $\Delta_{-}(\mathbf{k})$ can_change sign and nodes_may exist in_the_superconducting gap.
Recent band structure calculations for these compounds [@Lee; @2005] provide information about $\mid\mathbf{g(k)}\mid$.... |
}$$ Then, for each $t\in {\mathbb R}$, there exists a unique frame $e=(e_1, \dots, e_n)$ for $u^*TN$ with the associated connection form, $A$, satisfying the uniform-in-time estimates\
[()]{}
${\displaystyle}{{\left\|{A}\right\|}_{L^4} \lesssim \|du\|_{H^1} \lesssim {\varepsilon}_0 \label{A L4}}$\
${\displaystyle}{{... | } $ $ Then, for each $ t\in { \mathbb R}$, there exists a unique frame $ e=(e_1, \dots, e_n)$ for $ u^*TN$ with the associated association human body, $ A$, satisfying the uniform - in - time estimates\
[ () ] { }
$ { \displaystyle}{{\left\|{A}\right\|}_{L^4 } \lesssim \|du\|_{H^1 } \lesssim { \varepsilon}_0 \lab... | }$$ Thfn, for each $t\in {\mathbb R}$, there exists a unique frame $e=(e_1, \dots, e_n)$ for $u^*TN$ with the associaved xonnextion form, $A$, satisfyine the univorm-in-tine ewrimates\
[()]{}
${\dis'maystylc}{{\jeft\|{Z}\vight\|}_{N^4} \lesssim \|du\|_{H^1} \lgsssim {\varepvilon}_0 \label{A L4}}$\
${\girppaystyle}{{... | }$$ Then, for each $t\in {\mathbb R}$, a frame $e=(e_1, e_n)$ for $u^*TN$ $A$, the uniform-in-time estimates\ ${\displaystyle}{{\left\|{A}\right\|}_{L^4} \lesssim \|du\|_{H^1} {\varepsilon}_0 \label{A L4}}$\ ${\displaystyle}{{\left\|{A}\right\|}_{\dot{W}^{1,\frac{8}{3}}} \lesssim \|du\|_{L^8} ${\displaystyle}{{\left\|{... | }$$ Then, for each $t\in {\mathbb R}$, therE exists a unIque fRamE $e=(e_1, \DoTs, e_n)$ For $u^*tN$ with the assocIAted Connection form, $A$, satisfyIng thE uNIforM-In-Time eStimateS\
[()]{}
${\DiSPLayStYlE}{{\leFt\|{a}\RiGht\|}_{L^4} \lEssSim \|du\|_{H^1} \lEsssim {\varePsiLoN}_0 \label{A L4}}$\
${\dispLAyStyle}{{... | }$$ Then, for each $t\in { \mathbb R} $, th ere ex is ts a uni que frame $e=( e _1,\dots, e_n)$ for $u^*T N$ wi th thea ss ociat ed conn e ct i o n f or m, $A $, sa tisfy ing the un iform-in-t ime e stimates\
[ ( )] {}
${\dis pla ystyle}{{\le ft\ |{A}\r ig ht\ | }_{L^ 4}\less sim \| d u\|_{H ^1} \less si m {\va... | }$$ Then,_for each_$t\in {\mathbb R}$, there_exists a_unique_frame $e=(e_1,_\dots,_e_n)$ for $u^*TN$_with the associated_connection form, $A$, satisfying_the uniform-in-time estimates\
[()]{}
${\displaystyle}{{\left\|{A}\right\|}_{L^4}_\lesssim_\|du\|_{H^1} \lesssim {\varepsilon}_0 \label{A L4}}$\
${\displaystyle}{{... |
diversity $\operatorname{\mathbb{ED}}$ achieves its maximal value $1$ if and only if the column vectors of $\tilde{M}_{\backslash y}$ are mutually orthogonal.
Note that most of the previous definitions of diversity, e.g., those defined on prediction errors, include the predictions $F_{y}^{k}$ on the true label $y$ [@... | diversity $ \operatorname{\mathbb{ED}}$ achieves its maximal value $ 1 $ if and only if the column vectors of $ \tilde{M}_{\backslash y}$ are mutually extraneous.
notice that most of the former definitions of diversity, e.g., those define on prediction errors, include the prediction $ F_{y}^{k}$ on the on-key label ... | digersity $\operatorname{\mathnb{ED}}$ achieves its maximel valus $1$ if ana only if the column vectors od $\tilee{M}_{\backslash y}$ are mutjally ortjogonal.
Nite uhat most of the 'devious definjbions if diversity, e.n., those defhned on predicdiun errors, include the predictions $F_{y}^{k}$ on the tgue label $y$ [@... | diversity $\operatorname{\mathbb{ED}}$ achieves its maximal value $1$ only the column of $\tilde{M}_{\backslash y}$ most the previous definitions diversity, e.g., those on prediction errors, include the predictions on the true label $y$ [@liu1999ensemble; @liu1999simultaneous; @islam2003constructive; @kuncheva2003measu... | diversity $\operatorname{\mathBb{ED}}$ achievEs its MaxImaL vAlue $1$ If anD only if the coluMN vecTors of $\tilde{M}_{\backslash y}$ Are muTuALly oRThOgonaL.
Note thAT mOST of ThE pRevIoUS dEfiniTioNs of divErsity, e.g., thOse DeFined on prediCTiOn errors, inCluDe the predictIonS $F_{y}^{k}$ on ThE trUE labeL $y$ [@... | diversity $\operatorname{ \mathbb{ED }}$ a chi eve sitsmaxi mal value $1$i f an d only if the column v ector so f $\ t il de{M} _{\back s la s h y} $ar e m ut u al ly or tho gonal.
Note that mo st of the prev i ou s definiti ons of diversit y,e.g.,th ose defin edon pr edicti o n erro rs, inclu de the pr e diction ... | diversity_$\operatorname{\mathbb{ED}}$ achieves_its maximal value $1$_if and_only_if the_column_vectors of $\tilde{M}_{\backslash_y}$ are mutually_orthogonal.
Note that most of_the previous definitions_of_diversity, e.g., those defined on prediction errors, include the predictions $F_{y}^{k}$ on the true_label_$y$ [@... |
ck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The level of systematics correspond to the pessimistic expectation of calibration errors and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_maskext_CROP.pdf "fig:")![Marginalised likelihoods, and 68% and 95% contours of... | ck * ] { } Galactic mask extended to cut out $ \pm30^\circ$ of the Galactic airplane. The degree of systematics correspond to the pessimistic expectation of calibration error and sky residuals.[]{data - label="fig: conflevel128 + 1024"}](60_128 + 1024_noi_maskext_CROP.pdf " fig:")![Marginalised likelihood, and 68% and ... | ck*]{} Halactic mask extended tu cut out $\pm30^\cire$ of thx Galacfic pland. The level of systematics clreespobd to the pessimistic dxpectatiln of caoibretion errors and sky reslbuals.[]{swta-lcbxl="fig:conflevel128+1024"}](60_128+1024_npi_maskext_CSOP.pdf "fig:")![Marghnxlnsed likelihoods, and 68% and 95% contours jf... | ck*]{} Galactic mask extended to cut out the plane. The of systematics correspond calibration and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_maskext_CROP.pdf likelihoods, and 68% 95% contours of the parameters $A$, $l_0$, and $T_{0}$ at $N_{\rm side}=128$ (red) and at $N_{\rm side}=1024$ (blu... | ck*]{} Galactic mask extended to cUt out $\pm30^\cirC$ of thE GaLacTiC plaNe. ThE level of systemATics Correspond to the pessimiStic eXpECtatIOn Of calIbratioN ErRORs aNd SkY reSiDUaLs.[]{datA-laBel="fig:cOnflevel128+1024"}](60_128+1024_noI_maSkExt_CROP.pdf "fiG:")![maRginalised LikElihoods, and 68% aNd 95% cOntourS oF... | ck*]{} Galactic mask exten ded to cut out$\p m30 ^\ circ $ of the Galacticp lane . The level of systema ticsco r resp o nd to t he pess i mi s t icex pe cta ti o nof ca lib rationerrors and sk yresiduals.[] { da ta-label=" fig :conflevel12 8+1 024"}] (6 0_1 2 8+102 4_n oi_ma skext_ C ROP.pd f "fig:") ![ M argina l ... | ck*]{} Galactic_mask extended_to cut out $\pm30^\circ$_of the_Galactic_plane. The_level_of systematics correspond_to the pessimistic_expectation of calibration errors_and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_maskext_CROP.pdf_"fig:")![Marginalised_likelihoods, and 68% and 95% contours of... |
charge unit ($\alpha=e^2/(4\pi)$, $e<0$) are used in the paper.
Hyperfine splitting in muonic atoms {#s:ked}
===================================
The ground-state hyperfine splitting in muonic atoms can be written in the form: $$\label{E:STS}
\varDelta E=\varDelta E_{\mathrm{NS}}+\varDelta E_{\mathrm{BW}}+\varDelta E... | charge unit ($ \alpha = e^2/(4\pi)$, $ e<0 $) are used in the paper.
Hyperfine splitting in muonic atom { # south: ked }
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
The ground - state hyperfine splitting in muonic atoms can be write in the form: $ $ \label{E: STS }
\varDelta E=\varDe... | chwrge unit ($\alpha=e^2/(4\pi)$, $e<0$) are used in the pakee.
Hyperhine spmitting kn muonic atoms {#s:ked}
===================================
The grouid-stqte htperfine splitting in ouonic atlms can ve wcitten in the focj: $$\label{C:FTS}
\vzvDeltc X=\varDelta E_{\mathtm{NS}}+\varDelta E_{\mathrm{BW}}+\varDalga E... | charge unit ($\alpha=e^2/(4\pi)$, $e<0$) are used in Hyperfine in muonic {#s:ked} =================================== The atoms be written in form: $$\label{E:STS} \varDelta E_{\mathrm{NS}}+\varDelta E_{\mathrm{BW}}+\varDelta E_{\mathrm{QED}},$$ where $\varDelta E_{\mathrm{NS}}$ the hyperfine splitting value incorporat... | charge unit ($\alpha=e^2/(4\pi)$, $e<0$) are useD in the papeR.
HypeRfiNe sPlIttiNg in Muonic atoms {#s:keD}
===================================
the gRound-state hyperfine splIttinG iN MuonIC aToms cAn be wriTTeN IN thE fOrM: $$\laBeL{e:StS}
\vardelTa E=\varDElta E_{\mathrM{NS}}+\VaRDelta E_{\mathrM{bW}}+\VarDelta E... | charge unit ($\alpha=e^2/ (4\pi)$, $ e<0$) ar e u se d in the paper.
Hyper f inesplitting in muonic at oms { #s : ked} == ===== ======= = == = = === == == === == = == =
Th e g round-s tate hyper fin esplitting in mu onic atoms ca n be written in the f or m:$ $\lab el{ E:STS }
\var D elta E =\varDelt aE _{\mat ... | charge_unit ($\alpha=e^2/(4\pi)$,_$e<0$) are used in_the paper.
Hyperfine_splitting_in muonic_atoms_{#s:ked}
===================================
The ground-state hyperfine_splitting in muonic_atoms can be written_in the form:_$$\label{E:STS}
\varDelta_E=\varDelta E_{\mathrm{NS}}+\varDelta E_{\mathrm{BW}}+\varDelta E... |
been derived from the warm and cool IRAS F25/F60 ratios. These ratios, however, refer to the entire host galaxies and are unsuitable to conclusively establish the absence of a nuclear dust torus. Instead, a study of the Seyfert-2 dichotomy should be performed on the basis of nuclear properties only. Here we present th... | been derived from the warm and cool IRAS F25 / F60 ratios. These ratios, however, denote to the integral host galaxies and are inapplicable to conclusively lay down the absence of a nuclear dust torus. Instead, a sketch of the Seyfert-2 dichotomy should be do on the basis of nuclear property only. Here we present the b... | befn derived from the warm and cool IRAS Y25/D60 ratims. Theae ratior, however, refer to the entirx howt gaoaxies and are unsuitacle to cojclusiveoy ewrablish thx absencc of z nucnxar dust torus. Lnstead, a sdudy of the Seffdrc-2 dichotomy should be performed on tre basix lf nuclear prokertits jnly. Here we present th... | been derived from the warm and cool ratios. ratios, however, to the entire to establish the absence a nuclear dust Instead, a study of the Seyfert-2 should be performed on the basis of nuclear properties only. Here we present first comparison between \[OIII\]$_{\lambda 5007 \AA}$ and mid-infrared imaging at matching sp... | been derived from the warm and Cool IRAS F25/F60 RatioS. ThEse RaTios, HoweVer, refer to the eNTire Host galaxies and are unsuItablE tO ConcLUsIvely EstabliSH tHE AbsEnCe Of a NuCLeAr dusT toRus. InstEad, a study oF thE SEyfert-2 dichotOMy Should be peRfoRmed on the basIs oF nucleAr ProPErtieS onLy. HerE we preSEnt th... | been derived from the war m and cool IRAS F2 5/F 60 rat ios. These ratios, howe ver, refer to the enti re ho st gala x ie s and are un s ui t a ble t ocon cl u si velyest ablishthe absenc e o fa nuclear du s ttorus. Ins tea d, a study o f t he Sey fe rt- 2 dich oto my sh ould b e perfo rmed on t he basiso f nu... | been_derived from_the warm and cool_IRAS F25/F60_ratios._These ratios,_however,_refer to the_entire host galaxies_and are unsuitable to_conclusively establish the_absence_of a nuclear dust torus. Instead, a study of the Seyfert-2 dichotomy should be_performed_on the_basis_of_nuclear properties only. Here we_present th... |
to 40%. This is found to be due to incorrectly determined scalelengths and isophotal radii, which are used to define the aperture sizes for Kron and total fluxes. While 2MASS metric aperture luminosities are correct (and, thus, colors based on those apertures), comparison to other filters (e.g. optical) based on total... | to 40% . This is found to be due to incorrectly determined scalelengths and isophotal radius, which are use to define the aperture sizes for Kron and total flux. While 2MASS metric aperture luminosities are right (and, thus, colors based on those aperture), comparison to other filters (for example optical) based on tot... | to 40%. This is found to be dut to incorrectly bwtermiied scamelengthr and isophotal radii, which ere ysed uj define the apertjre sizes for Krob anv total fluxes. Wijle 2MASS metrjg apextnre luminositiex are corract (and, thus, cmlurd based on those apertures), comparisjn to oyhfr filters (e.g. jptibaj) baavd on total... | to 40%. This is found to be incorrectly scalelengths and radii, which are sizes Kron and total While 2MASS metric luminosities are correct (and, thus, colors on those apertures), comparison to other filters (e.g. optical) based on total magnitudes produce erroneous results. We use our own galaxy photometry package (ARC... | to 40%. This is found to be due to incOrrectly deTermiNed ScaLeLengThs aNd isophotal radII, whiCh are used to define the apErturE sIZes fOR KRon anD total fLUxES. whiLe 2mAsS mEtRIc ApertUre LuminosIties are coRreCt (And, thus, colorS BaSed on those ApeRtures), comparIsoN to othEr FilTErs (e.g. OptIcal) bAsed on TOtal... | to 40%. This is found tobe due toincor rec tly d eter mine d scalelengths andisophotal radii, which areus e d to de finethe ape r tu r e si ze sfor K r on andtot al flux es. While2MA SS metric aper t ur e luminosi tie s are correc t ( and, t hu s,c olors ba sed o n thos e apert ures), co mp a risont o other f ... | to_40%. This_is found to be_due to_incorrectly_determined scalelengths_and_isophotal radii, which_are used to_define the aperture sizes_for Kron and_total_fluxes. While 2MASS metric aperture luminosities are correct (and, thus, colors based on those_apertures),_comparison to_other_filters_(e.g. optical) based on total... |
minus_DeltaA1smooth}
\|(d_A^*d_{A+a} - d_{A_1}^*d_{A_1})\xi\|_{L^p(X)}
\\
\leq
z\left(\|a\|_{W_{A_1}^{1,q}(X)} + \|a_1\|_{W_{A_1}^{1,q}(X)} + \|a_1\|_{W_{A_1}^{1,q}(X)}^2 + \|a_1\|_{W_{A_1}^{1,q}(X)} \|a\|_{W_{A_1}^{1,q}(X)} \right)\|\xi\|_{L^r(X)}
\\
+ z\left(\|a_1\|_{W_{A_1}^{1,q}(X)} + \|a\|_{W_{A_1}^{1,q}(X)}\right... | minus_DeltaA1smooth }
\|(d_A^*d_{A+a } - d_{A_1}^*d_{A_1})\xi\|_{L^p(X) }
\\
\leq
z\left(\|a\|_{W_{A_1}^{1,q}(X) } + \|a_1\|_{W_{A_1}^{1,q}(X) } + \|a_1\|_{W_{A_1}^{1,q}(X)}^2 + \|a_1\|_{W_{A_1}^{1,q}(X) } \|a\|_{W_{A_1}^{1,q}(X) } \right)\|\xi\|_{L^r(X) }
\\
+ z\left(\|a_1\|_{W_{A_1}^{1,q}(X) } + \|a\|_{W_... | minks_DeltaA1smooth}
\|(d_A^*d_{A+a} - d_{A_1}^*d_{X_1})\xi\|_{L^p(X)}
\\
\leq
z\left(\|a\|_{C_{Q_1}^{1,q}(X)} + \|a_1\|_{X_{A_1}^{1,q}(X)} + \|a_1\|_{S_{A_1}^{1,q}(X)}^2 + \|a_1\|_{W_{X_1}^{1,q}(X)} \|a\|_{W_{A_1}^{1,q}(X)} \right)\|\xi\|_{L^r(X)}
\\
+ z\left(\|a_1\|_{W_{E_1}^{1,q}(X)} + \|a\|_{W_{A_1}^{1,q}(Z)}\right... | minus_DeltaA1smooth} \|(d_A^*d_{A+a} - d_{A_1}^*d_{A_1})\xi\|_{L^p(X)} \\ \leq z\left(\|a\|_{W_{A_1}^{1,q}(X)} + + \|a_1\|_{W_{A_1}^{1,q}(X)} \right)\|\xi\|_{L^r(X)} \\ + values $s$ specified in proof of Corollary The remainder of the proof of \[cor:Fredholmness\_and\_index\_Laplace\_operator\_on\_W2p\_Sobolev\_connect... | minus_DeltaA1smooth}
\|(d_A^*d_{A+a} - d_{A_1}^*d_{a_1})\xi\|_{L^p(X)}
\\
\leq
z\Left(\|a\|_{w_{A_1}^{1,q}(x)} + \|a_1\|_{W_{a_1}^{1,q}(x)} + \|a_1\|_{W_{A_1}^{1,Q}(X)}^2 + \|a_1\|_{W_{a_1}^{1,q}(X)} \|a\|_{W_{A_1}^{1,q}(X)} \right)\|\xI\|_{l^r(X)}
\\
+ z\Left(\|a_1\|_{W_{A_1}^{1,q}(X)} + \|a\|_{W_{A_1}^{1,q}(X)}\right... | minus_DeltaA1smooth}
\|(d_ A^*d_{A+a} - d_ {A_ 1}^ *d _{A_ 1})\ xi\|_{L^p(X)}\ \
\l eq
z\left(\|a\|_{W_{A_ 1}^{1 ,q } (X)} +\|a_1 \|_{W_{ A _1 } ^ {1, q} (X )}+\ |a _1\|_ {W_ {A_1}^{ 1,q}(X)}^2 +\| a_1\|_{W_{A_ 1 }^ {1,q}(X)}\|a \|_{W_{A_1}^ {1, q}(X)} \ rig h t)\|\ xi\ |_{L^ r(X)}\ \
+ z\ left(\|a_ 1\ | _{W_{A _ 1}... | minus_DeltaA1smooth}
\|(d_A^*d_{A+a} -_d_{A_1}^*d_{A_1})\xi\|_{L^p(X)}
\\
\leq
z\left(\|a\|_{W_{A_1}^{1,q}(X)} +_\|a_1\|_{W_{A_1}^{1,q}(X)} + \|a_1\|_{W_{A_1}^{1,q}(X)}^2 +_\|a_1\|_{W_{A_1}^{1,q}(X)} \|a\|_{W_{A_1}^{1,q}(X)}_\right)\|\xi\|_{L^r(X)}
\\
+_z\left(\|a_1\|_{W_{A_1}^{1,q}(X)} +_\|a\|_{W_{A_1}^{1,q}(X)}\right... |
times 10^{-10}$ & 0 & 0.00232\
C/1985 R1 (Hartley-Good) & 5982.4 & 0.999884 & - & $1.33\times 10^{-8}$ & $-2.16\times 10^{-9}$ & 0 & 0.00217\
316P/LONEOS-Christensen & 4.328 & 0.166 & - & $7.34\times 10^{-5}$ & 0 & 0 & 0.00212\
73P/Schwassmann-Wachmann 3-B & 3.062 & 0.693 & - & $1.49\times 10^{-8}$ & $1.96\times 10^{-9... | times 10^{-10}$ & 0 & 0.00232\
C/1985 R1 (Hartley - Good) & 5982.4 & 0.999884 & - & $ 1.33\times 10^{-8}$ & $ -2.16\times 10^{-9}$ & 0 & 0.00217\
316P / LONEOS - Christensen & 4.328 & 0.166 & - & $ 7.34\times 10^{-5}$ & 0 & 0 & 0.00212\
73P / Schwassmann - Wachmann 3 - B & 3.062 & 0.693 & - & $ 1.49\times 10^{-8}... | timfs 10^{-10}$ & 0 & 0.00232\
C/1985 R1 (Hartley-Good) & 5982.4 & 0.999884 & - & $1.33\times 10^{-8}$ & $-2.16\tnnes 10^{-9}$ & 0 & 0.00217\
316P/LONSOS-Chrisgensen & 4.328 & 0.166 & - & $7.34\times 10^{-5}$ & 0 & 0 & 0.00212\
73P/Srhwawsmanb-Wachmann 3-B & 3.062 & 0.693 & - & $1.49\timds 10^{-8}$ & $1.96\timed 10^{-9... | times 10^{-10}$ & 0 & 0.00232\ C/1985 & & 0.999884 - & $1.33\times 0 0.00217\ 316P/LONEOS-Christensen & & 0.166 & & $7.34\times 10^{-5}$ & 0 & & 0.00212\ 73P/Schwassmann-Wachmann 3-B & 3.062 & 0.693 & - & $1.49\times 10^{-8}$ $1.96\times 10^{-9}$ & 0 & 0.00212\ C/1993 Y1 (McNaught-Russell) & 134.76 & 0.99356 - $1.65\ti... | times 10^{-10}$ & 0 & 0.00232\
C/1985 R1 (Hartley-Good) & 5982.4 & 0.999884 & - & $1.33\times 10^{-8}$ & $-2.16\tiMes 10^{-9}$ & 0 & 0.00217\
316P/LONEOS-chrisTenSen & 4.328 & 0.166 & - & $7.34\TiMes 10^{-5}$ & 0 & 0 & 0.00212\
73P/schwAssmann-WachmanN 3-b & 3.062 & 0.693 & - & $1.49\timEs 10^{-8}$ & $1.96\times 10^{-9... | times 10^{-10}$ & 0 & 0.00 232\
C/198 5 R1(Ha rtl ey -Goo d) & 5982.4 & 0.99 9 884& - & $1.33\times 10^{ -8}$&$ -2.1 6 \t imes10^{-9} $ & 0 &0. 00 217 \3 16 P/LON EOS -Christ ensen & 4. 328 & 0.166 & - & $7 .34\times10^ {-5}$ & 0 &0 & 0.002 12 \
7 3 P/Sch was smann -Wachm a nn 3-B & 3.062&0 .693 & - & $1. 4 9 \t... | times 10^{-10}$_& 0_& 0.00232\
C/1985 R1 (Hartley-Good)_& 5982.4_&_0.999884 &_-_& $1.33\times 10^{-8}$_& $-2.16\times 10^{-9}$_& 0 & 0.00217\
316P/LONEOS-Christensen_& 4.328 &_0.166_& - & $7.34\times 10^{-5}$ & 0 & 0 & 0.00212\
73P/Schwassmann-Wachmann 3-B & 3.062_&_0.693 &_-_&_$1.49\times 10^{-8}$ & $1.96\times 10^{-9... |
{n+1}{2}$. In this paper, we prove that if $d^2 + d + 2 \leq n$ and if $d\geq 3$, then the spaces of rational curves are themselves rationally connected.'
address:
- |
Department of Mathematics\
Harvard University\
Cambridge MA 02138
- |
Department of Mathematics\
Massachusetts Institute of Technolo... | { n+1}{2}$. In this paper, we prove that if $ d^2 + d + 2 \leq n$ and if $ d\geq 3 $, then the spaces of intellectual curve are themselves rationally connected.'
address:
- |
Department of Mathematics\
Harvard University\
Cambridge MA 02138
- |
Department of Mathematics\
Massachusett... | {n+1}{2}$. Ij this paper, we prove thxt if $d^2 + d + 2 \leq n$ and if $d\gsq 3$, then the spaces of rational curvxs aee thtiselves rationally connectef.'
address:
- |
Eepartment of Mathciatidd\
Iarvard Universlty\
Cambsidge MA 02138
- |
Geoaxtment of Mathematics\
Massachusetes Instotkte of Technolj... | {n+1}{2}$. In this paper, we prove that + + 2 n$ and if of curves are themselves connected.' address: - Department of Mathematics\ Harvard University\ Cambridge 02138 - | Department of Mathematics\ Massachusetts Institute of Technology\ Cambridge MA 02139 - Joe Harris - Jason Starr bibliography: -'my.bib' title: 'Ratio... | {n+1}{2}$. In this paper, we prove that if $D^2 + d + 2 \leq n$ and iF $d\geq 3$, TheN thE sPaceS of rAtional curves aRE theMselves rationally conneCted.'
aDdREss:
- |
DEPaRtmenT of MathEMaTICs\
HArVaRd UNiVErSity\
CAmbRidge MA 02138
- |
department Of MAtHematics\
MassAChUsetts InstItuTe of Technolo... | {n+1}{2}$. In this paper,we prove t hat i f $ d^2 + d + 2 \ leq n$ and if$ d\ge q 3$, then the spacesof ra ti o nalc ur ves a re them s el v e s r at io nal ly co nnect ed. '
addre ss:
- |
D ep artment of M a th ematics\
Harvard Univ ers ity\
Ca m bridg e M A 021 38
- | De partmentof Mathem a tics\
M ass... | {n+1}{2}$. In_this paper,_we prove that if_$d^2 +_d_+ 2_\leq_n$ and if_$d\geq 3$, then_the spaces of rational_curves are themselves_rationally_connected.'
address:
- |
Department of Mathematics\
Harvard University\
__ Cambridge_MA_02138
-_|
Department_of Mathematics\
_Massachusetts Institute_of Technolo... |
[4] \xrightarrow[F_i]{\hspace*{.25em}{\varepsilon}_i'\hspace*{.1em}} \nolinebreak[4]}
(I_i, <_{_i}) { \nolinebreak[4] \xrightarrow[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}'\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}} \nolineb... | [ 4 ] \xrightarrow[F_i]{\hspace*{.25em}{\varepsilon}_i'\hspace*{.1em } } \nolinebreak[4 ] }
(I_i, < _ { _ i }) { \nolinebreak[4 ] \xrightarrow[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}'\hspace*{.1em } } \nolinebreak[4 ] }
\cdots { \nolinebreak[4 ] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}... | [4] \xgightarrow[F_i]{\hspace*{.25em}{\vareksilon}_i'\hspace*{.1em}} \uilinebceak[4]}
(J_i, <_{_i}) { \nulinebreak[4] \xrightarrow[F_{i+1}]{\hspece*{.25en}{\varekfilon}_{i+1}'\hspace*{.1em}} \nouinebreak[4]}
\cdots { \nioinebreak[4] \xrightavxow[F_k]{\gdpacz*{.25en}{\varepsilon}_k'\hskace*{.1em}} \noliteb... | [4] \xrightarrow[F_i]{\hspace*{.25em}{\varepsilon}_i'\hspace*{.1em}} \nolinebreak[4]} (I_i, <_{_i}) { \nolinebreak[4] \cdots \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}} (I_0, <_{_0}),$$ for $\widetilde{<_{_i}}$ equal to $<_{_i}$ ${\varepsilon}_{i+1}'':={\varepsilon}_{i+1}'$. Otherwi... | [4] \xrightarrow[F_i]{\hspace*{.25em}{\varePsilon}_i'\hspAce*{.1em}} \NolIneBrEak[4]}
(I_I, <_{_i}) { \noLinebreak[4] \xrighTArroW[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}'\HspacE*{.1eM}} \NoliNEbReak[4]}
\cDots { \nolINeBREak[4] \XrIgHtaRrOW[F_K]{\hspaCe*{.25eM}{\varepsIlon}_k'\hspacE*{.1em}} \NoLineb... | [4] \xrightarrow[F_i]{\hs pace*{.25e m}{\v are psi lo n}_i '\hs pace*{.1em}} \ noli nebreak[4]}
(I_i, < _{_i} ){ \n o li nebre ak[4] \ xr i g hta rr ow [F_ {i + 1} ]{\hs pac e*{.25e m}{\vareps ilo n} _{i+1}'\hspa c e* {.1em}} \ nol inebreak[4]}
\cdot s{ \ nolin ebr eak[4 ] \xr i ghtarr ow[F_k]{\ hs p ace*{. ... | [4] _\xrightarrow[F_i]{\hspace*{.25em}{\varepsilon}_i'\hspace*{.1em}} _\nolinebreak[4]}
(I_i,_<_{_i}) {__\nolinebreak[4] _\xrightarrow[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}'\hspace*{.1em}}_ \nolinebreak[4]}
_ \cdots {_ \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}}_ \nolineb... |
_{k=1}^{r}Q_k\gamma_k\right) \nonumber \\
&=& \prod_{j=1}^{\infty}\prod_{\rho=1}^{d(\nu)}\sum_n
\exp\!\left[n\left(-\beta E_j + i\sum_{k=1}^{r}
q_k^{(\rho)}\gamma_k\right) \right].
\label{hatZ}\end{aligned}$$ In the last step, we have expressed the trace in the basis of $n$ -particle Hamiltonian eigenstates. The $q_k... | _ { k=1}^{r}Q_k\gamma_k\right) \nonumber \\
& = & \prod_{j=1}^{\infty}\prod_{\rho=1}^{d(\nu)}\sum_n
\exp\!\left[n\left(-\beta E_j + i\sum_{k=1}^{r }
q_k^{(\rho)}\gamma_k\right) \right ].
\label{hatZ}\end{aligned}$$ In the last step, we have expressed the trace in the footing of $ n$ -particle Hamiltonian ei... | _{k=1}^{r}Q_n\gamma_k\right) \nonumber \\
&=& \pvod_{j=1}^{\infty}\prod_{\rho=1}^{b(\bu)}\sum_n
\xxp\!\left[h\left(-\betx E_j + i\sum_{k=1}^{r}
q_k^{(\rho)}\gamma_k\rigit) \rught].
\lqbel{hatZ}\end{aligned}$$ In ghe last dtep, we yave wxpressed vge tracc in fme bavms of $n$ -particlg Hamiltoniat eigenstates. Dhd $e_k... | _{k=1}^{r}Q_k\gamma_k\right) \nonumber \\ &=& \prod_{j=1}^{\infty}\prod_{\rho=1}^{d(\nu)}\sum_n \exp\!\left[n\left(-\beta E_j q_k^{(\rho)}\gamma_k\right) \label{hatZ}\end{aligned}$$ In last step, we the of $n$ -particle eigenstates. The $q_k^{(\rho)}$ the conserved charges, and the $\gamma_k$ the variables of the Carta... | _{k=1}^{r}Q_k\gamma_k\right) \nonumber \\
&=& \prOd_{j=1}^{\infty}\prOd_{\rho=1}^{D(\nu)}\Sum_N
\eXp\!\leFt[n\lEft(-\beta E_j + i\sum_{k=1}^{R}
Q_k^{(\rhO)}\gamma_k\right) \right].
\label{HatZ}\eNd{ALignED}$$ IN the lAst step, WE hAVE exPrEsSed ThE TrAce in The Basis of $N$ -particle HAmiLtOnian eigenstATeS. The $q_k... | _{k=1}^{r}Q_k\gamma_k\righ t) \nonumb er \\
&= & \ pr od_{ j=1} ^{\infty}\prod _ {\rh o=1}^{d(\nu)}\sum_n
\e xp\!\ le f t[n\ l ef t(-\b eta E_j + i\s um _{ k=1 }^ { r}
q_k^ {(\ rho)}\g amma_k\rig ht) \ right].
\lab e l{ hatZ}\end{ ali gned}$$ In t helast s te p,w e hav e e xpres sed th e trace in the b as i s of ... | _{k=1}^{r}Q_k\gamma_k\right) \nonumber_\\
&=& \prod_{j=1}^{\infty}\prod_{\rho=1}^{d(\nu)}\sum_n
\exp\!\left[n\left(-\beta_E_j + _i\sum_{k=1}^{r}
q_k^{(\rho)}\gamma_k\right) \right].
\label{hatZ}\end{aligned}$$_In_the last_step,_we have expressed_the trace in_the basis of $n$_-particle Hamiltonian eigenstates._The_$q_k... |
, DFG and GSI.
[10]{}
J. C. LeGuillou and J. Zinn-Justin, [*Large-Order Behaviour of Perturbation Theory*]{} (North-Holland, Amsterdam, 1990).
J. Zinn-Justin, [*Quantum Field Theory and Critical Phenomena*]{}, 3rd ed. (Clarendon Press, Oxford, 1996).
J. Fischer, Int. J. Mod. Phys. A [**12**]{}, 3625 (1997).
U. D. ... | , DFG and GSI.
[ 10 ] { }
J. C. LeGuillou and J. Zinn - Justin, [ * Large - Order Behaviour of Perturbation Theory * ] { } (North - Holland, Amsterdam, 1990).
J. Zinn - Justin, [ * Quantum Field Theory and Critical Phenomena * ] { }, 3rd ed. (Clarendon Press, Oxford, 1996).
J. Fischer, Int. J. Mod. Phys. ... | , DFH and GSI.
[10]{}
J. C. LeGuillou ana J. Zinn-Justin, [*Large-Orver Behzviour ow Perturbation Theory*]{} (North-Hlloand, Qmsterdam, 1990).
J. Zinn-Justin, [*Quantum Vield Thwory qnd Criticem Phenomena*]{}, 3rs ed. (Elerendon Press, Owford, 1996).
J. Fiswher, Int. J. Mod. Pfyd. A [**12**]{}, 3625 (1997).
U. D. ... | , DFG and GSI. [10]{} J. C. J. [*Large-Order Behaviour Perturbation Theory*]{} (North-Holland, Field and Critical Phenomena*]{}, ed. (Clarendon Press, 1996). J. Fischer, Int. J. Mod. A [**12**]{}, 3625 (1997). U. D. Jentschura, E. Weniger, and G. Soff, Asymptotic of Resummation and Perturbative Predictions, Los Alamos ... | , DFG and GSI.
[10]{}
J. C. LeGuillou and J. ZInn-Justin, [*LArge-ORdeR BeHaViouR of PErturbation TheORy*]{} (NoRth-Holland, Amsterdam, 1990).
J. ZiNn-JusTiN, [*quanTUm field theory aND CRITicAl phEnoMeNA*]{}, 3rD ed. (ClAreNdon PreSs, Oxford, 1996).
J. FIscHeR, Int. J. Mod. Phys. a [**12**]{}, 3625 (1997).
u. D. ... | , DFG and GSI.
[10]{}
J. C. LeGuil lou a ndJ.Zi nn-J usti n, [*Large-Ord e r Be haviour of Perturbatio n The or y *]{} (N orth- Holland , A m s ter da m, 19 90 ) .
J. Z inn -Justin , [*Quantu m F ie ld Theory an d C ritical Ph eno mena*]{}, 3r d e d. (Cl ar end o n Pre ss, Oxfo rd, 19 9 6).
J . Fischer ,I nt. J... | , DFG_and GSI.
[10]{}
J. C._LeGuillou and J. Zinn-Justin,_[*Large-Order Behaviour_of_Perturbation Theory*]{}_(North-Holland,_Amsterdam, 1990).
J. Zinn-Justin,_[*Quantum Field Theory_and Critical Phenomena*]{}, 3rd_ed. (Clarendon Press,_Oxford,_1996).
J. Fischer, Int. J. Mod. Phys. A [**12**]{}, 3625 (1997).
U. D. ... |
reflection coefficient $R=\gamma/\delta$. $R_\theta$ represents a wave plate which is used to rotate the horizontal polarization with an angle $\theta=arccos(\beta/\alpha)$. DL denotes a time-delay device which is used to make the two wavepackets of the two spatial modes arrive at PBS$_5$ (or PBS$_6$) in the same time... | reflection coefficient $ R=\gamma/\delta$. $ R_\theta$ represents a wave plate which is used to revolve the horizontal polarization with an slant $ \theta = arccos(\beta/\alpha)$. DL denotes a time - delay device which is use to cause the two wavepackets of the two spatial modes arrive at PBS$_5 $ (or PBS$_6 $) in the ... | revlection coefficient $R=\gaoma/\delta$. $R_\theta$ represxnts a save plage which is used to rotate tie hirizobtal polarization with an angle $\theta=arxcos(\ueta/\alpha)$. DL denotes a tlie-demwy dzvmce which is usgd to make tve two wavepacnegs of the two spatial modes arrive at PBS$_5$ (or PHS$_6$) in the same timt... | reflection coefficient $R=\gamma/\delta$. $R_\theta$ represents a wave is to rotate horizontal polarization with a device which is to make the wavepackets of the two spatial modes at PBS$_5$ (or PBS$_6$) in the same time. $D_i$ ($i=1,2,3$) represents a single-photon First, Alice splits the parameter of the spatial-mode... | reflection coefficient $R=\gamMa/\delta$. $R_\thEta$ rePreSenTs A wavE plaTe which is used tO RotaTe the horizontal polarizAtion WiTH an aNGlE $\thetA=arccos(\BEtA/\ALphA)$. Dl dEnoTeS A tIme-deLay Device wHich is used To mAkE the two wavepACkEts of the twO spAtial modes arRivE at PBS$_5$ (Or pBS$_6$) IN the sAme Time... | reflection coefficient $R =\gamma/\d elta$ . $ R_\ th eta$ rep resents a wave plat e which is used to rot ate t he hori z on tal p olariza t io n wit han an gl e $ \thet a=a rccos(\ beta/\alph a)$ .DL denotes a ti me-delay d evi ce which isuse d to m ak e t h e two wa vepac kets o f the t wo spatia lm odes a r ri... | reflection_coefficient $R=\gamma/\delta$._$R_\theta$ represents a wave_plate which_is_used to_rotate_the horizontal polarization_with an angle_$\theta=arccos(\beta/\alpha)$. DL denotes a_time-delay device which_is_used to make the two wavepackets of the two spatial modes arrive at PBS$_5$_(or_PBS$_6$) in_the_same_time... |
_N.ps "fig:"){width="34.50000%"}![\[fig:D10\]Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=10$ and five system sizes (left). The right plot shows a zoom into the region $D\in[80,160]$ for $N=84$. See comments in the text about the irregular approach to the $1/D=0$ limit. The r... | _ N.ps " fig:"){width="34.50000%"}![\[fig: D10\]Examples of the $ D$-dependence of the ground state chiral condensate for $ m / g=0.125 $, $ x=10 $ and five system size (leave). The right plot shows a zoom into the area $ D\in[80,160]$ for $ N=84$. See comments in the text about the irregular access to the $ 1 / D=0 $ ... | _N.ps "fig:"){width="34.50000%"}![\[fig:D10\]Examples of the $D$-dependencg if the grouns state zhiral condensate for $m/g=0.125$, $x=10$ aid fuve ststem sizes (left). The rkght plot shows a zoon into the csgion $D\lu[80,160]$ for $K=84$. See romments in the text aboud the irregulas xp'roach to the $1/D=0$ limit. The r... | _N.ps "fig:"){width="34.50000%"}![\[fig:D10\]Examples of the $D$-dependence of the chiral for $m/g=0.125$, and five system shows zoom into the $D\in[80,160]$ for $N=84$. comments in the text about the approach to the $1/D=0$ limit. The red band represents the uncertainty related to bond dimension, taken as explained in... | _N.ps "fig:"){width="34.50000%"}![\[fig:D10\]Examples of The $D$-dependEnce oF thE grOuNd stAte cHiral condensatE For $m/G=0.125$, $x=10$ and five system sizes (leFt). The RiGHt plOT sHows a Zoom intO ThE REgiOn $d\iN[80,160]$ foR $N=84$. sEe CommeNts In the teXt about the IrrEgUlar approach TO tHe $1/D=0$ limit. ThE r... | _N.ps "fig:"){width="34.50 000%"}![\[ fig:D 10\ ]Ex am ples ofthe $D$-depend e nceof the ground state ch iralco n dens a te for$m/g=0. 1 25 $ , $x =1 0$ an df iv e sys tem sizes(left). Th e r ig ht plot show s a zoom into th e region $D\ in[ 80,160 ]$ fo r $N=8 4$. Seecommen t s in t he text a bo u t thei rr... | _N.ps "fig:"){width="34.50000%"}![\[fig:D10\]Examples_of the_$D$-dependence of the ground_state chiral_condensate_for $m/g=0.125$,_$x=10$_and five system_sizes (left). The_right plot shows a_zoom into the_region_$D\in[80,160]$ for $N=84$. See comments in the text about the irregular approach to the_$1/D=0$_limit. The_r... |
)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|... | ) } $, (d) $ \delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $ \delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $ \delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $ \alpha{\left({0}\right)}=6\cdot2\pi KHz$, $ \gamma= 50\cdot2\pi M(Hz)^2 $ with $ \mu{\left({0}\right)}=-1$. The varying values of $ ... | )}$, (d) $\felta=0.01\cdot\alpha{\left({0}\right)}$, (t) $\delta=0.05\cdot\alpha{\lgfr({0}\right)}$, (f) $\delfa=0.1\cdot\aloha{\left({0}\right)}$. The experimentap paramtners are: $\alpha{\left({0}\rigft)}=6\cdot2\pi NHz$, $\gammq= 50\cdir2\pi M(Hz)^2$ wivg $\mu{\lefb({0}\xight)}=-1$. Bhe vcrbing values of $|\celta|=|... | )}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. parameters $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, 50\cdot2\pi M(Hz)^2$ with $|\delta|=|d\mu/dt|$ related to the of the inertial for slow change in $\mu$ the approximati... | )}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\Delta=0.05\cdot\aLpha{\lEft({0}\RigHt)}$, (F) $\delTa=0.1\cdOt\alpha{\left({0}\rigHT)}$. The Experimental parameters Are: $\alPhA{\Left({0}\RIgHt)}=6\cdoT2\pi KHz$, $\gAMmA= 50\CDot2\Pi m(HZ)^2$ wiTh $\MU{\lEft({0}\riGht)}=-1$. the varyIng values oF $|\deLtA|=|... | )}$, (d) $\delta=0.01\cdot \alpha{\le ft({0 }\r igh t) }$,(e)$\delta=0.05\c d ot\a lpha{\left({0}\right)} $, (f )$ \del t a= 0.1\c dot\alp h a{ \ l eft ({ 0} \ri gh t )} $. Th e e xperime ntal param ete rs are: $\alph a {\ left({0}\r igh t)}=6\cdot2\ piKHz$,$\ gam m a= 50 \cd ot2\p i M(Hz ) ^2$ wi th $\mu{\ le f t({0}... | )}$, (d)_$\delta=0.01\cdot\alpha{\left({0}\right)}$, (e)_$\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The_experimental parameters_are:_$\alpha{\left({0}\right)}=6\cdot2\pi KHz$,_$\gamma=_50\cdot2\pi M(Hz)^2$ with_$\mu{\left({0}\right)}=-1$. The varying_values of $|\delta|=|... |
ant {#ind}
==============================================
Let $f,g\in\mathbb{C}[x,y]$ be two polynomials which are algebraically independant. In this section, we will prove that the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is essentially concentrated in degree $0$. Finally we obtain a formula for the irregularity ... | ant { # ind }
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Let $ f, g\in\mathbb{C}[x, y]$ be two polynomials which are algebraically independant. In this section, we will rise that the complex $ f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is basically concentrated in degree $... | ant {#ind}
==============================================
Let $f,g\in\mathbb{C}[x,y]$ be two polynomials which are amgebraicxlly independant. In this secvion, we wull prove that the comolex $f_+(\matjcal{O}_{\matybb{C}^2}t^g)$ is essentially concentvcted jk degxex $0$. Finally we ontain a forkula for the isrdgblarity ... | ant {#ind} ============================================== Let $f,g\in\mathbb{C}[x,y]$ be two are independant. In section, we will is concentrated in degree Finally we obtain formula for the irregularity number at in terms of some geometric data associated with $f$ and $g$. The complex is essentially concentrated in deg... | ant {#ind}
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Let $f,g\in\mathbb{C}[x,y]$ be tWo polynomiAls whIch Are AlGebrAicaLly independant. iN thiS section, we will prove thaT the cOmPLex $f_+(\MAtHcal{O}_{\Mathbb{C}^2}E^G)$ iS ESseNtIaLly CoNCeNtratEd iN degree $0$. finally we oBtaIn A formula for tHE iRregularitY ... | ant {#ind}
=============== ========== ===== === === == ==== ====
Let $f,g\in\ m athb b{C}[x,y]$ be two poly nomia ls whic h a re al gebraic a ll y ind ep en dan t. In this se ction,we will pr ove t hat the comp l ex $f_+(\mat hca l{O}_{\mathb b{C }^2}e^ g) $ i s esse nti allyconcen t ratedin degree $ 0 $. Fin ... | ant {#ind}
==============================================
Let_$f,g\in\mathbb{C}[x,y]$ be_two polynomials which are_algebraically independant._In_this section,_we_will prove that_the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$_is essentially concentrated in_degree $0$. Finally_we_obtain a formula for the irregularity ... |
).
A. H. Safavi-Naeini, T. P.Mayer Alegre, J.Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Electromagnetically induced transparency and slow light with optomechanics, Nature (London) **472**, 69 (2011).
L. Tian, Robust photon entanglement via quantum interference in optomechanical ... | ).
A. H. Safavi - Naeini, T. P.Mayer Alegre, J.Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Electromagnetically induced transparency and slow light with optomechanics, Nature (London) * * 472 * *, 69 (2011).
L. Tian, full-bodied photon web via quantum hindrance in optomechanic... | ).
A. H. Safavi-Naeini, T. P.Mayer Auegre, J.Chan, M. Enxhenfixld, M. Wjnger, Q. Uin, J. T. Hill, D. E. Chang, and O. Pqintee, Electromagnetically knduced tgansparenxy aid slow light wivg optomcehanidd, Nacuce (London) **472**, 69 (2011).
L. Tlan, Robust [hoton entanglamdnc via quantum interference in optomeshanicak ... | ). A. H. Safavi-Naeini, T. P.Mayer Alegre, Eichenfield, Winger, Q. J. T. Hill, Painter, induced transparency and light with optomechanics, (London) **472**, 69 (2011). L. Tian, photon entanglement via quantum interference in optomechanical interfaces, Phys. Rev. Lett. **110**, 233602 Y. D. Wang and A. A. Clerk, Reservo... | ).
A. H. Safavi-Naeini, T. P.Mayer AlegRe, J.Chan, M. EiChenfIelD, M. WInGer, Q. lin, J. t. Hill, D. E. Chang, anD o. PaiNter, ElectromagneticallY induCeD TranSPaRency And slow LIgHT WitH oPtOmeChANiCs, NatUre (london) **472**, 69 (2011).
L. tian, Robust PhoToN entanglemenT ViA quantum inTerFerence in optOmeChanicAl ... | ).
A. H. Safavi-Naeini, T . P.MayerAlegr e,J.C ha n, M . Ei chenfield, M.W inge r, Q. Lin, J. T. Hill, D. E .C hang , a nd O. Painte r ,E l ect ro ma gne ti c al ly in duc ed tran sparency a ndsl ow light wit h o ptomechani cs, Nature (Lon don ) **47 2* *,6 9 (20 11) .
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L. Tian, Robust photon entanglement via quantum interference_in_optomechanical ... |
=0} =\left. \varphi'\frac{d}{dr}\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)
\right|_{r=0} = 0,$$ and so $$\begin{aligned}
\sum_{i=1}^{n-1}\frac{d^2}{dr^2}\left(\varphi\left.\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)\right)
\right|_{r=0} &= \sum_{i=1}^{n-1}\left.\left(\varphi'\frac{d^2}{dr^2}\left(\frac{L[\gamma_{i}(r,.)]}... | = 0 } = \left. \varphi'\frac{d}{dr}\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)
\right|_{r=0 } = 0,$$ and so $ $ \begin{aligned }
\sum_{i=1}^{n-1}\frac{d^2}{dr^2}\left(\varphi\left.\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)\right)
\right|_{r=0 } & = \sum_{i=1}^{n-1}\left.\left(\varphi'\frac{d^2}{dr^2}\left(\frac{L[\g... | =0} =\levt. \varphi'\frac{d}{dr}\left(\frac{U[\gamma_{i}(r,.)]}{2}\right)
\riyyt|_{r=0} = 0,$$ end so $$\gegin{aliened}
\sum_{i=1}^{n-1}\frac{d^2}{dr^2}\left(\varphi\leht.\ledt(\frax{L[\gamma_{i}(r,.)]}{2}\right)\right)
\rigft|_{r=0} &= \sum_{i=1}^{j-1}\left.\lefr(\varkhi'\frac{d^2}{dr^2}\left(\frar{M[\gamma_{i}(v,.)]}... | =0} =\left. \varphi'\frac{d}{dr}\left(\frac{L[\gamma_{i}(r,.)]}{2}\right) \right|_{r=0} = 0,$$ and \sum_{i=1}^{n-1}\frac{d^2}{dr^2}\left(\varphi\left.\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)\right) &= \sum_{i=1}^{n-1}\left.\left(\varphi'\frac{d^2}{dr^2}\left(\frac{L[\gamma_{i}(r,.)]}{2}\right) +\varphi''\left(\frac{d}... | =0} =\left. \varphi'\frac{d}{dr}\left(\frac{l[\gamma_{i}(r,.)]}{2}\riGht)
\riGht|_{R=0} = 0,$$ anD sO $$\begIn{alIgned}
\sum_{i=1}^{n-1}\frac{D^2}{Dr^2}\leFt(\varphi\left.\left(\frac{L[\gAmma_{i}(R,.)]}{2}\rIGht)\rIGhT)
\righT|_{r=0} &= \sum_{i=1}^{n-1}\LEfT.\LEft(\VaRpHi'\fRaC{D^2}{dR^2}\left(\FraC{L[\gamma_{I}(r,.)]}... | =0} =\left. \varphi'\frac{ d}{dr}\lef t(\fr ac{ L[\ ga mma_ {i}( r,.)]}{2}\righ t )
\r ight|_{r=0} = 0,$$ and so $ $\ b egin { al igned }
\sum_ { i= 1 } ^{n -1 }\ fra c{ d ^2 }{dr^ 2}\ left(\v arphi\left .\l ef t(\frac{L[\g a mm a_{i}(r,.) ]}{ 2}\right)\ri ght )
\rig ht |_{ r =0} & = \ sum_{ i=1}^{ n -1}\le ft.\left( ... | =0} =\left._\varphi'\frac{d}{dr}\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)
\right|_{r=0} =_0,$$ and so $$\begin{aligned}
\sum_{i=1}^{n-1}\frac{d^2}{dr^2}\left(\varphi\left.\left(\frac{L[\gamma_{i}(r,.)]}{2}\right)\right)
\right|_{r=0}_&= \sum_{i=1}^{n-1}\left.\left(\varphi'\frac{d^2}{dr^2}\left(\frac{L[\gamma_{i}(r,.)]}... |
\in \partial {\|\beta^\star\|}_\infty$ where $$\begin{aligned}
\partial {\|\beta^\star\|}_\infty
&= \bigl\{ g:~ \langle g, \beta^\star\rangle = {\|\beta^\star\|}_\infty,~ {\|g\|}_1\leq 1 \bigr\} \nonumber\\
&= \bigl\{ g:~ g_i = 0 \text{ if } {|\beta^\star_i|}<{\|\beta^\star\|}_\infty,~
{\|g\|}_1=1,~ g \circ \beta \geq... | \in \partial { \|\beta^\star\|}_\infty$ where $ $ \begin{aligned }
\partial { \|\beta^\star\|}_\infty
& = \bigl\ { g:~ \langle g, \beta^\star\rangle = { \|\beta^\star\|}_\infty,~ { \|g\|}_1\leq 1 \bigr\ } \nonumber\\
& = \bigl\ { g:~ g_i = 0 \text { if } { |\beta^\star_i|}<{\|\beta^\star\|}_\infty,~
{ \|g\|}_1=... | \in \oartial {\|\beta^\star\|}_\infty$ whtre $$\begin{aligned}
\pcetial {\|\ueta^\stad\|}_\infty
&= \bkgl\{ g:~ \langle g, \beta^\star\ranglx = {\|\bwta^\stqr\|}_\infty,~ {\|g\|}_1\leq 1 \bigr\} \nonjmber\\
&= \bigp\{ g:~ g_i = 0 \texu{ if } {|\beta^\star_i|}<{\|\bevz^\star\|}_\inncy,~
{\|g\|}_1=1,~ f \cire \ueta \geq... | \in \partial {\|\beta^\star\|}_\infty$ where $$\begin{aligned} \partial {\|\beta^\star\|}_\infty g:~ g, \beta^\star\rangle {\|\beta^\star\|}_\infty,~ {\|g\|}_1\leq 1 g_i 0 \text{ if {|\beta^\star_i|}<{\|\beta^\star\|}_\infty,~ {\|g\|}_1=1,~ g \beta \geq 0 \bigr\}.\label{eq:subdiff-linf}\end{aligned}$$ This time, that a... | \in \partial {\|\beta^\star\|}_\infty$ wheRe $$\begin{aliGned}
\pArtIal {\|\BeTa^\stAr\|}_\inFty
&= \bigl\{ g:~ \langle G, \Beta^\Star\rangle = {\|\beta^\star\|}_\inftY,~ {\|g\|}_1\leq 1 \BiGR\} \nonUMbEr\\
&= \bigL\{ g:~ g_i = 0 \texT{ If } {|\BETa^\sTaR_i|}<{\|\BetA^\sTAr\|}_\Infty,~
{\|G\|}_1=1,~ g \cIrc \beta \Geq... | \in \partial {\|\beta^\sta r\|}_\inft y$ wh ere $$ \b egin {ali gned}
\partial {\|\ beta^\star\|}_\infty
& = \bi gl \ { g: ~ \ langl e g, \b e ta ^ \ sta r\ ra ngl e= { \|\be ta^ \star\| }_\infty,~ {\ |g \|}_1\leq 1\ bi gr\} \nonu mbe r\\
&= \bigl \{g:~ g_ i= 0 \text { i f } { |\beta ^ \star_ i|}<{\|\b et a ^\star ... | \in \partial_{\|\beta^\star\|}_\infty$ where_$$\begin{aligned}
\partial {\|\beta^\star\|}_\infty
&= \bigl\{ g:~_\langle g,_\beta^\star\rangle_= {\|\beta^\star\|}_\infty,~_{\|g\|}_1\leq_1 \bigr\} \nonumber\\
&=_\bigl\{ g:~ g_i_= 0 \text{ if_} {|\beta^\star_i|}<{\|\beta^\star\|}_\infty,~
{\|g\|}_1=1,~_g_\circ \beta \geq... |
mathcal{R}|_g \geq (\omega_n -\epsilon_i )(\alpha \lambda_i^{-1})^4$$ for some sequence $\epsilon_i \to 0$. The main idea is to use the fact that, for $i\in \mathbb{N}$ sufficiently large, $(B^X(y_i,\alpha \lambda_i^{-1}),\lambda_i d,y_i)$ is arbitrarily close to a Euclidean ball in the pointed Gromov-Hausdroff sense, ... | mathcal{R}|_g \geq (\omega_n -\epsilon_i) (\alpha \lambda_i^{-1})^4$$ for some sequence $ \epsilon_i \to 0$. The main idea is to use the fact that, for $ i\in \mathbb{N}$ sufficiently big, $ (B^X(y_i,\alpha \lambda_i^{-1}),\lambda_i d, y_i)$ is randomly close to a Euclidean musket ball in the pointed Gromov - Hausdroff... | matjcal{R}|_g \geq (\omega_n -\epsilok_i )(\alpha \lambda_i^{-1})^4$$ for soke seqhence $\eprilon_i \to 0$. The main idea is vo uwe tht fact that, for $i\in \mathbb{N}$ dufficiebtly oarge, $(B^X(y_i,\empha \lambda_i^{-1}),\lzlbda_n v,y_i)$ is arbitrarlly close tm a Euclidean taul in the pointed Gromov-Hausdroff senfe, ... | mathcal{R}|_g \geq (\omega_n -\epsilon_i )(\alpha \lambda_i^{-1})^4$$ for $\epsilon_i 0$. The idea is to $i\in sufficiently large, $(B^X(y_i,\alpha d,y_i)$ is arbitrarily to a Euclidean ball in the Gromov-Hausdroff sense, and to then appeal to a volume convergence theorem for Riemannian with integral Ricci lower bounds... | mathcal{R}|_g \geq (\omega_n -\epsilon_I )(\alpha \lambDa_i^{-1})^4$$ foR soMe sEqUencE $\epsIlon_i \to 0$. The main IDea iS to use the fact that, for $i\iN \mathBb{n}$ SuffICiEntly Large, $(B^X(Y_I,\aLPHa \lAmBdA_i^{-1}),\lAmBDa_I d,y_i)$ iS arBitrariLy close to a eucLiDean ball in thE PoInted GromoV-HaUsdroff sense, ... | mathcal{R}|_g \geq (\omega _n -\epsil on_i)(\ alp ha \la mbda _i^{-1})^4$$ f o r so me sequence $\epsilon_ i \to 0 $ . Th e m ain i dea ist ou s e t he f act t h at , for $i \in \ma thbb{N}$ s uff ic iently large , $ (B^X(y_i,\ alp ha \lambda_i ^{- 1}),\l am bda _ i d,y _i) $ isarbitr a rily c lose to a E u cli... | mathcal{R}|_g \geq_(\omega_n -\epsilon_i_)(\alpha \lambda_i^{-1})^4$$ for some_sequence $\epsilon_i_\to_0$. The_main_idea is to_use the fact_that, for $i\in \mathbb{N}$_sufficiently large, $(B^X(y_i,\alpha_\lambda_i^{-1}),\lambda_i_d,y_i)$ is arbitrarily close to a Euclidean ball in the pointed Gromov-Hausdroff sense, ... |
2007**]{} (2007) 237, \[[[arXiv:0711.0023]{}](http://arxiv.org/abs/0711.0023)\].
B. B. Brandt, G. Endrödi, and S. Schmalzbauer, [*QCD phase diagram for nonzero isospin-asymmetry*]{}, [*Phys. Rev. D*]{} [**97**]{} (2018) 054514, \[[[arXiv:1712.08190]{}](http://arxiv.org/abs/1712.08190)\].
J. B. Kogut, M. A. Stephanov,... | 2007 * * ] { } (2007) 237, \[[[arXiv:0711.0023]{}](http://arxiv.org / abs/0711.0023)\ ].
B. B. Brandt, G. Endrödi, and S. Schmalzbauer, [ * QCD phase diagram for nonzero isospin - asymmetry * ] { }, [ * Phys. Rev. D * ] { } [ * * 97 * * ] { } (2018) 054514, \[[[arXiv:1712.08190]{}](http://arxiv.org / abs/171... | 2007**]{} (2007) 237, \[[[wrXiv:0711.0023]{}](http://arxiv.org/abs/0711.0023)\].
B. B. Brxndt, G. Endrödi, anb S. Schmelzbaued, [*QCD phxse diagram for nonzero isos'in-awymmeugy*]{}, [*Phys. Rev. D*]{} [**97**]{} (2018) 054514, \[[[arXix:1712.08190]{}](http://arxin.org/abs/1712.08190)\].
J. B. Joguu, M. A. Stephanov,... | 2007**]{} (2007) 237, \[[[arXiv:0711.0023]{}](http://arxiv.org/abs/0711.0023)\]. B. B. Brandt, and Schmalzbauer, [*QCD diagram for nonzero (2018) \[[[arXiv:1712.08190]{}](http://arxiv.org/abs/1712.08190)\]. J. B. M. A. Stephanov, Toublan, J. J. M. Verbaarschot, and Zhitnitsky, [*QCD-like theories at finite baryon densi... | 2007**]{} (2007) 237, \[[[arXiv:0711.0023]{}](http://arxiv.org/abs/0711.0023)\].
B. B. BraNdt, G. EndrödI, and S. schMalZbAuer, [*qCD pHase diagram for NOnzeRo isospin-asymmetry*]{}, [*Phys. rev. D*]{} [**97**]{} (2018) 054514, \[[[aRXIV:1712.08190]{}](httP://ArXiv.orG/abs/1712.08190)\].
J. B. KOGuT, m. a. StEpHaNov,... | 2007**]{} (2007) 237, \[[[ arXiv:0711 .0023 ]{} ](h tt p:// arxi v.org/abs/0711 . 0023 )\].
B. B. Brandt, G. Endr öd i , an d S . Sch malzbau e r, [ *QC Dph ase d i ag ram f ornonzero isospin-a sym me try*]{}, [*P h ys . Rev. D*] {}[**97**]{} ( 201 8) 054 51 4,\ [[[ar Xiv :1712 .08190 ] {}](ht tp://arxi v. o rg/ab... | 2007**]{} (2007)_237, \[[[arXiv:0711.0023]{}](http://arxiv.org/abs/0711.0023)\].
B. B. Brandt,_G. Endrödi, and S. Schmalzbauer, [*QCD_phase diagram_for_nonzero isospin-asymmetry*]{},_[*Phys._Rev. D*]{} [**97**]{}_(2018) 054514, \[[[arXiv:1712.08190]{}](http://arxiv.org/abs/1712.08190)\].
J. B. Kogut,_M. A. Stephanov,... |
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Ramani A., Grammaticos B., Tremblay S., Integrable system without the painlevé property, [J. Phys. A: Math... | - A Macsyma package for Painlevé analysis of ordinary differential equations, [ * Computer Physics Communications * ] { } [ * * 42 * * ] { } (1986), 359 - 383.
Ince E.L., average differential equation, Dover, New York, 1956.
Ramani A., Grammaticos B., Tremblay S., Integrable system without the painlevé property, ... | - A Macsyma package for Paiklevé analysis of ordinacy diffsrential equations, [*Computer Physics Rommynicaupons*]{} [**42**]{} (1986), 359-383.
Ince E.L., Ordinafy differvntial eqyatiibs, Dover, Nxs York, 1956.
Vcmani W., Grcmnaticos B., Tremnlay S., Intecrable system fighlut the painlevé property, [J. Phys. A: Mwth... | - A Macsyma package for Painlevé analysis differential [*Computer Physics [**42**]{} (1986), 359-383. Dover, York, 1956. Ramani Grammaticos B., Tremblay Integrable system without the painlevé property, Phys. A: Math. Gen]{} [**33**]{} (2000), 3045-3052. Tamizhmani K.M., Gammaticos B., Ramani A., all integrable evolutio... | - A Macsyma package for PainlevÉ analysis oF ordiNarY diFfErenTial Equations, [*CompuTEr PhYsics Communications*]{} [**42**]{} (1986), 359-383.
IncE E.L., OrDiNAry dIFfErentIal equaTIoNS, dovEr, neW YoRk, 1956.
rAmAni A., GRamMaticos b., Tremblay S., intEgRable system wIThOut the painLevÉ property, [J. PhYs. A: math... | - A Macsyma package for P ainlevé an alysi s o f o rd inar y di fferential equ a tion s, [*Computer PhysicsCommu ni c atio n s* ]{} [ **42**] { }( 1 986 ), 3 59- 38 3 .
Ince E. L., Ord inary diff ere nt ial equation s ,Dover, New Yo rk, 1956.
R ama ni A., G ram m atico s B ., Tr emblay S., In tegrablesy s tem wi ... | -_A Macsyma_package for Painlevé analysis_of ordinary_differential_equations, [*Computer_Physics_Communications*]{} [**42**]{} (1986),_359-383.
Ince E.L., Ordinary_differential equations, Dover, New_York, 1956.
Ramani A.,_Grammaticos_B., Tremblay S., Integrable system without the painlevé property, [J. Phys. A: Math... |
=0.3-0) = 0.75$. These are the predicted stellar mass growth factors of the ICL between $z=0.5-0$ and $z=0.3-0$ from [@Contini2014]. Substituting these values into the relevant equations above, we find $(\Delta M_{\star})_{\mathrm{ICL}} \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$.
We measure the amount of stellar mas... | = 0.3 - 0) = 0.75$. These are the predicted stellar mass growth divisor of the ICL between $ z=0.5 - 0 $ and $ z=0.3 - 0 $ from [ @Contini2014 ]. substitute these values into the relevant equations above, we find $ (\Delta M_{\star})_{\mathrm{ICL } } \sim 2 \times 10^{11 } \, \mathrm{M_{\odot}}$.
We quantify the amo... | =0.3-0) = 0.75$. Hhese are the predicted rtellar mass growth fartors or the ICU between $z=0.5-0$ and $z=0.3-0$ from [@Contiii2014]. Sybstiulting these values ingo the repevant ewuatmons above, we fiis $(\Delta M_{\star})_{\jwthrk{MCL}} \sim 2 \times 10^{11} \, \mathrm{M_{\ogot}}$.
We measure dhd cmount of stellar mas... | =0.3-0) = 0.75$. These are the predicted growth of the between $z=0.5-0$ and values the relevant equations we find $(\Delta \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$. measure the amount of stellar mass contained in the close companions of BCGs be $\sim 4 \times 10^{11} \, \mathrm{M_{\odot}}$, of which half is transf... | =0.3-0) = 0.75$. These are the predicted stellAr mass growTh facTorS of ThE ICL BetwEen $z=0.5-0$ and $z=0.3-0$ from [@CoNTini2014]. substituting these valueS into ThE ReleVAnT equaTions abOVe, WE FinD $(\DElTa M_{\StAR})_{\mAthrm{iCL}} \Sim 2 \timeS 10^{11} \, \mathrm{M_{\odOt}}$.
WE mEasure the amoUNt Of stellar mAs... | =0.3-0) = 0.75$. These are the predi ctedste lla rmass gro wth factors of theICL between $z=0.5-0$and $ z= 0 .3-0 $ f rom [ @Contin i 20 1 4 ].Su bs tit ut i ng thes e v alues i nto the re lev an t equationsa bo ve, we fin d $ (\Delta M_{\ sta r})_{\ ma thr m {ICL} } \ sim 2 \time s 10^{1 1} \, \ma th r m{M_{\ o ... | =0.3-0) =_0.75$. These_are the predicted stellar_mass growth_factors_of the_ICL_between $z=0.5-0$ and_$z=0.3-0$ from [@Contini2014]._Substituting these values into_the relevant equations_above,_we find $(\Delta M_{\star})_{\mathrm{ICL}} \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$.
We measure the amount of_stellar_mas... |
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\
\end{array} \right)
\left(
\be... | a_{11 } & a_{12 } & a_{13 } & a_{14 } & a_{15 } \\
a_{21 } & a_{22 } & a_{23 } & a_{24 } & a_{25 } \\
a_{31 } & a_{32 } & a_{33 } & a_{34 } & a_{35 } \\
a_{41 } & a_{42 } & a_{43 } & a_{44 } & a_{45 } \\
a_{51 } & a_{52 } & a_{53 } & a_{... |
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
a_{41} & a_{42} & a_{43} & e_{44} & a_{45} \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\
\enf{array} \rught)
\owft(
\be... | a_{11} & a_{12} & a_{13} & a_{14} \\ & a_{22} a_{23} & a_{24} a_{32} a_{33} & a_{34} a_{35} \\ a_{41} a_{42} & a_{43} & a_{44} & \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\ \end{array} \right) \begin{array}{c} f_1\\ f_3\\ f_{5+7} \\ f_{5-7} \\ f_9 \end{array} \right) \.$$ Similarly, by $\bar ( + \gamma_5 u(p,s) $,... |
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\
\End{array} \riGht)
\leFt(
\bE... |
a_{11} & a_{12} & a_ {13} & a_ {14} & a_ {1 5} \\
a_{21} & a _{22 } & a_{23} & a_{24} & a _ {25} \ \
a_ {31} & a _ {32 }& a _{ 3 3} & a _{3 4} & a_{35} \ \ a_ {41} & a_ { 42 } & a_{4 3}& a_{44} & a_{45} \\
a_{5 1}& a_{52} & a_ {53} & a _{ 5 4} & a_{55} \\ \end{array} \righ ... |
a_{11} &_ _ a_{12} & _ a_{13}_&_ a_{14}__& a_{15}_ _\\
a_{21} & _ a_{22}_&_ a_{23} & a_{24} & a_{25} \\
a_{31}_&_ __a_{32}_& a_{33} &_ a_{34} & _ a_{35}_ \\
a_{41} & a_{42}_&_ a_{43}_& a_{44} & a_{45} _ \\
a_{51} & _ a_{52} &___a_{53} & a_{54}_ & a_{55} _ \\
\end{array} \right)
\left(
\be... |
underset{( r_k\ge r_{ k+ 1})}{\longmapsto} { \ifthenelse{\equal{r_{ k+
2}}{}}{\renewcommand*{}{accepting}}{\renewcommand*{}{}}
\settowidth{\myautolength}{$ \max( b_k, b_{ k+ 1}- p_k)$}
\addtolength{\myautolength}{1em}
{\ifthenelse{\lengthtest{\the\myautolength>\minautolength}}
... | underset { (r_k\ge r _ { k+ 1})}{\longmapsto } { \ifthenelse{\equal{r _ { k+
2}}{}}{\renewcommand*{}{accepting}}{\renewcommand * { } { } }
\settowidth{\myautolength}{$ \max (b_k, b _ { k+ 1}- p_k)$ }
\addtolength{\myautolength}{1em }
{ \ifthenelse{\lengthtest{\the\myautolength>... | undfrset{( r_k\ge r_{ k+ 1})}{\longmapstu} { \ifthendlse{\equal{r_{ k+
2}}{}}{\renewcommaid*{}{acxeptibg}}{\renewcommand*{}{}}
\settowidgh{\myautolvngth}{$ \max( b_k, u_{ k+ 1}- p_k)$}
\addtolengvg{\myautolength}{1sl}
{\iftkeielse{\lengthtest{\jhe\myautolencth>\minautolengdh}}
... | underset{( r_k\ge r_{ k+ 1})}{\longmapsto} { \ifthenelse{\equal{r_{ \settowidth{\myautolength}{$ b_k, b_{ 1}- p_k)$} \addtolength{\myautolength}{1em} >=1pt] initial] (1) {$r_k$}; {}, right=\myautolength of (end) {$r_{ k+ 2}$}; \path[->] (1) node[above] {$ p_k+ p_{ k+ 1}$} node[below] {$ \max( b_k, b_{ k+ 1}- (end); \en... | underset{( r_k\ge r_{ k+ 1})}{\longmapsto} { \iFthenelse{\eQual{r_{ K+
2}}{}}{\reNewCoMmanD*{}{accEpting}}{\renewcomMAnd*{}{}}
\sEttowidth{\myautolength}{$ \mAx( b_k, b_{ K+ 1}- p_K)$}
\AddtOLeNgth{\mYautoleNGtH}{1EM}
{\ifThEnElsE{\lENgThtesT{\thE\myautoLength>\minaUtoLeNgth}}
... | underset{( r_k\ge r_{ k+ 1 })}{\longm apsto } { \ift henelse{\equal{r_{ k+ 2 } }{}} { \r enewc ommand* { }{ a c cep ti ng }}{ \r e ne wcomm and *{}{}}\settowidt h{\ my autolength}{ $ \ max( b_k,b_{ k+ 1}- p_k) $}\addto le ngt h {\mya uto lengt h}{1em }
{\ift henelse{\ le n gthtes t {\the\m ... | underset{( r_k\ge_r_{ k+_1})}{\longmapsto} { _ __ __ _ _ _ __ \ifthenelse{\equal{r_{ k+
2}}{}}{\renewcommand*{}{accepting}}{\renewcommand*{}{}}
\settowidth{\myautolength}{$ \max(_b_k,_b_{ k+_1}-_p_k)$}
\addtolength{\myautolength}{1em}
{\ifthenelse{\lengthtest{\the\myautolength>\minautolength}}
_ _ ... |
79 & 0.047 & -20 03 35.439 & 0.044 & 0.90 &0.03 & 9.22\
G10.34-0.14 & 18 09 01.456 & 0.010 & -20 05 07.703 & 0.010 & 5.33 &0.04 & 9.36\
G10.34-0.14 & 18 08 59.983 & 0.059 & -20 03 35.502 & 0.054 & 0.81 &0.03 & 9.36\
G10.34-0.14 & 18 09 01.457 & 0.007 & -20 05 07.698 & 0.007 & 9.37 &0.05 & 9.50\
G10.34-0.14 & 18 08 59.9... | 79 & 0.047 & -20 03 35.439 & 0.044 & 0.90 & 0.03 & 9.22\
G10.34 - 0.14 & 18 09 01.456 & 0.010 & -20 05 07.703 & 0.010 & 5.33 & 0.04 & 9.36\
G10.34 - 0.14 & 18 08 59.983 & 0.059 & -20 03 35.502 & 0.054 & 0.81 & 0.03 & 9.36\
G10.34 - 0.14 & 18 09 01.457 & 0.007 & -20 05 07.698 & 0.007 & 9.37 & 0.05 & 9.50\
G10.34... | 79 & 0.047 & -20 03 35.439 & 0.044 & 0.90 &0.03 & 9.22\
G10.34-0.14 & 18 09 01.456 & 0.010 & -20 05 07.703 & 0.010 & 5.33 &0.04 & 9.36\
G10.34-0.14 & 18 08 59.983 & 0.059 & -20 03 35.502 & 0.054 & 0.81 &0.03 & 9.36\
J10.34-0.14 & 18 09 01.457 & 0.007 & -20 05 07.698 & 0.007 & 9.37 &0.05 & 9.50\
G10.34-0.14 & 18 08 59.9... | 79 & 0.047 & -20 03 35.439 & &0.03 & G10.34-0.14 & 18 -20 07.703 & 0.010 5.33 &0.04 & G10.34-0.14 & 18 08 59.983 & & -20 03 35.502 & 0.054 & 0.81 &0.03 & 9.36\ G10.34-0.14 & 09 01.457 & 0.007 & -20 05 07.698 & 0.007 & 9.37 &0.05 9.50\ & 08 & 0.105 & -20 03 35.460 & 0.097 & 0.54 &0.04 & 9.50\ G10.34-0.14 & 18 01.457 & 0... | 79 & 0.047 & -20 03 35.439 & 0.044 & 0.90 &0.03 & 9.22\
G10.34-0.14 & 18 09 01.456 & 0.010 & -20 05 07.703 & 0.010 & 5.33 &0.04 & 9.36\
G10.34-0.14 & 18 08 59.983 & 0.059 & -20 03 35.502 & 0.054 & 0.81 &0.03 & 9.36\
G10.34-0.14 & 18 09 01.457 & 0.007 & -20 05 07.698 & 0.007 & 9.37 &0.05 & 9.50\
G10.34-0.14 & 18 08 59.9... | 79 & 0.047 & -20 03 35.439 & 0.044 & 0.90 &0 .03 & 9.2 2\
G 10.34-0.14 & 1 8 0901.456 & 0.010 & -20 0 5 07. 70 3 & 0 . 01 0 & 5 .33 &0. 0 4& 9.3 6\
G 10. 34 - 0. 14 &1808 59.9 83 & 0.059 &-2 0 03 35.502& 0 .054 & 0.8 1 & 0.03 & 9.36\
G1 0.34-0 .1 4 & 18 09 01 .457& 0.00 7 & -20 05 07.69 8& 0.007 & 9.37& ... | 79 &_0.047 &_-20 03 35.439 &_0.044 &_0.90_&0.03 &_9.22\
G10.34-0.14_& 18 09_01.456 & 0.010_& -20 05 07.703_& 0.010 &_5.33_&0.04 & 9.36\
G10.34-0.14 & 18 08 59.983 & 0.059 & -20 03 35.502 &_0.054_& 0.81_&0.03_&_9.36\
G10.34-0.14 & 18 09 01.457_& 0.007 & -20 05_07.698 &_0.007 & 9.37 &0.05 & 9.50\
G10.34-0.14 & 18_08_59.9... |
matrices $\sigma_i$ ($\tau_i$) operate in spin (particle-hole) space. Choosing the impurity spin ${\bf S}$ to point along the $z$ direction, the $4\times 4$ Hamiltonian separates into independent $2\times2$ Hamiltonians $${\cal H}_\pm = \xi_{\bf p} \tau_z (V\tau_z \pm J S) \delta({\bf r}) + \Delta \tau_x.
\label{Hpm}... | matrices $ \sigma_i$ ($ \tau_i$) operate in spin (particle - hole) quad. choose the impurity spin $ { \bf S}$ to point along the $ z$ direction, the $ 4\times 4 $ Hamiltonian separates into autonomous $ 2\times2 $ Hamiltonians $ $ { \cal H}_\pm = \xi_{\bf p } \tau_z (V\tau_z \pm J S) \delta({\bf r }) + \Delta \tau_x.
... | mahrices $\sigma_i$ ($\tau_i$) operaue in spin (particle-hole) vpace. Dhoosing the impurity spin ${\bf S}$ to plibt aling the $z$ direction, thd $4\times 4$ Jamiltonuan wwparates iifo indeizndenf $2\timzs2$ Hamiltonians $${\gal H}_\pm = \xi_{\tf p} \tau_z (V\tau_s \om J S) \delta({\bf r}) + \Delta \tau_x.
\label{Hpm}... | matrices $\sigma_i$ ($\tau_i$) operate in spin (particle-hole) the spin ${\bf to point along 4$ separates into independent Hamiltonians $${\cal H}_\pm \xi_{\bf p} \tau_z (V\tau_z \pm J \delta({\bf r}) + \Delta \tau_x. \label{Hpm}$$ A standard calculation [@SPientka2013] shows that $H_+$ has one subgap solution $[u_\eps... | matrices $\sigma_i$ ($\tau_i$) operate In spin (partIcle-hOle) SpaCe. chooSing The impurity spiN ${\Bf S}$ tO point along the $z$ directiOn, the $4\TiMEs 4$ HaMIlToniaN separaTEs INTo iNdEpEndEnT $2\TiMes2$ HaMilTonians $${\Cal H}_\pm = \xi_{\bf P} \taU_z (v\tau_z \pm J S) \delTA({\bF r}) + \Delta \tau_X.
\laBel{Hpm}... | matrices $\sigma_i$ ($\ta u_i$) oper ate i n s pin ( part icle -hole) space.C hoos ing the impurity spin${\bf S } $ to po int a long th e $ z $ di re ct ion ,t he $4\t ime s 4$ Ha miltoniansep ar ates into in d ep endent $2\ tim es2$ Hamilto nia ns $${ \c alH }_\pm =\xi_{ \bf p} \tau_z (V\tau_z \ p m J S) \del... | matrices_$\sigma_i$ ($\tau_i$)_operate in spin (particle-hole)_space. Choosing_the_impurity spin_${\bf_S}$ to point_along the $z$_direction, the $4\times 4$_Hamiltonian separates into_independent_$2\times2$ Hamiltonians $${\cal H}_\pm = \xi_{\bf p} \tau_z (V\tau_z \pm J S) \delta({\bf r})_+_\Delta \tau_x.
_\label{Hpm}... |
i & 0 & 1
\end{pmatrix} N^{-1}(z),$$ which gives us.
This completes the proof of Proposition \[prop:p-1\].
We conclude this section by evaluating $E(-1)$ and $E'(-1)$ for later use. The calculations are straightforward and cumbersome by combining and the asymptotics of $N(z)$ and $f(z)$ given in and. We omit the de... | i & 0 & 1
\end{pmatrix } N^{-1}(z),$$ which gives us.
This completes the proof of Proposition \[prop: p-1\ ].
We conclude this incision by measure $ E(-1)$ and $ E'(-1)$ for later use. The calculations are aboveboard and cumbersome by combining and the asymptotics of $ N(z)$ and $ f(z)$ give in and. We omit t... | i & 0 & 1
\end{pmatrix} N^{-1}(z),$$ which nives us.
This comklwtes tie proor of Prooosition \[prop:p-1\].
We conclude thms swctiob by evaluating $E(-1)$ and $D'(-1)$ for latvr use. Thw caoxulations ede stralyhtfodaard end cumbersome ny combininc and the asym[tutncs of $N(z)$ and $f(z)$ given in and. We omie the dr... | i & 0 & 1 \end{pmatrix} N^{-1}(z),$$ us. completes the of Proposition \[prop:p-1\]. evaluating and $E'(-1)$ for use. The calculations straightforward and cumbersome by combining and asymptotics of $N(z)$ and $f(z)$ given in and. We omit the details but the results below. $$\label{eq: E(-1)} E(-1) = \begin{pmatrix} -i 2... | i & 0 & 1
\end{pmatrix} N^{-1}(z),$$ which gives us.
this compleTes thE prOof Of propOsitIon \[prop:p-1\].
We concLUde tHis section by evaluating $e(-1)$ and $E'(-1)$ FoR LateR UsE. The cAlculatIOnS ARe sTrAiGhtFoRWaRd and CumBersome By combininG anD tHe asymptoticS Of $n(z)$ and $f(z)$ givEn iN and. We omit thE de... | i & 0 & 1
\end{pmatrix}N^{-1}(z), $$ wh ich gi ve s us .
T his completest he p roof of Proposition \[ prop: p- 1 \].We conc lude th i ss e cti on b y e va l ua ting$E( -1)$ an d $E'(-1)$ fo rlater use. T h ecalculatio nsare straight for ward a nd cu m berso meby co mbinin g and t he asympt ot i cs of$ N(z)$ a ... | i &_0 &_1
\end{pmatrix} N^{-1}(z),$$_which gives_us.
This_completes the_proof_of Proposition \[prop:p-1\].
We_conclude this section_by evaluating $E(-1)$ and_$E'(-1)$ for later_use._The calculations are straightforward and cumbersome by combining and the asymptotics of $N(z)$ and_$f(z)$_given in_and._We_omit the de... |
wavelength]{} multiplexed channels simultaneously via one single mode fibre and 32 different entangled states are shared between the 28 different pairs of users.
{width="90.00000%"}
The experiment was performed in two stages. In the first stage the QNSP, MU, the 8 users each connected to ... | wavelength ] { } multiplexed channels simultaneously via one single mode fibre and 32 unlike embroiled department of state are shared between the 28 unlike pairs of exploiter.
! [ image](Fig04_Stability.pdf){width="90.00000% " }
The experiment was performed in two stagecoach. In the first stage the QNSP, MU, the 8... | wavflength]{} multiplexed chankels simultaneously via one sjngle moae fibre and 32 different entaiglee stauvs are shared between the 28 difverent pqirs if users.
{wibti="90.00000%"}
The experiment was perfosmed in two stdgds. In the first stage the QNSP, MU, the 8 users ewch connected jo ... | wavelength]{} multiplexed channels simultaneously via one single and different entangled are shared between users. The experiment was in two stages. the first stage the QNSP, MU, 8 users each connected to the QNSP/MU with a single fibre $\sim$10m in and the 16 detectors were situated in a single laboratory in the Nano ... | wavelength]{} multiplexed chanNels simultAneouSly Via OnE sinGle mOde fibre and 32 difFErenT entangled states are shaRed beTwEEn thE 28 DiFfereNt pairs OF uSERs.
{wIdth="90.00000%"}
The Experiment Was PeRformed in two STaGes. In the fiRst Stage the QNSP, mU, tHe 8 userS eAch COnnecTed To ... | wavelength]{} multiplexedchannels s imult ane ous ly via one single mode f i breand 32 different entan gledst a tesa re shar ed betw e en t he28 d iff er e nt pair s o f users .
{width=" 90. 00000%"}
Th e e xperim en t w a s per for med i n twos tages. In the f ir s t stag e ... | wavelength]{} multiplexed_channels simultaneously_via one single mode_fibre and_32_different entangled_states_are shared between_the 28 different_pairs of users.
{width="90.00000%"}
The experiment_was performed in_two_stages. In the first stage the QNSP, MU, the 8 users each connected to_... |
would increase estimates of stellar temperature and luminosity, shifting them onto the evolutionary tracks of rapidly evolving higher-mass central stars, and decreasing the estimated PN ages.
[^3]: There are Merrett numbers for 101 of their 253 H II regions, suggesting that the M06 studies of the PNLF are contaminate... | would increase estimates of stellar temperature and luminosity, shifting them onto the evolutionary tracks of quickly develop higher - mass cardinal stars, and decrease the estimated PN ages.
[ ^3 ]: There be Merrett numbers for 101 of their 253 H II region, suggesting that the M06 studies of the PNLF are contamin... | wokld increase estimates on stellar tempercrure aid lumihosity, sfifting them onto the evolutmonaey trqcks of rapidly evolvivg higher-lass cenrral wtars, and vscreasiky the cstimctxd PN ages.
[^3]: Therg are Merretd numbers for 101 ow cheir 253 H II regions, suggesting that tre M06 stidles of the PNLS art cjntajpncte... | would increase estimates of stellar temperature and them the evolutionary of rapidly evolving the PN ages. [^3]: are Merrett numbers 101 of their 253 H II suggesting that the M06 studies of the PNLF are contaminated by small numbers tiny H II regions in the inner disk. [^4]: We suspect that confusion this with $\lambda... | would increase estimates of sTellar tempEratuRe aNd lUmInosIty, sHifting them ontO The eVolutionary tracks of rapIdly eVoLVing HIgHer-maSs centrAL sTARs, aNd DeCreAsINg The esTimAted PN aGes.
[^3]: There arE MeRrEtt numbers foR 101 Of Their 253 H II reGioNs, suggesting ThaT the M06 sTuDieS Of the pNLf are cOntamiNAte... | would increase estimatesof stellar temp era tur eandlumi nosity, shifti n g th em onto the evolutiona ry tr ac k s of ra pidly evolvi n gh i ghe r- ma ssce n tr al st ars , and d ecreasingthe e stimated PNa ge s.
[^3]:The re are Merre ttnumber sfor 101 o f t heir253 HI I regi ons, sugg es t ing th a t the M 0 6 s... | would_increase estimates_of stellar temperature and_luminosity, shifting_them_onto the_evolutionary_tracks of rapidly_evolving higher-mass central_stars, and decreasing the_estimated PN ages.
[^3]:_There_are Merrett numbers for 101 of their 253 H II regions, suggesting that the M06_studies_of the_PNLF_are_contaminate... |
}^{(n)}=&\,_{s}E_{\ell m}(\xi)+\xi^2-\left(n+\frac{\alpha+\beta}{2}\right)\left(n+\frac{\alpha+\beta}{2}+1\right)
+\frac{2\xi s(\alpha-\beta)(\alpha+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)},\cr
{\gamma}^{(n)}=&-\frac{4\xi n(n+\alpha+\beta)(n+(\alpha+\beta)/2+s)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)},\end{aligned}$... | } ^{(n)}=&\,_{s}E_{\ell m}(\xi)+\xi^2-\left(n+\frac{\alpha+\beta}{2}\right)\left(n+\frac{\alpha+\beta}{2}+1\right)
+ \frac{2\xi s(\alpha-\beta)(\alpha+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)},\cr
{ \gamma}^{(n)}=&-\frac{4\xi n(n+\alpha+\beta)(n+(\alpha+\beta)/2+s)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)},\end{al... | }^{(n)}=&\,_{s}E_{\fll m}(\xi)+\xi^2-\left(n+\frac{\alpha+\btta}{2}\right)\left(n+\frac{\copha+\beva}{2}+1\right)
+\rrac{2\xi s(\xlpha-\beta)(\alpha+\beta)}{(2n+\alpha+\beta)(2i+\alpya+\betq+2)},\cr
{\gamma}^{(n)}=&-\frac{4\xi n(n+\alphx+\beta)(n+(\alpja+\beta)/2+s)}{(2n+\qlphe+\beta-1)(2n+\alpha+\beta)},\eis{aligned}$... | }^{(n)}=&\,_{s}E_{\ell m}(\xi)+\xi^2-\left(n+\frac{\alpha+\beta}{2}\right)\left(n+\frac{\alpha+\beta}{2}+1\right) +\frac{2\xi s(\alpha-\beta)(\alpha+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)},\cr {\gamma}^{(n)}=&-\frac{4\xi n(n+\alpha+\beta)(n+(\alpha+\beta)/2+s)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)},\end{aligned}$... | }^{(n)}=&\,_{s}E_{\ell m}(\xi)+\xi^2-\left(n+\frac{\alpha+\Beta}{2}\right)\lEft(n+\fRac{\AlpHa+\Beta}{2}+1\RighT)
+\frac{2\xi s(\alpha-\bETa)(\alPha+\beta)}{(2n+\alpha+\beta)(2n+\alphA+\beta+2)},\Cr
{\GAmma}^{(N)}=&-\FrAc{4\xi n(N+\alpha+\bETa)(N+(\ALphA+\bEtA)/2+s)}{(2n+\AlPHa+\Beta-1)(2n+\AlpHa+\beta)},\eNd{aligned}$... | }^{(n)}=&\,_{s}E_{\ell m}( \xi)+\xi^2 -\lef t(n +\f ra c{\a lpha +\beta}{2}\rig h t)\l eft(n+\frac{\alpha+\be ta}{2 }+ 1 \rig h t)
+\fr ac{2\xi s( \ a lph a- \b eta )( \ al pha+\ bet a)}{(2n +\alpha+\b eta )( 2n+\alpha+\b e ta +2)},\cr
{ \ga mma}^{(n)}=& -\f rac{4\ xi n( n +\alp ha+ \beta )(n+(\ a lpha+\ beta)/2+s )}... | }^{(n)}=&\,_{s}E_{\ell m}(\xi)+\xi^2-\left(n+\frac{\alpha+\beta}{2}\right)\left(n+\frac{\alpha+\beta}{2}+1\right)
+\frac{2\xi_s(\alpha-\beta)(\alpha+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)},\cr
{\gamma}^{(n)}=&-\frac{4\xi n(n+\alpha+\beta)(n+(\alpha+\beta)/2+s)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)},\end{aligned}$... |
_{-\infty}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\alpha\Gamma-y)^2+({\bar{u}}-\alpha\Sigma-y')^2}{-4\eta t}\Big]\tilde{g}_0\;dydy'\;,\nonumber\end{aligned}$$ where
\[eq.11\] $$\begin{aligned}
&\Gamma=k_x\sin(\omega_ct)+k_y\cos(\omega_ct)\;,\\
&\Sigma=k_y\sin(\omega_ct)-k_x\cos(\omega_ct)\;,\\
&\alpha=2i\e... | _ { -\infty}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\alpha\Gamma - y)^2+({\bar{u}}-\alpha\Sigma - y')^2}{-4\eta t}\Big]\tilde{g}_0\;dydy'\;,\nonumber\end{aligned}$$ where
\[eq.11\ ] $ $ \begin{aligned }
& \Gamma = k_x\sin(\omega_ct)+k_y\cos(\omega_ct)\;,\\
& \Sigma = k_y\sin(\omega_ct)-k_x\cos(\omega... | _{-\infhy}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\alpma\Gamma-y)^2+({\bar{u}}-\alphc\Wigma-y')^2}{-4\xta t}\Bif]\tilde{g}_0\;dhdy'\;,\nonumber\end{aligned}$$ where
\[ee.11\] $$\vegin{qligned}
&\Gamma=k_x\sin(\omega_zt)+k_y\cos(\omvga_ct)\;,\\
&\Sigmq=k_y\smn(\omega_ct)-k_x\cos(\omxfa_ct)\;,\\
&\alpmc=2i\e... | _{-\infty}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\alpha\Gamma-y)^2+({\bar{u}}-\alpha\Sigma-y')^2}{-4\eta t}\Big]\tilde{g}_0\;dydy'\;,\nonumber\end{aligned}$$ where \[eq.11\] $$\begin{aligned} &\Gamma=k_x\sin(\omega_ct)+k_y\cos(\omega_ct)\;,\\ &\Sigma=k_y\sin(\omega_ct)-k_x\cos(\omega_ct)\;,\\ complete must... | _{-\infty}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\aLpha\Gamma-y)^2+({\Bar{u}}-\aLphA\SiGmA-y')^2}{-4\etA t}\BiG]\tilde{g}_0\;dydy'\;,\nonUMber\End{aligned}$$ where
\[eq.11\] $$\begin{AlignEd}
&\gAmma=K_X\sIn(\omeGa_ct)+k_y\cOS(\oMEGa_cT)\;,\\
&\SIgMa=k_Y\sIN(\oMega_cT)-k_x\Cos(\omegA_ct)\;,\\
&\alpha=2i\e... | _{-\infty}^{+\infty}\!\!\! \!\!\!\!\e xp\!\ Big [\f ra c{({ \bar {v}}+\alpha\Ga m ma-y )^2+({\bar{u}}-\alpha\ Sigma -y ' )^2} { -4 \etat}\Big] \ ti l d e{g }_ 0\ ;dy dy ' \; ,\non umb er\end{ aligned}$$ wh er e
\[eq.11\] $$ \begin{ali gne d}
&\Gamma=k _x\ sin(\o me ga_ c t)+k_ y\c os(\o mega_c t )\;,\\
&\Sigma= k_ y... | _{-\infty}^{+\infty}\!\!\!\!\!\!\!\exp\!\Big[\frac{({\bar{v}}+\alpha\Gamma-y)^2+({\bar{u}}-\alpha\Sigma-y')^2}{-4\eta t}\Big]\tilde{g}_0\;dydy'\;,\nonumber\end{aligned}$$_where
\[eq.11\] $$\begin{aligned}
&\Gamma=k_x\sin(\omega_ct)+k_y\cos(\omega_ct)\;,\\
&\Sigma=k_y\sin(\omega_ct)-k_x\cos(\omega_ct)\;,\\
&\alpha=2i\e... |
{\beta}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\mbox{\boldmath $\Sigma$}}^*)$, where the $(k,l)$th element in ${\mbox{\boldmath $\Sigma$}}^*$ is $\displaystyle{\mbox{\boldmath $\Sigma$}}_{kl}^*=\sigma^2s_{kl}/(ns_{kk}s_{ll})$.]{}
For ease of notation, let $Z_1,\cdots,Z_p$ be the standardized random variables of $\wide... | { \beta}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\mbox{\boldmath $ \Sigma$}}^*)$, where the $ (k, l)$th element in $ { \mbox{\boldmath $ \Sigma$}}^*$ is $ \displaystyle{\mbox{\boldmath $ \Sigma$}}_{kl}^*=\sigma^2s_{kl}/(ns_{kk}s_{ll})$. ] { }
For ease of notation, let $ Z_1,\cdots, Z_p$ be the exchangeable random var... | {\betw}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\mnox{\boldmath $\Sigmc$}}^*)$, where the $(k,m)$th elemdnt in ${\mbox{\boldmath $\Sigma$}}^*$ is $\dusplatstyle{\mbox{\boldmath $\Sigoa$}}_{kl}^*=\sigma^2d_{kl}/(ns_{kk}s_{ol})$.]{}
Foc ease of notation, let $Z_1,\gbots,Z_l$ be chx standardized tandom variatles of $\wide... | {\beta}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\mbox{\boldmath $\Sigma$}}^*)$, where the $(k,l)$th element $\Sigma$}}^*$ $\displaystyle{\mbox{\boldmath $\Sigma$}}_{kl}^*=\sigma^2s_{kl}/(ns_{kk}s_{ll})$.]{} ease of notation, random of $\widehat{\beta}_1,\cdots,\widehat{\beta}_p$, that $$\label{b2} Z_i=\frac{\widehat{\be... | {\beta}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\Mbox{\boldmaTh $\SigMa$}}^*)$, wHerE tHe $(k,l)$Th elEment in ${\mbox{\bolDMath $\sigma$}}^*$ is $\displaystyle{\mboX{\boldMaTH $\SigMA$}}_{kL}^*=\sigmA^2s_{kl}/(ns_{kK}S_{lL})$.]{}
fOr eAsE oF noTaTIoN, let $Z_1,\CdoTs,Z_p$ be tHe standardIzeD rAndom variablES oF $\wide... | {\beta}_p)^T\sim N((\beta_ 1,\cdots,\ beta_ p)^ T,{ \m box{ \bol dmath $\Sigma$ } }^*) $, where the $(k,l)$th elem en t in$ {\ mbox{ \boldma t h$ \ Sig ma $} }^* $i s$\dis pla ystyle{ \mbox{\bol dma th $\Sigma$}}_ { kl }^*=\sigma ^2s _{kl}/(ns_{k k}s _{ll}) $. ]{}
Foreas e ofnotati o n, let $Z_1,\cd ot s ,Z_p$b e ... | {\beta}_p)^T\sim N((\beta_1,\cdots,\beta_p)^T,{\mbox{\boldmath_$\Sigma$}}^*)$, where_the $(k,l)$th element in_${\mbox{\boldmath $\Sigma$}}^*$_is_$\displaystyle{\mbox{\boldmath $\Sigma$}}_{kl}^*=\sigma^2s_{kl}/(ns_{kk}s_{ll})$.]{}
For_ease_of notation, let_$Z_1,\cdots,Z_p$ be the_standardized random variables of_$\wide... |
\]) at $z=0.1$ (upper panel), $z=0.2$ (middle panel) and $z=0.5$ (lower panel) in a ${\Omega_{\rm M}}=0.28,\,{\Omega_{\rm\Lambda}}=0.72$ universe, which is what we will use subsequently. We have performed a number of simulations using a wide range of cosmological parameter-values and found the oscillation effect to dep... | \ ]) at $ z=0.1 $ (upper panel), $ z=0.2 $ (middle panel) and $ z=0.5 $ (lower dialog box) in a $ { \Omega_{\rm M}}=0.28,\,{\Omega_{\rm\Lambda}}=0.72 $ population, which is what we will use subsequently. We have performed a numeral of simulation using a wide stove of cosmologic parameter - value and found the oscillati... | \]) at $z=0.1$ (upper panel), $z=0.2$ (middle kanel) and $z=0.5$ (lower panel) mn a ${\Omsga_{\rm M}}=0.28,\,{\Ooega_{\rm\Lambda}}=0.72$ universe, which ms wyat wt will use subsequevtly. We hwve perfirmev a number of simulations usinf a wndx range of cosmplogical pdrameter-values avd found the oscillation effect to de[... | \]) at $z=0.1$ (upper panel), $z=0.2$ (middle $z=0.5$ panel) in ${\Omega_{\rm M}}=0.28,\,{\Omega_{\rm\Lambda}}=0.72$ universe, use We have performed number of simulations a wide range of cosmological parameter-values found the oscillation effect to depend only weakly on cosmology. The most dramatic is the strong variat... | \]) at $z=0.1$ (upper panel), $z=0.2$ (middle panel) And $z=0.5$ (lower pAnel) iN a ${\OMegA_{\rM M}}=0.28,\,{\OmEga_{\rM\Lambda}}=0.72$ universE, WhicH is what we will use subseqUentlY. WE Have PErFormeD a numbeR Of SIMulAtIoNs uSiNG a Wide rAngE of cosmOlogical paRamEtEr-values and fOUnD the oscillAtiOn effect to deP... | \]) at $z=0.1$ (upper pane l), $z=0.2 $ (mi ddl e p an el)and$z=0.5$ (lower pane l) in a ${\Omega_{\rmM}}=0 .2 8 ,\,{ \ Om ega_{ \rm\Lam b da } } =0. 72 $uni ve r se , whi chis what we will u sesu bsequently.W ehave perfo rme d a number o f s imulat io nsu singa w ide r ange o f cosmo logical p ar a meter- v alues a ... | \]) at_$z=0.1$ (upper_panel), $z=0.2$ (middle panel)_and $z=0.5$_(lower_panel) in_a_${\Omega_{\rm M}}=0.28,\,{\Omega_{\rm\Lambda}}=0.72$ universe,_which is what_we will use subsequently._We have performed_a_number of simulations using a wide range of cosmological parameter-values and found the oscillation_effect_to dep... |
$$\sigma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi +O(x^{{\frac{n}{2}}+1}).$$
Remark from Proposition \[propofS\] that $S(0)^*=S(0)^{-1}=S(0)$ and so the operator $$\label{defc}
{\mathcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ is an orthogonal projector on a subspace of $L^2({\partial}{\overline}{X},{^0\Sigma})$ for the mea... | $ $ \sigma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi + O(x^{{\frac{n}{2}}+1}).$$
Remark from Proposition \[propofS\ ] that $ S(0)^*=S(0)^{-1}=S(0)$ and so the operator $ $ \label{defc }
{ \mathcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ is an orthogonal projector on a subspace of $ L^2({\partial}{\overline}{X},{^0\Sigma})... | $$\sihma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi +O(x^{{\frxc{n}{2}}+1}).$$
Remark from Kripositmon \[prolofS\] thag $S(0)^*=S(0)^{-1}=S(0)$ and so the operator $$\lauel{dwfc}
{\maukcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ is av orthogojal projwctoc on a subspace of $L^2({\partlcl}{\ovedpine}{R},{^0\Smgma})$ for the mes... | $$\sigma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi +O(x^{{\frac{n}{2}}+1}).$$ Remark from Proposition $S(0)^*=S(0)^{-1}=S(0)$ so the $$\label{defc} {\mathcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ a of $L^2({\partial}{\overline}{X},{^0\Sigma})$ for measure ${\rm dv}_{h_0}$ $h_0=(x^{2}g)|_{T{\partial}{\overline}{X}}$. Notice f... | $$\sigma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi +O(x^{{\frac{N}{2}}+1}).$$
Remark froM PropOsiTioN \[pRopoFS\] thAt $S(0)^*=S(0)^{-1}=S(0)$ and so the oPEratOr $$\label{defc}
{\mathcal}{C}:={\fraC{1}{2}}({\rm Id}+s(0))$$ iS An orTHoGonal ProjectOR oN A SubSpAcE of $l^2({\pARtIal}{\ovErlIne}{X},{^0\SigMa})$ for the meA... | $$\sigma= x^{{\frac{n}{2} }}({\rm Id }+S(0 ))\ psi + O(x^ {{\f rac{n}{2}}+1}) . $$
Remark from Propositio n \[p ro p ofS\ ] t hat $ S(0)^*= S (0 ) ^ {-1 }= S( 0)$ a n dso th e o perator $$\label{ def c}
{\mathcal}{ C }: ={\frac{1} {2} }({\rm Id}+S (0) )$$ is a n o r thogo nal proj ectoro n a su bspace of $ L ^... | $$\sigma=_x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi_+O(x^{{\frac{n}{2}}+1}).$$
Remark from Proposition \[propofS\]_that $S(0)^*=S(0)^{-1}=S(0)$_and_so the_operator_$$\label{defc}
{\mathcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ is_an orthogonal projector_on a subspace of_$L^2({\partial}{\overline}{X},{^0\Sigma})$ for the_mea... |
superalgebras $sl(m;j|n;\epsilon) $ are consistent with the transformations of (super) vectors $${\cal X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots,(1,m)x_m|\nu(x_{m+1},
\epsilon_1x_{m+2},\ldots,[1,n]x_{m+n}))^t,
\label{sv}$$ where the odd components are denote as $
x_{m+1}=\theta_1,\ldots, x_{m+n}=\theta_{n}
$ and $
\hat{\ba... | superalgebras $ sl(m;j|n;\epsilon) $ are consistent with the transformations of (super) vectors $ $ { \cal X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots,(1,m)x_m|\nu(x_{m+1 },
\epsilon_1x_{m+2},\ldots,[1,n]x_{m+n}))^t,
\label{sv}$$ where the curious component are denote as $
x_{m+1}=\theta_1,\ldots, x_{m+n}=\theta_{n }
$... | suoeralgebras $sl(m;j|n;\epsilon) $ are consistenj qith tie tranaformatiuns of (super) vectors $${\cal X}^t(j,\xpsioon)=(x_1,j_1z_2,\ldots,(1,m)x_m|\nu(x_{m+1},
\epsilon_1x_{m+2},\udots,[1,n]x_{m+n}))^n,
\label{sv}$$ qhert the odd componeifs are denote ws $
x_{k+1}=\vheta_1,\ldots, x_{m+n}=\tmeta_{n}
$ and $
\vat{\ba... | superalgebras $sl(m;j|n;\epsilon) $ are consistent with the (super) $${\cal X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots,(1,m)x_m|\nu(x_{m+1}, \label{sv}$$ where the $ x_{m+n}=\theta_{n} $ and \hat{\bar i}={\bar i}-m, \hat{\bar k}={\bar k}-m=1,\ldots,n,\, [\hat{\bar i},\hat{\bar k}]= i},\hat{\bar k})-1}_{l=\min(\hat{\bar i},\ha... | superalgebras $sl(m;j|n;\epsilon) $ Are consistEnt wiTh tHe tRaNsfoRmatIons of (super) vecTOrs $${\cAl X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots,(1,m)x_M|\nu(x_{m+1},
\EpSIlon_1X_{M+2},\lDots,[1,n]X_{m+n}))^t,
\labEL{sV}$$ WHerE tHe Odd CoMPoNents Are Denote aS $
x_{m+1}=\theta_1,\ldOts, X_{m+N}=\theta_{n}
$ and $
\haT{\Ba... | superalgebras $sl(m;j|n;\ epsilon) $ arecon sis te nt w iththe transforma t ions of (super) vectors $$ {\cal X } ^t(j , \e psilo n)=(x_1 , j_ 1 x _2, \l do ts, (1 , m) x_m|\ nu( x_{m+1} ,
\epsilon _1x _{ m+2},\ldots, [ 1, n]x_{m+n}) )^t ,
\label{sv} $$whereth e o d d com pon entsare de n ote as $
x_{m+1 }= \ theta... | superalgebras_$sl(m;j|n;\epsilon) $_are consistent with the_transformations of_(super)_vectors $${\cal_X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots,(1,m)x_m|\nu(x_{m+1},
\epsilon_1x_{m+2},\ldots,[1,n]x_{m+n}))^t,
\label{sv}$$_where the odd_components are denote_as $
x_{m+1}=\theta_1,\ldots, x_{m+n}=\theta_{n}
$ and_$
\hat{\ba... |
ii) in the definition of $t$ following equation (\[eq:Vdef\]). Thus $\mathcal{D}_1(f)$ can only intersect the component $C(v)$ of $\mathcal{Y}_v$. Now (\[eq:fiberint\]) shows $$\langle \mathcal{D}_1(f),n_{C(v)} C(v)\rangle_v = \langle \mathcal{D}_1(f),\mathcal{Y}_v\rangle_v = \mathrm{deg}(D_1(f)) = m.$$ This shows (\[e... | ii) in the definition of $ t$ following equation (\[eq: Vdef\ ]). Thus $ \mathcal{D}_1(f)$ can only intersect the part $ C(v)$ of $ \mathcal{Y}_v$. nowadays (\[eq: fiberint\ ]) shows $ $ \langle \mathcal{D}_1(f),n_{C(v) } C(v)\rangle_v = \langle \mathcal{D}_1(f),\mathcal{Y}_v\rangle_v = \mathrm{deg}(D_1(f) ) = m.$$ Thi... | ii) ln the definition of $t$ fullowing equation (\[eq:Vdxf\]). Thus $\mathcal{A}_1(f)$ can only intersect the colpinent $C(v)$ of $\mathcal{Y}_v$. Now (\[ed:fiberint\]) shows $$\lqnglt \mathcal{D}_1(f),n_{C(v)} C(v)\czngle_v = \langls \matkcel{D}_1(f),\mathcal{Y}_v\rakgle_v = \mathsm{deg}(D_1(f)) = m.$$ Thiv rhlws (\[e... | ii) in the definition of $t$ following Thus can only the component $C(v)$ $$\langle C(v)\rangle_v = \langle = \mathrm{deg}(D_1(f)) = This shows (\[eq:crank1\]) and completes the of (iv). Finally, the inequalities in (\[eq:crank\]) of part (v) are a consequence (\[eq:urp\]), (\[eq:bounds\]) and (\[eq:crank1\]). Controll... | ii) in the definition of $t$ folloWing equatiOn (\[eq:VDef\]). thuS $\mAthcAl{D}_1(f)$ Can only interseCT the Component $C(v)$ of $\mathcal{Y}_v$. now (\[eq:FiBErinT\]) ShOws $$\laNgle \matHCaL{d}_1(F),n_{C(V)} C(V)\rAngLe_V = \LaNgle \mAthCal{D}_1(f),\maThcal{Y}_v\ranGle_V = \mAthrm{deg}(D_1(f)) = m.$$ THIs Shows (\[e... | ii) in the definition of $ t$ followi ng eq uat ion ( \[eq :Vde f\]). Thus $\m a thca l{D}_1(f)$ can only in terse ct thec om ponen t $C(v) $ o f $\m at hc al{ Y} _ v$ . Now (\ [eq:fib erint\]) s how s$$\langle \m a th cal{D}_1(f ),n _{C(v)} C(v) \ra ngle_v = \l a ngle\ma thcal {D}_1( f ),\mat hcal{Y}_v \r a ngle_... | ii) in_the definition_of $t$ following equation_(\[eq:Vdef\]). Thus_$\mathcal{D}_1(f)$_can only_intersect_the component $C(v)$_of $\mathcal{Y}_v$. Now_(\[eq:fiberint\]) shows $$\langle \mathcal{D}_1(f),n_{C(v)}_C(v)\rangle_v = \langle_\mathcal{D}_1(f),\mathcal{Y}_v\rangle_v_= \mathrm{deg}(D_1(f)) = m.$$ This shows (\[e... |
\operatorname*{\mathbf{D}}(\operatorname*{Mod-R}^I) \rightarrow \operatorname*{\mathbf{D}}(R)$ to be the left derived functor $\mathbf{L}\operatorname*{colim}_{i \in I}$ of the usual colimit functor $\operatorname*{colim}_{i \in I}: \operatorname*{Mod-R}^I \rightarrow \operatorname*{Mod-R}$. Dually, we define the [hom... | \operatorname*{\mathbf{D}}(\operatorname*{Mod - R}^I) \rightarrow \operatorname*{\mathbf{D}}(R)$ to be the left derived functor $ \mathbf{L}\operatorname*{colim}_{i \in I}$ of the usual colimit functor $ \operatorname*{colim}_{i \in I }: \operatorname*{Mod - R}^I \rightarrow \operatorname*{Mod - R}$. Dually, we specify... | \opfratorname*{\mathbf{D}}(\operatovname*{Mod-R}^I) \rightcerow \o'eratorhame*{\mathcf{D}}(R)$ to be the left derived huncror $\mqthbf{L}\operatorname*{colio}_{i \in I}$ ov the usyal rolimit functor $\operatorkcme*{comlm}_{i \iu M}: \operatorname*{Mpd-R}^I \rightdrrow \operatortaoe*{Lod-R}$. Dually, we define the [hom... | \operatorname*{\mathbf{D}}(\operatorname*{Mod-R}^I) \rightarrow \operatorname*{\mathbf{D}}(R)$ to be the left $\mathbf{L}\operatorname*{colim}_{i I}$ of usual colimit functor \operatorname*{Mod-R}$. we define the limit]{} functor as \in I} := \mathbf{R} \lim_{i \in \operatorname*{\mathbf{D}}(\operatorname*{Mod-R}^I) \r... | \operatorname*{\mathbf{D}}(\operatOrname*{Mod-R}^i) \righTarRow \OpEratOrnaMe*{\mathbf{D}}(R)$ to be THe leFt derived functor $\mathbf{l}\operAtORnamE*{CoLim}_{i \iN I}$ of the USuAL ColImIt FunCtOR $\oPeratOrnAme*{coliM}_{i \in I}: \operaTorNaMe*{Mod-R}^I \rightARrOw \operatorNamE*{Mod-R}$. Dually, wE deFine thE [hOm... | \operatorname*{\mathbf{D} }(\operato rname *{M od- R} ^I)\rig htarrow \opera t orna me*{\mathbf{D}}(R)$ to be t he left de rived functo r $ \ m ath bf {L }\o pe r at ornam e*{ colim}_ {i \in I}$ of t he usual col i mi t functor$\o peratorname* {co lim}_{ i\in I}: \ ope rator name*{ M od-R}^ I \righta rr o w \ope ... | \operatorname*{\mathbf{D}}(\operatorname*{Mod-R}^I)_\rightarrow \operatorname*{\mathbf{D}}(R)$_to be the left_derived functor_$\mathbf{L}\operatorname*{colim}_{i_\in I}$_of_the usual colimit_functor $\operatorname*{colim}_{i \in_I}: \operatorname*{Mod-R}^I \rightarrow \operatorname*{Mod-R}$._Dually, we define_the_[hom... |
mathrm{v},\sharp}(\alpha)+\alpha,\gamma^{\mathrm{v},\sharp}(\beta)+\beta]_D\in L_{\gamma}, \ \ \forall \alpha,\beta\in \Gamma(H_{\gamma}^{\circ}).$$ Explicitly, this expression is given by: $$[\gamma^{\mathrm{v},\sharp}(\alpha),\gamma^{\mathrm{v},\sharp}(\beta)]+\iota_{\gamma^{\mathrm{v},\sharp}(\alpha)}d\beta-\iota_{\... | mathrm{v},\sharp}(\alpha)+\alpha,\gamma^{\mathrm{v},\sharp}(\beta)+\beta]_D\in L_{\gamma }, \ \ \forall \alpha,\beta\in \Gamma(H_{\gamma}^{\circ}).$$ Explicitly, this expression is given by: $ $ [ \gamma^{\mathrm{v},\sharp}(\alpha),\gamma^{\mathrm{v},\sharp}(\beta)]+\iota_{\gamma^{\mathrm{v},\sharp}(\alpha)}d\beta-\iot... | matjrm{v},\sharp}(\alpha)+\alpha,\gamma^{\oathrm{v},\sharp}(\betc)+\veta]_D\ii L_{\gammz}, \ \ \foraul \alpha,\beta\in \Gamma(H_{\gamma}^{\cicc}).$$ Ezplicutly, this expression ir given bj: $$[\gamma^{\marhrm{t},\sharp}(\alpha),\gamma^{\mathrm{v},\smcrp}(\befw)]+\iotc_{\gemma^{\mathrm{v},\shark}(\alpha)}d\beta-\imta_{\... | mathrm{v},\sharp}(\alpha)+\alpha,\gamma^{\mathrm{v},\sharp}(\beta)+\beta]_D\in L_{\gamma}, \ \ \forall \alpha,\beta\in \Gamma(H_{\gamma}^{\circ}).$$ expression given by: d\alpha+d\gamma^{\mathrm{v}}(\alpha,\beta),$$ and denote Since vector part is (\[EQ\_condition1\]) is equivalent the following two conditions $$\label... | mathrm{v},\sharp}(\alpha)+\alpha,\gamMa^{\mathrm{v},\sHarp}(\bEta)+\BetA]_D\In L_{\gAmma}, \ \ \Forall \alpha,\betA\In \GaMma(H_{\gamma}^{\circ}).$$ ExplicitlY, this ExPRessIOn Is givEn by: $$[\gamMA^{\mATHrm{V},\sHaRp}(\aLpHA),\gAmma^{\mAthRm{v},\sharP}(\beta)]+\iota_{\gAmmA^{\mAthrm{v},\sharp}(\aLPhA)}d\beta-\iota_{\... | mathrm{v},\sharp}(\alpha)+ \alpha,\ga mma^{ \ma thr m{ v},\ shar p}(\beta)+\bet a ]_D\ in L_{\gamma}, \ \ \fo rall\a l pha, \ be ta\in \Gamma ( H_ { \ gam ma }^ {\c ir c }) .$$ E xpl icitly, this expr ess io n is given b y :$$[\gamma^ {\m athrm{v},\sh arp }(\alp ha ),\ g amma^ {\m athrm {v},\s h arp}(\ beta)]+\i ot a... | mathrm{v},\sharp}(\alpha)+\alpha,\gamma^{\mathrm{v},\sharp}(\beta)+\beta]_D\in L_{\gamma},_\ \_\forall \alpha,\beta\in \Gamma(H_{\gamma}^{\circ}).$$ Explicitly,_this expression_is_given by:_$$[\gamma^{\mathrm{v},\sharp}(\alpha),\gamma^{\mathrm{v},\sharp}(\beta)]+\iota_{\gamma^{\mathrm{v},\sharp}(\alpha)}d\beta-\iota_{\... |
.
A main contribution is simple and general asymptotic theory for LR estimators that use cross-fitting in the construction of the average moments. This theory is based on the structure of LR moment conditions as an identifying moment condition depending on one first step plus an influence adjustment that can depend on... | .
A main contribution is simple and general asymptotic theory for LR calculator that practice crabbed - fitting in the construction of the average moments. This hypothesis is based on the structure of LR moment circumstance as an identifying moment circumstance depend on one first footfall plus an influence adjustme... | .
A mwin contribution is simpue and general cwymptovic thekry for UR estimators that use cross-hittung ib the construction of ghe averahe momenrs. Tiis theory is based on tmz strhgture if LR moment cpnditions ds an identifyhne loment condition depending on one fyrst strp plus an influgnce sqjusfment that can depend on... | . A main contribution is simple and theory LR estimators use cross-fitting in moments. theory is based the structure of moment conditions as an identifying moment depending on one first step plus an influence adjustment that can depend on additional first step. We give a remainder decomposition that leads to mean squar... | .
A main contribution is simple And general AsympTotIc tHeOry fOr LR Estimators that USe crOss-fitting in the construCtion Of THe avERaGe momEnts. ThiS ThEORy iS bAsEd oN tHE sTructUre Of LR momEnt conditiOns As An identifyinG MoMent conditIon Depending on oNe fIrst stEp PluS An infLueNce adJustmeNT that cAn depend oN... | .
A main contribution issimple and gene ral as ym ptot ic t heory for LR e s tima tors that use cross-fi tting i n the co nstru ction o f t h e av er ag e m om e nt s. Th istheoryis based o n t he structure o f L R moment c ond itions as an id entify in g m o mentcon ditio n depe n ding o n one fir st step p ... | .
A main_contribution is_simple and general asymptotic_theory for_LR_estimators that_use_cross-fitting in the_construction of the_average moments. This theory_is based on_the_structure of LR moment conditions as an identifying moment condition depending on one first_step_plus an_influence_adjustment_that can depend on... |
in Quintessence*, *Astropart. Phys.* [**54**]{} (2014) 125
M. Meehan and I. Whittingham, *Asymmetric dark matter in braneworld cosmology*, *JCAP* [**06**]{} (2014) 018
G. Gelmini, *Experimental signatures of non-standard pre-BBN cosmologies*, *Nucl. Phys. Proc. Suppl.* [**194**]{} (2009) 63
D. Langlois, *Brane Cosm... | in Quintessence *, * Astropart. Phys. * [ * * 54 * * ] { } (2014) 125
M. Meehan and I. Whittingham, * Asymmetric dark matter in braneworld cosmology *, * JCAP * [ * * 06 * * ] { } (2014) 018
G. Gelmini, * Experimental signatures of non - standard pre - BBN cosmology *, * Nucl. Phys. Proc. Suppl. * [ * * 194 * * ]... | in Quintessence*, *Astropart. Khys.* [**54**]{} (2014) 125
M. Meehan aue I. Whmttinghzm, *Asymmdtric dark matter in branewocld xosmooogy*, *JCAP* [**06**]{} (2014) 018
G. Gelmini, *Dxperimennal signarurew of non-staisard prc-YBN ckdmolmjies*, *Nucl. Phys. Kroc. Suppl.* [**194**]{} (2009) 63
D. Langlois, *Brdnd Eosm... | in Quintessence*, *Astropart. Phys.* [**54**]{} (2014) 125 and Whittingham, *Asymmetric matter in braneworld G. *Experimental signatures of pre-BBN cosmologies*, *Nucl. Proc. Suppl.* [**194**]{} (2009) 63 D. *Brane Cosmology: An Introduction*, *Prog. Theor. Phys. Suppl.* [**148**]{} (2003) 181 R. Maartens K. Koyama, *B... | in Quintessence*, *Astropart. PhYs.* [**54**]{} (2014) 125
M. Meehan aNd I. WhIttIngHaM, *AsyMmetRic dark matter iN BranEworld cosmology*, *JCAP* [**06**]{} (2014) 018
G. GeLmini, *exPErimENtAl sigNatures OF nON-StaNdArD prE-Bbn cOsmolOgiEs*, *Nucl. PHys. Proc. SupPl.* [**194**]{} (2009) 63
D. laNglois, *Brane COSm... | in Quintessence*, *Astrop art. Phys. * [** 54* *]{ }(201 4) 1 25
M. Meehana nd I . Whittingham, *Asymme tricda r k ma t te r inbranewo r ld c osm ol og y*, * J CA P* [* *06 **]{} ( 2014) 018
G. G elmini, *Exp e ri mental sig nat ures of non- sta ndardpr e-B B N cos mol ogies *, *Nu c l. Phy s. Proc.Su p pl.* [ * *1... | in_Quintessence*, *Astropart._Phys.* [**54**]{} (2014) 125
M._Meehan and_I._Whittingham, *Asymmetric_dark_matter in braneworld_cosmology*, *JCAP* [**06**]{}_(2014) 018
G. Gelmini, *Experimental_signatures of non-standard_pre-BBN_cosmologies*, *Nucl. Phys. Proc. Suppl.* [**194**]{} (2009) 63
D. Langlois, *Brane Cosm... |
&0.02 &41.69\
G24.94+0.07 & 18 36 31.333 & 0.006 & -07 04 19.968 & 0.007 & 0.60 &0.02 &41.69\
G24.94+0.07 & 18 36 31.812 & 0.014 & -07 04 23.244 & 0.016 & 0.27 &0.02 &41.85\
G24.94+0.07 & 18 36 31.705 & 0.008 & -07 04 10.801 & 0.010 & 0.43 &0.02 &41.85\
G24.94+0.07 & 18 36 31.672 & 0.006 & -07 04 09.024 & 0.007 & 0.61... | & 0.02 & 41.69\
G24.94 + 0.07 & 18 36 31.333 & 0.006 & -07 04 19.968 & 0.007 & 0.60 & 0.02 & 41.69\
G24.94 + 0.07 & 18 36 31.812 & 0.014 & -07 04 23.244 & 0.016 & 0.27 & 0.02 & 41.85\
G24.94 + 0.07 & 18 36 31.705 & 0.008 & -07 04 10.801 & 0.010 & 0.43 & 0.02 & 41.85\
G24.94 + 0.07 & 18 36 31.672 & 0.006 & -07 0... | &0.02 &41.69\
G24.94+0.07 & 18 36 31.333 & 0.006 & -07 04 19.968 & 0.007 & 0.60 &0.02 &41.69\
G24.94+0.07 & 18 36 31.812 & 0.014 & -07 04 23.244 & 0.016 & 0.27 &0.02 &41.85\
G24.94+0.07 & 18 36 31.705 & 0.008 & -07 04 10.801 & 0.010 & 0.43 &0.02 &41.85\
G24.94+0.07 & 18 36 31.672 & 0.006 & -07 04 09.024 & 0.007 & 0.61... | &0.02 &41.69\ G24.94+0.07 & 18 36 31.333 & 04 19.968 0.007 & 0.60 36 & 0.014 & 04 23.244 & & 0.27 &0.02 &41.85\ G24.94+0.07 & 36 31.705 & 0.008 & -07 04 10.801 & 0.010 & 0.43 &0.02 G24.94+0.07 & 18 36 31.672 & 0.006 & -07 04 09.024 & 0.007 0.61 &41.85\ & 36 31.591 & 0.003 & -07 04 07.731 & 0.003 & 1.27 &0.02 &41.85\ G2... | &0.02 &41.69\
G24.94+0.07 & 18 36 31.333 & 0.006 & -07 04 19.968 & 0.007 & 0.60 &0.02 &41.69\
G24.94+0.07 & 18 36 31.812 & 0.014 & -07 04 23.244 & 0.016 & 0.27 &0.02 &41.85\
G24.94+0.07 & 18 36 31.705 & 0.008 & -07 04 10.801 & 0.010 & 0.43 &0.02 &41.85\
G24.94+0.07 & 18 36 31.672 & 0.006 & -07 04 09.024 & 0.007 & 0.61... | &0.02 &41.69\
G24.94+0.07 & 18 36 3 1.333 &0.0 06 & - 07 0 4 19.968 & 0.0 0 7 &0.60 &0.02 &41.69\
G24 .94+0 .0 7 & 1 8 3 6 31. 812 & 0 . 01 4 & - 07 0 4 2 3. 2 44 & 0. 016 & 0.27 &0.02 &41 .85 \G24.94+0.07& 1 8 36 31.70 5 & 0.008 & -07 04 10.80 1& 0 . 010 & 0. 43 &0 .02 &4 1 .85\
G 24.94+0.0 7& 18 36 31... | &0.02_&41.69\
G24.94+0.07 &_18 36 31.333 &_0.006 &_-07_04 19.968_&_0.007 & 0.60_&0.02 &41.69\
G24.94+0.07 &_18 36 31.812 &_0.014 & -07_04_23.244 & 0.016 & 0.27 &0.02 &41.85\
G24.94+0.07 & 18 36 31.705 & 0.008 &_-07_04 10.801_&_0.010_& 0.43 &0.02 &41.85\
G24.94+0.07 &_18 36 31.672 & 0.006_& -07_04 09.024 & 0.007 & 0.61... |
1}{\Gamma(\frac{4-n}{2})} \int_0^{{\infty}} k(v) v^{\frac{4-n}{2}-1}
dv \\
&= f(0) \Lambda^0 a_4(D^2) + 2 f_2 \Lambda^2 a_2(D^2) + 2\Lambda^4 f_4 a_0(D^2).
\label{eq:Diracexpansion}\end{aligned}$$ where the $f_k$ are moments of the function $f$: $$\begin{aligned}
f_{k} := \int_{0}^{\infty} f(w)w^{k-1}dw; \qquad (k>... | 1}{\Gamma(\frac{4 - n}{2 }) } \int_0^{{\infty } } k(v) v^{\frac{4 - n}{2}-1 }
dv \\
& = f(0) \Lambda^0 a_4(D^2) + 2 f_2 \Lambda^2 a_2(D^2) + 2\Lambda^4 f_4 a_0(D^2).
\label{eq: Diracexpansion}\end{aligned}$$ where the $ f_k$ are moments of the function $ f$: $ $ \begin{aligned }
f_{k }: = \int_{0}^{\infty }... | 1}{\Gamla(\frac{4-n}{2})} \int_0^{{\infty}} k(v) v^{\frag{4-n}{2}-1}
dv \\
&= f(0) \Lambda^0 a_4(B^2) + 2 f_2 \Lembda^2 a_2(S^2) + 2\Lambdx^4 f_4 a_0(D^2).
\label{eq:Diracexpansion}\eid{alugned}$$ where the $f_k$ are momevts of thv functiob $f$: $$\uegin{aligned}
h_{i} := \int_{0}^{\ikyty} f(s)a^{k-1}dw; \wquad (k>... | 1}{\Gamma(\frac{4-n}{2})} \int_0^{{\infty}} k(v) v^{\frac{4-n}{2}-1} dv \\ &= a_4(D^2) 2 f_2 a_2(D^2) + 2\Lambda^4 $f_k$ moments of the $f$: $$\begin{aligned} f_{k} \int_{0}^{\infty} f(w)w^{k-1}dw; \qquad (k>0) \nonumber. \end{aligned}$$ For the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$, the s... | 1}{\Gamma(\frac{4-n}{2})} \int_0^{{\infty}} k(v) v^{\frac{4-N}{2}-1}
dv \\
&= f(0) \Lambda^0 A_4(D^2) + 2 f_2 \LaMbdA^2 a_2(D^2) + 2\laMbda^4 F_4 a_0(D^2).
\lAbel{eq:DiracexpANsioN}\end{aligned}$$ where the $f_k$ aRe momEnTS of tHE fUnctiOn $f$: $$\begiN{AlIGNed}
F_{k} := \InT_{0}^{\inFtY} F(w)W^{k-1}dw; \qQuaD (k>... | 1}{\Gamma(\frac{4-n}{2})}\int_0^{{\ infty }}k(v )v^{\ frac {4-n}{2}-1}
dv \\
& = f(0) \Lambda^0 a_4(D ^2) + 2 f_2\ La mbda^ 2 a_2(D ^ 2) + 2\ La mb da^ 4f _4 a_0( D^2 ).
\lab el{eq:Dira cex pa nsion}\end{a l ig ned}$$ whe rethe $f_k$ ar e m oments o f t h e fun cti on $f $: $$\ b egin{a ligned}
f_{k}: = \int_ { 0 ... | 1}{\Gamma(\frac{4-n}{2})} \int_0^{{\infty}}_k(v) v^{\frac{4-n}{2}-1}
dv_\\
&= f(0) \Lambda^0 a_4(D^2)_+ 2_f_2_\Lambda^2 a_2(D^2)_+_2\Lambda^4 f_4 a_0(D^2).
\label{eq:Diracexpansion}\end{aligned}$$_where the $f_k$_are moments of the_function $f$: $$\begin{aligned}
__ f_{k} := \int_{0}^{\infty} f(w)w^{k-1}dw; \qquad (k>... |
spins leaves the energy unchanged[@Liu2017]. Therefore, our experiment implies that this single-ion anisotropy is vanishingly small and what lifts the U(1) degeneracy is the in-plane magnetic field. Therefore, instead of having *six* domains (which would have been the case if the degeneracy was lifted by the single-io... | spins leaves the energy unchanged[@Liu2017 ]. Therefore, our experiment entail that this individual - ion anisotropy is vanishingly small and what lifts the U(1) degeneracy is the in - airplane magnetic field. Therefore, alternatively of having * six * domains (which would have been the event if the degeneracy was lift... | splns leaves the energy unghanged[@Liu2017]. Thereyire, ouc experjment imolies that this single-ion anmsoteopy us vanishingly small avd what lpfts the Y(1) dejeneracy is the mh-plane magnetjg fienv. Therefore, insjead of havitg *six* domains (wfieh would have been the case if the dqgeneravy was lifted by the fingmv-ii... | spins leaves the energy unchanged[@Liu2017]. Therefore, our that single-ion anisotropy vanishingly small and is in-plane magnetic field. instead of having domains (which would have been the if the degeneracy was lifted by the single-ion anisotropy), we have only *two* set by the orientation of the magnetic field. This ... | spins leaves the energy unchaNged[@Liu2017]. TheReforE, ouR exPeRimeNt imPlies that this sINgle-Ion anisotropy is vanishiNgly sMaLL and WHaT liftS the U(1) deGEnERAcy Is ThE in-PlANe MagneTic Field. ThErefore, insTeaD oF having *six* doMAiNs (which wouLd hAve been the caSe iF the deGeNerACy was LifTed by The sinGLe-io... | spins leaves the energy u nchanged[@ Liu20 17] . T he refo re,our experiment impl ies that this single-i on an is o trop y i s van ishingl y s m a llan dwha tl if ts th e U (1) deg eneracy is th ein-plane mag n et ic field.The refore, inst ead of ha vi ng* six*dom ains(which wouldhave been t h e case if thed e ge... | spins_leaves the_energy unchanged[@Liu2017]. Therefore, our_experiment implies_that_this single-ion_anisotropy_is vanishingly small_and what lifts_the U(1) degeneracy is_the in-plane magnetic_field._Therefore, instead of having *six* domains (which would have been the case if the_degeneracy_was lifted_by_the_single-io... |
.\end{gathered}$$
The $S$ orbit function $S_{\lambda}(x)$, $\lambda\in P^{++}$ is defined as $$\begin{gathered}
\label{def_s-function1}
S_\lambda(x) := \sum_{\mu\in W_\lambda(G)} (-1)^{p(\mu)}e^{2\pi i \l\mu, x\r},\qquad
x\in\R^n,\end{gathered}$$ where $p(\mu)$ is the number of reflections necessary to obtain $\mu$ fr... | .\end{gathered}$$
The $ S$ orbit function $ S_{\lambda}(x)$, $ \lambda\in P^{++}$ is defined as $ $ \begin{gathered }
\label{def_s - function1 }
S_\lambda(x): = \sum_{\mu\in W_\lambda(G) } (-1)^{p(\mu)}e^{2\pi i \l\mu, x\r},\qquad
x\in\R^n,\end{gathered}$$ where $ p(\mu)$ is the number of expression necessary ... | .\end{hathered}$$
The $S$ orbit funcuion $S_{\lambda}(x)$, $\lamyea\in P^{++}$ is derined as $$\begin{gathered}
\label{def_s-functmon1}
S_\oambdq(x) := \sum_{\mu\in W_\lambda(G)} (-1)^{p(\ou)}e^{2\pi i \l\lu, x\r},\qquqd
x\ii\R^n,\end{gathered}$$ wisre $p(\mu)$ is ths numyec of reflectionx necessarf to obtain $\mu$ ff... | .\end{gathered}$$ The $S$ orbit function $S_{\lambda}(x)$, $\lambda\in defined $$\begin{gathered} \label{def_s-function1} := \sum_{\mu\in W_\lambda(G)} where is the number reflections necessary to $\mu$ from $\lambda$. Of course the $\mu$ can be obtained by different successions of reflections, but all routes from to $... | .\end{gathered}$$
The $S$ orbit functIon $S_{\lambda}(X)$, $\lambDa\iN P^{++}$ iS dEfinEd as $$\Begin{gathered}
\lABel{dEf_s-function1}
S_\lambda(x) := \sum_{\Mu\in W_\LaMBda(G)} (-1)^{P(\Mu)}E^{2\pi i \l\Mu, x\r},\qquAD
x\IN\r^n,\eNd{GaTheReD}$$ WhEre $p(\mU)$ is The numbEr of reflecTioNs Necessary to oBTaIn $\mu$ fr... | .\end{gathered}$$
The $S$ orbit fun ction $S _{\ la mbda }(x) $, $\lambda\in P^{+ +}$ is defined as $$\b egin{ ga t here d }\labe l{def_s - fu n c tio n1 }S_\ la m bd a(x):=\sum_{\ mu\in W_\l amb da (G)} (-1)^{p ( \m u)}e^{2\pi i\l\mu, x\r}, \qq uad
x\ in \R^ n ,\end {ga there d}$$ w h ere $p (\mu)$ is t h e numb ... | .\end{gathered}$$
The $S$_orbit function_$S_{\lambda}(x)$, $\lambda\in P^{++}$ is_defined as_$$\begin{gathered}
\label{def_s-function1}
S_\lambda(x)_:= \sum_{\mu\in_W_\lambda(G)}_(-1)^{p(\mu)}e^{2\pi i \l\mu,_x\r},\qquad
x\in\R^n,\end{gathered}$$ where $p(\mu)$_is the number of_reflections necessary to_obtain_$\mu$ fr... |
part of the proof of Theorem \[NonObtuse\] is dealing with the $\sqrt{k}$ internal corners that propagate through the spiral; since we have no bound for the number $N$ of windings of the spiral in terms of $n$, this could produce arbitrarily many new vertices. Thus propagation paths of the internal corners must be ben... | part of the proof of Theorem \[NonObtuse\ ] is dealing with the $ \sqrt{k}$ internal corner that spread through the spiral; since we have no bound for the phone number $ N$ of windings of the spiral in condition of $ n$, this could produce arbitrarily many fresh vertices. Thus generation paths of the internal corners m... | pagt of the proof of Theortm \[NonObtuse\] is dgaoing wmth the $\sqrt{k}$ ivternal corners that propagave tyrougy the spiral; since we fave no blund for the bumber $N$ oh windinnf of bhe s'ical in terms of $n$, this cogld produce artigrcrily many new vertices. Thus propagaeion payhd of the intergal bownera must be ben... | part of the proof of Theorem \[NonObtuse\] with $\sqrt{k}$ internal that propagate through no for the number of windings of spiral in terms of $n$, this produce arbitrarily many new vertices. Thus propagation paths of the internal corners must bent to terminate earlier. Consider the case of paths that start near the en... | part of the proof of Theorem \[NoNObtuse\] is dEalinG wiTh tHe $\Sqrt{K}$ intErnal corners thAT proPagate through the spiral; Since We HAve nO BoUnd foR the numBEr $n$ OF wiNdInGs oF tHE sPiral In tErms of $n$, This could pRodUcE arbitrarily MAnY new verticEs. THus propagatiOn pAths of ThE inTErnal CorNers mUst be bEN... | part of the proof of Theo rem \[NonO btuse \]isde alin g wi th the $\sqrt{ k }$ i nternal corners that p ropag at e thr o ug h the spiral ; s i n cewe h ave n o b oundfor the nu mber $N$ o f w in dings of the sp iral in te rms of $n$, thi s c ould p ro duc e arbi tra rilymany n e w vert ices. Thu sp ropaga t ... | part_of the_proof of Theorem \[NonObtuse\]_is dealing_with_the $\sqrt{k}$_internal_corners that propagate_through the spiral;_since we have no_bound for the_number_$N$ of windings of the spiral in terms of $n$, this could produce arbitrarily_many_new vertices._Thus_propagation_paths of the internal corners_must be ben... |
described below. The transmission not only depends on the phase, frequency, and amplitude of the oscillation, but also on the coefficients $a_n$ and $b_n$ determined by the state selectors, and the horizontal velocity distribution $\eta_v$. Here, we assume $a_n\approx b_n$, which is valid in our experiment within $\si... | described below. The transmission not only depend on the phase, frequency, and amplitude of the cycle, but also on the coefficients $ a_n$ and $ b_n$ determine by the country selectors, and the horizontal velocity distribution $ \eta_v$. Here, we bear $ a_n\approx b_n$, which is valid in our experiment within $ \sim 10... | dedcribed below. The transmlssion not only bwpends on ths phase, wrequency, and amplitude of tie owcillqtion, but also on the zoefficiejts $a_n$ abd $b_i$ determined by vge statc selsgtors, end the horizonjal velocity distribution $\atx_v$. Here, we assume $a_n\approx b_n$, which if valid ij our experimegt wptrin $\ap... | described below. The transmission not only depends phase, and amplitude the oscillation, but and determined by the selectors, and the velocity distribution $\eta_v$. Here, we assume b_n$, which is valid in our experiment within $\sim 10\%$. Then, from $\sum_n and $\sum_n |C_{n}(\tau_1)|^2=|C_{2}(\tau_1)|^2=1$ after the... | described below. The transmisSion not onlY depeNds On tHe PhasE, freQuency, and ampliTUde oF the oscillation, but also On the CoEFficIEnTs $a_n$ aNd $b_n$ detERmINEd bY tHe StaTe SElEctorS, anD the horIzontal velOciTy Distribution $\ETa_V$. Here, we assUme $A_n\approx b_n$, whIch Is valiD iN ouR ExperImeNt witHin $\si... | described below. The tran smission n ot on lydep en ds o n th e phase, frequ e ncy, and amplitude of theoscil la t ion, bu t als o on th e c o e ffi ci en ts$a _ n$ and$b_ n$ dete rmined bythe s tate selecto r s, and the h ori zontal veloc ity distr ib uti o n $\e ta_ v$. H ere, w e assum e $a_n\ap pr o x b_n$ ... | described_below. The_transmission not only depends_on the_phase,_frequency, and_amplitude_of the oscillation,_but also on_the coefficients $a_n$ and_$b_n$ determined by_the_state selectors, and the horizontal velocity distribution $\eta_v$. Here, we assume $a_n\approx b_n$, which_is_valid in_our_experiment_within $\si... |
, or the modes $n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ in a, of $10^5$ simulation trajectories. For finite-size correction we use $n_c = 10\tau_d$ in a and $n_c = 22\tau_d^{4/5}$ in b. Time is in units of $1/k_1^-$.[]{data-label="fig:kz"}](fig4){width="\linewidth"}
When testing these predictions using simulations of a spa... | , or the modes $ n_*^{(1)}<n_c$ and $ n_*^{(2)}>n_c$ in a, of $ 10 ^ 5 $ simulation trajectories. For finite - size correction we practice $ n_c = 10\tau_d$ in a and $ n_c = 22\tau_d^{4/5}$ in b. Time is in unit of $ 1 / k_1 ^ -$.[]{data - label="fig: kz"}](fig4){width="\linewidth " }
When testing these predictions ... | , or the modes $n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ ik a, of $10^5$ simulation trajxctoriea. For fivite-size correction we use $n_r = 10\tqu_d$ ib a and $n_c = 22\tau_d^{4/5}$ in b. Gime is ij units if $1/k_1^-$.[]{vata-label="fig:kz"}](fij4){sidth="\likzwidtg"}
Ahen vesting these ptedictions uving simulatiots oy a spa... | , or the modes $n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ of simulation trajectories. finite-size correction we a $n_c = 22\tau_d^{4/5}$ b. Time is units of $1/k_1^-$.[]{data-label="fig:kz"}](fig4){width="\linewidth"} When testing these using simulations of a spatially extended physical system, the finite size of the causes a... | , or the modes $n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ in a, of $10^5$ sImulation tRajecTorIes. foR finIte-sIze correction wE Use $n_C = 10\tau_d$ in a and $n_c = 22\tau_d^{4/5}$ in b. TiMe is iN uNIts oF $1/K_1^-$.[]{dAta-laBel="fig:kZ"}](FiG4){WIdtH="\lInEwiDtH"}
whEn tesTinG these pRedictions UsiNg Simulations oF A sPa... | , or the modes $n_*^{(1)}< n_c$ and $ n_*^{ (2) }>n _c $ in a,of $10^5$ simu l atio n trajectories. For fi nite- si z e co r re ction we use $n _ c =10 \t au_ d$ in a an d $ n_c = 2 2\tau_d^{4 /5} $in b. Time i s i n units of $1 /k_1^-$.[]{d ata -label =" fig : kz"}] (fi g4){w idth=" \ linewi dth"}
Wh en testin ... | , or_the modes_$n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ in_a, of_$10^5$_simulation trajectories._For_finite-size correction we_use $n_c =_10\tau_d$ in a and_$n_c = 22\tau_d^{4/5}$_in_b. Time is in units of $1/k_1^-$.[]{data-label="fig:kz"}](fig4){width="\linewidth"}
When testing these predictions using simulations of a_spa... |
)
,\omega\right) =1$$ whenever $m$ is sufficiently large and $\varepsilon$ is sufficiently small.
The proof of next theorem, being a slight modification of that of Theorem 1, will be omitted.
Suppose that $\mathcal{M}$ is a finitely generated von Neumann algebra with a tracial state $\tau$. Suppose that $\{\mathcal{... | )
, \omega\right) = 1$$ whenever $ m$ is sufficiently large and $ \varepsilon$ is sufficiently small.
The proof of next theorem, being a slender alteration of that of Theorem 1, will be omitted.
Suppose that $ \mathcal{M}$ is a finitely generated von Neumann algebra with a tracial country $ \tau$. Suppose that... | )
,\omeha\right) =1$$ whenever $m$ is rufficiently latgw and $\tarepsimon$ is sjfficiently small.
The proof oh nezt thtjrem, being a slighg modificwtion of thau of Theorem 1, will be omibced.
Suliose chet $\mathcal{M}$ is s finitely generated von Ndulann algebra with a tracial state $\twu$. Supppsf that $\{\mathcal{... | ) ,\omega\right) =1$$ whenever $m$ is sufficiently $\varepsilon$ sufficiently small. proof of next of of Theorem 1, be omitted. Suppose $\mathcal{M}$ is a finitely generated von algebra with a tracial state $\tau$. Suppose that $\{\mathcal{N}_{i}\}_{i=1}^{\infty}$ is an ascending sequence von Neumann subalgebras of $\m... | )
,\omega\right) =1$$ whenever $m$ is suffIciently laRge anD $\vaRepSiLon$ iS sufFiciently small.
tHe prOof of next theorem, being a SlighT mODifiCAtIon of That of THEoREM 1, wiLl Be OmiTtED.
SUpposE thAt $\mathcAl{M}$ is a finiTelY gEnerated von NEUmAnn algebra WitH a tracial staTe $\tAu$. SuppOsE thAT $\{\mathCal{... | )
,\omega\right) =1$$ whe never $m$is su ffi cie nt ly l arge and $\varepsi l on$is sufficiently small.
The p r oofo fnexttheorem , b e i ngasl igh tm od ifica tio n of th at of Theo rem 1 , will be om i tt ed.
Suppo sethat $\mathc al{ M}$ is a fi n itely ge nerat ed von Neuman n algebra w i th a t r acial ... | )
,\omega\right) _=1$$ whenever_$m$ is sufficiently large_and $\varepsilon$_is_sufficiently small.
The_proof_of next theorem,_being a slight_modification of that of_Theorem 1, will_be_omitted.
Suppose that $\mathcal{M}$ is a finitely generated von Neumann algebra with a tracial state_$\tau$._Suppose that_$\{\mathcal{... |
\]) tends to the Fick’s free diffusion expression (\[fick\]) [@EGELSTAFF]
### The mode coupling theory \[sec\_mct\]
The Laplace transform of the intermediate scattering function can be generally written as:
$$\begin{aligned}
\tilde{S}_s(Q,s)=\frac{1}{s+Q^2\tilde{U}(Q,s)} \nonumber\end{aligned}$$
being $\tilde{U}(Q,... | \ ]) tends to the Fick ’s free diffusion expression (\[fick\ ]) [ @EGELSTAFF ]
# # # The mood pair theory \[sec\_mct\ ]
The Laplace transform of the intermediate scattering routine can be generally written as:
$ $ \begin{aligned }
\tilde{S}_s(Q, s)=\frac{1}{s+Q^2\tilde{U}(Q, s) } \nonumber\end{aligned}$$
... | \]) tejds to the Fick’s free dinfusion expression (\[fick\]) [@EGELSFAFF]
### The mode coupling theory \[sec\_mct\]
Vhe Oaplaxe transform of the ingermediatv scatterung hunction can be jsneralln wrifben av:
$$\uegin{aligned}
\tilce{S}_s(Q,s)=\frac{1}{v+Q^2\tilde{U}(Q,s)} \nongmcex\end{aligned}$$
being $\tilde{U}(Q,... | \]) tends to the Fick’s free diffusion [@EGELSTAFF] The mode theory \[sec\_mct\] The scattering can be generally as: $$\begin{aligned} \tilde{S}_s(Q,s)=\frac{1}{s+Q^2\tilde{U}(Q,s)} being $\tilde{U}(Q,s)$ a generalized frequency and dependent diffusion coefficient. The mode coupling theory provides a self consistent ex... | \]) tends to the Fick’s free diffusIon expressIon (\[fiCk\]) [@EgELsTaFF]
### THe moDe coupling theoRY \[sec\_Mct\]
The Laplace transform Of the InTErmeDIaTe scaTtering FUnCTIon CaN bE geNeRAlLy wriTteN as:
$$\begiN{aligned}
\tiLde{s}_s(q,s)=\frac{1}{s+Q^2\tildE{u}(Q,S)} \nonumber\eNd{aLigned}$$
being $\tIldE{U}(Q,... | \]) tends to the Fick’s fr ee diffusi on ex pre ssi on (\[ fick \]) [@EGELSTAF F ]
# ## The mode coupling t heory \ [ sec\ _ mc t\]
The Lap l ac e tra ns fo rmof th e int erm ediatescattering fu nc tion can beg en erally wri tte n as:
$$\be gin {align ed }
\ t ilde{ S}_ s(Q,s )=\fra c {1}{s+ Q^2\tilde {U } (Q,s)} ... | \]) tends_to the_Fick’s free diffusion expression_(\[fick\]) [@EGELSTAFF]
###_The_mode coupling_theory_\[sec\_mct\]
The Laplace transform_of the intermediate_scattering function can be_generally written as:
$$\begin{aligned}
\tilde{S}_s(Q,s)=\frac{1}{s+Q^2\tilde{U}(Q,s)}_\nonumber\end{aligned}$$
being_$\tilde{U}(Q,... |
hand, because ${\widetilde}{G}$ is a subgroup of $G_{\mathbb{R}}$, we have that $S(t)$ is also a one parameter subgroup of $G_{\mathbb{R}}$, therefore the curve $\gamma(t)=S(t)\cdot \gamma(0)$ also gives a geodesic in $D$. Since geodesics on ${\widetilde}{D}$ are also geodesics on $D$, we have proved ${\widetilde}{D}$... | hand, because $ { \widetilde}{G}$ is a subgroup of $ G_{\mathbb{R}}$, we have that $ S(t)$ is also a one parameter subgroup of $ G_{\mathbb{R}}$, consequently the curvature $ \gamma(t)=S(t)\cdot \gamma(0)$ also gives a geodesic in $ D$. Since geodesic on $ { \widetilde}{D}$ are also geodesic on $ D$, we have proved $ {... | hajd, because ${\widetilde}{G}$ is a subgroup of $Y_{\nathbb{C}}$, we habe that $R(t)$ is also a one parameter snbgriup od $G_{\mathbb{R}}$, therefore tfe curve $\hamma(t)=S(t)\xdot \tamma(0)$ also gives a geodealc in $V$. Since geodesigs on ${\widethlde}{D}$ are also gdobesics on $D$, we have proved ${\widetilde}{Q}$... | hand, because ${\widetilde}{G}$ is a subgroup of have $S(t)$ is a one parameter curve \gamma(0)$ also gives geodesic in $D$. geodesics on ${\widetilde}{D}$ are also geodesics $D$, we have proved ${\widetilde}{D}$ is totally geodesic in $D$. We recall that to the discussion in Section 2.3, we have the following commutat... | hand, because ${\widetilde}{G}$ is a sUbgroup of $G_{\MathbB{R}}$, wE haVe That $s(t)$ is Also a one parameTEr suBgroup of $G_{\mathbb{R}}$, therefOre thE cURve $\gAMmA(t)=S(t)\cDot \gammA(0)$ AlSO GivEs A gEodEsIC iN $D$. SinCe gEodesicS on ${\widetilDe}{D}$ ArE also geodesiCS oN $D$, we have prOveD ${\widetilde}{D}$... | hand, because ${\widetild e}{G}$ isa sub gro upof $G_ {\ma thbb{R}}$, weh avethat $S(t)$ is also aone p ar a mete r s ubgro up of $ G _{ \ m ath bb {R }}$ ,t he refor e t he curv e $\gamma( t)= S( t)\cdot \gam m a( 0)$ also g ive s a geodesic in $D$.Si nce geode sic s on${\wid e tilde} {D}$ areal s o geod e sics o... | hand,_because ${\widetilde}{G}$_is a subgroup of_$G_{\mathbb{R}}$, we_have_that $S(t)$_is_also a one_parameter subgroup of_$G_{\mathbb{R}}$, therefore the curve_$\gamma(t)=S(t)\cdot \gamma(0)$ also_gives_a geodesic in $D$. Since geodesics on ${\widetilde}{D}$ are also geodesics on $D$, we_have_proved ${\widetilde}{D}$... |
& & $\nwarrow $ & \\
$\mathcal{\mathbf{P}}^{c}$ & & $\overset{\Phi }{\rightarrow }$ & & $\mathcal{\mathbf{P}}$%
\end{tabular}$$ Considering $\mathcal{\mathbf{P}}^{c}$ and $\mathcal{\mathbf{P}} $ as base spaces, the map $\overline{\Phi
}$ is induced by $\Phi $. Hence it is a bundle map. We need to check that it i... | & & $ \nwarrow $ & \\
$ \mathcal{\mathbf{P}}^{c}$ & & $ \overset{\Phi } { \rightarrow } $ & & $ \mathcal{\mathbf{P}}$%
\end{tabular}$$ Considering $ \mathcal{\mathbf{P}}^{c}$ and $ \mathcal{\mathbf{P } } $ as base spaces, the map $ \overline{\Phi
} $ is induce by $ \Phi $. therefore it is a bundle map. ... | & & $\nwarrow $ & \\
$\mathcal{\mathnf{P}}^{c}$ & & $\overset{\Pku }{\righvarrow }$ & & $\mathzal{\mathbf{P}}$%
\end{tabular}$$ Considecing $\mathxal{\mathbf{P}}^{c}$ and $\mathcau{\mathbf{P}} $ as base spares, the map $\overline{\Phi
}$ lf insmced yy $\Phi $. Hence it is a bundne map. We need tu eheck that it i... | & & $\nwarrow $ & \\ $\mathcal{\mathbf{P}}^{c}$ $\overset{\Phi }$ & $\mathcal{\mathbf{P}}$% \end{tabular}$$ Considering base the map $\overline{\Phi is induced by $. Hence it is a bundle We need to check that it is equivariant, i.e., $$\overline{\Phi }\left( \widetilde{\overline{u}}\cdot \gamma = \overline{\Phi }(\wide... | & & $\nwarrow $ & \\
$\mathcal{\mathbf{P}}^{c}$ & & $\oveRset{\Phi }{\rigHtarrOw }$ & & $\mAthCaL{\matHbf{P}}$%
\End{tabular}$$ ConsIDeriNg $\mathcal{\mathbf{P}}^{c}$ and $\maThcal{\MaTHbf{P}} $ AS bAse spAces, the MAp $\OVErlInE{\PHi
}$ iS iNDuCed by $\phi $. hence it Is a bundle mAp. WE nEed to check thAT iT i... | & & $\nwarrow $ & \\
$ \mathcal{\ mathb f{P }}^ {c }$ & &$\overset{\Phi }{\r ightarrow }$ & & $\ma thcal {\ m athb f {P }}$%\end{ta b ul a r }$$ C on sid er i ng $\ma thc al{\mat hbf{P}}^{c }$an d $\mathcal{ \ ma thbf{P}} $ as base spaces , t he map $ \ov e rline {\P hi
} $ is i n ducedby $\Phi$. Hencei t ... | &_ &_$\nwarrow $ & _\\
$\mathcal{\mathbf{P}}^{c}$_&_ &_$\overset{\Phi_}{\rightarrow }$ &_ & $\mathcal{\mathbf{P}}$%
\end{tabular}$$_Considering $\mathcal{\mathbf{P}}^{c}$ and $\mathcal{\mathbf{P}}_$ as base_spaces,_the map $\overline{\Phi
}$ is induced by $\Phi $. Hence it is a bundle map._We_need to_check_that_it i... |
,258}$Fm, are well-known examples where model calculations deviate from experimental data [@Flynn+72; @Hoffman+89; @Gonnenwein+99]. The origin of this transition has long been debated [@Warda+02; @Bonneau+06]. Improvements to model calculations can be made by increasing the smoothing range of the Strutinsky shell-corre... | , 258}$Fm, are well - known examples where model calculations deviate from experimental datum [ @Flynn+72; @Hoffman+89; @Gonnenwein+99 ]. The beginning of this transition has long been debated [ @Warda+02; @Bonneau+06 ]. improvement to mannequin calculations can be made by increase the polish range of the Strutinsky sh... | ,258}$Fm, wre well-known examples wmere model calculations deviafe from dxperimental data [@Flynn+72; @Hofflab+89; @Gonbenwein+99]. The origin of ghis trandition hqs libg been deuzted [@Wavba+02; @Bohkeau+06]. Nm'rovements to mpdel calcunations can be mxdz by increasing the smoothing range jf the Xtgutinsky shell-sorrt... | ,258}$Fm, are well-known examples where model calculations experimental [@Flynn+72; @Hoffman+89; The origin of debated @Bonneau+06]. Improvements to calculations can be by increasing the smoothing range of Strutinsky shell-correction procedure [@Albertsson+2019a; @Albertsson+2019b] or by applying Langevin dynamics [@Us... | ,258}$Fm, are well-known examples wheRe model calCulatIonS deViAte fRom eXperimental datA [@flynN+72; @Hoffman+89; @Gonnenwein+99]. The oRigin Of THis tRAnSitioN has lonG BeEN DebAtEd [@warDa+02; @bOnNeau+06]. IMprOvementS to model caLcuLaTions can be maDE bY increasinG thE smoothing raNge Of the STrUtiNSky shEll-Corre... | ,258}$Fm, are well-known e xamples wh ere m ode l c al cula tion s deviate from expe rimental data [@Flynn+ 72; @ Ho f fman + 89 ; @Go nnenwei n +9 9 ] . T he o rig in of this tr ansitio n has long be en debated [@W a rd a+02; @Bon nea u+06]. Impro vem ents t omod e l cal cul ation s canb e made by incre as i ng ... | ,258}$Fm, are_well-known examples_where model calculations deviate_from experimental_data_[@Flynn+72; @Hoffman+89;_@Gonnenwein+99]._The origin of_this transition has_long been debated [@Warda+02;_@Bonneau+06]. Improvements to_model_calculations can be made by increasing the smoothing range of the Strutinsky shell-corre... |
\begin{pmatrix}
-\mu-v_Fk & 0 & 0 & -\Delta^{h}_{\vec{k}} \\
0 & -\mu+v_Fk & \Delta^{h}_{\vec{k}} & 0 \\
0 &\Delta^{h*}_{\vec{k}} & \mu- v_Fk &0 \\
-\Delta^{h*}_{\vec{k}} &0& 0&\mu+v_Fk
\end{pmatrix},\end{aligned}$$ provided that the holon pairing parameter $\Delta^{h}_{\vec{k}}$ is p-wave like, i.... | \begin{pmatrix }
-\mu - v_Fk & 0 & 0 & -\Delta^{h}_{\vec{k } } \\
0 & -\mu+v_Fk & \Delta^{h}_{\vec{k } } & 0 \\
0 & \Delta^{h*}_{\vec{k } } & \mu- v_Fk & 0 \\
-\Delta^{h*}_{\vec{k } } & 0 & 0&\mu+v_Fk
\end{pmatrix},\end{aligned}$$ provided that the holon pairing parameter $ \Delta^{h}_{\ve... | \bfgin{pmatrix}
-\mu-v_Fk & 0 & 0 & -\Delta^{h}_{\vec{k}} \\
0 & -\mu+v_Hk & \Delfa^{h}_{\vec{k}} & 0 \\
0 &\Delta^{h*}_{\vec{k}} & \mu- v_Fk &0 \\
-\Deluc^{h*}_{\vec{k}} &0& 0&\mu+v_Fk
\end{pmxtrix},\end{apigned}$$ peovived that the holon pairiky parzletex $\Velta^{h}_{\vec{k}}$ is p-eave like, h.... | \begin{pmatrix} -\mu-v_Fk & 0 & 0 & 0 -\mu+v_Fk & & 0 \\ &0 -\Delta^{h*}_{\vec{k}} &0& 0&\mu+v_Fk provided that the pairing parameter $\Delta^{h}_{\vec{k}}$ is p-wave like, $\Delta^h_{-\vec{k}} = -\Delta^h_{\vec{k}}$. The spectrum of quasiparticles consists of two decoupled branches, $$\begin{aligned} \epsilon_{\pm,\ve... | \begin{pmatrix}
-\mu-v_Fk & 0 & 0 & -\Delta^{h}_{\veC{k}} \\
0 & -\mu+v_Fk & \DelTa^{h}_{\veC{k}} & 0 \\
0 &\DEltA^{h*}_{\Vec{k}} & \Mu- v_FK &0 \\
-\Delta^{h*}_{\vec{k}} &0& 0&\mu+v_fK
\end{Pmatrix},\end{aligned}$$ proviDed thAt THe hoLOn PairiNg paramETeR $\dEltA^{h}_{\VeC{k}}$ iS p-WAvE like, I.... | \begin{pmatrix}
-\mu -v_Fk & 0& 0 & -\ Del ta ^{h} _{\v ec{k}} \\
0 & - \mu+v_Fk & \Delta^{h}_ {\vec {k } } & 0\\
0 &\D e lt a ^ {h* }_ {\ vec {k } }& \mu - v _Fk &0\\
-\D elt a^ {h*}_{\vec{k } }&0& 0&\mu+ v_F k
\end{pma tri x},\en d{ ali g ned}$ $ p rovid ed tha t the h olon pair in g param e te... | _\begin{pmatrix}
_ -\mu-v_Fk &_0 &_0_& -\Delta^{h}_{\vec{k}}_\\
_ _0 & -\mu+v_Fk_& \Delta^{h}_{\vec{k}} &_0 \\
__ 0 &\Delta^{h*}_{\vec{k}} & \mu- v_Fk &0 \\
-\Delta^{h*}_{\vec{k}} &0& 0&\mu+v_Fk
__\end{pmatrix},\end{aligned}$$ provided_that_the_holon pairing parameter $\Delta^{h}_{\vec{k}}$ is_p-wave like, i.... |
isospin-0 vectors could possibly be connected to the $Y(3940)\rightarrow J/\psi\omega$ and $Y(4140)\rightarrow J/\psi \phi$ since both $J/\psi$ and $\phi$ have isospin-0 and are very close to the two vector threshold. Second, it is striking that there are at least two structures observed in the same $J/\psi\phi$ spect... | isospin-0 vectors could possibly be connected to the $ Y(3940)\rightarrow J/\psi\omega$ and $ Y(4140)\rightarrow J/\psi \phi$ since both $ J/\psi$ and $ \phi$ have isospin-0 and are very close to the two vector brink. Second, it is strike that there are at least two structures observed in the like $ J/\psi\phi$ spectru... | islspin-0 vectors could posslbly be connecteb to thx $Y(3940)\righfarrow J/\osi\omega$ and $Y(4140)\rightarrow J/\psm \phu$ sinxe both $J/\psi$ and $\phi$ hxve isosppn-0 and arw vecy close to the vso vector thrsdholb. Wecond, it is sjriking that there are at nexsc two structures observed in the samq $J/\psi\pni$ spect... | isospin-0 vectors could possibly be connected to J/\psi\omega$ $Y(4140)\rightarrow J/\psi since both $J/\psi$ are close to the vector threshold. Second, is striking that there are at two structures observed in the same $J/\psi\phi$ spectrum from the same exclusive $B$ The two structures do not need to arise from the sa... | isospin-0 vectors could possibLy be connecTed to The $y(3940)\riGhTarrOw J/\pSi\omega$ and $Y(4140)\rigHTarrOw J/\psi \phi$ since both $J/\psi$ And $\phI$ hAVe isOSpIn-0 and Are very CLoSE To tHe TwO veCtOR tHreshOld. second, iT is strikinG thAt There are at leASt Two structuRes Observed in thE saMe $J/\psi\PhI$ spECt... | isospin-0 vectors could p ossibly be conn ect edto the $Y( 3940)\rightarr o w J/ \psi\omega$ and $Y(414 0)\ri gh t arro w J /\psi \phi$s in c e bo th $ J/\ ps i $and $ \ph i$ have isospin-0 an dare very clo s eto the two ve ctor thresho ld. Secon d, it is st rik ing t hat th e re are at least t w o stru ... | isospin-0_vectors could_possibly be connected to_the $Y(3940)\rightarrow_J/\psi\omega$_and $Y(4140)\rightarrow_J/\psi_\phi$ since both_$J/\psi$ and $\phi$_have isospin-0 and are_very close to_the_two vector threshold. Second, it is striking that there are at least two structures_observed_in the_same_$J/\psi\phi$_spect... |
Note that, if we take the alphabet $[k]$ to be a set of colours, the definition of a picture is analogous to that of a coloured pattern [@ref22].
Early generating/accepting systems for 2D languages comprise $2 \times 2$ tiles [@g31], 2D automata [@ref109], two-dimensional on-line tessellation acceptors [@tesselation]... | Note that, if we take the alphabet $ [ k]$ to be a set of coloring material, the definition of a movie is analogous to that of a coloured pattern [ @ref22 ].
Early render / accepting systems for 2D linguistic process incorporate $ 2 \times 2 $ tiles [ @g31 ], 2D automata [ @ref109 ], two - dimensional on - line tess... | Nohe that, if we take the auphabet $[k]$ to be a set mf colkurs, the definition of a picture is enaligous to that of a coloured pattern [@gef22].
Early tenecating/accepting systems njr 2D pangbajes comprise $2 \tlmes 2$ tiles [@g31], 2D automata [@sew109], cwo-dimensional on-line tessellation asceptorx [@hesselation]... | Note that, if we take the alphabet be set of the definition of that a coloured pattern Early generating/accepting systems 2D languages comprise $2 \times 2$ [@g31], 2D automata [@ref109], two-dimensional on-line tessellation acceptors [@tesselation], and 2D grammars. More a generating system was introduced by Varricchi... | Note that, if we take the alphabEt $[k]$ to be a seT of coLouRs, tHe DefiNitiOn of a picture is ANaloGous to that of a coloured pAtterN [@rEF22].
EarLY gEneraTing/accEPtING syStEmS foR 2D LAnGuageS coMprise $2 \tImes 2$ tiles [@g31], 2d auToMata [@ref109], two-diMEnSional on-liNe tEssellation aCcePtors [@tEsSelATion]... | Note that, if we take the alphabet$[k]$ to be a set ofcolours, the d e fini tion of a picture is a nalog ou s tot ha t ofa colou r ed p att er n[@r ef 2 2] .
Ea rly genera ting/accep tin gsystems for2 Dlanguagescom prise $2 \ti mes 2$ ti le s [ @ g31], 2D auto mata [ @ ref109 ], two-di me n sional on-line ... | Note_that, if_we take the alphabet_$[k]$ to_be_a set_of_colours, the definition_of a picture_is analogous to that_of a coloured_pattern_[@ref22].
Early generating/accepting systems for 2D languages comprise $2 \times 2$ tiles [@g31], 2D automata_[@ref109],_two-dimensional on-line_tessellation_acceptors_[@tesselation]... |
adiabatic stage, the outflowing mass creates the over-pressurized bubble of hot gas. The mass of the bubble $M_{\textrm{b}}$ is determined by the conservation of mass law: $$\label{mass_e}
\frac{{\textrm{d}}M_{\textrm{b}}}{{\textrm{d}}t}=\dot{M}_{\textrm{w}} + 4\pi R^2 \rho_{\textrm{o}} \left( v_{\textrm{s}} - v_{\tex... | adiabatic stage, the outflowing mass creates the all over - pressurize bubble of hot accelerator. The multitude of the bubble $ M_{\textrm{b}}$ is determined by the conservation of batch law: $ $ \label{mass_e }
\frac{{\textrm{d}}M_{\textrm{b}}}{{\textrm{d}}t}=\dot{M}_{\textrm{w } } + 4\pi R^2 \rho_{\textrm{o } } \le... | adlabatic stage, the outfloding mass creatgs the oter-presaurized cubble of hot gas. The mass oh thw bubvle $M_{\textrm{b}}$ is determkned by tje conseevatmon of mass law: $$\label{mass_e}
\frac{{\fcxtrm{b}}M_{\vextrm{b}}}{{\textrm{d}}t}=\cot{M}_{\textrm{f}} + 4\pi R^2 \rho_{\texdro{o}} \left( v_{\textrm{s}} - v_{\tex... | adiabatic stage, the outflowing mass creates the of gas. The of the bubble conservation mass law: $$\label{mass_e} + 4\pi R^2 \left( v_{\textrm{s}} - v_{\textrm{o}}\right),$$ where the term on the right hand side represents the mass of cold gas reheated SN explosions and stellar winds and the second term defines the am... | adiabatic stage, the outflowiNg mass creaTes thE ovEr-pReSsurIzed Bubble of hot gas. tHe maSs of the bubble $M_{\textrm{b}}$ iS deteRmINed bY ThE consErvatioN Of MASs lAw: $$\LaBel{MaSS_e}
\Frac{{\tExtRm{d}}M_{\texTrm{b}}}{{\textrm{D}}t}=\dOt{m}_{\textrm{w}} + 4\pi R^2 \rHO_{\tExtrm{o}} \left( V_{\teXtrm{s}} - v_{\tex... | adiabatic stage, the outf lowing mas s cre ate s t he ove r-pr essurized bubb l e of hot gas. The mass ofthe b ub b le $ M _{ \text rm{b}}$ is d ete rm in edby th e con ser vationof mass la w:$$ \label{mass_ e }\frac{{\te xtr m{d}}M_{\tex trm {b}}}{ {\ tex t rm{d} }t} =\dot {M}_{\ t extrm{ w}} + 4\p iR ^2 \rh o _{\t... | adiabatic_stage, the_outflowing mass creates the_over-pressurized bubble_of_hot gas._The_mass of the_bubble $M_{\textrm{b}}$ is_determined by the conservation_of mass law:_$$\label{mass_e}
\frac{{\textrm{d}}M_{\textrm{b}}}{{\textrm{d}}t}=\dot{M}_{\textrm{w}}_+ 4\pi R^2 \rho_{\textrm{o}} \left( v_{\textrm{s}} - v_{\tex... |
situations.
Acknowledgements {#acknowledgements.unnumbered}
----------------
Thanks to Ralph Kaufmann for bringing our attention to the references [@Bun] and [@Day]. We thank Paul Balmer for pointing us to [@BKS] and subsequent discussions of the case of homological functors, see Section \[sec:kuenneth\]. We thank t... | situations.
Acknowledgements { # acknowledgements.unnumbered }
----------------
Thanks to Ralph Kaufmann for bringing our attention to the reference point [ @Bun ] and [ @Day ]. We thank Paul Balmer for target us to [ @BKS ] and subsequent discussions of the case of homological functors, witness Section \[sec... | sihuations.
Acknowledgements {#acknowledgemenjs.ynnumbxred}
----------------
Thahks to Rxlph Kaufmann for bringing onr artentuon to the references [@Cun] and [@Dwy]. We thqnk Kaul Balmer for pointing mf to [@NKS] aud subsequent dixcussions mf the case of humllogical functors, see Section \[sec:kuegneth\]. Wr hhank t... | situations. Acknowledgements {#acknowledgements.unnumbered} ---------------- Thanks to Ralph bringing attention to references [@Bun] and for us to [@BKS] subsequent discussions of case of homological functors, see Section We thank the referee for her/his suggestions that considerably improved the exposition. Notation -... | situations.
AcknowledgementS {#acknowledGemenTs.uNnuMbEred}
----------------
thanKs to Ralph KaufmANn foR bringing our attention tO the rEfERencES [@BUn] and [@day]. We thANk pAUl BAlMeR foR pOInTing uS to [@bKS] and sUbsequent dIscUsSions of the caSE oF homologicAl fUnctors, see SeCtiOn \[sec:kUeNneTH\]. We thAnk T... | situations.
Acknowledgem ents {#ack nowle dge men ts .unn umbe red}
--------- - ---- --
Thanks to Ralph Ka ufman nf or b r in gingour att e nt i o n t oth e r ef e re nces[@B un] and [@Day]. W e t ha nk Paul Balm e rfor pointi ngus to [@BKS] an d subs eq uen t disc uss ionsof the case o f homolog ic a l func t or... | situations.
Acknowledgements_{#acknowledgements.unnumbered}
----------------
Thanks to_Ralph Kaufmann for bringing_our attention_to_the references_[@Bun]_and [@Day]. We_thank Paul Balmer_for pointing us to_[@BKS] and subsequent_discussions_of the case of homological functors, see Section \[sec:kuenneth\]. We thank t... |
functions on a convex domain. In this case one can use the expectation-maximization (EM) algorithm [@Vardi] as discussed in [@Zhang]. A problem is that due to stopping criteria and numerical precision, one cannot expect to find the exact optimum. We show in App. \[sect:bad estimation\] that one can compensate for this... | functions on a convex domain. In this case one can use the anticipation - maximization (EM) algorithm [ @Vardi ] as discourse in [ @Zhang ]. A problem is that due to stopping criteria and numeral precision, one cannot expect to discover the exact optimum. We show in App. \[sect: regretful estimation\ ] that one c... | fujctions on a convex domaln. In this case one can use tge expecgation-maximization (EM) algorivhm [@Vqrdi] qs discussed in [@Zhang]. A problem ps that dye ti stopping rditeria and nhlerieao precision, ong cannot expact to find tha dxcct optimum. We show in App. \[sect:bad eseimatiom\] hhat one can cjmpemfate for this... | functions on a convex domain. In this can the expectation-maximization algorithm [@Vardi] as is due to stopping and numerical precision, cannot expect to find the exact We show in App. \[sect:bad estimation\] that one can compensate for this problem maintain validity of the computed $p$-value. Conclusion {#sect:conclus... | functions on a convex domain. IN this case oNe can Use The ExPectAtioN-maximization (Em) AlgoRithm [@Vardi] as discussed iN [@ZhanG]. A PRoblEM iS that Due to stOPpING crItErIa aNd NUmEricaL prEcision, One cannot eXpeCt To find the exaCT oPtimum. We shOw iN App. \[sect:bad eStiMation\] ThAt oNE can cOmpEnsatE for thIS... | functions on a convex dom ain. In th is ca seone c an u se t he expectation - maxi mization (EM) algorith m [@V ar d i] a s d iscus sed in[ @Z h a ng] .Apro bl e mis th atdue tostopping c rit er ia and numer i ca l precisio n,one cannot e xpe ct tofi ndt he ex act opti mum. W e showin App. \ [s e ct:bad estimat i ... | functions_on a_convex domain. In this_case one_can_use the_expectation-maximization_(EM) algorithm [@Vardi] as_discussed in [@Zhang]. A_problem is that due_to stopping criteria_and_numerical precision, one cannot expect to find the exact optimum. We show in App. \[sect:bad_estimation\]_that one_can_compensate_for this... |
].
Besides this strong $\Delta J = 2$ propensity rule, one can see from table \[tab:rates2\] and figures \[fig:ratecompare\], \[fig:rate12\] that the rod-like interaction drives large $\Delta J$ transfers. For instance, for T $> 20$ K, rates for $\Delta J > 6$ are generally larger than rates for $\Delta J = 1$, and ra... | ].
Besides this strong $ \Delta J = 2 $ propensity rule, one can see from mesa \[tab: rates2\ ] and visualize \[fig: ratecompare\ ], \[fig: rate12\ ] that the rod - like interaction drives big $ \Delta J$ transfers. For instance, for T $ > 20 $ K, rate for $ \Delta J > 6 $ are broadly larger than rate for $ \Del... | ].
Besldes this strong $\Delta J = 2$ propensity rolw, one ran see from tacle \[tab:rates2\] and figures \[fig:retecimpart\], \[fig:rate12\] that the fod-like ijteractiin dcives large $\Delte J$ transfers. Rlr iusvance, for T $> 20$ K, tates for $\Denta J > 6$ are getefaply larger than rates for $\Delta J = 1$, and ra... | ]. Besides this strong $\Delta J = rule, can see table \[tab:rates2\] and rod-like drives large $\Delta transfers. For instance, T $> 20$ K, rates for J > 6$ are generally larger than rates for $\Delta J = 1$, rates for $\Delta J > 8$ are only one order of magnitude below for J 2$. behaviour is likely to emphasize the ... | ].
Besides this strong $\Delta J = 2$ prOpensity ruLe, one Can See FrOm taBle \[tAb:rates2\] and figuREs \[fiG:ratecompare\], \[fig:rate12\] thaT the rOd-LIke iNTeRactiOn driveS LaRGE $\DeLtA J$ TraNsFErS. For iNstAnce, for t $> 20$ K, rates for $\delTa j > 6$ are generallY LaRger than raTes For $\Delta J = 1$, and Ra... | ].
Besides this strong $\ Delta J =2$ pr ope nsi ty rul e, o ne can see fro m tab le \[tab:rates2\] andfigur es \[fi g :r ateco mpare\] , \ [ f ig: ra te 12\ ]t ha t the ro d-likeinteractio n d ri ves large $\ D el ta J$ tran sfe rs. For inst anc e, for T $> 20$ K , r atesfor $\ D elta J > 6$ are g e nerall y ... | ].
Besides this_strong $\Delta_J = 2$ propensity_rule, one_can_see from_table \[tab:rates2\]_and figures \[fig:ratecompare\],_\[fig:rate12\] that the_rod-like interaction drives large_$\Delta J$ transfers._For_instance, for T $> 20$ K, rates for $\Delta J > 6$ are generally larger_than_rates for_$\Delta_J_= 1$, and ra... |
ederna F. Olsson, A.S. Levine and H.B. Schioth, [*Analysis of the network of feeding neuroregulators using the Allen Brain Atlas*]{}, Neurosci. Biobehav. Rev. [**[32]{}**]{}, 945–956 (2008).
H.-W. Dong, *The Allen reference atlas: a digital brain atlas of the C57BL/6J male mouse*, Wiley, 2007.
P. Grange and P.P. Mitr... | ederna F. Olsson, A.S. Levine and H.B. Schioth, [ * Analysis of the network of feeding neuroregulators using the Allen Brain Atlas * ] { }, Neurosci. Biobehav. Rev. [ * * [ 32 ] { } * * ] { }, 945–956 (2008).
H.-W. Dong, * The Allen reference point atlas: a digital mind atlas of the C57BL/6J male mouse *, Wiley, 200... | edegna F. Olsson, A.S. Levine akd H.B. Schioth, [*Ancoysis mf the network of feeding neuroregulators nsint the Allen Brain Atlas*]{}, Neufosci. Biohehav. Rec. [**[32]{}**]{}, 945–956 (2008).
I.-W. Dong, *The Allei referekee atmws: a vigital brain ajlas of the W57BL/6J male mousa*, Dipey, 2007.
P. Grange and P.P. Mitr... | ederna F. Olsson, A.S. Levine and H.B. of network of neuroregulators using the Rev. 945–956 (2008). H.-W. *The Allen reference a digital brain atlas of the male mouse*, Wiley, 2007. P. Grange and P.P. Mitra, [*Computational neuroanatomy and gene Optimal sets of marker genes for brain regions*]{}, IEEE, in CISS 2012, 46... | ederna F. Olsson, A.S. Levine and H.b. Schioth, [*AnAlysiS of The NeTworK of fEeding neuroregULatoRs using the Allen Brain AtLas*]{}, NeUrOSci. BIObEhav. REv. [**[32]{}**]{}, 945–956 (2008).
H.-W. DonG, *thE aLleN rEfEreNcE AtLas: a dIgiTal braiN atlas of thE C57Bl/6J Male mouse*, WilEY, 2007.
P. grange and P.p. MiTr... | ederna F. Olsson, A.S. Lev ine and H. B. Sc hio th, [ *Ana lysi s of the netwo r k of feeding neuroregulato rs us in g the Al len B rain At l as * ] {}, N eu ros ci . B iobeh av. Rev. [ **[32]{}** ]{} ,945–956 (200 8 ).
H.-W. Do ng, *The Allenref erence a tla s : a d igi tal b rain a t las of the C57B L/ 6 J... | ederna F._Olsson, A.S._Levine and H.B. Schioth,_[*Analysis of_the_network of_feeding_neuroregulators using the_Allen Brain Atlas*]{},_Neurosci. Biobehav. Rev. [**[32]{}**]{},_945–956 (2008).
H.-W. Dong,_*The_Allen reference atlas: a digital brain atlas of the C57BL/6J male mouse*, Wiley, 2007.
P._Grange_and P.P._Mitr... |
Q$ such that $Qf(t_i,\cdot) :=Q_{t_i}f(t_i,\cdot)$ where for $1\leq i\leq n$ and $g:{\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$, $Q_{t_i}g(x) := q_{t_i}(x)g(x)$.
\[thm1\] Assume that for all $1\leq i\leq j\leq k\leq n$ the following holds:
- [Right-invertibility]{}: ${\mathcal{W}}_{t_i,t_j}{\mathcal{W}}_... | Q$ such that $ Qf(t_i,\cdot): = Q_{t_i}f(t_i,\cdot)$ where for $ 1\leq i\leq n$ and $ g:{\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$, $ Q_{t_i}g(x): = q_{t_i}(x)g(x)$.
\[thm1\ ] Assume that for all $ 1\leq i\leq j\leq k\leq n$ the following holds:
- [ Right - invertibility ] { }: $ { \mathcal{W}}_{t_... | Q$ skch that $Qf(t_i,\cdot) :=Q_{t_i}f(t_i,\gdot)$ where for $1\lgq i\leq i$ and $g:{\snsuremagh{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$, $Q_{r_i}g(x) := q_{t_i}(x)g(x)$.
\[thm1\] Assume that for all $1\peq i\leq j\lew k\leq n$ thx followlug homfs:
- [Cight-invertibillty]{}: ${\mathcal{F}}_{t_i,t_j}{\mathcal{W}}_... | Q$ such that $Qf(t_i,\cdot) :=Q_{t_i}f(t_i,\cdot)$ where for n$ $g:{\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$, $Q_{t_i}g(x) q_{t_i}(x)g(x)$. \[thm1\] Assume j\leq n$ the following - [Right-invertibility]{}: ${\mathcal{W}}_{t_i,t_j}{\mathcal{W}}_{t_j,t_i}K_{t_i}=K_{t_i}$; [Semigroup property]{}: ${\mathcal{W... | Q$ such that $Qf(t_i,\cdot) :=Q_{t_i}f(t_i,\cdOt)$ where for $1\Leq i\lEq n$ And $G:{\eNsurEmatH{\mathbb{R}}}\to{\ensuREmatH{\mathbb{R}}}$, $Q_{t_i}g(x) := q_{t_i}(x)g(x)$.
\[thm1\] assumE tHAt foR AlL $1\leq i\Leq j\leq K\LeQ N$ The FoLlOwiNg HOlDs:
- [RigHt-iNvertibIlity]{}: ${\mathcAl{W}}_{T_i,T_j}{\mathcal{W}}_... | Q$ such that $Qf(t_i,\cdot ) :=Q_{t_i }f(t_ i,\ cdo t) $ wh erefor $1\leq i\l e q n$ and $g:{\ensuremath{\ mathb b{ R }}}\ t o{ \ensu remath{ \ ma t h bb{ R} }} $,$Q _ {t _i}g( x):= q_{t _i}(x)g(x) $.
\ [thm1\] Assu m ethat for a ll$1\leq i\leq j\ leq k\ le q n $ thefol lowin g hold s :
- [Right-i nv e rtibil i ty... | Q$ such_that $Qf(t_i,\cdot)_:=Q_{t_i}f(t_i,\cdot)$ where for $1\leq_i\leq n$_and_$g:{\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$, $Q_{t_i}g(x)_:=_q_{t_i}(x)g(x)$.
\[thm1\] Assume that_for all $1\leq_i\leq j\leq k\leq n$_the following holds:
-__ [Right-invertibility]{}: ${\mathcal{W}}_{t_i,t_j}{\mathcal{W}}_... |
e_2$ in QD$_2$ are prepared in $|+\rangle_{e_1}$ and $|+\rangle_{e_2}$, respectively. Here $|\pm\rangle_e=\frac{1}{\sqrt{2}}(|\uparrow\rangle\pm|\downarrow\rangle)_e$.
After the two photons $A$ and $B$ in the hyperentangled Bell state pass through the quantum circuit shown in Fig. \[figure3.10\]a in sequence, the stat... | e_2 $ in QD$_2 $ are prepared in $ |+\rangle_{e_1}$ and $ |+\rangle_{e_2}$, respectively. Here $ |\pm\rangle_e=\frac{1}{\sqrt{2}}(|\uparrow\rangle\pm|\downarrow\rangle)_e$.
After the two photons $ A$ and $ B$ in the hyperentangled Bell state pass through the quantum racing circuit usher in Fig. \[figure3.10\]a in ... | e_2$ ij QD$_2$ are prepared in $|+\rannle_{e_1}$ and $|+\rangle_{e_2}$, respecvively. Gere $|\pm\rxngle_e=\frac{1}{\sqrt{2}}(|\uparrow\rangle\pl|\diwnareow\rangle)_e$.
After the twu photons $A$ and $B$ in uhe hyperentanglev Bell sbcte pzds tkriugh the quantom circuit svown in Fig. \[figgrd3.10\]a in sequence, the stat... | e_2$ in QD$_2$ are prepared in $|+\rangle_{e_1}$ respectively. $|\pm\rangle_e=\frac{1}{\sqrt{2}}(|\uparrow\rangle\pm|\downarrow\rangle)_e$. After two photons $A$ Bell pass through the circuit shown in \[figure3.10\]a in sequence, the state of system $ABe_1e_2$ evolves to $$\begin{aligned} %eq.11 % Eq. 60 |\phi^\pm\rang... | e_2$ in QD$_2$ are prepared in $|+\rangle_{e_1}$ And $|+\rangle_{e_2}$, RespeCtiVelY. HEre $|\pM\ranGle_e=\frac{1}{\sqrt{2}}(|\upARrow\Rangle\pm|\downarrow\ranglE)_e$.
AftEr THe twO PhOtons $a$ and $B$ in THe HYPerEnTaNglEd bElL statE paSs throuGh the quantUm cIrCuit shown in FIG. \[fIgure3.10\]a in seQueNce, the stat... | e_2$ in QD$_2$ are prepare d in $|+\r angle _{e _1} $and$|+\ rangle_{e_2}$, resp ectively. Here $|\pm\r angle _e = \fra c {1 }{\sq rt{2}}( | \u p a rro w\ ra ngl e\ p m| \down arr ow\rang le)_e$.
A fte rthe two phot o ns $A$ and $ B$in the hyper ent angled B ell state pa ss th rought he qua ntum circ ui t shown in... | e_2$ in_QD$_2$ are_prepared in $|+\rangle_{e_1}$ and_$|+\rangle_{e_2}$, respectively._Here_$|\pm\rangle_e=\frac{1}{\sqrt{2}}(|\uparrow\rangle\pm|\downarrow\rangle)_e$.
After the_two_photons $A$ and_$B$ in the_hyperentangled Bell state pass_through the quantum_circuit_shown in Fig. \[figure3.10\]a in sequence, the stat... |
(\pi_\ast\Sigma_g, { {\textstyle {V_T\over M}} },
{ {\textstyle {1\over M}} } \pi_\ast B_T+{ {\textstyle {1\over M}} }\check{B}_M\right),$$ where $\check{B}_M\in\Htz\cap \left(\pi_\ast \Htzt^{\Z_M}\right)^\perp$ is a fixed primitive lattice vector with $\no{\check{B}_M}=-2M^2$.
The following application of Prop. \[we... | ( \pi_\ast\Sigma_g, { { \textstyle { V_T\over M } } },
{ { \textstyle { 1\over M } } } \pi_\ast B_T+ { { \textstyle { 1\over M } } } \check{B}_M\right),$$ where $ \check{B}_M\in\Htz\cap \left(\pi_\ast \Htzt^{\Z_M}\right)^\perp$ is a fixed primitive lattice vector with $ \no{\check{B}_M}=-2M^2$.
The following lotio... | (\pi_\adt\Sigma_g, { {\textstyle {V_T\ovtr M}} },
{ {\textstyle {1\over M}} } \pi_\ast B_T+{ {\textrtyle {1\over M}} }\check{B}_M\right),$$ whxre $\xheck{V}_M\in\Htz\cap \left(\pi_\ast \Hgzt^{\Z_M}\righn)^\perp$ is q fieed primitive lavfice vegcor wjbh $\no{\ehxck{B}_M}=-2M^2$.
The folloeing appliwation of Prop. \[fe... | (\pi_\ast\Sigma_g, { {\textstyle {V_T\over M}} }, { M}} \pi_\ast B_T+{ {1\over M}} }\check{B}_M\right),$$ a primitive lattice vector $\no{\check{B}_M}=-2M^2$. The following of Prop. \[wendland:ZMorb\] is a helpful see Sect. \[wendland:SI\]: Let $a,\,b,\,c\in\Z$ such that $$Q_{a,b,c}:=\left(\begin{array}{cc} 8a&4b\\4b&8... | (\pi_\ast\Sigma_g, { {\textstyle {V_T\oveR M}} },
{ {\textstylE {1\over m}} } \pi_\Ast b_T+{ {\TextStylE {1\over M}} }\check{B}_M\rIGht),$$ wHere $\check{B}_M\in\Htz\cap \lefT(\pi_\asT \HTZt^{\Z_M}\RIgHt)^\perP$ is a fixED pRIMitIvE lAttIcE VeCtor wIth $\No{\check{b}_M}=-2M^2$.
The follOwiNg Application oF prOp. \[we... | (\pi_\ast\Sigma_g, { {\tex tstyle {V_ T\ove r M }}},
{{\te xtstyle {1\ove r M}} } \pi_\ast B_T+{ {\te xtsty le {1\o v er M}}}\check { B} _ M \ri gh t) ,$$ w h er e $\c hec k{B}_M\ in\Htz\cap \l ef t(\pi_\ast \ H tz t^{\Z_M}\r igh t)^\perp$ is afixedpr imi t ive l att ice v ectorw ith $\ no{\check {B } _M}=-2 M ... | (\pi_\ast\Sigma_g, {_{\textstyle {V_T\over_M}} },
{ {\textstyle_{1\over M}}_}_\pi_\ast B_T+{_{\textstyle_{1\over M}} }\check{B}_M\right),$$_where $\check{B}_M\in\Htz\cap \left(\pi_\ast_\Htzt^{\Z_M}\right)^\perp$ is a fixed_primitive lattice vector_with_$\no{\check{B}_M}=-2M^2$.
The following application of Prop. \[we... |
{\epsilon\over 2}+ \eta+ {\epsilon\over 2}=\epsilon+ \eta.$$ Clearly, the pervious relation for each $n\in\Bbb Z$ is true. Since $\epsilon>0$ is a small arbitrary number then for each $n\in\Bbb Z$ we have $r\Big(F_{\sigma_n}(x), F_{\sigma_n}(y)\Big)\leq \eta$. Whereof $\mathcal{F}$ is expansive relative to $\sigma$ wit... | { \epsilon\over 2}+ \eta+ { \epsilon\over 2}=\epsilon+ \eta.$$ Clearly, the pervious relation for each $ n\in\Bbb Z$ is true. Since $ \epsilon>0 $ is a small arbitrary numeral then for each $ n\in\Bbb Z$ we take $ r\Big(F_{\sigma_n}(x), F_{\sigma_n}(y)\Big)\leq \eta$. Whereof $ \mathcal{F}$ is expansive relative to $ \... | {\epsllon\over 2}+ \eta+ {\epsilon\ovev 2}=\epsilon+ \eta.$$ Clgaely, thx pervikus relagion for each $n\in\Bbb Z$ is trne. Sunce $\tisilon>0$ is a small arbktrary nulber theb foc each $n\in\Bbb Z$ xs have $v\Yig(F_{\sjnma_n}(x), H_{\sigma_n}(y)\Big)\leq \gta$. Whereof $\kathcal{F}$ is ex[avsnve relative to $\sigma$ wit... | {\epsilon\over 2}+ \eta+ {\epsilon\over 2}=\epsilon+ \eta.$$ Clearly, relation each $n\in\Bbb is true. Since number for each $n\in\Bbb we have $r\Big(F_{\sigma_n}(x), \eta$. Whereof $\mathcal{F}$ is expansive relative $\sigma$ with constant expansive $\eta$ hence we obtain $x= y$ and so $r(x, So $$\begin{aligned} \begi... | {\epsilon\over 2}+ \eta+ {\epsilon\over 2}=\Epsilon+ \eta.$$ clearLy, tHe pErViouS relAtion for each $n\iN\bbb Z$ Is true. Since $\epsilon>0$ is a sMall aRbITrarY NuMber tHen for eACh $N\IN\BbB Z$ We HavE $r\bIg(f_{\sigmA_n}(x), f_{\sigma_n}(Y)\Big)\leq \eta$. wheReOf $\mathcal{F}$ is EXpAnsive relaTivE to $\sigma$ wit... | {\epsilon\over 2}+ \eta+ { \epsilon\o ver 2 }=\ eps il on+\eta .$$ Clearly, t h e pe rvious relation for ea ch $n \i n \Bbb Z$ is t rue. Si n ce $ \ep si lo n>0 $i sa sma llarbitra ry numberthe nfor each $n\ i n\ Bbb Z$ wehav e $r\Big(F_{ \si gma_n} (x ),F _{\si gma _n}(y )\Big) \ leq \e ta$. Wher eo f $\mat h cal{F}... | {\epsilon\over 2}+_\eta+ {\epsilon\over_2}=\epsilon+ \eta.$$ Clearly, the_pervious relation_for_each $n\in\Bbb_Z$_is true. Since_$\epsilon>0$ is a_small arbitrary number then_for each $n\in\Bbb_Z$_we have $r\Big(F_{\sigma_n}(x), F_{\sigma_n}(y)\Big)\leq \eta$. Whereof $\mathcal{F}$ is expansive relative to $\sigma$ wit... |
j_0}}\Bigr) \mbox { as $$} \nonumber \\
&= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_o\psi_k\Bigr) \mbox { as $\psi_0=1$} \nonumber \\
&=0 \mbox{ due to orthogonality of the Fourier basis... | j_0}}\Bigr) \mbox { as $ $ } \nonumber \\
& = \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1 } \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1 } } { \lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0 - 1}\Bigl(\psi_o\psi_k\Bigr) \mbox { as $ \psi_0=1 $ } \nonumber \\
& = 0 \mbox { due to orthogonality o... | j_0}}\Bihr) \mbox { as $$} \nonumber \\
&= \exp \Bigl(\frac{2{\pi}n{h_1}{x_1}}{\lambva_1} \Bigr) \times \caots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}e_{d-1}} {\lqmbda_{e-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_u\psi_k\Bigr) \mbox { aw $\psm_0=1$} \nonumber \\
&=0 \mbox{ due to orthogkkalitv if the Fourier basis... | j_0}}\Bigr) \mbox { as $$} \nonumber \\ \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \times \cdots \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_o\psi_k\Bigr) \\ \mbox{ due to of the Fourier functions} \nonumber\end{aligned}$$ When n wild cards present, the extension is ... | j_0}}\Bigr) \mbox { as $$} \nonumber \\
&= \exp \BigL(\frac{2{\pi}i{j_1}{x_1}}{\LambdA_1} \BiGr) \tImEs \cdOts \tImes \exp \Bigl(\fraC{2{\Pi}i{j_{D-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambDa_0-1}\BigL(\pSI_o\psI_K\BIgr) \mbOx { as $\psi_0=1$} \NOnUMBer \\
&=0 \MbOx{ Due To ORtHogonAliTy of the fourier basIs... | j_0}}\Bigr) \mbox { as $$ } \nonumbe r \\
&= \e xp \Bi gl(\ frac{2{\pi}i{j _ 1}{x _1}}{\lambda_1} \Bigr) \tim es \cdo t s\time s \exp\ Bi g l (\f ra c{ 2{\ pi } i{ j_{d- 1}} x_{d-1} } {\lambda _{d -1 }}\Bigr)\sum _ {k =0}^{\lamb da_ 0-1}\Bigl(\p si_ o\psi_ k\ Big r ) \mb ox{ as$\psi_ 0 =1$} \ nonumber\\ &=0 \m b ox... | j_0}}\Bigr) _\mbox {_as $$} \nonumber \\_
&= \exp_\Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1}_\Bigr) \times_\cdots_\times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}}_{\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_o\psi_k\Bigr) \mbox {_as $\psi_0=1$} \nonumber \\
&=0_\mbox{ due to_orthogonality_of the Fourier basis... |
toward this X-ray filament, our qualitative analysis suggests that this filament is also matched by the galaxy distribution.
Finally, we consider the structure to the north. In contrast to the southeastern filament, here our spectroscopic coverage extends well to the north of the northern border of the X-ray field of... | toward this X - ray filament, our qualitative analysis suggest that this fibril is also matched by the galax distribution.
Finally, we regard the structure to the union. In contrast to the southeastern filament, here our spectroscopic coverage extend well to the north of the northern molding of the X - ray field of ... | toaard this X-ray filament, uur qualitative analysms suggssts thag this filament is also matcied vy tht galaxy distributiun.
Finally, we consuder rhe structnde to tmz norfm. In eoitrast to the sputheastert filament, hera uux spectroscopic coverage extends welj to thr jorth of the njrthtrn borsvr of the X-ray field of... | toward this X-ray filament, our qualitative analysis this is also by the galaxy structure the north. In to the southeastern here our spectroscopic coverage extends well the north of the northern border of the X-ray field of view. Considering the distribution of spectroscopically confirmed cluster galaxies (shown in the... | toward this X-ray filament, our QualitativE analYsiS suGgEsts That This filament is ALso mAtched by the galaxy distrIbutiOn.
fInalLY, wE consIder the STrUCTurE tO tHe nOrTH. IN contRasT to the sOutheasterN fiLaMent, here our sPEcTroscopic cOveRage extends wEll To the nOrTh oF The noRthErn boRder of THe X-ray Field of... | toward this X-ray filamen t, our qua litat ive an al ysis sug gests that thi s fil ament is also matchedby th eg alax y d istri bution.
F i n all y, w e c on s id er th e s tructur e to the n ort h. In contrast to the south eas tern filamen t,here o ur sp e ctros cop ic co verage extend s well to t h e nort ... | toward_this X-ray_filament, our qualitative analysis_suggests that_this_filament is_also_matched by the_galaxy distribution.
Finally, we_consider the structure to_the north. In_contrast_to the southeastern filament, here our spectroscopic coverage extends well to the north of_the_northern border_of_the_X-ray field of... |
n$-dimensional sphere and $\operatorname{B}$ is the beta function defined as $$\operatorname{B}{\left(}a,b{\right)}=\frac{\Gamma{\left(}a{\right)}\Gamma{\left(}b{\right)}}{\Gamma{\left(}a+b{\right)}}\quad\forall a,b>0.$$
We let $\operatorname{P}$ be the Schouten tensor defined as $$\operatorname{P}:=\frac{1}{n-2}{\lef... | n$-dimensional sphere and $ \operatorname{B}$ is the beta function defined as $ $ \operatorname{B}{\left(}a, b{\right)}=\frac{\Gamma{\left(}a{\right)}\Gamma{\left(}b{\right)}}{\Gamma{\left(}a+b{\right)}}\quad\forall a, b>0.$$
We let $ \operatorname{P}$ be the Schouten tensor define as $ $ \operatorname{P}:=\frac{1}{... | n$-dilensional sphere and $\opevatorname{B}$ is thg veta fnnction defined as $$\operatorname{B}{\left(}a,b{\right)}=\hrac{\Tamma{\oeft(}a{\right)}\Gamma{\left(}b{\rieht)}}{\Gamma{\lvft(}a+b{\righr)}}\quav\forall a,b>0.$$
We let $\operatovuame{P}$ ne thz Wchouten tensot defined as $$\operatorname{P}:=\xrxc{1}{u-2}{\lef... | n$-dimensional sphere and $\operatorname{B}$ is the beta as a,b>0.$$ We $\operatorname{P}$ be the and be the Bach whose coordinates are by $$\operatorname{B}_{ij}:=g^{ab}g^{cd}\operatorname{P}_{ac}\operatorname{W}_{ibjd}+g^{ab}{\left(}\operatorname{P}_{ij;ab}-\operatorname{P}_{ia;jb}{\right)}$$ in Einstein’s summation ... | n$-dimensional sphere and $\operAtorname{B}$ iS the bEta FunCtIon dEfinEd as $$\operatornaME{B}{\leFt(}a,b{\right)}=\frac{\Gamma{\left(}A{\righT)}\GAMma{\lEFt(}B{\righT)}}{\Gamma{\lEFt(}A+B{\RigHt)}}\QuAd\fOrALl A,b>0.$$
We lEt $\oPeratorName{P}$ be the schOuTen tensor defINeD as $$\operatoRnaMe{P}:=\frac{1}{n-2}{\lef... | n$-dimensional sphere and$\operator name{ B}$ is t he b etafunction defin e d as $$\operatorname{B}{\l eft(} a, b {\ri g ht )}=\f rac{\Ga m ma { \ lef t( }a {\r ig h t) }\Gam ma{ \left(} b{\right)} }{\ Ga mma{\left(}a + b{ \right)}}\ qua d\forall a,b >0. $$
We l et$ \oper ato rname {P}$ b e the S chouten t en s o... | n$-dimensional sphere_and $\operatorname{B}$_is the beta function_defined as_$$\operatorname{B}{\left(}a,b{\right)}=\frac{\Gamma{\left(}a{\right)}\Gamma{\left(}b{\right)}}{\Gamma{\left(}a+b{\right)}}\quad\forall_a,b>0.$$
We let_$\operatorname{P}$_be the Schouten_tensor defined as_$$\operatorname{P}:=\frac{1}{n-2}{\lef... |
width="49.70000%"}
We also investigate the evolution for the Newtonian potential, $\Phi$, as a function of $a$ and this is shown in. We see that for a designer $\mathcal{F}(\mathcal{K})$ model which mimics a $\Lambda$CDM background the evolution is now sensitive to the scale, where $K_0= k/H_0$, unlike the case of a c... | width="49.70000% " }
We also investigate the evolution for the Newtonian potential, $ \Phi$, as a function of $ a$ and this is testify in. We examine that for a designer $ \mathcal{F}(\mathcal{K})$ model which mimics a $ \Lambda$CDM backdrop the evolution is now sensible to the scale, where $ K_0= k / H_0 $, unlike ... | widhh="49.70000%"}
We also investigate tht evolution for tkw Newtmnian lotentiau, $\Phi$, as a function of $a$ and tyis iw shown in. We see that for a dedigner $\mqthcel{F}(\mathcal{K})$ model which mimics w $\Lakuda$CDM backgroukd the evolgtion is now sanricive to the scale, where $K_0= k/H_0$, unlike ehe casr lf a c... | width="49.70000%"} We also investigate the evolution for potential, as a of $a$ and see for a designer model which mimics $\Lambda$CDM background the evolution is now to the scale, where $K_0= k/H_0$, unlike the case of a cosmological constant. amplitude of $\Phi$ grows with respect to $\Lambda$CDM for large scales, wh... | width="49.70000%"}
We also investigate the Evolution fOr the newTonIaN potEntiAl, $\Phi$, as a functiON of $a$ And this is shown in. We see tHat foR a DEsigNEr $\MathcAl{F}(\mathCAl{k})$ MOdeL wHiCh mImICs A $\LambDa$CdM backgRound the evOluTiOn is now sensiTIvE to the scalE, whEre $K_0= k/H_0$, unlike The Case of A c... | width="49.70000%"}
We als o investig ate t heevo lu tion for the Newtonian pote ntial, $\Phi$, as a fu nctio no f $a $ a nd th is is s h ow n in. W esee t h at fora d esigner $\mathcal {F} (\ mathcal{K})$ mo del whichmim ics a $\Lamb da$ CDM ba ck gro u nd th e e volut ion is now se nsitive t ot he sca l e, whe... | width="49.70000%"}
We also_investigate the_evolution for the Newtonian_potential, $\Phi$,_as_a function_of_$a$ and this_is shown in._We see that for_a designer $\mathcal{F}(\mathcal{K})$_model_which mimics a $\Lambda$CDM background the evolution is now sensitive to the scale, where_$K_0=_k/H_0$, unlike_the_case_of a c... |
^{0}$ are gauge equivalent to this first term.
In term of the induced representations, $P^{0}$ being rigid means that all the representations obtained from Maurer-Cartan elements of the form (\[eq:MC\_element\]) are equivalent as representations to the representation $$T_{g}^{P^{0}}\psi(x)=P^{0}(x)\psi(\varphi_{g}^{-1... | ^{0}$ are gauge equivalent to this first term.
In term of the induce representation, $ P^{0}$ being rigid mean that all the representations receive from Maurer - Cartan elements of the phase (\[eq: MC\_element\ ]) are equivalent as representations to the representation $ $ T_{g}^{P^{0}}\psi(x)=P^{0}(x)\psi(\varphi_{... | ^{0}$ arf gauge equivalent to thls first term.
In jeem of vhe indhced repfesentations, $P^{0}$ being rigid mxans that all the representatiovs obtainvd from Mqurec-Cartan elements of the njrm (\[se:MC\_enxment\]) are equivslent as rapresentations tu che representation $$T_{g}^{P^{0}}\psi(x)=P^{0}(x)\psi(\varphy_{g}^{-1... | ^{0}$ are gauge equivalent to this first term the induced $P^{0}$ being rigid obtained Maurer-Cartan elements of form (\[eq:MC\_element\]) are as representations to the representation $$T_{g}^{P^{0}}\psi(x)=P^{0}(x)\psi(\varphi_{g}^{-1}(x)).$$ following theorem gives cohomological conditions answering the questions men... | ^{0}$ are gauge equivalent to this fIrst term.
In Term oF thE inDuCed rEpreSentations, $P^{0}$ beiNG rigId means that all the repreSentaTiONs obTAiNed frOm MaureR-caRTAn eLeMeNts Of THe Form (\[eQ:MC\_Element\]) Are equivalEnt As RepresentatiONs To the repreSenTation $$T_{g}^{P^{0}}\psi(X)=P^{0}(x)\Psi(\varPhI_{g}^{-1... | ^{0}$ are gauge equivalent to this f irstter m.
I n te rm o f the inducedr epre sentations, $P^{0}$ be ing r ig i d me a ns that all th e r e p res en ta tio ns ob taine d f rom Mau rer-Cartan el em ents of thef or m (\[eq:MC \_e lement\]) ar e e quival en t a s repr ese ntati ons to the re presentat io n $$T_{ ... | ^{0}$ are_gauge equivalent_to this first term.
In_term of_the_induced representations,_$P^{0}$_being rigid means_that all the_representations obtained from Maurer-Cartan_elements of the_form_(\[eq:MC\_element\]) are equivalent as representations to the representation $$T_{g}^{P^{0}}\psi(x)=P^{0}(x)\psi(\varphi_{g}^{-1... |
$K and continuous to grow up to $S_m =0.8 * (2RLn4)$ at T = 40K. Similar behavior is observed in the $x=1$ sample (i.e. Ce$_2$Ni$_2$Sn) but with $S_(T)$ increasing in a more monotonous manner. Compared with the $S_m(T)$ behavior of a sharp energy distribution of levels, provided by the results depicted in Fig. \[F5\]a,... | $ K and continuous to grow up to $ S_m = 0.8 * (2RLn4)$ at T = 40K. Similar demeanor is observe in the $ x=1 $ sample (i.e. Ce$_2$Ni$_2$Sn) but with $ S_(T)$ increase in a more flat manner. Compared with the $ S_m(T)$ behavior of a astute department of energy distribution of levels, provided by the results depicted in ... | $K ajd continuous to grow up to $S_m =0.8 * (2RLn4)$ at T = 40K. Smmilar gehavior is observed in the $x=1$ sample (i.w. Ce$_2$Nu$_2$Sn) but with $S_(T)$ increaring in a more mobotoious manner. Compeded witm the $D_m(T)$ yeiavior of a shatp energy divtribution of nexeps, provided by the results depicted in Fig. \[G5\]a,... | $K and continuous to grow up to * at T 40K. Similar behavior sample Ce$_2$Ni$_2$Sn) but with increasing in a monotonous manner. Compared with the $S_m(T)$ of a sharp energy distribution of levels, provided by the results depicted in \[F5\]a, it becomes evident an overlap of the side levels of the ground first CFE This ... | $K and continuous to grow up to $S_M =0.8 * (2RLn4)$ at T = 40K. SiMilar BehAviOr Is obServEd in the $x=1$ sample (I.E. Ce$_2$NI$_2$Sn) but with $S_(T)$ increasing In a moRe MOnotONoUs manNer. CompAReD WIth ThE $S_M(T)$ bEhAViOr of a ShaRp energY distributIon Of Levels, providED bY the resultS dePicted in Fig. \[F5\]A,... | $K and continuous to growup to $S_m =0.8 *(2R Ln 4)$at T = 40K. Simila r beh avior is observed in t he $x =1 $ sam p le (i.e . Ce$_2 $ Ni $ _ 2$S n) b utwi t h$S_(T )$increas ing in a m ore m onotonous ma n ne r. Compare d w ith the $S_m (T) $ beha vi oro f a s har p ene rgy di s tribut ion of le ve l s, pro v ... | $K and_continuous to_grow up to $S_m_=0.8 *_(2RLn4)$_at T_=_40K. Similar behavior_is observed in_the $x=1$ sample (i.e._Ce$_2$Ni$_2$Sn) but with_$S_(T)$_increasing in a more monotonous manner. Compared with the $S_m(T)$ behavior of a sharp_energy_distribution of_levels,_provided_by the results depicted in_Fig. \[F5\]a,... |
\] (u22) edge\[normalEdge, <-\] (u21) ++(2, 0) node\[normalNode\] (u23) edge\[normalEdge, <-\] (u22); (s) ++(1.5, -1.5) node\[normalNode\] (u31) edge\[normalEdge, <-\] (s) ++(2, 0) node\[normalNode\] (u32) edge\[normalEdge, <-\] (u31) ++(2, 0) node\[normalNode\] (u33) edge\[normalEdge, <-\] (u32) ++(2, 1... | \ ] (u22) edge\[normalEdge, & lt;-\ ] (u21) + + (2, 0) node\[normalNode\ ] (u23) edge\[normalEdge, & lt;-\ ] (u22); (s) + + (1.5, -1.5) node\[normalNode\ ] (u31) edge\[normalEdge, & lt;-\ ] (s) + + (2, 0) node\[normalNode\ ] (u32) edge\[normalEdge, & lt;-\ ] (u31) + + (2, 0) node\[normalNode\ ] (u33) edge\[normalEdge, ... | \] (u22) fdge\[normalEdge, <-\] (u21) ++(2, 0) noae\[normalNode\] (u23) gdte\[normelEdge, &mt;-\] (u22); (s) ++(1.5, -1.5) node\[normalNode\] (u31) edge\[normalXdge, <-\] (s) ++(2, 0) node\[normalNode\] (u32) edee\[normalEfge, <-\] (u31) ++(2, 0) niee\[normalNovs\] (u33) edgc\[uormamCdge, &nv;-\] (u32) ++(2, 1... | \] (u22) edge\[normalEdge, <-\] (u21) ++(2, 0) edge\[normalEdge, (u22); (s) -1.5) node\[normalNode\] (u31) node\[normalNode\] edge\[normalEdge, <-\] (u31) 0) node\[normalNode\] (u33) <-\] (u32) ++(2, 1.5) node\[labeledNode\] (v) edge\[normalEdge, bend left=15, <-\] (s) edge\[normalEdge, <-\] (u13) edge\[... | \] (u22) edge\[normalEdge, <-\] (u21) ++(2, 0) node\[norMalNode\] (u23) edGe\[norMaledgE, &lT;-\] (u22); (s) ++(1.5, -1.5) nOde\[nOrmalNode\] (u31) edge\[NOrmaLEdge, <-\] (s) ++(2, 0) node\[normalNode\] (U32) edge\[NoRMalEDGe, ≪-\] (u31) ++(2, 0) noDe\[normaLnoDE\] (U33) edGe\[NoRmaLEDGe, ≪-\] (u32) ++(2, 1... | \] (u22) edge\[normalEdge, <-\] ( u21)++( 2,0) nod e\[n ormalNode\] (u 2 3) e dge\[normalEdge, <- \] (u 22 ) ; (s ) + +(1.5 , -1.5) no d e \[n or ma lNo de \ ](u31) ed ge\[nor malEdge, & lt; -\ ] (s) ++(2,0 )node\[norm alN ode\] (u32)edg e\[nor ma lEd g e, &l t;- \] (u 31) ++ ( 2, 0)node\[nor ma l Node\] (u33... | \] (u22)_edge\[normalEdge, <-\]_(u21) ++(2, 0) node\[normalNode\]_(u23) edge\[normalEdge,_<-\]_(u22); (s)_++(1.5,_-1.5) node\[normalNode\] (u31)_edge\[normalEdge, <-\] (s)_++(2, 0) node\[normalNode\] (u32)_edge\[normalEdge, <-\] (u31)_++(2,_0) node\[normalNode\] (u33) edge\[normalEdge, <-\] (u32) ++(2, 1... |
Figs. \[gasout\] and \[metalsout\] show how efficiently mass and metals, respectively, are ejected by winds and deposited into the IGM. The upper panel of Fig. \[gasout\] shows the average fraction of mass ejected by winds $M_{\textrm{out}}$, defined as the sum of the mass currently in winds, plus the mass deposited i... | Figs. \[gasout\ ] and \[metalsout\ ] show how efficiently mass and metals, respectively, are ejected by wind and situate into the IGM. The upper panel of Fig. \[gasout\ ] shows the average fraction of multitude ejected by winds $ M_{\textrm{out}}$, define as the union of the mass presently in wind, plus the mass situat... |
Figd. \[gasout\] and \[metalsout\] smow how efficienjlt mass and mstals, rerpectively, are ejected by wiids qnd dtiosited into the IGM. Ghe upper panel od Fij. \[gasout\] shows tis averanz fradbion mh mass ejected ny winds $M_{\taxtrm{out}}$, definad ad the sum of the mass currently in rinds, pkud the mass depjsittd y... | Figs. \[gasout\] and \[metalsout\] show how efficiently metals, are ejected winds and deposited panel Fig. \[gasout\] shows average fraction of ejected by winds $M_{\textrm{out}}$, defined as sum of the mass currently in winds, plus the mass deposited into the $M_{\textrm{IGM}}$, over time. Similarly, the upper panel o... |
Figs. \[gasout\] and \[metalsout\] shoW how efficiEntly MasS anD mEtalS, resPectively, are ejECted By winds and deposited intO the IgM. tHe upPEr Panel Of Fig. \[gaSOuT\] SHowS tHe AveRaGE fRactiOn oF mass ejEcted by winDs $M_{\TeXtrm{out}}$, definED aS the sum of tHe mAss currently In wInds, plUs The MAss dePosIted i... |
Figs. \[gasout\] and \[me talsout\]showhow ef fi cien tlymass and metal s , re spectively, are ejecte d bywi n ds a n ddepos ited in t ot h e I GM .The u p pe r pan elof Fig. \[gasout\ ] s ho ws the avera g efraction o f m ass ejectedbywinds$M _{\ t extrm {ou t}}$, defin e d as t he sum of t h e mass current l y ... |
Figs. \[gasout\]_and \[metalsout\]_show how efficiently mass_and metals,_respectively,_are ejected_by_winds and deposited_into the IGM._The upper panel of_Fig. \[gasout\] shows_the_average fraction of mass ejected by winds $M_{\textrm{out}}$, defined as the sum of the_mass_currently in_winds,_plus_the mass deposited i... |
$(Q_{11}):\ 0=C_{11}{}^2 \{-2 (C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2-3)+8 \varepsilon (C_{11}{}^2)^2$ $-4 (C_{12}{}^2)^2+9 C_{12}{}^2+5\}$, $(Q_{12}):\ 0=(C_{11}{}^1)^2-3 C_{11}{}^1 C_{12}{}^2-2\varepsilon (C_{11}{}^2)^2+2 (C_{12}{}^2)^2-C_{12}{}^2-1$.
Case 1a {#case-1a.unnumbered}
-------
If $C_{11}{}^2=0$, Equati... | $ (Q_{11}):\ 0 = C_{11}{}^2 \{-2 (C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2 - 3)+8 \varepsilon (C_{11}{}^2)^2 $ $ -4 (C_{12}{}^2)^2 + 9 C_{12}{}^2 + 5\}$, $ (Q_{12}):\ 0=(C_{11}{}^1)^2 - 3 C_{11}{}^1 C_{12}{}^2 - 2\varepsilon (C_{11}{}^2)^2 + 2 (C_{12}{}^2)^2 - C_{12}{}^2 - 1$.
Case 1a { # case-1a.unnumbered }
------... | $(Q_{11}):\ 0=F_{11}{}^2 \{-2 (C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2-3)+8 \varepsilon (C_{11}{}^2)^2$ $-4 (C_{12}{}^2)^2+9 C_{12}{}^2+5\}$, $(Q_{12}):\ 0=(C_{11}{}^1)^2-3 C_{11}{}^1 C_{12}{}^2-2\varepsilon (C_{11}{}^2)^2+2 (C_{12}{}^2)^2-C_{12}{}^2-1$.
Cass 1a {#case-1x.unnumbered}
-------
If $C_{11}{}^2=0$, Equati... | $(Q_{11}):\ 0=C_{11}{}^2 \{-2 (C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2-3)+8 \varepsilon (C_{12}{}^2)^2+9 $(Q_{12}):\ 0=(C_{11}{}^1)^2-3 C_{12}{}^2-2\varepsilon (C_{11}{}^2)^2+2 (C_{12}{}^2)^2-C_{12}{}^2-1$. $C_{11}{}^2=0$, ($Q_{11}$) is trivial we obtain $(Q_{12}):\ C_{11}{}^1 C_{12}{}^2+2 (C_{12}{}^2)^2-C_{12}{}^2-1$ $... | $(Q_{11}):\ 0=C_{11}{}^2 \{-2 (C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2-3)+8 \varepsilon (C_{11}{}^2)^2$ $-4 (C_{12}{}^2)^2+9 C_{12}{}^2+5\}$, $(Q_{12}):\ 0=(C_{11}{}^1)^2-3 C_{11}{}^1 C_{12}{}^2-2\varEpsilon (C_{11}{}^2)^2+2 (C_{12}{}^2)^2-C_{12}{}^2-1$.
case 1a {#CasE-1a.uNnUmbeRed}
-------
IF $C_{11}{}^2=0$, Equati... | $(Q_{11}):\ 0=C_{11}{}^2\{-2 (C_{1 1}{}^ 1)^ 2+C _{ 11}{ }^1(6 C_{12}{}^2- 3 )+8\varepsilon (C_{11}{}^ 2)^2$ $ - 4 (C _ {1 2}{}^ 2)^2+9C _{ 1 2 }{} ^2 +5 \}$ ,$ (Q _{12} ):\ 0=(C_{ 11}{}^1)^2 -3C_ {11}{}^1 C_{ 1 2} {}^2-2\var eps ilon (C_{11} {}^ 2)^2+2 ( C_{ 1 2}{}^ 2)^ 2-C_{ 12}{}^ 2 -1$.
Case 1a { #c a se-1a. u ... | $(Q_{11}):\_0=C_{11}{}^2 \{-2_(C_{11}{}^1)^2+C_{11}{}^1 (6 C_{12}{}^2-3)+8 \varepsilon_(C_{11}{}^2)^2$ $-4_(C_{12}{}^2)^2+9_C_{12}{}^2+5\}$, $(Q_{12}):\_0=(C_{11}{}^1)^2-3_C_{11}{}^1 C_{12}{}^2-2\varepsilon (C_{11}{}^2)^2+2_(C_{12}{}^2)^2-C_{12}{}^2-1$.
Case 1a {#case-1a.unnumbered}
-------
If_$C_{11}{}^2=0$, Equati... |
DynamicFeedback2018; @hauswirthTimescaleSeparationAutonomous2019] which stipulate that the interconnection of *fast decaying* plant dynamics and *slow* optimization dynamics is asymptotically stable. The results in this section can be generalized to a dynamic plant accordingly.
[^11]: The penalty $d^2_{\mathcal{U}}$ i... | DynamicFeedback2018; @hauswirthTimescaleSeparationAutonomous2019 ] which stipulate that the interconnection of * fast decaying * plant dynamics and * slow * optimization dynamics is asymptotically static. The result in this section can be generalized to a active plant accordingly.
[ ^11 ]: The punishment $ d^2_{\mat... | DynwmicFeedback2018; @hauswirthTioescaleSeparationAutonmmous2019] shich stkpulate that the interconnecvion of *fqst decaying* plant dynxmics and *slow* oprimieation dynamics is asymptoticalmn staylx. The results ik this secthon can be genarxlnzed to a dynamic plant accordingly.
[^11]: Ehe penslhy $d^2_{\mathcal{U}}$ i... | DynamicFeedback2018; @hauswirthTimescaleSeparationAutonomous2019] which stipulate that the interconnection decaying* dynamics and optimization dynamics is this can be generalized a dynamic plant [^11]: The penalty $d^2_{\mathcal{U}}$ is illustrative the context of autonomous optimization, however, it is not generally p... | DynamicFeedback2018; @hauswirthTImescaleSeParatIonautOnOmouS2019] whiCh stipulate thaT The iNterconnection of *fast deCayinG* pLAnt dYNaMics aNd *slow* oPTiMIZatIoN dYnaMiCS iS asymPtoTically Stable. The rEsuLtS in this sectiON cAn be generaLizEd to a dynamic PlaNt accoRdIngLY.
[^11]: The pEnaLty $d^2_{\mAthcal{u}}$ I... | DynamicFeedback2018; @haus wirthTimes caleS epa rat io nAut onom ous2019] which stip ulate that the interco nnect io n of* fa st de caying* pl a n t d yn am ics a n d*slow * o ptimiza tion dynam ics i s asymptotic a ll y stable.The results inthi s sect io n c a n begen erali zed to a dyna mic plant a c cordin g ly... | DynamicFeedback2018; @hauswirthTimescaleSeparationAutonomous2019]_which stipulate_that the interconnection of_*fast decaying*_plant_dynamics and_*slow*_optimization dynamics is_asymptotically stable. The_results in this section_can be generalized_to_a dynamic plant accordingly.
[^11]: The penalty $d^2_{\mathcal{U}}$ i... |
\delta \Gamma = \frac{\langle || \delta \mathbf{u}^{\text{NA}} ||^2 \rangle}{\ell^2 \delta \gamma^2},$$ in which $\ell$ is the typical bond length of the network, and $\delta \mathbf{u}^{\text{NA}} = \mathbf{u} - \mathbf{u}^{\text{affine}}$ is the nonaffine displacement of a node that is caused by applying an infinites... | \delta \Gamma = \frac{\langle || \delta \mathbf{u}^{\text{NA } } ||^2 \rangle}{\ell^2 \delta \gamma^2},$$ in which $ \ell$ is the typical bond length of the network, and $ \delta \mathbf{u}^{\text{NA } } = \mathbf{u } - \mathbf{u}^{\text{affine}}$ is the nonaffine translation of a lymph node that is caused by applying ... | \delha \Gamma = \frac{\langle || \deuta \mathbf{u}^{\text{UQ}} ||^2 \ranjle}{\ell^2 \selta \gaoma^2},$$ in which $\ell$ is the typiral vond oength of the network, xnd $\delta \mathbf{u}^{\rext{IA}} = \mathbf{u} - \matigf{u}^{\text{affine}}$ ls thz ionaffine displscement of a node that iv zabsed by applying an infinites... | \delta \Gamma = \frac{\langle || \delta \mathbf{u}^{\text{NA}} \delta in which is the typical and \mathbf{u}^{\text{NA}} = \mathbf{u} \mathbf{u}^{\text{affine}}$ is the displacement of a node that is by applying an infinitesimal shear strain $\delta \gamma$. To better illustrate this parameter, show the nonaffine displ... | \delta \Gamma = \frac{\langle || \delta \Mathbf{u}^{\texT{NA}} ||^2 \raNglE}{\elL^2 \dElta \GammA^2},$$ in which $\ell$ is tHE typIcal bond length of the netWork, aNd $\DElta \MAtHbf{u}^{\tExt{NA}} = \maTHbF{U} - \MatHbF{u}^{\TexT{aFFiNe}}$ is tHe nOnaffinE displacemEnt Of A node that is cAUsEd by applyiNg aN infinites... | \delta \Gamma = \frac{\lan gle || \de lta \ mat hbf {u }^{\ text {NA}} ||^2 \ra n gle} {\ell^2 \delta \gamma^ 2},$$ i n whi c h$\ell $ is th e t y p ica lbo ndle n gt h ofthe networ k, and $\d elt a\mathbf{u}^{ \ te xt{NA}} =\ma thbf{u} - \m ath bf{u}^ {\ tex t {affi ne} }$ is the n o naffin e displac em e nt ofa ... | \delta \Gamma_= \frac{\langle_|| \delta \mathbf{u}^{\text{NA}} ||^2_\rangle}{\ell^2 \delta_\gamma^2},$$_in which_$\ell$_is the typical_bond length of_the network, and $\delta_\mathbf{u}^{\text{NA}} = \mathbf{u}_-_\mathbf{u}^{\text{affine}}$ is the nonaffine displacement of a node that is caused by applying an_infinites... |
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