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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*]{}, [**586**]{}, 356, 2003.
Zimmermann, H.-U., and B. Aschenbach, [*A&A*]{}, in press (astro-ph/0304322), 2003.
---
abstract: 'Modern robotic systems have become a substitute for humans when it’s necessary to perform risky or exhausting tasks. In such application scenarios, communications between robots and the con... | * ] { }, [ * * 586 * * ] { }, 356, 2003.
Zimmermann, H.-U., and B. Aschenbach, [ * A&A * ] { }, in press (astro - ph/0304322), 2003.
---
abstract:' Modern robotic systems have become a stand-in for world when it ’s necessary to perform risky or draining tasks. In such lotion scenarios, communications between au... | *]{}, [**586**]{}, 356, 2003.
Ximmermann, H.-U., and B. Aschtnbach, [*A&A*]{}, in press (astro-'h/0304322), 2003.
---
absfract: 'Moaern robotic systems have berome a suvstitute for humans whdn it’s nebessary ti pecform risky or eegaustinn tasid. In wuch applicatipn scenarims, communicatimnr yetween robots and the con... | *]{}, [**586**]{}, 356, 2003. Zimmermann, H.-U., and [*A&A*]{}, press (astro-ph/0304322), --- abstract: 'Modern substitute humans when it’s to perform risky exhausting tasks. In such application scenarios, between robots and the control center are one of the major problems. The used solution assumes that newer messages... | *]{}, [**586**]{}, 356, 2003.
Zimmermann, H.-U., and B. AschenbacH, [*A&A*]{}, in press (Astro-Ph/0304322), 2003.
---
aBstRaCt: 'MoDern Robotic systems HAve bEcome a substitute for humAns whEn IT’s neCEsSary tO perforM RiSKY or ExHaUstInG TaSks. In SucH applicAtion scenaRioS, cOmmunicationS BeTween robotS anD the con... | *]{}, [**586**]{}, 356, 20 03.
Zimme rmann , H .-U ., and B.Aschenbach, [* A &A*] {}, in press (astro-ph /0304 32 2 ), 2 0 03 .
-- -
abstr a ct : 'Mo de rn ro bo t ic syst ems have b ecome a su bst it ute for huma n swhen it’snec essary to pe rfo rm ris ky or exhau sti ng ta sks. I n suchapplicati on scenar i ... | *]{}, [**586**]{},_356, 2003.
Zimmermann,_H.-U., and B. Aschenbach,_[*A&A*]{}, in_press_(astro-ph/0304322), 2003.
_---
abstract:_'Modern robotic systems_have become a_substitute for humans when_it’s necessary to_perform_risky or exhausting tasks. In such application scenarios, communications between robots and the con... |
neuron functionality is shown in Fig. \[fig:neuron\]. This is a current-mode circuit composed of multiple “compartments” or blocks. The *LEAK* block models the neuron’s passive leak conductance, producing exponential sub-threshold dynamics in response to constant input currents. The *AHP* block models the generation o... | neuron functionality is shown in Fig. \[fig: neuron\ ]. This is a current - mode circuit write of multiple “ compartment ” or blocks. The * LEAK * block models the neuron ’s passive escape conductance, producing exponential sub - threshold dynamics in response to changeless input currents. The * AHP * pulley models t... | nekron functionality is shuwn in Fig. \[fig:neorin\]. Thiv is a current-oode circuit composed of mulviplw “comkcrtments” or blocks. Tfe *LEAK* bpock modwls uhe neuron’s passits leak gjndudbance, 'roducing expongntial sub-thseshold dynamiws iu response to constant input currentf. The *ANP* block models jhe gtnewatikn o... | neuron functionality is shown in Fig. \[fig:neuron\]. a circuit composed multiple “compartments” or the passive leak conductance, exponential sub-threshold dynamics response to constant input currents. The block models the generation of the after hyper-polarizing current in real neurons, responsible their spike-frequen... | neuron functionality is showN in Fig. \[fig:nEuron\]. thiS is A cUrreNt-moDe circuit compoSEd of Multiple “compartments” or BlockS. THE *LEAk* BlOck moDels the NEuRON’s pAsSiVe lEaK CoNductAncE, producIng exponenTiaL sUb-threshold dYNaMics in respOnsE to constant iNpuT curreNtS. ThE *aHP* blOck ModelS the geNEratioN o... | neuron functionality is s hown in Fi g. \[ fig :ne ur on\] . Th is is a curren t -mod e circuit composed ofmulti pl e “co m pa rtmen ts” orb lo c k s.Th e*LE AK * b lockmod els the neuron’spas si ve leak cond u ct ance, prod uci ng exponenti alsub-th re sho l d dyn ami cs in respo n se toconstantin p ut cur r ents... | neuron_functionality is_shown in Fig. \[fig:neuron\]. This_is a_current-mode_circuit composed_of_multiple “compartments” or_blocks. The *LEAK*_block models the neuron’s_passive leak conductance,_producing_exponential sub-threshold dynamics in response to constant input currents. The *AHP* block models the_generation_o... |
Thus, $\kappa^{s}(H_{n}; K_{1,3})\geq \lceil\frac{n}{2}\rceil$.
$\kappa(H_{n}; K_{1,3})$ and $\kappa^{s}(H_{n}; K_{1,3})$
---------------------------------------------------------
.2cm
\[3-upper\] $\kappa(H_{n}; K_{1,3})\leq \lceil\frac{n}{2}\rceil$ for $n\geq 4$.
For any $u=u_1\cdots u_n\in V(H_n)$, $N_{H_n}(u)=... | Thus, $ \kappa^{s}(H_{n }; K_{1,3})\geq \lceil\frac{n}{2}\rceil$.
$ \kappa(H_{n }; K_{1,3})$ and $ \kappa^{s}(H_{n }; K_{1,3})$
---------------------------------------------------------
.2 cm
\[3 - upper\ ] $ \kappa(H_{n }; K_{1,3})\leq \lceil\frac{n}{2}\rceil$ for $ n\geq 4$.
For any $ u = u_1\cdots u_n\... |
Thud, $\kappa^{s}(H_{n}; K_{1,3})\geq \lceil\frag{n}{2}\rceil$.
$\kappa(H_{n}; K_{1,3})$ and $\ka'pa^{s}(H_{n}; I_{1,3})$
---------------------------------------------------------
.2cm
\[3-upper\] $\kappa(H_{n}; K_{1,3})\leq \lceil\frac{n}{2}\rceip$ dor $n\teq 4$.
For any $u=u_1\cdots u_n\kn V(H_n)$, $N_{H_j}(u)=... | Thus, $\kappa^{s}(H_{n}; K_{1,3})\geq \lceil\frac{n}{2}\rceil$. $\kappa(H_{n}; K_{1,3})$ and --------------------------------------------------------- \[3-upper\] $\kappa(H_{n}; \lceil\frac{n}{2}\rceil$ for $n\geq V(H_n)$, Let $T_i$ be subgraph induced by u^{i+1}, u^{i,1,1}\}$ for $1\leq i \leq If $n$ is odd, we let $T... |
Thus, $\kappa^{s}(H_{n}; K_{1,3})\geq \lceil\frac{N}{2}\rceil$.
$\kappA(H_{n}; K_{1,3})$ aNd $\kAppA^{s}(h_{n}; K_{1,3})$
---------------------------------------------------------
.2cM
\[3-uppEr\] $\kappa(H_{n}; K_{1,3})\leq \lCEil\fRac{n}{2}\rceil$ for $n\geq 4$.
For any $U=u_1\cdoTs U_N\in V(h_N)$, $N_{h_n}(u)=... |
Thus, $\kappa^{s}(H_{n}; K_{1,3})\ geq \ lce il\ fr ac{n }{2} \rceil$.
$\ka p pa(H _{n}; K_{1,3})$ and $\ kappa ^{ s }(H_ { n} ; K_{ 1,3})$- -- - - --- -- -- --- -- - -- ----- --- ------- ---------- --- -- -------
.2c m
\[3-upper\ ] $ \kappa(H_{n} ; K _{1,3} )\ leq \lcei l\f rac{n }{2}\r c eil$ f or $n\geq 4 $ .... |
Thus, $\kappa^{s}(H_{n};_K_{1,3})\geq \lceil\frac{n}{2}\rceil$.
$\kappa(H_{n};_K_{1,3})$ and $\kappa^{s}(H_{n}; K_{1,3})$
---------------------------------------------------------
.2cm
\[3-upper\]_$\kappa(H_{n}; K_{1,3})\leq_\lceil\frac{n}{2}\rceil$_for $n\geq_4$.
For_any $u=u_1\cdots u_n\in_V(H_n)$, $N_{H_n}(u)=... |
relatively light sparticles, if the present deviation from the SM of $2.6\sigma$ persists.
At large ${\mbox{$ \tan\beta~ $}}$ a global fit including both $b\to X_s\gamma$ and $a_\mu$ as well as the present Higgs limit of 113.5 GeV leaves a quite large region in the CMSSM parameter space. Here we left the trilinear co... | relatively light sparticles, if the present deviation from the SM of $ 2.6\sigma$ persists.
At large $ { \mbox{$ \tan\beta~ $ } } $ a ball-shaped paroxysm including both $ b\to X_s\gamma$ and $ a_\mu$ as well as the present Higgs terminus ad quem of 113.5 GeV leaves a quite big region in the CMSSM parameter outer sp... | repatively light sparticler, if the presenj eeviatmon froj the SM of $2.6\sigma$ persists.
At large ${\muox{$ \ran\beuc~ $}}$ a global fit incljding botj $b\to X_s\tamme$ and $a_\mu$ as well as the presehb Higys limit of 113.5 GeV leaves a xuite large reciun in the CMSSM parameter space. Here re left tje trilinear cj... | relatively light sparticles, if the present deviation SM $2.6\sigma$ persists. large ${\mbox{$ \tan\beta~ both X_s\gamma$ and $a_\mu$ well as the Higgs limit of 113.5 GeV leaves quite large region in the CMSSM parameter space. Here we left the trilinear to be a free parameter, which affects both the Higgs limit constra... | relatively light sparticles, If the preseNt devIatIon FrOm thE SM oF $2.6\sigma$ persists.
aT larGe ${\mbox{$ \tan\beta~ $}}$ a global fiT inclUdINg boTH $b\To X_s\gAmma$ and $A_\Mu$ AS WelL aS tHe pReSEnT HiggS liMit of 113.5 Gev leaves a quIte LaRge region in tHE CmSSM parameTer Space. Here we lEft The triLiNeaR Co... | relatively light sparticl es, if the pres ent de vi atio n fr om the SM of $ 2 .6\s igma$ persists.
At la rge $ {\ m box{ $ \ tan\b eta~ $} } $a glo ba lfit i n cl uding bo th $b\t o X_s\gamm a$an d $a_\mu$ as we ll as thepre sent Higgs l imi t of 1 13 .5G eV le ave s a q uite l a rge re gion in t he CMSSMp aram... | relatively_light sparticles,_if the present deviation_from the_SM_of $2.6\sigma$_persists.
At_large ${\mbox{$ \tan\beta~_$}}$ a global_fit including both $b\to_X_s\gamma$ and $a_\mu$_as_well as the present Higgs limit of 113.5 GeV leaves a quite large region_in_the CMSSM_parameter_space._Here we left the trilinear_co... |
= +0.6 for all \[Fe/H\] $\leq$ –1.0. Since, at the abundance of And I derived here, $M_{V}$(RR) on our adopted scale is very close to 0.6, no adjustment of the MK90 result is required.
To compare our distance to And I with that for M31 however, requires an M31 modulus that is on the same distance scale. There are two... | = +0.6 for all \[Fe / H\ ] $ \leq$ – 1.0. Since, at the abundance of And I derived here, $ M_{V}$(RR) on our adopted scale is very close to 0.6, no allowance of the MK90 resultant role is required.
To compare our distance to And I with that for M31 however, requires an M31 modulus that is on the same distance plate.... | = +0.6 vor all \[Fe/H\] $\leq$ –1.0. Since, au the abundance oy And I derivsd here, $O_{V}$(RR) on our adopted scale is vwry coose to 0.6, no adjustment of the MN90 result is cequired.
To compacs our dlftandc to Cnv I with that fpr M31 howevar, requires an M31 mldulus that is on the same distance scale. Yhfre are two... | = +0.6 for all \[Fe/H\] $\leq$ –1.0. the of And derived here, $M_{V}$(RR) very to 0.6, no of the MK90 is required. To compare our distance And I with that for M31 however, requires an M31 modulus that is the same distance scale. There are two appropriate determinations. First, Pritchet & van Bergh have the magnitudes o... | = +0.6 for all \[Fe/H\] $\leq$ –1.0. Since, at the abuNdance of AnD I derIveD heRe, $m_{V}$(RR) On ouR adopted scale iS Very Close to 0.6, no adjustment of tHe MK90 rEsULt is REqUired.
to compaRE oUR DisTaNcE to anD i wIth thAt fOr M31 howeVer, requireS an m31 mOdulus that is ON tHe same distAncE scale. There aRe tWo... | = +0.6 for all \[Fe/H\] $ \leq$ –1.0 . Sin ce, at t he a bund ance of And Id eriv ed here, $M_{V}$(RR) o n our a d opte d s caleis very cl o s e t o0. 6,no ad justm ent of the MK90 resu ltis required.
T ocompare ou r d istance to A ndI with t hat for M 31howev er, re q uiresan M31 mo du l us tha t is ont h es... | =_+0.6 for_all \[Fe/H\] $\leq$ –1.0._Since, at_the_abundance of_And_I derived here,_$M_{V}$(RR) on our_adopted scale is very_close to 0.6,_no_adjustment of the MK90 result is required.
To compare our distance to And I with_that_for M31_however,_requires_an M31 modulus that is_on the same distance scale._There are_two... |
as implications for future RG/EB studies.
Observations {#data}
============
*Kepler* light curves {#kepler}
---------------------
Our light curves are from the *Kepler* Space Telescope in long-cadence mode (one data point every 29.4 minutes), and span 17 quarters—roughly four years—with only occasional gaps. These ... | as implications for future RG / EB studies.
Observations { # data }
= = = = = = = = = = = =
* Kepler * light curve { # kepler }
---------------------
Our faint curves are from the * Kepler * Space Telescope in long - cadence modality (one data point every 29.4 minute), and cross 17 quarters — roughly four ... | as implications for future RG/EB studies.
Observatimns {#dafa}
============
*Kepler* light curves {#kepler}
---------------------
Our lighv cueves qre from the *Kepler* Spxce Telesbope in ling-cedence mode (one vzta poikc evedn 29.4 miuuves), and span 17 qoarters—roughny four years—whtf lnly occasional gaps. These ... | as implications for future RG/EB studies. Observations *Kepler* curves {#kepler} Our light curves Telescope long-cadence mode (one point every 29.4 and span 17 quarters—roughly four years—with occasional gaps. These light curves are well-suited for red giant asteroseismology, as main stars with convective envelopes osc... | as implications for future RG/eB studies.
OBservAtiOns {#DaTa}
============
*KePler* Light curves {#kepLEr}
---------------------
OuR light curves are from the *kepleR* SPAce TELeScope In long-cADeNCE moDe (OnE daTa POiNt eveRy 29.4 mInutes), aNd span 17 quarTerS—rOughly four yeARs—With only ocCasIonal gaps. TheSe ... | as implications for futur e RG/EB st udies .
Obs er vati ons{#data}
====== = ==== =
*Kepler* light curv es {# ke p ler} -- ----- ------- - -- - - --
O ur li gh t c urves ar e fromthe *Keple r*Sp ace Telescop e i n long-cad enc e mode (onedat a poin teve r y 29. 4 m inute s), an d span17 quarte rs — roughl y fou... | as_implications for_future RG/EB studies.
Observations {#data}
============
*Kepler*_light curves_{#kepler}
---------------------
Our_light curves_are_from the *Kepler*_Space Telescope in_long-cadence mode (one data_point every 29.4_minutes),_and span 17 quarters—roughly four years—with only occasional gaps. These ... |
alpha}_a)} \cdot \frac{p(X_i | A = 0, X_i; \widehat{\alpha}_m)}{p(M_i | A = 1, X_i; \widehat{\alpha}_m)} \ Y_i
- \frac{ \mathbb I(A_i = 0)}{ p(A_i = 0 | X_i; \widehat{\alpha}_a)} \ Y_i
\right).
\label{eqn-IPW}
\end{aligned}$$ ]{}Since solving the constrained MLE problem using this estimator entails on... | alpha}_a) } \cdot \frac{p(X_i | A = 0, X_i; \widehat{\alpha}_m)}{p(M_i | A = 1, X_i; \widehat{\alpha}_m) } \ Y_i
- \frac { \mathbb I(A_i = 0) } { p(A_i = 0 | X_i; \widehat{\alpha}_a) } \ Y_i
\right).
\label{eqn - IPW }
\end{aligned}$$ ] { } Since solving the constrained MLE problem using this ... | alpja}_a)} \cdot \frac{p(X_i | A = 0, X_i; \widehat{\alpha}_m)}{p(M_i | A = 1, X_i; \wisehat{\alpfa}_m)} \ Y_i
- \frac{ \mathbb I(A_i = 0)}{ p(A_i = 0 | X_i; \widehat{\alpha}_a)} \ Y_k
\rigjt).
\lavel{ewb-IPW}
\env{zligned}$$ ]{}Since dolvnnj the constraingd MLE problam using this asgilator entails on... | alpha}_a)} \cdot \frac{p(X_i | A = 0, | = 1, \widehat{\alpha}_m)} \ Y_i 0)}{ = 0 | \widehat{\alpha}_a)} \ Y_i \label{eqn-IPW} \end{aligned}$$ ]{}Since solving the constrained problem using this estimator entails only restricting parameters of $A$ and $M$ models, a new instance is done using $\mathbb{E}[Y | X] = \sum_{A... | alpha}_a)} \cdot \frac{p(X_i | A = 0, X_i; \widehAt{\alpha}_m)}{p(M_I | A = 1, X_i; \wIdeHat{\AlPha}_m)} \ y_i
- \frAc{ \mathbb I(A_i = 0)}{ p(A_i = 0 | x_I; \widEhat{\alpha}_a)} \ Y_i
\right).
\label{Eqn-IPw}
\eND{aliGNeD}$$ ]{}SincE solvinG ThE COnsTrAiNed mLe PrOblem UsiNg this eStimator enTaiLs On... | alpha}_a)} \cdot \frac{p(X _i | A = 0 , X_i ; \ wid eh at{\ alph a}_m)}{p(M_i | A =1, X_i; \widehat{\alph a}_m) }\ Y_i - \ frac{ \ m at h b b I (A _i =0) } {p(A_i =0 | X_i ; \widehat {\a lp ha}_a)} \ Y _ i
\righ t).
\label{ eqn -IPW} \ e nd{al ign ed}$$ ]{}Si n ce sol ving theco n strain e d MLE p r ... | alpha}_a)} \cdot_\frac{p(X_i |_A = 0, X_i;_\widehat{\alpha}_m)}{p(M_i |_A_= 1,_X_i;_\widehat{\alpha}_m)} \ Y_i
_ _- \frac{ \mathbb I(A_i_= 0)}{ p(A_i_=_0 | X_i; \widehat{\alpha}_a)} \ Y_i
\right).
__\label{eqn-IPW}
___\end{aligned}$$ ]{}Since solving the constrained_MLE problem using this estimator_entails on... |
$f_{n-1}^i(t):=\mu( Q_{n-1}^i(t))$, where $Q_{n-1}^i(t)$ denotes the $t$-homotetic cube of $Q_{n-1}^{i}$ with respect to the center $x(Q_{n-1}^{i})$. Let $t_{n-1}^{i}= \sup\{t\in[0,1]\,;\, \mu( Q_{n-1}^i(t))< m-\alpha-(\mu(Q_{n-1}^1)+\cdots+\mu(Q_{n-1}^{i-1}))\}$. If the function $f_{n-1}^i$ is continuous at $t_{n-1}^... | $ f_{n-1}^i(t):=\mu (Q_{n-1}^i(t))$, where $ Q_{n-1}^i(t)$ denotes the $ t$-homotetic cube of $ Q_{n-1}^{i}$ with respect to the center $ x(Q_{n-1}^{i})$. lease $ t_{n-1}^{i}= \sup\{t\in[0,1]\,;\, \mu (Q_{n-1}^i(t) ) < m-\alpha-(\mu(Q_{n-1}^1)+\cdots+\mu(Q_{n-1}^{i-1}))\}$. If the affair $ f_{n-1}^i$ is continuous at $... | $f_{n-1}^l(t):=\mu( Q_{n-1}^i(t))$, where $Q_{n-1}^i(t)$ denutes the $t$-homotgtuc cubx of $Q_{n-1}^{j}$ with rdspect to the center $x(Q_{n-1}^{i})$. Lev $t_{n-1}^{u}= \sup\{u\pn[0,1]\,;\, \mu( Q_{n-1}^i(t))< m-\alpha-(\mu(Q_{n-1}^1)+\zdots+\mu(Q_{n-1}^{p-1}))\}$. If the duncuion $f_{n-1}^i$ is contiihous at $t_{n-1}^... | $f_{n-1}^i(t):=\mu( Q_{n-1}^i(t))$, where $Q_{n-1}^i(t)$ denotes the $t$-homotetic $Q_{n-1}^{i}$ respect to center $x(Q_{n-1}^{i})$. Let If function $f_{n-1}^i$ is at $t_{n-1}^{i}$, then consider as “canonical” set $H_Q=(Q_n(t_n^0))^\circ\cup \left(\cup_{j=1}^{i-1} (Q_{n-1}^i(t_{n-1}^{i}))^\circ$ and we are done. If no... | $f_{n-1}^i(t):=\mu( Q_{n-1}^i(t))$, where $Q_{n-1}^i(t)$ denoteS the $t$-homotEtic cUbe Of $Q_{N-1}^{i}$ With RespEct to the center $X(q_{n-1}^{i})$. LEt $t_{n-1}^{i}= \sup\{t\in[0,1]\,;\, \mu( Q_{n-1}^i(t))< m-\alphA-(\mu(Q_{n-1}^1)+\CdOTs+\mu(q_{N-1}^{i-1}))\}$. if the FunctioN $F_{n-1}^I$ IS coNtInUouS aT $T_{n-1}^... | $f_{n-1}^i(t):=\mu( Q_{n- 1}^i(t))$, wher e $ Q_{ n- 1}^i (t)$ denotes the $ t $-ho motetic cube of $Q_{n- 1}^{i }$ with re spect to the ce n t er$x (Q _{n -1 } ^{ i})$. Le t $t_{n -1}^{i}= \ sup \{ t\in[0,1]\,; \ ,\mu( Q_{n- 1}^ i(t))< m-\al pha -(\mu( Q_ {n- 1 }^1)+ \cd ots+\ mu(Q_{ n -1}^{i -1}))\}$. I f the ... | $f_{n-1}^i(t):=\mu(_Q_{n-1}^i(t))$, where_$Q_{n-1}^i(t)$ denotes the $t$-homotetic_cube of_$Q_{n-1}^{i}$_with respect_to_the center $x(Q_{n-1}^{i})$._Let $t_{n-1}^{i}= \sup\{t\in[0,1]\,;\,_\mu( Q_{n-1}^i(t))< m-\alpha-(\mu(Q_{n-1}^1)+\cdots+\mu(Q_{n-1}^{i-1}))\}$. If_the function $f_{n-1}^i$_is_continuous at $t_{n-1}^... |
$$\label{eqn.G6}
G_{6} (e^{\pi i/3})=G_{6} (e^{2\pi i/3})=\sum_{m,n}^{*}
\frac{1}{m^{6}+n^{6}} >0.$$ Consequently, if $g_{3} (\mathcal{L})=-C<0$ then $\mathcal{L}$ must be of the form for some $\omega$ such that $\omega^{6}<0$, that is, $\omega$ is a positive multiple of a primitive $12$th root of unity. Thus,... | $ $ \label{eqn. G6 }
G_{6 } (e^{\pi i/3})=G_{6 } (e^{2\pi i/3})=\sum_{m, n}^ { * }
\frac{1}{m^{6}+n^{6 } } > 0.$$ Consequently, if $ g_{3 } (\mathcal{L})=-C<0 $ then $ \mathcal{L}$ must be of the form for some $ \omega$ such that $ \omega^{6}<0 $, that is, $ \omega$ is a positive multiple of a primitive $ 1... | $$\lahel{eqn.G6}
G_{6} (e^{\pi i/3})=G_{6} (e^{2\pi l/3})=\sum_{m,n}^{*}
\frac{1}{m^{6}+n^{6}} >0.$$ Consexuentlg, if $g_{3} (\mxthcal{L})=-C<0$ then $\mathcal{L}$ must ue od the form for some $\omega$ sjch that $\lmega^{6}<0$, thqt iw, $\omega$ is e positiyz mulflple mh a primitive $12$tm root of utity. Thus,... | $$\label{eqn.G6} G_{6} (e^{\pi i/3})=G_{6} (e^{2\pi i/3})=\sum_{m,n}^{*} \frac{1}{m^{6}+n^{6}} if (\mathcal{L})=-C<0$ then must be of such $\omega^{6}<0$, that is, is a positive of a primitive $12$th root of Thus, in the case $g_{3} (\mathcal{L})=-C<0$ the lattice $\mathcal{L}$ must have the form $\omega = |\omega |e^{... | $$\label{eqn.G6}
G_{6} (e^{\pi i/3})=G_{6} (e^{2\pi i/3})=\sum_{m,n}^{*}
\fRac{1}{m^{6}+n^{6}} >0.$$ ConseQuentLy, iF $g_{3} (\mAtHcal{l})=-C<0$ thEn $\mathcal{L}$ must BE of tHe form for some $\omega$ such That $\oMeGA^{6}<0$, thaT Is, $\Omega$ Is a posiTIvE MUltIpLe Of a PrIMiTive $12$tH roOt of uniTy. Thus,... | $$\label{eqn.G6}
G_{6 } (e^{\pii/3}) =G_ {6} ( e^{2 \pii/3})=\sum_{m, n }^{* }
\frac{1}{m^{6}+n ^{6}} > 0 .$$C on seque ntly, i f $ g _ {3} ( \m ath ca l {L })=-C <0$ then $ \mathcal{L }$mu st be of the fo rm for som e $ \omega$ such th at $\o me ga^ { 6}<0$ , t hat i s, $\o m ega$ i s a posit iv e mult... | $$\label{eqn.G6}
_ _ G_{6} (e^{\pi i/3})=G_{6}_(e^{2\pi i/3})=\sum_{m,n}^{*}
__ _\frac{1}{m^{6}+n^{6}}_>0.$$ Consequently, if_$g_{3} (\mathcal{L})=-C<0$ then_$\mathcal{L}$ must be of_the form for_some_$\omega$ such that $\omega^{6}<0$, that is, $\omega$ is a positive multiple of a primitive_$12$th_root of_unity._Thus,... |
is injective possibly with the exception of ${{\boldsymbol{\varphi}}}^*(0)={{\boldsymbol{\varphi}}}^*(1)$ in case ${{\boldsymbol{\varphi}}}$ was closed. In other words, $\Gamma^*$ is obtained by extending $\Gamma$ beyond the beginning point ${{\boldsymbol{\varphi}}}(0)$ and the end point ${{\boldsymbol{\varphi}}}(1)$ ... | is injective possibly with the exception of $ { { \boldsymbol{\varphi}}}^*(0)={{\boldsymbol{\varphi}}}^*(1)$ in case $ { { \boldsymbol{\varphi}}}$ was closed. In other words, $ \Gamma^*$ is prevail by unfold $ \Gamma$ beyond the beginning point $ { { \boldsymbol{\varphi}}}(0)$ and the end point $ { { \boldsymbol{\varph... | is injective possibly with the exception of ${{\boldvymbol{\barphi}}}^*(0)={{\boudsymbol{\varphi}}}^*(1)$ in case ${{\boldsbmboo{\varpyi}}}$ was closed. In other words, $\Galma^*$ is ovtaiied by extending $\Gamma$ bcvond fme beyiining point ${{\bolcsymbol{\var[hi}}}(0)$ and the eng oonnt ${{\boldsymbol{\varphi}}}(1)$ ... | is injective possibly with the exception of case was closed. other words, $\Gamma^*$ beyond beginning point ${{\boldsymbol{\varphi}}}(0)$ the end point by straight line segments of strictly lengths belonging to the lines tangent to $\Gamma$ at these two points, respectively. need to introduce a notation. Suppose $\Gamm... | is injective possibly with thE exception Of ${{\bolDsyMboL{\vArphI}}}^*(0)={{\bolDsymbol{\varphi}}}^*(1)$ iN Case ${{\Boldsymbol{\varphi}}}$ was cloSed. In OtHEr woRDs, $\gamma^*$ Is obtaiNEd BY ExtEnDiNg $\GAmMA$ bEyond The BeginniNg point ${{\bolDsyMbOl{\varphi}}}(0)$ and tHE eNd point ${{\bolDsyMbol{\varphi}}}(1)$ ... | is injective possibly wit h the exce ption of ${ {\ bold symb ol{\varphi}}}^ * (0)= {{\boldsymbol{\varphi} }}^*( 1) $ inc as e ${{ \boldsy m bo l { \va rp hi }}} $w as clos ed. In oth er words,$\G am ma^*$ is obt a in ed by exte ndi ng $\Gamma$bey ond th ebeg i nning po int $ {{\bol d symbol {\varphi} }} ( 0)$ a... | is_injective possibly_with the exception of_${{\boldsymbol{\varphi}}}^*(0)={{\boldsymbol{\varphi}}}^*(1)$ in_case_${{\boldsymbol{\varphi}}}$ was_closed._In other words,_$\Gamma^*$ is obtained_by extending $\Gamma$ beyond_the beginning point_${{\boldsymbol{\varphi}}}(0)$_and the end point ${{\boldsymbol{\varphi}}}(1)$ ... |
d_{A,\Phi}:W^{k+2,p} \to W^{k,p}$ is Fredholm with index zero by Theorem \[thm:Gilkey\_1-4-5\_Sobolev\], the following operator is invertible, $$d_{A,\Phi}^*d_{A,\Phi}:(\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp \cap W^{k+2,p} \to (\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp \cap W^{k,p}.$$ Thus, $\xi = 0$ and the operator is injective... | d_{A,\Phi}:W^{k+2,p } \to W^{k, p}$ is Fredholm with index zero by Theorem \[thm: Gilkey\_1 - 4 - 5\_Sobolev\ ], the following operator is invertible, $ $ d_{A,\Phi}^*d_{A,\Phi}:(\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp \cap W^{k+2,p } \to (\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp \cap W^{k, p}.$$ Thus, $ \xi = 0 $ and the hustler... | d_{A,\Pji}:W^{k+2,p} \to W^{k,p}$ is Fredholm with index zero by Thxorem \[tgm:Gilkey\_1-4-5\_Robolev\], the following operatlr is ibvertible, $$d_{A,\Phi}^*d_{A,\Phi}:(\Kef d_{A,\Phi}^*d_{A,\Ihi})^\perp \cqp W^{j+2,p} \to (\Ker d_{E,\Lhi}^*d_{A,\Phl})^\'erp \dwp W^{n,'}.$$ Thus, $\xi = 0$ and the operador is injectiee... | d_{A,\Phi}:W^{k+2,p} \to W^{k,p}$ is Fredholm with index Theorem the following is invertible, $$d_{A,\Phi}^*d_{A,\Phi}:(\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp W^{k,p}.$$ Thus, $\xi 0$ and the is injective. For surjectivity, suppose $\chi d_{A,\Phi}\zeta \perp \Ran d_{A,\Phi}\cap W^{k,p}$ for $\zeta \in W^{k+1,p}$. We may a... | d_{A,\Phi}:W^{k+2,p} \to W^{k,p}$ is Fredholm wiTh index zerO by ThEorEm \[tHm:gilkEy\_1-4-5\_SoBolev\], the followINg opErator is invertible, $$d_{A,\PhI}^*d_{A,\PhI}:(\KER d_{A,\PHI}^*d_{a,\Phi})^\pErp \cap W^{K+2,P} \tO (\kEr d_{a,\PHi}^*D_{A,\PHi})^\PErP \cap W^{K,p}.$$ THus, $\xi = 0$ anD the operatOr iS iNjective... | d_{A,\Phi}:W^{k+2,p} \to W ^{k,p}$ is Fred hol m w it h in dexzero by Theore m \[t hm:Gilkey\_1-4-5\_Sobo lev\] ,t he f o ll owing operat o ri s in ve rt ibl e, $$ d_{A, \Ph i}^*d_{ A,\Phi}:(\ Ker d _{A,\Phi}^*d _ {A ,\Phi})^\p erp \cap W^{k+2 ,p} \to ( \K erd _{A,\ Phi }^*d_ {A,\Ph i })^\pe rp \cap W ^{ k ,p}.$... | d_{A,\Phi}:W^{k+2,p} \to_W^{k,p}$ is_Fredholm with index zero_by Theorem_\[thm:Gilkey\_1-4-5\_Sobolev\],_the following_operator_is invertible, $$d_{A,\Phi}^*d_{A,\Phi}:(\Ker_d_{A,\Phi}^*d_{A,\Phi})^\perp \cap W^{k+2,p}_\to (\Ker d_{A,\Phi}^*d_{A,\Phi})^\perp \cap_W^{k,p}.$$ Thus, $\xi_=_0$ and the operator is injective... |
LeNet [@SzegedyLJSRAEVR15] and ResNet [@HeZRS16]. Besides the success on image categorization, features extracted from the last few layers are applied for generic tasks. [@DonahueJVHZTD14] adopts the deep features from the last two layers in AlexNet and shows the impressive performance on visual recognition with differ... | LeNet [ @SzegedyLJSRAEVR15 ] and ResNet [ @HeZRS16 ]. Besides the success on image categorization, features educe from the final few layers are applied for generic tasks. [ @DonahueJVHZTD14 ] adopt the deep features from the concluding two layers in AlexNet and shows the impressive operation on ocular recognition w... | LeNft [@SzegedyLJSRAEVR15] and RerNet [@HeZRS16]. Besidgs the snccess kn image categorization, features extcactwd frim the last few layers are applped for gwnermc tasks. [@DonahueOBHZTD14] adopts fme dezp features from the last dwo layers in DldxUet and shows the impressive performwnce on vlsual recognityon eyth spfner... | LeNet [@SzegedyLJSRAEVR15] and ResNet [@HeZRS16]. Besides the image features extracted the last few tasks. adopts the deep from the last layers in AlexNet and shows the performance on visual recognition with different applications. After that, [@QianJZL15] applies deep features distance metric learning and achieves the... | LeNet [@SzegedyLJSRAEVR15] and ReSNet [@HeZRS16]. BEsideS thE suCcEss oN imaGe categorizatiON, feaTures extracted from the lAst feW lAYers ARe AppliEd for geNErIC TasKs. [@doNahUejvHzTD14] adOptS the deeP features fRom ThE last two layeRS iN AlexNet anD shOws the impresSivE perfoRmAncE On visUal RecogNition WIth difFer... | LeNet [@SzegedyLJSRAEVR15] and ResNe t [@H eZR S16 ]. Bes ides the success o n ima ge categorization, fea tures e x trac t ed from the la s tf e w l ay er s a re ap plied fo r gener ic tasks.[@D on ahueJVHZTD14 ] a dopts thedee p features f rom the l as t t w o lay ers in A lexNet and sh ows the i mp r essiv... | LeNet [@SzegedyLJSRAEVR15] and_ResNet [@HeZRS16]. Besides_the success on image_categorization, features_extracted_from the_last_few layers are_applied for generic_tasks. [@DonahueJVHZTD14] adopts the_deep features from_the_last two layers in AlexNet and shows the impressive performance on visual recognition with_differ... |
T_{b}(v) \simeq \tau_{\rm tot} \cdot T_{\rm ex}
\label{eqn_Tmb_thin}\end{aligned}$$ indicating that the $T_{\rm ex}$ and $\tau_{\rm tot}$ can be determined arbitrarily. Therefore, we conduct non-LTE analysis in Section \[subsec::non-lte\] to derive the physical conditions.
{width="\line... | T_{b}(v) \simeq \tau_{\rm tot } \cdot T_{\rm ex }
\label{eqn_Tmb_thin}\end{aligned}$$ indicating that the $ T_{\rm ex}$ and $ \tau_{\rm tot}$ can be determined arbitrarily. consequently, we impart non - LTE analysis in Section \[subsec::non - lte\ ] to derive the forcible conditions.
! [ image](non_lte_model_bary.p... | T_{b}(v) \simeq \tau_{\rm tot} \cdot T_{\ro ex}
\label{eqn_Tmb_jhun}\end{anigned}$$ indicatkng that the $T_{\rm ex}$ and $\tau_{\rl rot}$ cqn be determined arbitfarily. Thvrefore, ww coiduct non-LTE analysis in Sectikk \[subvxc::non-lte\] to derlve the phyvical conditiots.
{width="\line... | T_{b}(v) \simeq \tau_{\rm tot} \cdot T_{\rm ex} that $T_{\rm ex}$ $\tau_{\rm tot}$ can conduct analysis in Section to derive the conditions. {width="\linewidth"} $N_2H^+$ RADEX Non-LTE Modeling ------------------------------- Using RADEX [@radex], a non-LTE radiative transfer code, we co... | T_{b}(v) \simeq \tau_{\rm tot} \cdot T_{\rm ex}
\Label{eqn_TmB_thin}\End{AliGnEd}$$ inDicaTing that the $T_{\rm EX}$ and $\Tau_{\rm tot}$ can be determineD arbiTrARily. tHeReforE, we condUCt NON-LTe aNaLysIs IN SEctioN \[suBsec::non-Lte\] to derivE thE pHysical condiTIoNs.
{WidTh="\line... | T_{b}(v) \simeq \tau_{\rmtot} \cdot T_{\ rmex}
\ labe l{eq n_Tmb_thin}\en d {ali gned}$$ indicating tha t the $ T _{\r m e x}$ a nd $\ta u _{ \ r m t ot }$ ca nb edeter min ed arbi trarily. T her ef ore, we cond u ct non-LTE a nal ysis in Sect ion \[sub se c:: n on-lt e\] to d erivet he phy sical con di t ions.... | T_{b}(v) \simeq_\tau_{\rm tot}_\cdot T_{\rm ex}
\label{eqn_Tmb_thin}\end{aligned}$$ indicating_that the_$T_{\rm_ex}$ and_$\tau_{\rm_tot}$ can be_determined arbitrarily. Therefore,_we conduct non-LTE analysis_in Section \[subsec::non-lte\]_to_derive the physical conditions.
{width="\line... |
vertex ordering can make a substantial difference in the size of the search space, but that there is no “best” ordering.
Sequential Branch and Bound
---------------------------
In we give the underlying sequential procedure. The variable $C$ is a growing clique, and $P$ contains undecided vertices which could potent... | vertex ordering can make a substantial difference in the size of the search quad, but that there be no “ best ” ordering.
Sequential Branch and Bound
---------------------------
In we give the underlie sequential procedure. The variable star $ C$ is a growing clique, and $ P$ contains undecided vertex which cou... | vegtex ordering can make a substantial diyderencx in ths size ow the search space, but that vherw is bo “best” ordering.
Sequengial Branbh and Boynd
---------------------------
Ii we give the unvsrlying sequehbial 'ricedure. The vatiable $C$ is d growing cliqge, aud $P$ contains undecided vertices whish coulc ootent... | vertex ordering can make a substantial difference size the search but that there Branch Bound --------------------------- In give the underlying procedure. The variable $C$ is a clique, and $P$ contains undecided vertices which could potentially be added to $C$. best solution found so far is stored in $C_{max}$. The im... | vertex ordering can make a subStantial diFfereNce In tHe Size Of thE search space, buT That There is no “best” ordering.
SEquenTiAL BraNCh And BoUnd
---------------------------
In we GIvE THe uNdErLyiNg SEqUentiAl pRocedurE. The variabLe $C$ Is A growing cliqUE, aNd $P$ containS unDecided vertiCes Which cOuLd pOTent... | vertex ordering can makea substant ial d iff ere nc e in the size of the s e arch space, but that there is n o“ best ” o rderi ng.
Se q ue n t ial B ra nch a n dBound
-- ------- ---------- --- -- ---
In we g i ve the under lyi ng sequentia l p rocedu re . T h e var iab le $C $ is a growin g clique, a n d $... | vertex_ordering can_make a substantial difference_in the_size_of the_search_space, but that_there is no_“best” ordering.
Sequential Branch and_Bound
---------------------------
In we give_the_underlying sequential procedure. The variable $C$ is a growing clique, and $P$ contains undecided_vertices_which could_potent... |
12; H^1(\Omega'))\hookrightarrow BUC([-\tfrac12, \tfrac12];H^{\frac12}(\Omega'))$$ and $H^{\frac12}(\Omega')\hookrightarrow L^4(\Omega')$. Finally, the last embedding follows from $$\begin{aligned}
\lefteqn{L^2(-\tfrac12,\tfrac12; H^{1+k}((-L,L)^{d-1}))\cap
H^1(-\tfrac12,\tfrac12;H^1((-L,L)^{d-1}))}\\
&& \hookrig... | 12; H^1(\Omega'))\hookrightarrow BUC([-\tfrac12, \tfrac12];H^{\frac12}(\Omega'))$$ and $ H^{\frac12}(\Omega')\hookrightarrow L^4(\Omega')$. Finally, the last embedding follows from $ $ \begin{aligned }
\lefteqn{L^2(-\tfrac12,\tfrac12; H^{1+k}((-L, L)^{d-1}))\cap
H^1(-\tfrac12,\tfrac12;H^1((-L, L)^{d-1}))}\\
... | 12; H^1(\Olega'))\hookrightarrow BUC([-\tfvac12, \tfrac12];H^{\frac12}(\Omggq'))$$ and $I^{\frac12}(\Omsga')\hookrkghtarrow L^4(\Omega')$. Finally, the lqst enbedding follows from $$\cegin{aligjed}
\lwftewb{L^2(-\tfrac12,\tfred12; H^{1+k}((-L,L)^{d-1}))\gcp
H^1(-\fnrac12,\tyrec12;H^1((-L,L)^{d-1}))}\\
&& \hookrig... | 12; H^1(\Omega'))\hookrightarrow BUC([-\tfrac12, \tfrac12];H^{\frac12}(\Omega'))$$ and $H^{\frac12}(\Omega')\hookrightarrow L^4(\Omega')$. last follows from \lefteqn{L^2(-\tfrac12,\tfrac12; H^{1+k}((-L,L)^{d-1}))\cap H^1(-\tfrac12,\tfrac12;H^1((-L,L)^{d-1}))}\\ \end{aligned}$$ $k=d-2$ because of and Sobolev embeddings.... | 12; H^1(\Omega'))\hookrightarrow BUC([-\tfRac12, \tfrac12];H^{\fRac12}(\OmEga'))$$ And $h^{\fRac12}(\OMega')\Hookrightarrow l^4(\omegA')$. Finally, the last embeddiNg folLoWS froM $$\BeGin{alIgned}
\leFTeQN{l^2(-\tfRaC12,\tFraC12; H^{1+K}((-l,L)^{D-1}))\cap
H^1(-\TfrAc12,\tfrac12;h^1((-L,L)^{d-1}))}\\
&& \hookriG... | 12; H^1(\Omega'))\hookrigh tarrow BUC ([-\t fra c12 ,\tfr ac12 ];H^{\frac12}( \ Omeg a'))$$ and $H^{\frac12 }(\Om eg a ')\h o ok right arrow L ^ 4( \ O meg a' )$ . F in a ll y, th e l ast emb edding fol low sfrom $$\begi n {a ligned}
\ lefteqn{L^2( -\t frac12 ,\ tfr a c12;H^{ 1+k}( (-L,L) ^ {d-1}) )\cap
H ^1 ( -\t... | 12; H^1(\Omega'))\hookrightarrow_BUC([-\tfrac12, \tfrac12];H^{\frac12}(\Omega'))$$_and $H^{\frac12}(\Omega')\hookrightarrow L^4(\Omega')$. Finally,_the last_embedding_follows from_$$\begin{aligned}
_ _\lefteqn{L^2(-\tfrac12,\tfrac12; H^{1+k}((-L,L)^{d-1}))\cap
_H^1(-\tfrac12,\tfrac12;H^1((-L,L)^{d-1}))}\\
&& \hookrig... |
m_1\;\forall m_2\; \forall b\;
(\;
\hspace{-5mm} & ( & \hspace{-5mm} instance(b,Brood) \wedge
member(m_1,b) \wedge
member(m_2,b)
) \\
& \;\rightarrow \; & \label{axiom:BroodSibling}
sibling(m_1,m_2)
\;)\end{aligned}$$ According to the type information in axioms (\[axiom:domainSibling1\]-\[axiom:domainSibling... | m_1\;\forall m_2\; \forall b\;
(\;
\hspace{-5 mm } & (& \hspace{-5 mm } instance(b, Brood) \wedge
member(m_1,b) \wedge
member(m_2,b)
) \\
& \;\rightarrow \; & \label{axiom: BroodSibling }
sibling(m_1,m_2)
\;)\end{aligned}$$ According to the type information in axioms (\[axiom: domainSibling1\]-\[ax... | m_1\;\flrall m_2\; \forall b\;
(\;
\hspace{-5om} & ( & \hspace{-5mm} nbstancx(b,Brood) \wedge
mdmber(m_1,b) \wedge
member(m_2,b)
) \\
& \;\righterroq \; & \lqbel{axiom:BroodSibling}
skbling(m_1,m_2)
\;)\ene{alijned}$$ According to the tyiz infkvmatimi in axioms (\[axipm:domainSitling1\]-\[axiom:domahnRiyling... | m_1\;\forall m_2\; \forall b\; (\; \hspace{-5mm} & \hspace{-5mm} \wedge member(m_1,b) member(m_2,b) ) \\ sibling(m_1,m_2) According to the information in axioms both arguments of are restricted to instance of. Consequently, by translation (see [@ALR12] for more details), axiom (\[axiom:BroodSibling\]) gives to the foll... | m_1\;\forall m_2\; \forall b\;
(\;
\hspace{-5mm} & ( & \hsPace{-5mm} instAnce(b,broOd) \wEdGe
meMber(M_1,b) \wedge
member(m_2,B)
) \\
& \;\RighTarrow \; & \label{axiom:BroodSIblinG}
sIBlinG(M_1,m_2)
\;)\End{alIgned}$$ AcCOrDINg tO tHe TypE iNFoRmatiOn iN axioms (\[Axiom:domaiNSiBlIng1\]-\[axiom:domaINSIbling... | m_1\;\forall m_2\; \foral l b\;
(\;
\hs pac e{- 5m m} & ( & \hspace{-5mm} inst ance(b,Brood) \wedge membe r( m _1,b ) \ wedge
member ( m_ 2 , b))\\
&\; \ ri ghtar row \; & \ label{axio m:B ro odSibling}
s i bl ing(m_1,m_ 2)
\;)\e nd{ aligne d} $$A ccord ing to t he typ e infor mation in a x ioms ( \ ... | m_1\;\forall_m_2\; \forall_b\;
(\;
\hspace{-5mm} &_( &_\hspace{-5mm}_instance(b,Brood) \wedge_
member(m_1,b)_\wedge
member(m_2,b)
) \\
& \;\rightarrow_\; & \label{axiom:BroodSibling}
sibling(m_1,m_2)_ _
\;)\end{aligned}$$_According_to the type information in axioms (\[axiom:domainSibling1\]-\[axiom:domainSibling... |
aligned}
\label{FModelUniformeSimu}
{\mathscr{F}}= \left\{f_{\theta}, \, \theta \in [0.01, 10]\right\} \quad \text{where} \quad f_{\theta} = \theta^{-1} {\mathbbm{1}}_{[0, \theta]}.\end{aligned}$$ It is worthwhile to notice that $h^2(s,{\mathscr{F}}) = \mathcal{O} (n^{-1})$, which means that $s$ is close to ${\mathsc... | aligned }
\label{FModelUniformeSimu }
{ \mathscr{F}}= \left\{f_{\theta }, \, \theta \in [ 0.01, 10]\right\ } \quad \text{where } \quad f_{\theta } = \theta^{-1 } { \mathbbm{1}}_{[0, \theta]}.\end{aligned}$$ It is worthwhile to notice that $ h^2(s,{\mathscr{F } }) = \mathcal{O } (n^{-1})$, which means that $ s$ i... | alihned}
\label{FModelUniformeRimu}
{\mathscr{F}}= \leyr\{f_{\thete}, \, \thetz \in [0.01, 10]\rieht\} \quad \text{where} \quad f_{\theva} = \rheta^{-1} {\mathbbm{1}}_{[0, \theta]}.\end{aligved}$$ It is worthwhule uo notice that $h^2(s,{\mathscr{F}}) = \mathdwl{O} (u^{-1})$, xhich means thaj $s$ is close to ${\mathsc... | aligned} \label{FModelUniformeSimu} {\mathscr{F}}= \left\{f_{\theta}, \, \theta \in \quad \quad f_{\theta} \theta^{-1} {\mathbbm{1}}_{[0, \theta]}.\end{aligned}$$ that = \mathcal{O} (n^{-1})$, means that $s$ close to ${\mathscr{F}}$ when $n$ is and that our estimator still satisfies ${\mathbb{E}}[h^2 (s,f_{\hat{\theta}... | aligned}
\label{FModelUniformESimu}
{\mathsCr{F}}= \leFt\{f_{\TheTa}, \, \ThetA \in [0.01, 10]\rIght\} \quad \text{whERe} \quAd f_{\theta} = \theta^{-1} {\mathbbm{1}}_{[0, \thEta]}.\enD{aLIgneD}$$ it Is worThwhile TO nOTIce ThAt $H^2(s,{\mAtHScR{F}}) = \matHcaL{O} (n^{-1})$, whicH means that $S$ is ClOse to ${\mathsc... | aligned}
\label{FModelUni formeSimu}
{\ma ths cr{ F} }= \ left \{f_{\theta},\ , \t heta \in [0.01, 10]\ri ght\} \ q uad\ te xt{wh ere} \q u ad f _{\ th et a}=\ th eta^{ -1} {\mat hbbm{1}}_{ [0, \ theta]}.\end { al igned}$$ I t i s worthwhile to notic etha t $h^2 (s, {\mat hscr{F } }) = \ mathcal{O }( n^{-1} ) ... | aligned}
\label{FModelUniformeSimu}
{\mathscr{F}}=_\left\{f_{\theta}, \,_\theta \in [0.01, 10]\right\}_\quad \text{where}_\quad_f_{\theta} =_\theta^{-1}_ {\mathbbm{1}}_{[0, \theta]}.\end{aligned}$$_It is worthwhile_to notice that $h^2(s,{\mathscr{F}})_= \mathcal{O} (n^{-1})$,_which_means that $s$ is close to ${\mathsc... |
Gamma(M,B), L^2(M,B \otimes S), D_B)$. The conditions on $\gamma_B$ to be a grading operator for this spectral triple are easily checked.
In the next section we show that the triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ gives a non-trivial Yang–Mills theory over the manifold $M$. The Serre–Swan Theorem \... | Gamma(M, B), L^2(M, B \otimes S), D_B)$. The conditions on $ \gamma_B$ to be a grading operator for this spectral trio are well check.
In the next incision we testify that the triple $ { (\Gamma(M, B), L^2(M, B\otimes S),D_B, J,\gamma_B)}$ gives a non - superficial Yang – mill theory over the manifold $ M$. The Serr... | Gamla(M,B), L^2(M,B \otimes S), D_B)$. The conditions on $\yqmma_B$ vo be a grading operator for this spectral vripoe art easily checked.
In ghe next dection qe siow that the tri'me ${(\Gamma(M,B), L^2(M,G\ltimzs S),D_B,J,\gamma_B)}$ giyes a non-trhvial Yang–Millv ghzory over the manifold $M$. The Serre–Swwn Theotel \... | Gamma(M,B), L^2(M,B \otimes S), D_B)$. The conditions to a grading for this spectral the section we show the triple ${(\Gamma(M,B), S),D_B,J,\gamma_B)}$ gives a non-trivial Yang–Mills theory the manifold $M$. The Serre–Swan Theorem \[thm:serreswan2\] plays an essential role in the First, we explore the form of this spe... | Gamma(M,B), L^2(M,B \otimes S), D_B)$. The conDitions on $\gAmma_B$ To bE a gRaDing OperAtor for this speCTral Triple are easily checked.
in the NeXT secTIoN we shOw that tHE tRIPle ${(\gaMmA(M,B), l^2(M,b\OtImes S),d_B,J,\Gamma_B)}$ gIves a non-trIviAl yang–Mills theORy Over the manIfoLd $M$. The Serre–SWan theoreM \... | Gamma(M,B), L^2(M,B \otime s S), D_B) $. Th e c ond it ions on$\gamma_B$ tob e agrading operator for t his s pe c tral tr ipleare eas i ly c hec ke d.
I nt he next se ction w e show tha t t he triple ${(\ G am ma(M,B), L ^2( M,B\otimes S ),D _B,J,\ ga mma _ B)}$giv es anon-tr i vial Y ang–Mills t h eory o v er t... | Gamma(M,B), L^2(M,B_\otimes S),_D_B)$. The conditions on_$\gamma_B$ to_be_a grading_operator_for this spectral_triple are easily_checked.
In the next section_we show that_the_triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ gives a non-trivial Yang–Mills theory over the manifold $M$. The_Serre–Swan_Theorem \... |
N=2^q$ for a positive integer $q$, then the $\beta_j$’s can be chosen to be the design matrix for a saturated model of a $2^q$ factorial design in which the levels of the factors are set at $\pm1$ \[@BHH, Chapter 5\]. In addition, assume that $X_1,\ldots,X_m$ are random diagonal matrices of size $N$ and $Y_{jk}$, $j=1,... | N=2^q$ for a positive integer $ q$, then the $ \beta_j$ ’s can be chosen to be the design matrix for a saturated model of a $ 2^q$ factorial invention in which the degree of the factors are set at $ \pm1 $ \[@BHH, Chapter 5\ ]. In addition, simulate that $ X_1,\ldots, X_m$ are random diagonal matrices of size $ N$ and ... | N=2^q$ vor a positive integer $q$, then the $\beta_j$’s can bx choseh to be ghe design matrix for a satucatee modtj of a $2^q$ factorial design ij which rhe owvels of tis factovf ars set et $\pm1$ \[@BHH, Chaptgr 5\]. In addithon, assume thad $B_1,\lbots,X_m$ are random diagonal matrices jf size $N$ and $Y_{jk}$, $j=1,... | N=2^q$ for a positive integer $q$, then can chosen to the design matrix a factorial design in the levels of factors are set at $\pm1$ \[@BHH, 5\]. In addition, assume that $X_1,\ldots,X_m$ are random diagonal matrices of size $N$ $Y_{jk}$, $j=1,\ldots,N; k=1,\ldots,m$ are random variables such that all the diagonal ele... | N=2^q$ for a positive integer $q$, theN the $\beta_j$’s Can be ChoSen To Be thE desIgn matrix for a sATuraTed model of a $2^q$ factorial dEsign In WHich THe LevelS of the fACtORS arE sEt At $\pM1$ \[@Bhh, CHapteR 5\]. In AdditioN, assume thaT $X_1,\lDoTs,X_m$ are randoM DiAgonal matrIceS of size $N$ and $Y_{Jk}$, $j=1,... | N=2^q$ for a positive inte ger $q$, t hen t he$\b et a_j$ ’s c an be chosen t o bethe design matrix fora sat ur a tedm od el of a $2^q $ f a c tor ia ldes ig n i n whi chthe lev els of the fa ct ors are seta t$\pm1$ \[@ BHH , Chapter 5\ ].In add it ion , assu methat$X_1,\ l dots,X _m$ are r an d om dia g onal ma ... | N=2^q$ for_a positive_integer $q$, then the_$\beta_j$’s can_be_chosen to_be_the design matrix_for a saturated_model of a $2^q$_factorial design in_which_the levels of the factors are set at $\pm1$ \[@BHH, Chapter 5\]. In addition,_assume_that $X_1,\ldots,X_m$_are_random_diagonal matrices of size $N$_and $Y_{jk}$, $j=1,... |
=
\operatorname{Supp}G_{1}/E_{1} \cup \cdots \cup \operatorname{Supp}G_{n}/E_{n}.$$ Therefore $${\operatorname{grade}\,}G/E =
\min_{1 \leq i \leq n} \{ {\operatorname{grade}\,}G_{i}/E_{i} \} \geq 2 \iff {\operatorname{grade}\,}G_{i}/E_{i} \geq
2, \; 1 \leq i \leq n,$$ proving the equivalence.
We observe ... | =
\operatorname{Supp}G_{1}/E_{1 } \cup \cdots \cup \operatorname{Supp}G_{n}/E_{n}.$$ Therefore $ $ { \operatorname{grade}\,}G / E =
\min_{1 \leq i \leq n } \ { { \operatorname{grade}\,}G_{i}/E_{i } \ } \geq 2 \iff { \operatorname{grade}\,}G_{i}/E_{i } \geq
2, \; 1 \leq i \leq n,$$ proving the equival... | =
\operatorname{Supp}G_{1}/E_{1} \cup \cdots \cup \opercrornamx{Supp}G_{n}/S_{n}.$$ Therewore $${\operatorname{grade}\,}G/E =
\mun_{1 \lew i \leq n} \{ {\operatornamd{grade}\,}G_{i}/E_{p} \} \geq 2 \idf {\okeratorname{grade}\,}G_{m}/S_{i} \geq
2, \; 1 \peq n \oeq n,$$ proving jhe equivaletce.
We observe ... | = \operatorname{Supp}G_{1}/E_{1} \cup \cdots \cup \operatorname{Supp}G_{n}/E_{n}.$$ Therefore \min_{1 i \leq \{ {\operatorname{grade}\,}G_{i}/E_{i} \} 2, 1 \leq i n,$$ proving the We observe that a direct sum ideals cannot be a [complete intersection]{} module. \[ds1\] Let $R$ be a [Noetherian and $E = {{E}_1 {\oplus}\... | =
\operatorname{Supp}G_{1}/E_{1} \cup \cdotS \cup \operatOrnamE{SuPp}G_{N}/E_{N}.$$ TheRefoRe $${\operatorname{GRade}\,}g/E =
\min_{1 \leq i \leq n} \{ {\operatornAme{grAdE}\,}g_{i}/E_{i} \} \GEq 2 \Iff {\opEratornAMe{GRAde}\,}g_{i}/e_{i} \Geq
2, \; 1 \LeQ I \lEq n,$$ prOviNg the eqUivalence.
WE obSeRve ... | =
\operatorname{Supp} G_{1}/E_{1 } \cu p \ cdo ts \cu p \o peratorname{Su p p}G_ {n}/E_{n}.$$ Therefore $${\ op e rato r na me{gr ade}\,} G /E =
\m in_ {1 \l eq i\le q n} \{ {\operato rna me {grade}\,}G_ { i} /E_{i} \}\ge q 2 \iff {\o per atorna me {gr a de}\, }G_ {i}/E _{i} \ g eq
2, \; 1 \ l eq i \ l ... | =
_ _ \operatorname{Supp}G_{1}/E_{1} \cup \cdots_\cup \operatorname{Supp}G_{n}/E_{n}.$$_Therefore_$${\operatorname{grade}\,}G/E =
__ \min_{1_\leq i \leq_n} \{ {\operatorname{grade}\,}G_{i}/E_{i} \}_\geq 2 \iff_{\operatorname{grade}\,}G_{i}/E_{i}_\geq
2, \; 1 \leq i \leq n,$$ proving the_equivalence.
We_observe ... |
\left[e^{2 {{\sigma_{\! \! x}}}^2k^2 r} (3 r^2 + 6r -25)-(3r^2-6r-25)\right] \\
\notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \frac{12r}{({{\sigma_{\! \! x}}}k)^4} \left[(3r^3-12r^2-19r-28)- e^{2{{\sigma_{\! \! x}}}^2k^2 r} (3 r^3+12r^2-19r+28)\right] \\
\notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac... | \left[e^{2 { { \sigma_{\! \! x}}}^2k^2 r } (3 r^2 + 6r -25)-(3r^2 - 6r-25)\right ] \\
\notag & \qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \frac{12r}{({{\sigma_{\! \! x}}}k)^4 } \left[(3r^3 - 12r^2 - 19r-28)- e^{2{{\sigma_{\! \! x}}}^2k^2 r } (3 r^3 + 12r^2 - 19r+28)\right ] \\
\notag & \qquad\qquad\qquad\qquad... | \levt[e^{2 {{\sigma_{\! \! x}}}^2k^2 r} (3 r^2 + 6r -25)-(3r^2-6r-25)\vight] \\
\notag &\qquab\wquad\qxuad\qqhad\qquad\dquad\qquad+ \frac{12r}{({{\sigma_{\! \! x}}}k)^4} \lxft[(3r^3-12e^2-19r-28)- e^{2{{\sugma_{\! \! x}}}^2k^2 r} (3 r^3+12r^2-19r+28)\right] \\
\votag &\qquwd\qquad\qwuad\wwuad\qquad\qquad\qquad+\frac... | \left[e^{2 {{\sigma_{\! \! x}}}^2k^2 r} (3 r^2 -25)-(3r^2-6r-25)\right] \notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \! x}}}k)^4} \left[(3r^3-12r^2-19r-28)- r^3+12r^2-19r+28)\right] \notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{48}{({{\sigma_{\! \! \left[e^{2{{\sigma_{\! \! x}}}^2k^2 }(3r^2-12 r+7)-(3... | \left[e^{2 {{\sigma_{\! \! x}}}^2k^2 r} (3 r^2 + 6r -25)-(3r^2-6r-25)\right] \\
\notAg &\qquad\qquAd\qquAd\qQuaD\qQuad\QquaD\qquad+ \frac{12r}{({{\sigMA_{\! \! x}}}k)^4} \lEft[(3r^3-12r^2-19r-28)- e^{2{{\sigma_{\! \! x}}}^2k^2 r} (3 r^3+12r^2-19r+28)\righT] \\
\notaG &\qQUad\qQUaD\qquaD\qquad\qQUaD\QQuaD\qQuAd+\fRaC... | \left[e^{2 {{\sigma_{\! \ ! x}}}^2k^ 2 r}(3r^2 + 6r-25) -(3r^2-6r-25)\ r ight ] \\
\notag &\qquad\qq uad\q qu a d\qq u ad \qqua d\qquad \ qq u a d+\f ra c{1 2r } {( {{\si gma _{\! \! x}}}k)^4} \ le ft[(3r^3-12r ^ 2- 19r-28)- e ^{2 {{\sigma_{\! \! x}}}^ 2k ^2r } (3r^3 +12r^ 2-19r+ 2 8)\rig ht] \\
\ no t ag &\q ... | \left[e^{2_{{\sigma_{\! \!_x}}}^2k^2 r} (3 r^2_+ 6r_-25)-(3r^2-6r-25)\right]_\\
\notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+_\frac{12r}{({{\sigma_{\!_\! x}}}k)^4} _\left[(3r^3-12r^2-19r-28)- e^{2{{\sigma_{\! \!_x}}}^2k^2 r} (3 r^3+12r^2-19r+28)\right]_ \\
\notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac... |
in time polynomial in $n$. Furthermore, if $K$ is any field possessing $D_P$ distinct ${D_P}{^{\text{\underline{th}}}}$ roots of unity, then a ${\mathtt{3CNFSAT}}$ instance $B(y)\!:=C_1(y)\wedge \cdots \wedge
C_k(y)$ has a satisfying assignment iff the univariate polynomial system $F_B\!:=\!({{\mathcal{P}}}_P(C_1), \... | in time polynomial in $ n$. Furthermore, if $ K$ is any field possessing $ D_P$ distinct $ { D_P}{^{\text{\underline{th}}}}$ root of oneness, then a $ { \mathtt{3CNFSAT}}$ instance $ B(y)\!:=C_1(y)\wedge \cdots \wedge
C_k(y)$ has a satisfying appointment iff the univariate polynomial system $ F_B\!:=\!({{\mathcal{P}}... | in time polynomial in $n$. Fuvthermore, if $K$ is any fmeld poasessing $D_P$ distinct ${D_P}{^{\text{\underline{vh}}}}$ riots if unity, then a ${\mathtt{3ZNFSAT}}$ indtance $B(t)\!:=C_1(y)\wtdge \cdots \wedge
R_i(y)$ has a satianying essignment iff jhe univariade polynomial vyrtzm $F_B\!:=\!({{\mathcal{P}}}_P(C_1), \... | in time polynomial in $n$. Furthermore, if any possessing $D_P$ ${D_P}{^{\text{\underline{th}}}}$ roots of $B(y)\!:=C_1(y)\wedge \wedge C_k(y)$ has satisfying assignment iff univariate polynomial system $F_B\!:=\!({{\mathcal{P}}}_P(C_1), \ldots,{{\mathcal{P}}}_P(C_k))$ has root $\zeta\!\in\!K$ satisfying $\zeta^{D_P}-1... | in time polynomial in $n$. FurtheRmore, if $K$ is Any fiEld PosSeSsinG $D_P$ dIstinct ${D_P}{^{\text{\uNDerlIne{th}}}}$ roots of unity, then a ${\MathtT{3CnfSAT}}$ INsTance $b(y)\!:=C_1(y)\wedGE \cDOTs \wEdGe
c_k(y)$ HaS A sAtisfYinG assignMent iff the UniVaRiate polynomIAl System $F_B\!:=\!({{\maThcAl{P}}}_P(C_1), \... | in time polynomial in $n$ . Furtherm ore,if$K$ i s an y fi eld possessing $D_P $ distinct ${D_P}{^{\t ext{\ un d erli n e{ th}}} }$ root s o f uni ty ,the na $ {\mat htt {3CNFSA T}}$ insta nce $ B(y)\!:=C_1( y )\ wedge \cdo ts\wedge
C_k( y)$ has a s ati s fying as signm ent if f the u nivariate p o lynomi a ... | in_time polynomial_in $n$. Furthermore, if_$K$ is_any_field possessing_$D_P$_distinct ${D_P}{^{\text{\underline{th}}}}$ roots_of unity, then_a ${\mathtt{3CNFSAT}}$ instance $B(y)\!:=C_1(y)\wedge_\cdots \wedge
C_k(y)$_has_a satisfying assignment iff the univariate polynomial system $F_B\!:=\!({{\mathcal{P}}}_P(C_1), \... |
n |c_i|^p\right)^{1/p}\leq \left(\sum_{k=0}^\infty \lambda_k^p\right)^{1/p}\leq \left(\frac{1}{1-\delta^{p}}\right)^{1/p}<1+\e$$ for $\delta$ sufficiently small.
This “greedy algorithm” especifies a unique decomposition for each $f$ in the unit sphere of $F$. However it does not guarantee any kind of homogeneity in ... | n |c_i|^p\right)^{1 / p}\leq \left(\sum_{k=0}^\infty \lambda_k^p\right)^{1 / p}\leq \left(\frac{1}{1-\delta^{p}}\right)^{1 / p}<1+\e$$ for $ \delta$ sufficiently small.
This “ greedy algorithm ” especifies a unique decomposition for each $ f$ in the unit of measurement sector of $ F$. However it does not guarant... | n |c_l|^p\right)^{1/p}\leq \left(\sum_{k=0}^\infuy \lambda_k^p\right)^{1/k}\lwq \lefv(\frac{1}{1-\demta^{p}}\righg)^{1/p}<1+\e$$ for $\delta$ sufficiently slaol.
Thiw “greedy algorithm” espdcifies a unique eeconposition for each $n$ in fme unnt sphere of $F$. Hpwever it goes not guarattde any kind of homogeneity in ... | n |c_i|^p\right)^{1/p}\leq \left(\sum_{k=0}^\infty \lambda_k^p\right)^{1/p}\leq \left(\frac{1}{1-\delta^{p}}\right)^{1/p}<1+\e$$ for $\delta$ This algorithm” especifies unique decomposition for sphere $F$. However it not guarantee any of homogeneity in these decompositions. To this, let $S_0$ be a maximal subset of the... | n |c_i|^p\right)^{1/p}\leq \left(\sum_{k=0}^\inftY \lambda_k^p\rIght)^{1/p}\Leq \LefT(\fRac{1}{1-\dElta^{P}}\right)^{1/p}<1+\e$$ for $\delTA$ sufFiciently small.
This “greeDy algOrIThm” eSPeCifieS a uniquE DeCOMpoSiTiOn fOr EAcH $f$ in tHe uNit spheRe of $F$. HowevEr iT dOes not guaranTEe Any kind of hOmoGeneity in ... | n |c_i|^p\right)^{1/p}\leq \left(\s um_{k =0} ^\i nf ty \lam bda_k^p\right) ^ {1/p }\leq \left(\frac{1}{1 -\del ta ^ {p}} \ ri ght)^ {1/p}<1 + \e $ $ fo r$\ del ta $ s uffic ien tly sma ll.
This“gr ee dy algorithm ” e specifiesa u nique decomp osi tion f or ea c h $f$ in theunit s p here o f $F$. Ho we v er it... | n |c_i|^p\right)^{1/p}\leq_ \left(\sum_{k=0}^\infty_ \lambda_k^p\right)^{1/p}\leq \left(\frac{1}{1-\delta^{p}}\right)^{1/p}<1+\e$$ for_$\delta$ sufficiently_small.
This_“greedy algorithm”_especifies_a unique decomposition_for each $f$_in the unit sphere_of $F$. However_it_does not guarantee any kind of homogeneity in ... |
between vertices of the first type. As a consequence, every vertex of the first type admits [*at most*]{} one incoming edge, [*i.e.*]{} $\Gamma_A$ can be only of the second type in Figure 15; any of the vertices of $\Gamma_A$ may further have [*at most*]{} one outgoing edge.
We briefly discuss the coloring of an admi... | between vertices of the first type. As a consequence, every vertex of the first character admit [ * at most * ] { } one incoming edge, [ * i.e. * ] { } $ \Gamma_A$ can be only of the second character in Figure 15; any of the vertex of $ \Gamma_A$ may further have [ * at most * ] { } one extroverted boundary.
We brie... | behween vertices of the fivst type. As a couwequenre, everg vertex of the first type admits [*at mist*]{} obe incoming edge, [*i.e.*]{} $\Gaoma_A$ can he only if tie second type ii Figure 15; any kn the tertices of $\Gamka_A$ may fusther have [*at kort*]{} one outgoing edge.
We briefly discusf the cpllring of an adii... | between vertices of the first type. As every of the type admits [*at $\Gamma_A$ be only of second type in 15; any of the vertices of may further have [*at most*]{} one outgoing edge. We briefly discuss the coloring an admissible graph. We choose a system of coordinates on $\mathfrak g$ which adapted the submanifolds k^... | between vertices of the first Type. As a conSequeNce, EveRy VertEx of The first type adMIts [*aT most*]{} one incoming edge, [*i.e.*]{} $\gamma_a$ cAN be oNLy Of the Second tYPe IN figUrE 15; aNy oF tHE vErticEs oF $\Gamma_A$ May further HavE [*aT most*]{} one outgOInG edge.
We briEflY discuss the cOloRing of An AdmI... | between vertices of the f irst type. As a co nse qu ence , ev ery vertex oft he f irst type admits [*atmost* ]{ } one in comin g edge, [* i . e.* ]{ }$\G am m a_ A$ ca n b e onlyof the sec ond t ype in Figur e 1 5; any ofthe vertices of $\ Gamma_ A$ ma y furt her have [*atm ost*]{ } one out go i ng edg e .
... | between_vertices of_the first type. As_a consequence,_every_vertex of_the_first type admits_[*at most*]{} one_incoming edge, [*i.e.*]{} $\Gamma_A$_can be only_of_the second type in Figure 15; any of the vertices of $\Gamma_A$ may further_have_[*at most*]{}_one_outgoing_edge.
We briefly discuss the coloring_of an admi... |
Similarly, *meet-irreducible elements* cannot be written as an infimum of other elements, and are such that they are covered by a single element. We denote by $\mathcal{M}(L)$ the set of meet-irreducible elements of $L$. *Co-atoms* are meet-irreducible elements covered by $\top$.
For any $x\in L$, we say that $x$ *has... | Similarly, * meet - irreducible elements * cannot be written as an infimum of other elements, and are such that they are cover by a individual element. We denote by $ \mathcal{M}(L)$ the set of meet - irreducible component of $ L$. * Co - atoms * are meet - irreducible element cover by $ \top$.
For any $ x\in L$, we... | Simllarly, *meet-irreducible euements* cannot yw writven as zn infimjm of other elements, and are sych tyat they are covered bh a singlv element. We venote by $\mathcal{M}(L)$ the set of leet-nrceducible elemekts of $L$. *Co-dtoms* are meet-hrfebucible elements covered by $\top$.
For agy $x\in K$, ae say that $x$ *ras... | Similarly, *meet-irreducible elements* cannot be written as of elements, and such that they element. denote by $\mathcal{M}(L)$ set of meet-irreducible of $L$. *Co-atoms* are meet-irreducible elements by $\top$. For any $x\in L$, we say that $x$ *has a complement $L$* if there exists $x'\in L$ such that $x\wedge x'=\bo... | Similarly, *meet-irreducible eLements* canNot be WriTteN aS an iNfimUm of other elemeNTs, anD are such that they are covEred bY a SInglE ElEment. we denotE By $\MAThcAl{m}(L)$ The SeT Of Meet-iRreDucible Elements of $l$. *Co-AtOms* are meet-irREdUcible elemEntS covered by $\toP$.
FoR any $x\iN L$, We sAY that $X$ *haS... | Similarly, *meet-irreducib le element s* ca nno t b ewrit tenas an infimumo f ot her elements, and aresuchth a t th e yare c overedb ya sin gl eele me n t. We d eno te by $ \mathcal{M }(L )$ the set ofm ee t-irreduci ble elements of $L $. *Co -a tom s * are me et-ir reduci b le ele ments cov er e d by $ \ top$.
F ... | Similarly, *meet-irreducible_elements* cannot_be written as an_infimum of_other_elements, and_are_such that they_are covered by_a single element. We_denote by $\mathcal{M}(L)$_the_set of meet-irreducible elements of $L$. *Co-atoms* are meet-irreducible elements covered by $\top$.
For any_$x\in_L$, we_say_that_$x$ *has... |
[Bilder/Observations/cont08.pdf]{} (3,89)
[Bilder/Observations/dop08.pdf]{} (3,89)
[Bilder/dopmean\_spot/dopmean2010\_07\_06.pdf]{} (3,89)
[Bilder/dopmean\_moat/dopmean2010\_07\_06\_mask.pdf]{} (3,71)
[Bilder/dopmean\_moat/dopmean2010\_09\_25.pdf]{} (3,89)
[Bilder/dopmean\_moat/dopmean2010\_10\_17.pdf]{} (3,89)
... | [ Bilder / Observations / cont08.pdf ] { } (3,89)
[ Bilder / Observations / dop08.pdf ] { } (3,89)
[ Bilder / dopmean\_spot / dopmean2010\_07\_06.pdf ] { } (3,89)
[ Bilder / dopmean\_moat / dopmean2010\_07\_06\_mask.pdf ] { } (3,71)
[ Bilder / dopmean\_moat / dopmean2010\_09\_25.pdf ] { } (3,89)
[ Bilder... |
[Bilfer/Observations/cont08.pdf]{} (3,89)
[Bllder/Observations/dop08.pdf]{} (3,89)
[Bilded/dopmean\_rpot/dopmean2010\_07\_06.pdf]{} (3,89)
[Bilder/dopmean\_loqt/dopnean2010\_07\_06\_mask.pdf]{} (3,71)
[Bilder/dopmdan\_moat/doimean2010\_09\_25.pdf]{} (3,89)
[Vildtr/dopmean\_moat/dopmxzn2010\_10\_17.pdf]{} (3,89)
... | [Bilder/Observations/cont08.pdf]{} (3,89) [Bilder/Observations/dop08.pdf]{} (3,89) [Bilder/dopmean\_spot/dopmean2010\_07\_06.pdf]{} (3,89) [Bilder/dopmean\_moat/dopmean2010\_07\_06\_mask.pdf]{} (3,89) (3,89) [Bilder/dopmean\_moat/dopmeanAR11131\_1.pdf]{} [Bilder/dopmean\_moat/dopmean2011\_01\_02.pdf]{} (3,71) Data The ... |
[Bilder/Observations/cont08.pdf]{} (3,89)
[bilder/ObseRvatiOns/Dop08.PdF]{} (3,89)
[BilDer/dOpmean\_spot/dopmEAn2010\_07\_06.pdF]{} (3,89)
[Bilder/dopmean\_moat/dopmEan2010\_07\_06\_maSk.PDf]{} (3,71)
[BiLDeR/dopmEan\_moat/DOpMEAn2010\_09\_25.pDf]{} (3,89)
[biLdeR/dOPmEan\_moAt/dOpmean2010\_10\_17.pDf]{} (3,89)
... |
[Bilder/Observations/con t08.pdf]{} (3,8 9)
[B il der/ Obse rvations/dop08 . pdf] {} (3,89)
[Bilder/dop mean\ _s p ot/d o pm ean20 10\_07\ _ 06 . p df] {} ( 3,8 9)
[ Bilde r/d opmean\ _moat/dopm ean 20 10\_07\_06\_ m as k.pdf]{} ( 3,7 1)
[Bilder/ dop mean\_ mo at/ d opmea n20 10\_0 9\_25. p df]{}(3,89)
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[Bilder/Observations/cont08.pdf]{} (3,89)
[Bilder/Observations/dop08.pdf]{}_(3,89)
[Bilder/dopmean\_spot/dopmean2010\_07\_06.pdf]{} (3,89)
[Bilder/dopmean\_moat/dopmean2010\_07\_06\_mask.pdf]{}_(3,71)
[Bilder/dopmean\_moat/dopmean2010\_09\_25.pdf]{} (3,89)
[Bilder/dopmean\_moat/dopmean2010\_10\_17.pdf]{} (3,89)
... |
T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right) \right\},$$ and so in total, we have $$\label{eqn:final AHat eqn}
\widehat A(T,{\mathbf v})(\varphi_1\otimes \varphi_2) = \sum_{{\mathbf y}\text{ mod } \Gamma} \sum_m \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1) (\varphi_1'' \otimes \varphi_2... | T_{12' } & T_2 \end{smallmatrix } \right) }, \varphi_1' \right) \right\},$$ and so in total, we have $ $ \label{eqn: final AHat eqn }
\widehat A(T,{\mathbf v})(\varphi_1\otimes \varphi_2) = \sum_{{\mathbf y}\text { mod } \Gamma } \sum_m \widehat Z_{{\mathbf y}}(m, { \mathbf v}_1) (\varphi_1" \otime... | T_{12'} & T_2 \end{smallmatrix} \right)}, \yarphi_1' \right) \rigkr\},$$ and vo in fotal, we have $$\label{eqn:final AHat eqn}
\widehat A(T,{\matfbf v})(\varpji_1\otimes \varkhi_2) = \sum_{{\mathbf y}\txst{ mod } \Gamma} \suk_n \widehat Z_{{\matmbf y}}(m, {\mathtf v}_1) (\varphi_1'' \othmds \varphi_2... | T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right) so total, we $$\label{eqn:final AHat eqn} \sum_{{\mathbf mod } \Gamma} \widehat Z_{{\mathbf y}}(m, v}_1) (\varphi_1'' \otimes \varphi_2'')\cdot \widehat{\omega}^{n - \mathrm{rk}}(T_2) - 1} \cdot \left\{ \varphi_2'({\mathbf y}) \, r_{{\mathbf y}}\left( {\left(... | T_{12'} & T_2 \end{smallmatrix} \right)}, \varpHi_1' \right) \rigHt\},$$ and So iN toTaL, we hAve $$\lAbel{eqn:final AHAT eqn}
\Widehat A(T,{\mathbf v})(\varphi_1\OtimeS \vARphi_2) = \SUm_{{\MathbF y}\text{ mOD } \GAMMa} \sUm_M \wIdeHaT z_{{\mAthbf Y}}(m, {\mAthbf v}_1) (\vArphi_1'' \otimeS \vaRpHi_2... | T_{12'} & T_2 \end{smallm atrix} \ri ght)} , \ var ph i_1' \ri ght) \right\}, $ $ an d so in total, we have $$\l ab e l{eq n :f inalAHat eq n } \ wideh atA(T,{\m athbf v})( \va rp hi_1\otimes\ va rphi_2) =\su m_{{\mathbfy}\ text{mo d } \Gamm a} \su m_m \w i dehatZ_{{\math bf y}}(m, {\mathb f v} _1)(\varphi_1... | T_{12'}_& T_2_\end{smallmatrix} \right)}, \varphi_1' \right)_\right\},$$ and_so_in total,_we_have $$\label{eqn:final AHat_eqn}
_ _ __ \widehat A(T,{\mathbf v})(\varphi_1\otimes \varphi_2) = \sum_{{\mathbf y}\text{ mod }_\Gamma}_ _\sum_m_\widehat_Z_{{\mathbf y}}(m, {\mathbf v}_1) (\varphi_1''_\otimes \varphi_2... |
asi-linear saturation"). This scenario is indeed what we observe: the evolution of $\langle | \delta {\mbox{\boldmath{$B$}}}_\perp |^2 \rangle$ and $\Lambda_{\rm f}$ is shown in Fig. \[fig:fhs-boxavg\]; note $\langle \Lambda_{\rm f} \rangle_{\rm max} \propto S^{2/3}$ (inset in Fig. \[fig:fhs-boxavg\]b). To test the ide... | asi - linear saturation "). This scenario is indeed what we observe: the evolution of $ \langle | \delta { \mbox{\boldmath{$B$}}}_\perp |^2 \rangle$ and $ \Lambda_{\rm f}$ is testify in Fig. \[fig: fhs - boxavg\ ]; notice $ \langle \Lambda_{\rm f } \rangle_{\rm max } \propto S^{2/3}$ (inset in Fig. \[fig: fhs - box... | asi-pinear saturation"). This sgenario is indeeb what xe obsedve: the dvolution of $\langle | \delta {\muox{\bildmauk{$B$}}}_\perp |^2 \rangle$ and $\Lxmbda_{\rm f}$ is showb in Dig. \[fig:fhs-boxavg\]; nobz $\lanfpe \Lcmuda_{\rm f} \rangle_{\rk max} \propdo S^{2/3}$ (inset in Xie. \[fng:fhs-boxavg\]b). To test the ide... | asi-linear saturation"). This scenario is indeed what the of $\langle \delta {\mbox{\boldmath{$B$}}}_\perp |^2 shown Fig. \[fig:fhs-boxavg\]; note \Lambda_{\rm f} \rangle_{\rm \propto S^{2/3}$ (inset in Fig. \[fig:fhs-boxavg\]b). test the idea [@sckrh08; @rsrc11] that, during the secular phase, the average $B$ by parti... | asi-linear saturation"). This scEnario is inDeed wHat We oBsErve: The eVolution of $\langLE | \delTa {\mbox{\boldmath{$B$}}}_\perp |^2 \ranGle$ anD $\LAMbda_{\RM f}$ Is shoWn in Fig. \[FIg:FHS-boXaVg\]; NotE $\lANgLe \LamBda_{\Rm f} \rangLe_{\rm max} \proPto s^{2/3}$ (iNset in Fig. \[fig:FHs-Boxavg\]b). To tEst The ide... | asi-linear saturation"). T his scenar io is in dee dwhat weobserve: the e v olut ion of $\langle | \del ta {\ mb o x{\b o ld math{ $B$}}}_ \ pe r p |^ 2\r ang le $ a nd $\ Lam bda_{\r m f}$ is s how nin Fig. \[fi g :f hs-boxavg\ ];note $\langl e \ Lambda _{ \rm f} \r ang le_{\ rm max } \prop to S^{2/3 }$ (inset in... | asi-linear saturation")._This scenario_is indeed what we_observe: the_evolution_of $\langle_|_\delta {\mbox{\boldmath{$B$}}}_\perp |^2_\rangle$ and $\Lambda_{\rm_f}$ is shown in_Fig. \[fig:fhs-boxavg\]; note $\langle_\Lambda_{\rm_f} \rangle_{\rm max} \propto S^{2/3}$ (inset in Fig. \[fig:fhs-boxavg\]b). To test the ide... |
to mode coupling behavior, identified with the onset of non-activated motions[@stevenson.2006]. Reconfiguration events of the more extended type are more susceptible to fluctuations in the local driving force, even away from the crossover. These ramified or “stringy” reconfiguration events thus dominate the low barrie... | to mode coupling behavior, identified with the onset of non - activated motions[@stevenson.2006 ]. Reconfiguration event of the more extensive type are more susceptible to fluctuations in the local driving violence, even away from the crossover. These complexify or “ stringy ” reconfiguration events therefore predomina... | to mode coupling behavior, ldentified with jhw onsev of noh-activatdd motions[@stevenson.2006]. Reconfignratuon ecents of the more extevded type are morw suwxeptible to fluctuations ln thz oocal driving norce, even dway from the wrusdover. These ramified or “stringy” recjnfigurstlon events thuf dokynats the low barrie... | to mode coupling behavior, identified with the non-activated Reconfiguration events the more extended fluctuations the local driving even away from crossover. These ramified or “stringy” reconfiguration thus dominate the low barrier tail of the activation energy distribution. When the distribution of reconfiguration pr... | to mode coupling behavior, ideNtified witH the oNseT of NoN-actIvatEd motions[@steveNSon.2006]. REconfiguration events of The moRe EXtenDEd Type aRe more sUScEPTibLe To FluCtUAtIons iN thE local dRiving forcE, evEn Away from the cROsSover. These RamIfied or “strinGy” rEconfiGuRatIOn eveNts Thus dOminatE The low Barrie... | to mode coupling behavior , identifi ed wi ththe o nset ofnon-activatedm otio ns[@stevenson.2006]. R econf ig u rati o nevent s of th e m o r e e xt en ded t y pe aremor e susce ptible toflu ct uations in t h elocal driv ing force, even aw ay fro mthe cross ove r. Th ese ra m ifiedor “strin gy ” recon f igurat... | to_mode coupling_behavior, identified with the_onset of_non-activated_motions[@stevenson.2006]. Reconfiguration_events_of the more_extended type are_more susceptible to fluctuations_in the local_driving_force, even away from the crossover. These ramified or “stringy” reconfiguration events thus dominate_the_low barrie... |
0.064$\pm$0.012 & 0.12$^{+0.18}_{-0.12}$ & 0.228 & 0.07$^{+0.16}_{-0.07}$ & 0.239\
&0.835 &$0.87^{+0.07}_{-0.12}$ & burst & 0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$ & 0.146 & 0.17$^{+0.11}_{-0.17}$ & 0.082\
&0.966 &$0.82^{+0.16}_{-0.18}$ & burst & 0.006$\pm$0.122 & 1.60$^{+0.05}_{-1.27}$ & 0.683 & 0.38$^{+0.25}_{-0.24... | 0.064$\pm$0.012 & 0.12$^{+0.18}_{-0.12}$ & 0.228 & 0.07$^{+0.16}_{-0.07}$ & 0.239\
& 0.835 & $ 0.87^{+0.07}_{-0.12}$ & burst & 0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$ & 0.146 & 0.17$^{+0.11}_{-0.17}$ & 0.082\
& 0.966 & $ 0.82^{+0.16}_{-0.18}$ & burst & 0.006$\pm$0.122 & 1.60$^{+0.05}_{-1.27}$ & 0.683 & 0.38$^{+0.2... | 0.064$\pm$0.012 & 0.12$^{+0.18}_{-0.12}$ & 0.228 & 0.07$^{+0.16}_{-0.07}$ & 0.239\
&0.835 &$0.87^{+0.07}_{-0.12}$ & burst & 0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$ & 0.146 & 0.17$^{+0.11}_{-0.17}$ & 0.082\
&0.966 &$0.82^{+0.16}_{-0.18}$ & burst & 0.006$\pm$0.122 & 1.60$^{+0.05}_{-1.27}$ & 0.683 & 0.38$^{+0.25}_{-0.24... | 0.064$\pm$0.012 & 0.12$^{+0.18}_{-0.12}$ & 0.228 & 0.07$^{+0.16}_{-0.07}$ &0.835 & burst 0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$ 0.082\ &$0.82^{+0.16}_{-0.18}$ & burst 0.006$\pm$0.122 & 1.60$^{+0.05}_{-1.27}$ 0.683 & 0.38$^{+0.25}_{-0.24}$ & 0.831\ &1.600 & Sa & 0.012$\pm$0.006 & 0.90$^{+0.17}_{-0.20}$ & 0.823 & 0.96$... | 0.064$\pm$0.012 & 0.12$^{+0.18}_{-0.12}$ & 0.228 & 0.07$^{+0.16}_{-0.07}$ & 0.239\
&0.835 &$0.87^{+0.07}_{-0.12}$ & burst & 0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$ & 0.146 & 0.17$^{+0.11}_{-0.17}$ & 0.082\
&0.966 &$0.82^{+0.16}_{-0.18}$ & burst & 0.006$\pm$0.122 & 1.60$^{+0.05}_{-1.27}$ & 0.683 & 0.38$^{+0.25}_{-0.24... | 0.064$\pm$0.012 & 0.12$^{ +0.18}_{-0 .12}$ &0.2 28 & 0 .07$ ^{+0.16}_{-0.0 7 }$ & 0.239\
&0.835 &$0.87^ {+0.0 7} _ {-0. 1 2} $ & b urst &0 .0 9 1 $\p m$ 0. 090 & 0. 17$^{ +0. 11}_{-0 .17}$ & 0. 146 & 0.17$^{+0.1 1 }_ {-0.17}$ & 0. 082\
&0.966&$0 .82^{+ 0. 16} _ {-0.1 8}$ & bu rst &0 .006$\ pm$0.122&1 .60$^{ + ... | 0.064$\pm$0.012_& 0.12$^{+0.18}_{-0.12}$_& 0.228 & 0.07$^{+0.16}_{-0.07}$_& 0.239\
&0.835_&$0.87^{+0.07}_{-0.12}$_& burst_&_0.091$\pm$0.090 & 0.17$^{+0.11}_{-0.17}$_& 0.146 &_0.17$^{+0.11}_{-0.17}$ & 0.082\
&0.966 &$0.82^{+0.16}_{-0.18}$_& burst &_0.006$\pm$0.122_& 1.60$^{+0.05}_{-1.27}$ & 0.683 & 0.38$^{+0.25}_{-0.24... |
a look on the general form for the scattering matrix.
The general form for diagrams contributing to the scattering matrix is $$\begin{split}
{\cal M} &\sim \Big(A+B q^2 + \ldots + (\alpha_1 \kappa^2 + \alpha_2 e^2) \frac{1}{q^2} \\
&+\beta_1 e^2\kappa^2\ln(-q^2) + \beta_2 e^2\kappa^2 \frac{m}{\sqrt{-q^2}} + \ldots\Bi... | a look on the general form for the scattering matrix.
The cosmopolitan human body for diagrams contributing to the scatter matrix is $ $ \begin{split }
{ \cal M } & \sim \Big(A+B q^2 + \ldots + (\alpha_1 \kappa^2 + \alpha_2 e^2) \frac{1}{q^2 } \\
& + \beta_1 e^2\kappa^2\ln(-q^2) + \beta_2 e^2\kappa^2 \frac{m}{\s... | a pook on the general form for the scattetibg matcix.
The feneral worm for diagrams contributiig ti the scattering matrix is $$\cegin{splin}
{\cal M} &\sin \Bij(A+B q^2 + \ldots + (\al'ga_1 \kappa^2 + \alpgw_2 e^2) \yrec{1}{q^2} \\
&+\beta_1 e^2\kappa^2\kn(-q^2) + \beta_2 a^2\kappa^2 \frac{m}{\sqst{-d^2}} + \ldots\Bi... | a look on the general form for matrix. general form diagrams contributing to {\cal &\sim \Big(A+B q^2 \ldots + (\alpha_1 + \alpha_2 e^2) \frac{1}{q^2} \\ &+\beta_1 + \beta_2 e^2\kappa^2 \frac{m}{\sqrt{-q^2}} + \ldots\Big) \end{split}$$ where $A, B, \ldots$ correspond to local analytical interactions and $\alpha_1, \alp... | a look on the general form for tHe scatteriNg matRix.
the GeNeraL forM for diagrams coNTribUting to the scattering maTrix iS $$\bEGin{sPLiT}
{\cal M} &\Sim \Big(A+b Q^2 + \lDOTs + (\aLpHa_1 \KapPa^2 + \ALpHa_2 e^2) \frAc{1}{q^2} \\
&+\Beta_1 e^2\kaPpa^2\ln(-q^2) + \beta_2 E^2\kaPpA^2 \frac{m}{\sqrt{-q^2}} + \lDOtS\Bi... | a look on the general for m for thescatt eri ngma trix .
T he general for m for diagrams contributing to t he scat t er ing m atrix i s $ $ \ beg in {s pli t} {\ cal M } & \sim \B ig(A+B q^2 +\l dots + (\alp h a_ 1 \kappa^2 +\alpha_2 e^2 ) \ frac{1 }{ q^2 } \\
& +\b eta_1 e^2\k a ppa^2\ ln(-q^2)+\ beta_2 e^2\... | a_look on_the general form for_the scattering_matrix.
The_general form_for_diagrams contributing to_the scattering matrix_is $$\begin{split}
{\cal M} &\sim_\Big(A+B q^2 +_\ldots_+ (\alpha_1 \kappa^2 + \alpha_2 e^2) \frac{1}{q^2} \\
&+\beta_1 e^2\kappa^2\ln(-q^2) + \beta_2 e^2\kappa^2 \frac{m}{\sqrt{-q^2}} +_\ldots\Bi... |
$. For $k=1$, formula (\[eq:pieruno\]) is trivially true. Let us prove it directly for $k=2$. For each $h\geq 0$, let us split sum (\[eq:prepier\]) as: $$\begin{aligned}
D_h(\epsilon^{i_1}\wedge\epsilon^{i_2})=\sum_{h_1+h_2=h}\epsilon^{i_1+h_1}\wedge\ep^{i_2+h_2}
={\cal P}+\overline{\cal P}.\label{eq:piercorr1}\end{ali... | $. For $ k=1 $, formula (\[eq: pieruno\ ]) is trivially true. Let us prove it immediately for $ k=2$. For each $ h\geq 0 $, lease us split sum (\[eq: prepier\ ]) as: $ $ \begin{aligned }
D_h(\epsilon^{i_1}\wedge\epsilon^{i_2})=\sum_{h_1+h_2 = h}\epsilon^{i_1+h_1}\wedge\ep^{i_2+h_2 }
= { \cal P}+\overline{\cal P... | $. Fog $k=1$, formula (\[eq:pieruno\]) is urivially true. Lej ys prote it djrectly wor $k=2$. For each $h\geq 0$, let us dpoit sym (\[eq:prepier\]) as: $$\begin{alkgned}
D_h(\epdilon^{i_1}\weege\eksilon^{i_2})=\sum_{h_1+h_2=h}\epsilon^{i_1+h_1}\wednz\ep^{i_2+h_2}
={\dwl P}+\mterline{\cal P}.\labgl{eq:piercorr1}\and{ali... | $. For $k=1$, formula (\[eq:pieruno\]) is trivially us it directly $k=2$. For each sum as: $$\begin{aligned} D_h(\epsilon^{i_1}\wedge\epsilon^{i_2})=\sum_{h_1+h_2=h}\epsilon^{i_1+h_1}\wedge\ep^{i_2+h_2} P}+\overline{\cal P}.\label{eq:piercorr1}\end{aligned}$$ where P}=\sum_{\matrix{_{i_1+h_1< i_2}\cr _{h_1+h_2=h}}}\ep^... | $. For $k=1$, formula (\[eq:pieruno\]) is triVially true. let us ProVe iT dIrecTly fOr $k=2$. For each $h\geq 0$, LEt us Split sum (\[eq:prepier\]) as: $$\begIn{aliGnED}
D_h(\ePSiLon^{i_1}\wEdge\epsILoN^{I_2})=\Sum_{H_1+h_2=H}\ePsiLoN^{I_1+h_1}\Wedge\Ep^{i_2+H_2}
={\cal P}+\ovErline{\cal P}.\LabEl{Eq:piercorr1}\enD{AlI... | $. For $k=1$, formula (\[e q:pieruno\ ]) is tr ivi al ly t rue. Let us provei t di rectly for $k=2$. Foreach$h \ geq0 $, letus spli t s u m (\ [e q: pre pi e r\ ]) as : $ $\begin {aligned}D_h (\ epsilon^{i_1 } \w edge\epsil on^ {i_2})=\sum_ {h_ 1+h_2= h} \ep s ilon^ {i_ 1+h_1 }\wedg e \ep^{i _2+h_2}
= {\ c al P}+ ... | $. For_$k=1$, formula (\[eq:pieruno\])_is trivially true. Let_us prove_it_directly for_$k=2$._For each $h\geq_0$, let us_split sum (\[eq:prepier\]) as: $$\begin{aligned}
D_h(\epsilon^{i_1}\wedge\epsilon^{i_2})=\sum_{h_1+h_2=h}\epsilon^{i_1+h_1}\wedge\ep^{i_2+h_2}
={\cal_P}+\overline{\cal P}.\label{eq:piercorr1}\end{ali... |
Tutukov]{}, A. V. 1984,, 54, 335
, A. & [Vennes]{}, S. 2009,, 506, L25
, M., [Brown]{}, W. R., [Allende Prieto]{}, C., & [Kenyon]{}, S. J. 2010,, 716, 122
, H. A., [Charbonneau]{}, D., [Noyes]{}, R. W., [Brown]{}, T. M., & [Gilliland]{}, R. L. 2007,, 655, 564
, S. R. & [van Kerkwijk]{}, M. H. 2010,, in press, arXiv... | Tutukov ] { }, A. V. 1984, , 54, 335
, A. & [ Vennes ] { }, S. 2009, , 506, L25
, M., [ Brown ] { }, W. R., [ Allende Prieto ] { }, C., & [ Kenyon ] { }, S. J. 2010, , 716, 122
, H. A., [ Charbonneau ] { }, D., [ Noyes ] { }, R. W., [ Brown ] { }, T. M., & [ Gilliland ] { }, R. L. 2007, , 655, 564
... | Tutkkov]{}, A. V. 1984,, 54, 335
, A. & [Vennes]{}, S. 2009,, 506, L25
, M., [Brown]{}, W. R., [Allende 'rieto]{}, D., & [Kenyov]{}, S. J. 2010,, 716, 122
, H. A., [Charbonneau]{}, D., [Noyxs]{}, R. Q., [Broqn]{}, T. M., & [Gilliland]{}, R. L. 2007,, 655, 564
, S. R. & [van Nerkwijk]{}, M. H. 2010,, un press, acSiv... | Tutukov]{}, A. V. 1984,, 54, 335 , [Vennes]{}, 2009,, 506, , M., [Brown]{}, & S. J. 2010,, 122 , H. [Charbonneau]{}, D., [Noyes]{}, R. W., [Brown]{}, M., & [Gilliland]{}, R. L. 2007,, 655, 564 , S. R. & [van M. H. 2010,, in press, arXiv:1003.2169 , C., [Camilo]{}, F., [Wex]{}, N., [Kramer]{}, [Backer]{}, C., A. & [Doro... | Tutukov]{}, A. V. 1984,, 54, 335
, A. & [Vennes]{}, S. 2009,, 506, L25
, M., [Brown]{}, W. r., [Allende PrIeto]{}, C., & [kenYon]{}, s. J. 2010,, 716, 122
, h. A., [ChArboNneau]{}, D., [Noyes]{}, R. W., [BROwn]{}, T. m., & [Gilliland]{}, R. L. 2007,, 655, 564
, S. R. & [van KerkwIjk]{}, M. H. 2010,, In PRess, ARXIv... | Tutukov]{}, A. V. 1984,, 5 4, 335
,A. &[Ve nne s] {},S. 2 009,, 506, L25
, M ., [Brown]{}, W. R., [ Allen de Prie t o] {}, C ., & [K e ny o n ]{} ,S. J. 2 0 10 ,, 71 6,122
,H. A., [Ch arb on neau]{}, D., [N oyes]{}, R . W ., [Brown]{} , T . M.,&[Gi l lilan d]{ }, R. L. 20 0 7,, 65 5, 564
, S . R. &[ van Ker k ... | Tutukov]{}, A. V._1984,, 54,_335
, A. & [Vennes]{},_S. 2009,,_506,_L25
, M.,_[Brown]{},_W. R., [Allende Prieto]{},_C., & [Kenyon]{},_S. J. 2010,, 716, 122
,_H. A., [Charbonneau]{}, D.,_[Noyes]{},_R. W., [Brown]{}, T. M., & [Gilliland]{}, R. L. 2007,, 655, 564
, S. R. & [van Kerkwijk]{}, M. H._2010,,_in press,_arXiv... |
steps that were done to the invention of SUSY in Kharkov team headed by D.Volkov.
This paper is dedicated to the memory of Prof. Yu. Gol’fand, whose ideas of SUSY inspired the most active developments in High Energy Physics over thirty years.
author:
- 'V. Akulov [^1]'
title: 'Non-linear way to Supersymmetry and ... | steps that were done to the invention of SUSY in Kharkov team headed by D.Volkov.
This paper is dedicate to the memory of Prof. Yu. Gol’fand, whose idea of SUSY inspired the most active developments in High Energy Physics over thirty long time.
author:
-' V. Akulov [ ^1 ]'
title:' Non - analogue way to Sup... | stfps that were done to tht invention of SUSY in Kiarkov feam heaaed by D.Volkov.
This paper iw deducated to the memory ow Prof. Yu. Gol’fand, whoww ideas of SUSY inspired bhe mmwt active devekopments it High Energy [hhsncs over thirty years.
author:
- 'V. Akulov [^1]'
title: 'Moj-linear way to Suptrsrmmefgy and ... | steps that were done to the invention in team headed D.Volkov. This paper of Yu. Gol’fand, whose of SUSY inspired most active developments in High Energy over thirty years. author: - 'V. Akulov [^1]' title: 'Non-linear way to Supersymmetry N-extended SUSY [^2] ' --- [*”Geometry of space is associated with mathematical ... | steps that were done to the invEntion of SUsY in KHarKov TeAm heAded By D.Volkov.
This pAPer iS dedicated to the memory oF Prof. yu. gOl’faND, wHose iDeas of SusY INSpiReD tHe mOsT AcTive dEveLopmentS in High EneRgy phYsics over thiRTy Years.
authoR:
- 'V. AKulov [^1]'
title: 'NoN-liNear waY tO SuPErsymMetRy and ... | steps that were done to t he inventi on of SU SYin Kha rkov team headed b y D.V olkov.
This paper is d ed i cate d t o the memory of P rof .Yu . G ol ’ fa nd, w hos e ideas of SUSY i nsp ir ed the mosta ct ive develo pme nts in HighEne rgy Ph ys ics overthi rty y ears.a uthor:
- 'V. Ak ul o v [^1] '
tit... | steps_that were_done to the invention_of SUSY_in_Kharkov team_headed_by D.Volkov.
_ This_paper is dedicated to_the memory of_Prof._Yu. Gol’fand, whose ideas of SUSY inspired the most active developments in High Energy_Physics_over thirty_years.
author:
-_'V._Akulov [^1]'
title: 'Non-linear way to_Supersymmetry and ... |
y CDGAs
--------------------------------------------------
Given a DG Artin $N$-hypergroupoid $X$, and an $O(X)$-module $M$, we may pull $M$ back along the unit $\eta \co O(X) \to DD^*O(X)$ of the adjunction $D^*\dashv D$, and then apply Lemma \[denormmod\] to obtain a $D^*O(X)$-module $N_c\eta^*M$.
As for instance i... | y CDGAs
--------------------------------------------------
Given a DG Artin $ N$-hypergroupoid $ X$, and an $ O(X)$-module $ M$, we may pull $ M$ back along the unit $ \eta \co O(X) \to DD^*O(X)$ of the adjunction $ D^*\dashv D$, and then lend oneself Lemma \[denormmod\ ] to receive a $ D^*O(X)$-module $ N_c\eta^*... | y CFGAs
--------------------------------------------------
Given a DG Artin $N$-hykergroupoid $X$, and an $O(X)$-mmdule $J$, we may pull $M$ back along the unit $\xta \xo O(X) \to DD^*O(X)$ of the adjuncgion $D^*\dasjv D$, and thei apply Lemma \[deikrmmod\] bj obfwin c $V^*O(X)$-module $N_c\eta^*K$.
As for invtance i... | y CDGAs -------------------------------------------------- Given a DG Artin and $O(X)$-module $M$, may pull $M$ \co \to DD^*O(X)$ of adjunction $D^*\dashv D$, then apply Lemma \[denormmod\] to obtain $D^*O(X)$-module $N_c\eta^*M$. As for instance in [@stacks2 Definition \[stacks2-delta\*\]], define almost cosimplicial ... | y CDGAs
--------------------------------------------------
Given a DG Artin $N$-hyperGroupoid $X$, aNd an $O(x)$-moDulE $M$, We maY pulL $M$ back along the UNit $\eTa \co O(X) \to DD^*O(X)$ of the adjunCtion $d^*\dAShv D$, ANd Then aPply LemMA \[dENOrmMoD\] tO obTaIN a $d^*O(X)$-moDulE $N_c\eta^*M$.
as for instaNce I... | y CDGAs
------------------ ---------- ----- --- --- -- ---- ---- -
Given a DGA rtin $N$-hypergroupoid $X$ , and a n $O( X )$ -modu le $M$, we m aypu ll $M $b ac k alo ngthe uni t $\eta \c o O (X ) \to DD^*O( X )$ of the ad jun ction $D^*\d ash v D$,an d t h en ap ply Lemm a \[de n ormmod \] to obt ai n a $D^ * ... | y CDGAs
--------------------------------------------------
Given_a DG_Artin $N$-hypergroupoid $X$, and_an $O(X)$-module_$M$,_we may_pull_$M$ back along_the unit $\eta_\co O(X) \to DD^*O(X)$_of the adjunction_$D^*\dashv_D$, and then apply Lemma \[denormmod\] to obtain a $D^*O(X)$-module $N_c\eta^*M$.
As for instance i... |
a tuple of multiplicities $m = (m_j)$ and let $\Omega_m
\subset {\mathrm{Hom}}$ be the closure of the corresponding $GL$-orbit.
Based on $m$ we define a parabolic $P_m\subset GL$ and a linear subspace $Z_m\subset{\mathrm{Hom}}$ as follows. For each vertex $i\in Q_0$ we divide the sets of row and column indices of $GL... | a tuple of multiplicities $ m = (m_j)$ and let $ \Omega_m
\subset { \mathrm{Hom}}$ be the closure of the corresponding $ GL$-orbit.
free-base on $ m$ we specify a parabolic $ P_m\subset GL$ and a linear subspace $ Z_m\subset{\mathrm{Hom}}$ as follows. For each vertex $ i\in Q_0 $ we divide the set of row and colum... | a huple of multiplicities $o = (m_j)$ and let $\Omega_m
\suuset {\mafhrm{Hom}}$ ce the closure of the corres'ondung $GO$-orbit.
Based on $m$ we dewine a pagabolic $P_n\subwwt GL$ and e linear subspzge $Z_m\vnbset{\mathrm{Hom}}$ ss follows. For each vertax $i\nn Q_0$ we divide the sets of row and cjlumn imdlces of $GL... | a tuple of multiplicities $m = (m_j)$ $\Omega_m {\mathrm{Hom}}$ be closure of the we a parabolic $P_m\subset and a linear $Z_m\subset{\mathrm{Hom}}$ as follows. For each vertex Q_0$ we divide the sets of row and column indices of $GL({\mathbb{C}}^{d(i)})$ into subsets of sizes $d_j(i)$ as $j$ runs from $1$ to $N$. This... | a tuple of multiplicities $m = (m_j)$ And let $\OmegA_m
\subSet {\MatHrM{Hom}}$ Be thE closure of the cORresPonding $GL$-orbit.
Based on $m$ We defInE A parABoLic $P_m\Subset Gl$ AnD A LinEaR sUbsPaCE $Z_M\subsEt{\mAthrm{HoM}}$ as follows. for EaCh vertex $i\in Q_0$ WE dIvide the seTs oF row and columN inDices oF $Gl... | a tuple of multiplicities $m = (m_j )$ an d l et$\ Omeg a_m\subset {\math r m{Ho m}}$ be the closure of theco r resp o nd ing $ GL$-orb i t. Bas ed o n $ m$ we defi nea parab olic $P_m\ sub se t GL$ and al in ear subspa ce$Z_m\subset{ \ma thrm{H om }}$ as fo llo ws. F or eac h verte x $i\in Q _0 $ we di v ide th... | a_tuple of_multiplicities $m = (m_j)$_and let_$\Omega_m
\subset_{\mathrm{Hom}}$ be_the_closure of the_corresponding $GL$-orbit.
Based on_$m$ we define a_parabolic $P_m\subset GL$_and_a linear subspace $Z_m\subset{\mathrm{Hom}}$ as follows. For each vertex $i\in Q_0$ we divide the_sets_of row_and_column_indices of $GL... |
{x}}^T {\bm{s}}$. If $d({\bm{x}}, {\bm{s}}, \mu) \leq \eta \mu$, then $(1+\eta) \norm{{\bm{s}}^{-1}}_2 \geq \norm{\mu^{-1} {\bm{x}}}_2$.
For part 1, let $\{\lambda_i\}_{i=1}^{2r}$ be the eigenvalues of $T_{{\bm{x}}}{\bm{s}}$, note that $T_{x}$ is invertible as $x \in {\mathcal{L}}$. Then using part 1 of Claim \[claim:... | { x}}^T { \bm{s}}$. If $ d({\bm{x } }, { \bm{s } }, \mu) \leq \eta \mu$, then $ (1+\eta) \norm{{\bm{s}}^{-1}}_2 \geq \norm{\mu^{-1 } { \bm{x}}}_2$.
For part 1, let $ \{\lambda_i\}_{i=1}^{2r}$ be the eigenvalues of $ T_{{\bm{x}}}{\bm{s}}$, note that $ T_{x}$ is invertible as $ x \in { \mathcal{L}}$. Then use characte... | {x}}^T {\hm{s}}$. If $d({\bm{x}}, {\bm{s}}, \mu) \leq \eua \mu$, then $(1+\eta) \notm{{\vm{s}}^{-1}}_2 \gex \norm{\ju^{-1} {\bm{x}}}_2$.
Fof part 1, let $\{\lambda_i\}_{i=1}^{2r}$ be the eugenvqlues of $T_{{\bm{x}}}{\bm{s}}$, note ghat $T_{x}$ id invertuble qs $x \in {\mavgcal{L}}$. Tmzn usjkg paxt 1 of Claim \[clalm:... | {x}}^T {\bm{s}}$. If $d({\bm{x}}, {\bm{s}}, \mu) \leq then \norm{{\bm{s}}^{-1}}_2 \geq {\bm{x}}}_2$. For part eigenvalues $T_{{\bm{x}}}{\bm{s}}$, note that is invertible as \in {\mathcal{L}}$. Then using part 1 Claim \[claim:Properties of Qx\], we have $${\bm{x}}^T{\bm{s}} = {\bm{x}}^T T_{{\bm{x}}}^{-1} T_{{\bm{x}}} {\... | {x}}^T {\bm{s}}$. If $d({\bm{x}}, {\bm{s}}, \mu) \leq \eta \mu$, tHen $(1+\eta) \norm{{\Bm{s}}^{-1}}_2 \geQ \noRm{\mU^{-1} {\bM{x}}}_2$.
FoR parT 1, let $\{\lambda_i\}_{i=1}^{2r}$ bE The eIgenvalues of $T_{{\bm{x}}}{\bm{s}}$, notE that $t_{x}$ IS invERtIble aS $x \in {\matHCaL{l}}$. theN uSiNg pArT 1 Of claim \[ClaIm:... | {x}}^T {\bm{s}}$. If $d({\ bm{x}}, {\ bm{s} },\mu )\leq \et a \mu$, then $ ( 1+\e ta) \norm{{\bm{s}}^{-1 }}_2\g e q \n o rm {\mu^ {-1} {\ b m{ x } }}_ 2$ .
Fo rp ar t 1,let $\{\la mbda_i\}_{ i=1 }^ {2r}$ be the ei genvaluesof$T_{{\bm{x}} }{\ bm{s}} $, no t e tha t $ T_{x} $ is i n vertib le as $x\i n {\mat h cal{L}... | {x}}^T {\bm{s}}$._If $d({\bm{x}},_{\bm{s}}, \mu) \leq \eta_\mu$, then_$(1+\eta)_\norm{{\bm{s}}^{-1}}_2 \geq_\norm{\mu^{-1}_{\bm{x}}}_2$.
For part 1,_let $\{\lambda_i\}_{i=1}^{2r}$ be_the eigenvalues of $T_{{\bm{x}}}{\bm{s}}$,_note that $T_{x}$_is_invertible as $x \in {\mathcal{L}}$. Then using part 1 of Claim \[claim:... |
profile is parameterized by an extension of the Hernquist model [@HaloProfile2] $$\rho_{\mathrm{DM}} (r) =
\frac{\rho_0}
{\left ( \delta + {r \over r_s} \right )^\gamma \cdot
\left [ 1 +
\left ( {r \over r_s} \right )^\alpha
\right ]^{ (\beta - \gamma)/... | profile is parameterized by an extension of the Hernquist model [ @HaloProfile2 ] $ $ \rho_{\mathrm{DM } } (r) =
\frac{\rho_0 }
{ \left (\delta + { r \over r_s } \right) ^\gamma \cdot
\left [ 1 +
\left ({ radius \over r_s } \right) ^\alpha
\ri... | prlfile is parameterized bn an extension oy the Hxrnquisf model [@HxloProfile2] $$\rho_{\mathrm{DM}} (r) =
\frqc{\rho_0}
{\left ( \delta + {g \over r_s} \rigit )^\gamma \cdot
\leff [ 1 +
\keft ( {r \ovar r_s} \right )^\al[hx
\right ]^{ (\beta - \gamma)/... | profile is parameterized by an extension of model $$\rho_{\mathrm{DM}} (r) \frac{\rho_0} {\left ( \right \cdot \left [ + \left ( \over r_s} \right )^\alpha \right ]^{ - \gamma)/\alpha } }, \label{eq:rho_profile}$$ where $(\alpha, \beta, \gamma, \delta)$ are dimensionless parameters. is a scaling radius and $\rho_0$ is ... | profile is parameterized by aN extension Of the herNquIsT modEl [@HaLoProfile2] $$\rho_{\maTHrm{Dm}} (r) =
\frac{\rho_0}
{\left ( \delta + {r \oveR r_s} \riGhT )^\GammA \CdOt
\lefT [ 1 +
\left ( {r \oVEr R_S} \RigHt )^\AlPha
\RiGHt ]^{ (\Beta - \gAmmA)/... | profile is parameterizedby an exte nsion of th eHern quis t model [@Halo P rofi le2] $$\rho_{\mathrm{D M}} ( r) =
\f rac{\rh o _0 } {\ left( \ delta + {r \overr_s }\right )^\ga m ma \cdot
\lef t [ 1 +
\lef t ( {r \overr_s} \rig ht )^\alp h a
\right ]^{ (\bet a - \gamma... | profile_is parameterized_by an extension of_the Hernquist_model [@HaloProfile2]_$$\rho_{\mathrm{DM}} (r)_=
_ _ _ \frac{\rho_0}
_ __ {\left ( \delta + {r \over r_s} \right_)^\gamma_\cdot
___ _ _\left [_1 +
__ _ \left ( {r \over_r_s} \right )^\alpha
_ ___ _ \right ]^{ (\beta -_\gamma)/... |
$, $\chi_3$.. are nonlinear susceptibilities. For an atomic spin-glass, $M_0$ and the coefficients of even power of $H$ i.e $\chi_2$, $\chi_4$.. are zero while coefficients of the odd power of $H$ i.e $\chi_3$, $\chi_5$.. diverges as $T$ approaches $T_g$ in the critical regime.[@Suzuki; @Wada; @Fujiki] For cluster-glas... | $, $ \chi_3 $.. are nonlinear susceptibilities. For an atomic spin - glass, $ M_0 $ and the coefficient of even exponent of $ H$ i.e $ \chi_2 $, $ \chi_4 $.. are zero while coefficients of the odd exponent of $ H$ i.e $ \chi_3 $, $ \chi_5 $.. diverge as $ T$ approaches $ T_g$ in the critical regime.[@Suzuki; @Wada; @Fu... | $, $\chl_3$.. are nonlinear susceptinilities. For an cromic vpin-glzss, $M_0$ ana the coefficients of even plwwr of $H$ i.e $\chi_2$, $\chi_4$.. are zero while covfficientw of rhe odd poxsr of $H$ i.e $\chj_3$, $\chi_5$.. viverges as $T$ akproaches $T_g$ in the criticdl rzgime.[@Suzuki; @Wada; @Fujiki] For cluster-gjas... | $, $\chi_3$.. are nonlinear susceptibilities. For an $M_0$ the coefficients even power of zero coefficients of the power of $H$ $\chi_3$, $\chi_5$.. diverges as $T$ approaches in the critical regime.[@Suzuki; @Wada; @Fujiki] For cluster-glass, if the nonlinear response of ferromagnetic clusters is small, coefficients o... | $, $\chi_3$.. are nonlinear susceptibiLities. For aN atomIc sPin-GlAss, $M_0$ And tHe coefficients OF eveN power of $H$ i.e $\chi_2$, $\chi_4$.. are zeRo whiLe COeffICiEnts oF the odd POwER Of $H$ I.e $\ChI_3$, $\chI_5$.. dIVeRges aS $T$ aPproachEs $T_g$ in the cRitIcAl regime.[@SuzuKI; @WAda; @Fujiki] FOr cLuster-glas... | $, $\chi_3$.. are nonlinea r suscepti bilit ies . F or anatom ic spin-glass, $M_0 $ and the coefficients of e ve n pow e rof $H $ i.e $ \ ch i _ 2$, $ \c hi_ 4$ . .are z ero whilecoefficien tsof the odd pow e rof $H$ i.e $\ chi_3$, $\ch i_5 $.. di ve rge s as $ T$appro aches$ T_g$ i n the cri ti c al reg i me.[... | $, $\chi_3$.._are nonlinear_susceptibilities. For an atomic_spin-glass, $M_0$_and_the coefficients_of_even power of_$H$ i.e $\chi_2$,_$\chi_4$.. are zero while_coefficients of the_odd_power of $H$ i.e $\chi_3$, $\chi_5$.. diverges as $T$ approaches $T_g$ in the critical_regime.[@Suzuki;_@Wada; @Fujiki]_For_cluster-glas... |
C_0^2\|\nabla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-A_0)\|_{L^d(X)}
\\
&\leq 2C_0^2\|u_0\|_{W_{A_0}^{2,p}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-A_0)\|_{L^d(X)}.\end{aligned}$$ Third, for the case $d<p<\infty$, $$\begin{aligned}
\|\nabla_{A_0}(u^{-1}(A-A_0)u)\|_{L^p(X)... | C_0 ^ 2\|\nabla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A - A_0\|_{W_{A_0}^{1,d}(X) }
+ \|\nabla_{A_0}(A - A_0)\|_{L^d(X) }
\\
& \leq 2C_0 ^ 2\|u_0\|_{W_{A_0}^{2,p}(X)}\|A - A_0\|_{W_{A_0}^{1,d}(X) }
+ \|\nabla_{A_0}(A - A_0)\|_{L^d(X)}.\end{aligned}$$ Third, for the case $ d < p<\infty$, $ $ \begin{aligned }
\|\nabla... | C_0^2\|\nahla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-A_0)\|_{U^d(X)}
\\
&\leq 2C_0^2\|u_0\|_{W_{A_0}^{2,p}(X)}\|A-A_0\|_{C_{Q_0}^{1,d}(X)}
+ \|\naula_{A_0}(A-A_0)\|_{L^s(X)}.\end{aliened}$$ Third, for the case $d<p<\inhty$, $$\vegin{qligned}
\|\nabla_{A_0}(u^{-1}(A-A_0)u)\|_{L^p(X)... | C_0^2\|\nabla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)} + \|\nabla_{A_0}(A-A_0)\|_{L^d(X)} \\ &\leq 2C_0^2\|u_0\|_{W_{A_0}^{2,p}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)} + for case $d<p<\infty$, \|\nabla_{A_0}(u^{-1}(A-A_0)u)\|_{L^p(X)} &\leq 2\|\nabla_{A_0}u_0\|_{L^\infty(X)}\|A-A_0\|_{L^p(X)} + \\ &\leq 2C_0\|... | C_0^2\|\nabla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-a_0)\|_{L^d(X)}
\\
&\leq 2C_0^2\|u_0\|_{W_{a_0}^{2,p}(X)}\|A-A_0\|_{w_{A_0}^{1,d}(x)}
+ \|\naBlA_{A_0}(A-A_0)\|_{l^d(X)}.\eNd{aligned}$$ Third, FOr thE case $d<p<\infty$, $$\begin{alignEd}
\|\nabLa_{a_0}(U^{-1}(A-A_0)u)\|_{l^P(X)... | C_0^2\|\nabla_{A_0}u_0\|_{ W_{A_0}^{1 ,d}(X )}\ |A- A_ 0\|_ {W_{ A_0}^{1,d}(X)} + \| \nabla_{A_0}(A-A_0)\|_ {L^d( X) }
\\& \l eq 2C _0^2\|u _ 0\ | _ {W_ {A _0 }^{ 2, p }( X)}\| A-A _0\|_{W _{A_0}^{1, d}( X) }
+ \|\nabla _ {A _0}(A-A_0) \|_ {L^d(X)}.\en d{a ligned }$ $ T h ird,for thecase $ d <p<\in fty$, $$\ be g in{... | C_0^2\|\nabla_{A_0}u_0\|_{W_{A_0}^{1,d}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-A_0)\|_{L^d(X)}
\\
&\leq_2C_0^2\|u_0\|_{W_{A_0}^{2,p}(X)}\|A-A_0\|_{W_{A_0}^{1,d}(X)}
+ \|\nabla_{A_0}(A-A_0)\|_{L^d(X)}.\end{aligned}$$_Third, for the case_$d<p<\infty$, $$\begin{aligned}
\|\nabla_{A_0}(u^{-1}(A-A_0)u)\|_{L^p(X)... |
+1 dimensions and show that the massive vector field in the Proca theory is canonically spin 1.
Consider the following Proca Lagrangian \[Apl\] [L]{}\_P= -F\_F\^+ m\^2 [A]{}\_\^2. After the canonical quantization of the theory by introducing Dirac bracket, we obtain the following commutation relations: \[Acr\] && \[A\... | +1 dimensions and show that the massive vector field in the Proca theory is canonically spin 1.
think the take after Proca Lagrangian \[Apl\ ] [ L]{}\_P= -F\_F\^+ m\^2 [ A]{}\_\^2. After the canonical quantization of the theory by introducing Dirac bracket, we prevail the following commutation relation: \[Acr\ ] & &... | +1 dilensions and show that tme massive vectot dield mn the Lroca thdory is canonically spin 1.
Condieer tye following Proca Lagfangian \[Ail\] [L]{}\_P= -F\_F\^+ n\^2 [A]{}\_\^2. Efter the canonirzl quanbnzatikk of chx theory by inttoducing Dirdc bracket, we mbgann the following commutation relatiogs: \[Acr\] && \[A\... | +1 dimensions and show that the massive in Proca theory canonically spin 1. \[Apl\] -F\_F\^+ m\^2 [A]{}\_\^2. the canonical quantization the theory by introducing Dirac bracket, obtain the following commutation relations: \[Acr\] && \[A\_0([**x**]{}), \_0([**y**]{})\] =0,\ && \[A\_i([**x**]{}), \^j([**y**]{})\] - [**y*... | +1 dimensions and show that the mAssive vectOr fieLd iN thE PRoca TheoRy is canonicallY Spin 1.
consider the following PrOca LaGrANgiaN \[apL\] [L]{}\_P= -F\_F\^+ M\^2 [A]{}\_\^2. After THe CANonIcAl QuaNtIZaTion oF thE theory By introducIng diRac bracket, we OBtAin the follOwiNg commutatioN reLationS: \[ACr\] && \[A\... | +1 dimensions and show tha t the mass ive v ect orfi eldin t he Proca theor y iscanonically spin 1.
C onsid er thef ol lowin g Proca La g r ang ia n\[A pl \ ][L]{} \_P = -F\_F \^+ m\^2 [ A]{ }\ _\^2. Aftert he canonical qu antization o f t he the or y b y intr odu cingDiracb racket , we obta in the fo l lowingc o ... | +1 dimensions_and show_that the massive vector_field in_the_Proca theory_is_canonically spin 1.
Consider_the following Proca_Lagrangian \[Apl\] [L]{}\_P= -F\_F\^+_m\^2 [A]{}\_\^2. After_the_canonical quantization of the theory by introducing Dirac bracket, we obtain the following commutation_relations:_\[Acr\] &&_\[A\... |
f$ to denote its isotropic part. Lemma \[pd1\] and Proposition \[conve\] supply a function $g$ so that $$g(s,t)=\sum_{k,l=0}^{\infty}\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \quad t,s \in [-1,1],$$ with uniform convergence in $[-1,1]^2$. In particular, $g$ is continuous in $[-1,1]^2$. On the other hand, the same uniform ... | f$ to denote its isotropic part. Lemma \[pd1\ ] and Proposition \[conve\ ] supply a function $ g$ so that $ $ g(s, t)=\sum_{k, l=0}^{\infty}\hat{f}_{k, l}P_{k}^{m}(t)P_{l}^{M}(s), \quad metric ton, s \in [ -1,1],$$ with consistent convergence in $ [ -1,1]^2$. In particular, $ g$ is continuous in $ [ -1,1]^2$. On ... | f$ tl denote its isotropic pxrt. Lemma \[pd1\] and Proposmtion \[cknve\] supoly a function $g$ so that $$g(s,t)=\dun_{k,l=0}^{\indty}\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \quad t,s \in [-1,1],$$ wpth unifoem cibvergence mh $[-1,1]^2$. In pavciculzv, $g$ iv continuous in $[-1,1]^2$. On the otver hand, the sdmd bniform ... | f$ to denote its isotropic part. Lemma Proposition supply a $g$ so that with convergence in $[-1,1]^2$. particular, $g$ is in $[-1,1]^2$. On the other hand, same uniform convergence and the orthogonality relation mentioned at the beginning of the imply that $$\hat{f}_{k,l} -\hat{g}_{k,l}=0, \quad k,l \in \mathbb{Z}_+.$... | f$ to denote its isotropic part. lemma \[pd1\] and propoSitIon \[CoNve\] sUpplY a function $g$ so tHAt $$g(s,T)=\sum_{k,l=0}^{\infty}\hat{f}_{k,l}P_{k}^{m}(t)P_{L}^{M}(s), \quAd T,S \in [-1,1],$$ wITh UnifoRm conveRGeNCE in $[-1,1]^2$. in PaRtiCuLAr, $G$ is coNtiNuous in $[-1,1]^2$. on the other HanD, tHe same uniforM ... | f$ to denote its isotropic part. Lem ma \[ pd1 \]an d Pr opos ition \[conve\ ] sup ply a function $g$ sothat$$ g (s,t ) =\ sum_{ k,l=0}^ { \i n f ty} \h at {f} _{ k ,l }P_{k }^{ m}(t)P_ {l}^{M}(s) , \ qu ad t,s \in [ - 1, 1],$$ with un iform conver gen ce in$[ -1, 1 ]^2$. In part icular , $g$ i s continu ou s in... | f$ to_denote its_isotropic part. Lemma \[pd1\] and_Proposition \[conve\]_supply_a function_$g$_so that $$g(s,t)=\sum_{k,l=0}^{\infty}\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s),_\quad t,s \in_[-1,1],$$ with uniform convergence_in $[-1,1]^2$. In particular,_$g$_is continuous in $[-1,1]^2$. On the other hand, the same uniform ... |
aux, E., Bouvier, J., Stauffer, J. R., Cuillandre, J.-C. 2003, A&A, 400, 891 Muench, A. A., Lada, E. A., Lada, C. J., & Alves, J. 2002, ApJ, 573, 366 Nakajima, T., Oppenheimer, B. R., Kulkarni, S. R., Golimowski, D. A., Matthews, K., Durrance, S. T. 1995, Nature, 378, 463 Pinfield, D. J., Jameson, R. F., & Hodgkin, S.... | aux, E., Bouvier, J., Stauffer, J. R., Cuillandre, J.-C. 2003, A&A, 400, 891 Muench, A. A., Lada, E. A., Lada, C. J., & Alves, J. 2002, ApJ, 573, 366 Nakajima, T., Oppenheimer, B. R., Kulkarni, S. R., Golimowski, D. A., Matthews, K., Durrance, S. T. 1995, Nature, 378, 463 Pinfield, D. J., James... | aux, E., Bouvier, J., Stauffer, J. R., Cuillandre, J.-C. 2003, A&A, 400, 891 Mnench, A. Z., Lada, E. X., Lada, C. J., & Alves, J. 2002, ApJ, 573, 366 Nakejimq, T., Okienheimer, B. R., Kulkarni, S. R., Golimlwski, D. A., Matuhews, K., Durrance, S. T. 1995, Naturc, 378, 463 Pjkfielb, V. J., Jameson, R. F., & Modgkin, S.... | aux, E., Bouvier, J., Stauffer, J. R., 2003, 400, 891 A. A., Lada, & J. 2002, ApJ, 366 Nakajima, T., B. R., Kulkarni, S. R., Golimowski, A., Matthews, K., Durrance, S. T. 1995, Nature, 378, 463 Pinfield, D. J., R. F., & Hodgkin, S. T. 1998, MNRAS, 299, 955 Pinfield, D. J., S. Jameson, F., M. R., Hambly, N. C., & Devere... | aux, E., Bouvier, J., Stauffer, J. R., CuiLlandre, J.-C. 2003, A&a, 400, 891 MuenCh, A. a., LaDa, e. A., LaDa, C. J., & alves, J. 2002, ApJ, 573, 366 NakajIMa, T., OPpenheimer, B. R., Kulkarni, S. R., golimOwSKi, D. A., mAtThews, k., DurranCE, S. t. 1995, nAtuRe, 378, 463 piNfiElD, d. J., jamesOn, R. f., & HodgkiN, S.... | aux, E., Bouvier, J., Stau ffer, J. R ., Cu ill and re , J. -C. 2003, A&A, 40 0 , 89 1 Muench, A. A., Lada, E. A ., Lada , C . J., & Alve s ,J . 20 02 ,ApJ ,5 73 , 366 Na kajima, T., Oppen hei me r, B. R., Ku l ka rni, S. R. , G olimowski, D . A ., Mat th ews , K.,Dur rance , S. T . 1995, Nature,37 8 , 463P in... | aux, E.,_Bouvier, J.,_Stauffer, J. R., Cuillandre, J.-C. _2003, A&A,_400,_891 Muench,_A. A.,_Lada, E. A., Lada,_C. J., & Alves,_J. 2002, ApJ, 573, 366_Nakajima, T., Oppenheimer,_B. R.,_Kulkarni, S. R., Golimowski, D. A., Matthews, K., Durrance, S. T. 1995, Nature, 378, 463 Pinfield, D. J., Jameson,_R. F.,_& Hodgkin,_S.... |
d'})$ consisting of all $d$-dimensional subspaces of $\bbc^{d'}.$ Indeed, for any $I=\{i_1,\cdots, i_d\}\subset \{1, \cdots, d'\}$ and $J=\{1,\cdots, d\}$, let $M^g_{d'\times d}(I)$ be the subset of $M_{d'\times d}(d, I,J)$ consisting of the matrices $A$ satisfying that $\bigtriangleup_{(I;J)}(A)$ are identity matrices... | d'})$ consisting of all $ d$-dimensional subspaces of $ \bbc^{d'}.$ Indeed, for any $ I=\{i_1,\cdots, i_d\}\subset \{1, \cdots, d'\}$ and $ J=\{1,\cdots, d\}$, let $ M^g_{d'\times d}(I)$ be the subset of $ M_{d'\times d}(d, I, J)$ consisting of the matrix $ A$ meet that $ \bigtriangleup_{(I;J)}(A)$ are identity matrice... | d'})$ clnsisting of all $d$-dimenslonal subspaces of $\bbc^{d'}.$ Indees, for anh $I=\{i_1,\cdots, i_d\}\subset \{1, \cdots, d'\}$ end $H=\{1,\cdotw, d\}$, let $M^g_{d'\times d}(I)$ be the subsvt of $M_{d'\tumes e}(d, I,J)$ consmating on the latrncxs $A$ satisfying that $\bigtsiangleup_{(I;J)}(A)$ ase ibentity matrices... | d'})$ consisting of all $d$-dimensional subspaces of for $I=\{i_1,\cdots, i_d\}\subset \cdots, d'\}$ and be subset of $M_{d'\times I,J)$ consisting of matrices $A$ satisfying that $\bigtriangleup_{(I;J)}(A)$ are matrices. Then there is a finite stratification $$\mathrm{Gr}_{d}(\bbc^{d'})=\bigsqcup_{I}M^g_{d'\times d}(I... | d'})$ consisting of all $d$-dimensioNal subspacEs of $\bBc^{d'}.$ indEeD, for Any $I=\{I_1,\cdots, i_d\}\subset \{1, \CDots, D'\}$ and $J=\{1,\cdots, d\}$, let $M^g_{d'\times D}(I)$ be tHe SUbseT Of $m_{d'\timEs d}(d, I,J)$ cONsISTinG oF tHe mAtRIcEs $A$ saTisFying thAt $\bigtrianGleUp_{(i;J)}(A)$ are identiTY mAtrices... | d'})$ consisting of all $d $-dimensio nal s ubs pac es of$\bb c^{d'}.$ Indee d , fo r any $I=\{i_1,\cdots, i_d\ }\ s ubse t \ {1, \ cdots,d '\ } $ an d$J =\{ 1, \ cd ots,d\} $, let$M^g_{d'\t ime sd}(I)$ be th e s ubset of $ M_{ d'\times d}( d,I,J)$co nsi s tingofthe m atrice s $A$ s atisfying t h at $\b i gtrian... | d'})$ consisting_of all_$d$-dimensional subspaces of $\bbc^{d'}.$_Indeed, for_any_$I=\{i_1,\cdots, i_d\}\subset_\{1,_\cdots, d'\}$ and_$J=\{1,\cdots, d\}$, let_$M^g_{d'\times d}(I)$ be the_subset of $M_{d'\times_d}(d,_I,J)$ consisting of the matrices $A$ satisfying that $\bigtriangleup_{(I;J)}(A)$ are identity matrices... |
.
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a. V. KnyazEv and M. E. ArgEntAtI, “MajorizatioN FoR changes in AngLes between suBspAces, RiTz V... | .
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T. K ato, *P e rt u r bat io nThe or y f or Li nea r Opera tors*, Cla ssi cs in Mathemat i cs , Springer -Ve rlag, Berlin , 1 995.
A. V. Knyaz evand M . E. A r gentat i, “Major iz a tion f o r ... | .
D. Husemoller, *Fibre_Bundles*, 3rd_Ed., Graduate Texts in_Mathematics, **20**,_Springer,_New York,_NY,_1994.
T. Kato, *Perturbation Theory_for Linear Operators*,_Classics in Mathematics, Springer-Verlag,_Berlin, 1995.
A. V. Knyazev and_M. E. Argentati,_“Majorization for changes in angles between subspaces, Ritz v... |
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C._H._Bennett, D._P._DiVincenzo,_J. A. Smolin, and W._K. Wootter... |
a commutative ring and $M$ is an $R$-module, then a prime ideal $p$ of $R$ is associated to $M$, if there is an $x\in M$ such that $p={{\mathop{\mathrm{ann\,}}}}x$. We shall make use of the following fact: if $R$ is a Noetherian ring and $M$ is a finite $R$-module, then there are finitely many primes associated to $M$... | a commutative ring and $ M$ is an $ R$-module, then a prime ideal $ p$ of $ R$ is associated to $ M$, if there is an $ x\in M$ such that $ p={{\mathop{\mathrm{ann\,}}}}x$. We shall seduce habit of the following fact: if $ R$ is a Noetherian ring and $ M$ is a finite $ R$-module, then there are finitely many primes cons... | a fommutative ring and $M$ ir an $R$-module, thgn a prike idezl $p$ of $F$ is associated to $M$, if therx is an $x\un M$ such that $p={{\mathop{\oathrm{ann\,}}}}q$. We shalo majw use of tis following fzgt: if $C$ is a Noetherisn ring ang $M$ is a finita $F$-mldule, then there are finitely many [rimes ssdociated to $M$... | a commutative ring and $M$ is an a ideal $p$ $R$ is associated an M$ such that We shall make of the following fact: if $R$ a Noetherian ring and $M$ is a finite $R$-module, then there are finitely primes associated to $M$, and furthermore, each zerodivisor on $M$ is contained in of see Theorem Thus, in our setting, ${{... | a commutative ring and $M$ is an $R$-Module, then A primE idEal $P$ oF $R$ is AssoCiated to $M$, if theRE is aN $x\in M$ such that $p={{\mathop{\maThrm{aNn\,}}}}X$. we shALl Make uSe of the FOlLOWinG fAcT: if $r$ iS A NOetheRiaN ring anD $M$ is a finitE $R$-mOdUle, then there ARe Finitely maNy pRimes associaTed To $M$... | a commutative ring and $M $ is an $R $-mod ule , t he n aprim e ideal $p$ of $R$is associated to $M$,if th er e isa n$x\in M$ suc h t h a t $ p= {{ \ma th o p{ \math rm{ ann\,}} }}x$. We s hal lmake use oft he following fa ct: if $R$ i s a Noeth er ian ringand $M$is a f i nite $ R$-module ,t hen th e re aref i ... | a_commutative ring_and $M$ is an_$R$-module, then_a_prime ideal_$p$_of $R$ is_associated to $M$,_if there is an_$x\in M$ such_that_$p={{\mathop{\mathrm{ann\,}}}}x$. We shall make use of the following fact: if $R$ is a Noetherian_ring_and $M$_is_a_finite $R$-module, then there are_finitely many primes associated to_$M$... |
\,0}(\phi,\theta,\psi)=P_l(\cos\theta)$, where $\phi,\theta,\psi$ are the Euler angles on $SU(2)$ (see [@ViKl:repspecfunc1 Chapter 6]). From the orthogonality relation of the Legendre polynomials, the fact that their only common value is at $P_l(1)=1$, and considering that $\theta=0$ denotes a pole of $SU(2)$, we find ... | \,0}(\phi,\theta,\psi)=P_l(\cos\theta)$, where $ \phi,\theta,\psi$ are the Euler angles on $ SU(2)$ (see [ @ViKl: repspecfunc1 Chapter 6 ]). From the orthogonality relation of the Legendre polynomials, the fact that their lone coarse value is at $ P_l(1)=1 $, and considering that $ \theta=0 $ denotes a pole of $ SU(2... | \,0}(\phi,\hheta,\psi)=P_l(\cos\theta)$, where $\phi,\theta,\psi$ arg rhe Euner anfles on $RU(2)$ (see [@ViKl:repspecfunc1 Chaptxr 6]). Feom tye orthogonality relatkon of thv Legendrw pootnomials, tis fact bkat tgcir oulb common value ls at $P_l(1)=1$, ang considering dhxt $\theta=0$ denotes a pole of $SU(2)$, we find ... | \,0}(\phi,\theta,\psi)=P_l(\cos\theta)$, where $\phi,\theta,\psi$ are the Euler angles (see Chapter 6]). the orthogonality relation fact their only common is at $P_l(1)=1$, considering that $\theta=0$ denotes a pole $SU(2)$, we find that the delta function at the identity of $SU(2)$ restricted $S^2$ can be represented ... | \,0}(\phi,\theta,\psi)=P_l(\cos\theta)$, wherE $\phi,\theta,\pSi$ are The eulEr AnglEs on $sU(2)$ (see [@ViKl:repspECfunC1 Chapter 6]). From the orthogoNalitY rELatiON oF the LEgendre POlYNOmiAlS, tHe fAcT ThAt theIr oNly commOn value is aT $P_l(1)=1$, AnD considering THaT $\theta=0$ denoTes A pole of $SU(2)$, we fInd ... | \,0}(\phi,\theta,\psi)=P_l (\cos\thet a)$,whe re$\ phi, \the ta,\psi$ are t h e Eu ler angles on $SU(2)$(see[@ V iKl: r ep specf unc1 Ch a pt e r 6] ). F rom t h eortho gon ality r elation of th eLegendre pol y no mials, the fa ct that thei r o nly co mm onv alueisat $P _l(1)= 1 $, and consider in g that$ \theta... | \,0}(\phi,\theta,\psi)=P_l(\cos\theta)$, where_$\phi,\theta,\psi$ are_the Euler angles on_$SU(2)$ (see_[@ViKl:repspecfunc1_Chapter 6]). From_the_orthogonality relation of_the Legendre polynomials,_the fact that their_only common value_is_at $P_l(1)=1$, and considering that $\theta=0$ denotes a pole of $SU(2)$, we find ... |
^{*}$ is $$M(s) =\int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\mathbb P dt.$$ This has been verified in Section \[secorl\] (see formulae (\[Orlicz1\]) and (\[OrliczMin\])).
For the next lemma we need the following simple claim.
\[simcl\] Let $(x_i)_{i=1}^n$ be a sequence. Then for every $j\leq n-k$ one has $$\ope... | ^{*}$ is $ $ M(s) = \int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\mathbb P dt.$$ This has been verified in Section \[secorl\ ] (see formulae (\[Orlicz1\ ]) and (\[OrliczMin\ ]) ).
For the next lemma we necessitate the take after simple claim.
\[simcl\ ] Let $ (x_i)_{i=1}^n$ be a succession. Then for every $... | ^{*}$ is $$M(s) =\int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\oathbb P dt.$$ This has bxen verjfied in Section \[secorl\] (see formulae (\[Lroicz1\]) qnd (\[OrliczMin\])).
For the ndxt lemma we need the dollowing simple claim.
\[sijgl\] Lec $(e_i)_{i=1}^n$ be a sequekce. Then fos every $j\leq n-n$ unz has $$\ope... | ^{*}$ is $$M(s) =\int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\mathbb P dt.$$ This verified Section \[secorl\] formulae (\[Orlicz1\]) and we the following simple \[simcl\] Let $(x_i)_{i=1}^n$ a sequence. Then for every $j\leq one has $$\operatornamewithlimits{k-max}_{1\leq i\leq n}|x_{i}| \leq \operatornamewithlimi... | ^{*}$ is $$M(s) =\int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\mAthbb P dt.$$ ThIs has BeeN veRiFied In SeCtion \[secorl\] (see FOrmuLae (\[Orlicz1\]) and (\[OrliczMin\])).
FOr the NeXT lemMA wE need The follOWiNG SimPlE cLaiM.
\[sIMcL\] Let $(x_I)_{i=1}^n$ Be a sequEnce. Then foR evErY $j\leq n-k$ one haS $$\OpE... | ^{*}$ is $$M(s) =\int_{0}^ {s}\int_{\ frac{ 1}{ t}\ le q|\x i_1| }|\xi_1|d\math b b Pdt.$$ This has been ve rifie di n Se c ti on \[ secorl\ ] ( s e e f or mu lae ( \ [O rlicz 1\] ) and ( \[OrliczMi n\] )) .
For the n e xt lemma wenee d the follow ing simpl ecla i m.
\ [si mcl\] Let $ ( x_i)_{ i=1}^n$ b ea sequ... | ^{*}$ is_$$M(s) =\int_{0}^{s}\int_{\frac{1}{t}\leq|\xi_1|}|\xi_1|d\mathbb_P dt.$$ This has_been verified_in_Section \[secorl\] (see_formulae_(\[Orlicz1\]) and (\[OrliczMin\])).
For_the next lemma_we need the following_simple claim.
\[simcl\] Let_$(x_i)_{i=1}^n$_be a sequence. Then for every $j\leq n-k$ one has $$\ope... |
text{int}}$ (Eq. \[Hint\]) leads to $$\begin{split}
& \hat{\Sigma}^<(t_1,t_2) = \hbar^2 |g|^2 \big[\\
& F_{\text{m}}(t_2-t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}\hat{a}^{\dagger} \\
& + F_{\text{p}}(\tau)(t_2-t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}^{\dagger}\hat{a} \\
& + F_{\text{p}}(\t... | text{int}}$ (Eq. \[Hint\ ]) leads to $ $ \begin{split }
& \hat{\Sigma}^<(t_1,t_2) = \hbar^2 |g|^2 \big[\\
& F_{\text{m}}(t_2 - t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}\hat{a}^{\dagger } \\
& + F_{\text{p}}(\tau)(t_2 - t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}^{\dagger}\hat{a } \\ ... | texh{int}}$ (Eq. \[Hint\]) leads to $$\begln{split}
& \hat{\Sigma}^<(j_1,t_2) = \hbar^2 |g|^2 \big[\\
& F_{\text{m}}(g_2-t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}\iat{a}^{\eaggee} \\
& + F_{\text{p}}(\tau)(t_2-t_1) \hat{c}\hag{a}^{\dagger}\hwt{G}^<(t_1,t_2) \har{c}^{\dajger}\hat{a} \\
& + F_{\text{'}}(\f... | text{int}}$ (Eq. \[Hint\]) leads to $$\begin{split} & \hbar^2 \big[\\ & \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}\hat{a}^{\dagger} \\ \\ + F_{\text{p}}(\tau)(t_2-t_1) \hat{c}^{\dagger}\hat{a}\hat{G}^<(t_1,t_2) \\ & + \hat{c}^{\dagger}\hat{a}\hat{G}^<(t_1,t_2) \hat{c}^{\dagger}\hat{a} \big],\\ \end{split} \labe... | text{int}}$ (Eq. \[Hint\]) leads to $$\begin{Split}
& \hat{\SiGma}^<(t_1,t_2) = \HbaR^2 |g|^2 \bIg[\\
& f_{\texT{m}}(t_2-t_1) \Hat{c}\hat{a}^{\dagger}\HAt{G}^<(t_1,T_2) \hat{c}\hat{a}^{\dagger} \\
& + F_{\text{p}}(\tAu)(t_2-t_1) \hAt{C}\Hat{a}^{\DAgGer}\haT{G}^<(t_1,t_2) \hat{C}^{\DaGGEr}\hAt{A} \\
& + F_{\TexT{p}}(\T... | text{int}}$ (Eq. \[Hint\]) leads to$$\be gin {sp li t}
& \ha t{\Sigma}^<(t_ 1 ,t_2 ) = \hbar^2 |g|^2 \big [\\
& F _{\t e xt {m}}( t_2-t_1 ) \ h a t{c }\ ha t{a }^ { \d agger }\h at{G}^< (t_1,t_2)\ha t{ c}\hat{a}^{\ d ag ger} \\
&+ F _{\text{p}}( \ta u)(t_2 -t _1) \hat{ c}\ hat{a }^{\da g ger}\h at{G}^<(t _1 , t_2) ... | text{int}}$ (Eq. \[Hint\])_leads to_$$\begin{split}
& \hat{\Sigma}^<(t_1,t_2) = \hbar^2_|g|^2 \big[\\
&__F_{\text{m}}(t_2-t_1) \hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2)_\hat{c}\hat{a}^{\dagger}_\\
& + F_{\text{p}}(\tau)(t_2-t_1)_\hat{c}\hat{a}^{\dagger}\hat{G}^<(t_1,t_2) \hat{c}^{\dagger}\hat{a} \\
&_+ F_{\text{p}}(\t... |
o\], we have that $F_{\lambda}(k+1) \leq
F_{\mu}(k)$ if and only if $$\label{AAA}
e^{-\lambda} \lambda^{k+1} \leq (k+1) \int_{\mu} ^{\lambda} e^{-t} t^k dt.$$ By the change of variables, $t=\lambda s$, we see that (\[AAA\]) is equivalent to $$\label{AAAA}
1 \leq (k+1) \int_p ^1 e^{(1-s)\lambda}s^k ds.$$ The right side ... | o\ ], we have that $ F_{\lambda}(k+1) \leq
F_{\mu}(k)$ if and only if $ $ \label{AAA }
e^{-\lambda } \lambda^{k+1 } \leq (k+1) \int_{\mu } ^{\lambda } e^{-t } t^k dt.$$ By the change of variables, $ t=\lambda s$, we visualize that (\[AAA\ ]) is equivalent to $ $ \label{AAAA }
1 \leq (k+1) \int_p ^1 e^{(1 - s)\lam... | o\], wf have that $F_{\lambda}(k+1) \leq
N_{\mu}(k)$ if and only if $$\labxl{AAA}
e^{-\lzmbda} \laobda^{k+1} \leq (k+1) \int_{\mu} ^{\lambda} e^{-t} v^k dr.$$ By uke change of variablds, $t=\lambdw s$, we swe tiat (\[AAA\]) is equivement to $$\label{ZWAA}
1 \nxq (k+1) \int_p ^1 e^{(1-s)\lakbda}s^k ds.$$ Dhe right side ... | o\], we have that $F_{\lambda}(k+1) \leq F_{\mu}(k)$ only $$\label{AAA} e^{-\lambda} \leq (k+1) \int_{\mu} the of variables, $t=\lambda we see that is equivalent to $$\label{AAAA} 1 \leq \int_p ^1 e^{(1-s)\lambda}s^k ds.$$ The right side of (\[AAAA\]) is increasing in $\lambda$. $\mu'/\lambda' = p$ and $\lambda' > \lam... | o\], we have that $F_{\lambda}(k+1) \leq
F_{\mu}(K)$ if and only If $$\labEl{AaA}
e^{-\LaMbda} \LambDa^{k+1} \leq (k+1) \int_{\mu} ^{\laMBda} e^{-T} t^k dt.$$ By the change of variAbles, $T=\lAMbda S$, We See thAt (\[AAA\]) is EQuIVAleNt To $$\LabEl{aaAa}
1 \leq (k+1) \Int_P ^1 e^{(1-s)\lambDa}s^k ds.$$ The rIghT sIde ... | o\], we have that $F_{\lam bda}(k+1)\leqF_{ \mu }( k)$if a nd only if $$\ l abel {AAA}
e^{-\lambda} \la mbda^ {k + 1} \ l eq (k+1 ) \int_ { \m u } ^{ \l am bda }e ^{ -t} t ^kdt.$$ B y the chan geof variables,$ t= \lambda s$ , w e see that ( \[A AA\])is eq u ivale ntto $$ \label { AAAA}1 \leq (k +1 ) \int_ p ^1 e^... | o\], we_have that_$F_{\lambda}(k+1) \leq
F_{\mu}(k)$ if and_only if_$$\label{AAA}
e^{-\lambda}_\lambda^{k+1} \leq_(k+1)_\int_{\mu} ^{\lambda} e^{-t}_t^k dt.$$ By_the change of variables,_$t=\lambda s$, we_see_that (\[AAA\]) is equivalent to $$\label{AAAA}
1 \leq (k+1) \int_p ^1 e^{(1-s)\lambda}s^k ds.$$ The right_side_... |
\left( \sum_i \hat{s_i} \right).$$ This will follow if we can prove that, for positive real numbers $t_1, \dots, t_k$, $$\label{eqn:to_prove_about_phi_flipped}
t_1\dots t_k \Phi(t^{-1}_1, \dots, t^{-1}_k) {\leqslant}\frac{\sigma_k}{k}(t_1 + \dots + t_k).$$
To prove (\[eqn:to\_prove\_about\_phi\_flipped\]), first obse... | \left (\sum_i \hat{s_i } \right).$$ This will follow if we can prove that, for positive real number $ t_1, \dots, t_k$, $ $ \label{eqn: to_prove_about_phi_flipped }
t_1\dots t_k \Phi(t^{-1}_1, \dots, t^{-1}_k) { \leqslant}\frac{\sigma_k}{k}(t_1 + \dots + t_k).$$
To rise (\[eqn: to\_prove\_about\_phi\_flipped\ ]), ... | \levt( \sum_i \hat{s_i} \right).$$ This will follow if we can prove that, fof positive real numbers $t_1, \dovs, t_j$, $$\labtj{eqn:to_prove_about_phk_flipped}
t_1\fots t_k \Phi(t^{-1}_1, \eots, t^{-1}_k) {\leqslant}\frac{\sigmz_n}{k}(t_1 + \vots + t_k).$$
To provg (\[eqn:to\_prove\_dbout\_phi\_flippeg\]), wixst obse... | \left( \sum_i \hat{s_i} \right).$$ This will follow can that, for real numbers $t_1, \Phi(t^{-1}_1, t^{-1}_k) {\leqslant}\frac{\sigma_k}{k}(t_1 + + t_k).$$ To (\[eqn:to\_prove\_about\_phi\_flipped\]), first observe that equality holds the case that all of the $t_i$ are equal. Indeed, when $t_i = the relation follows fr... | \left( \sum_i \hat{s_i} \right).$$ This wilL follow if wE can pRovE thAt, For pOsitIve real numbers $T_1, \Dots, T_k$, $$\label{eqn:to_prove_about_Phi_flIpPEd}
t_1\dOTs T_k \Phi(T^{-1}_1, \dots, t^{-1}_k) {\LEqSLAnt}\FrAc{\SigMa_K}{K}(t_1 + \Dots + t_K).$$
To Prove (\[eqN:to\_prove\_abOut\_PhI\_flipped\]), firsT ObSe... | \left( \sum_i \hat{s_i} \ right).$$Thiswil l f ol lowif w e can prove th a t, f or positive real numbe rs $t _1 , \do t s, t_k$ , $$\la b el { e qn: to _p rov e_ a bo ut_ph i_f lipped}
t_1\dotst_k \ Phi(t^{-1}_1 , \ dots, t^{- 1}_ k) {\leqslan t}\ frac{\ si gma _ k}{k} (t_ 1 + \ dots + t_k).$ $
To pro ve (\[eqn ... | \left(_\sum_i \hat{s_i}_\right).$$ This will follow_if we_can_prove that,_for_positive real numbers_$t_1, \dots, t_k$,_$$\label{eqn:to_prove_about_phi_flipped}
t_1\dots t_k \Phi(t^{-1}_1, \dots,_t^{-1}_k) {\leqslant}\frac{\sigma_k}{k}(t_1 +_\dots_+ t_k).$$
To prove (\[eqn:to\_prove\_about\_phi\_flipped\]), first obse... |
vertices.
Pairwise beta model
-------------------
The tilted Dirichlet model has been successfully used for applications [e.g. @coles1991], although it suffers from a lack of interpretability of the parameters. @cooley2010 proposed a similar model but with easily interpretable parameters. The definition of their mod... | vertices.
Pairwise beta model
-------------------
The tilted Dirichlet model has been successfully used for application [ for example @coles1991 ], although it suffers from a lack of interpretability of the parameters. @cooley2010 aim a like model but with easily explainable parameters. The definition of their ... | vegtices.
Pairwise beta modeu
-------------------
The tilted Dirnxhlet kodel gas been successfully used for appliratiins [e.t. @coles1991], although it suwfers frol a lack of mnterpretability of the icramefcrs. @cmiley2010 proposed s similar kodel but with exsnly interpretable parameters. The defynition ov their mod... | vertices. Pairwise beta model ------------------- The tilted has successfully used applications [e.g. @coles1991], lack interpretability of the @cooley2010 proposed a model but with easily interpretable parameters. definition of their model is based on a geometric approach. Specifically, they considered symmetric pairw... | vertices.
Pairwise beta model
-------------------
the tilted DIrichLet ModEl Has bEen sUccessfully useD For aPplications [e.g. @coles1991], althOugh iT sUFferS FrOm a laCk of intERpRETabIlItY of ThE PaRametErs. @Cooley2010 pRoposed a siMilAr Model but with EAsIly interprEtaBle parameterS. ThE definItIon OF theiR moD... | vertices.
Pairwise betamodel
---- ----- --- --- -- --
Thetilted Dirichl e t mo del has been successfu lly u se d for ap plica tions [ e .g . @co le s1 991 ], al thoug h i t suffe rs from alac kof interpret a bi lity of th e p arameters. @ coo ley201 0pro p oseda s imila r mode l but w ith easil yi nterpr e table p ... | vertices.
Pairwise_beta model
-------------------
The_tilted Dirichlet model has_been successfully_used_for applications_[e.g._@coles1991], although it_suffers from a_lack of interpretability of_the parameters. @cooley2010_proposed_a similar model but with easily interpretable parameters. The definition of their mod... |
9
0.0316 9
0.0100 24
: Ratio of the wind X-ray luminosities as a function of wind momentum ratio, $\eta$, for colliding wind systems with equal, spatially invariant wind speeds, equal abundances, and near to the adiabatic limit.[]{data-label="tab:ad_lim"}
$L_{1}/L_{2}$ in-between these limits {... | 9
0.0316 9
0.0100 24
: Ratio of the wind X - ray luminosities as a affair of fart momentum ratio, $ \eta$, for colliding wind system with equal, spatially invariant tip speeds, equal abundance, and near to the adiabatic limit.[]{data - label="tab: ad_lim " }
$ L_{1}/L_{2}$ in - between... | 9
0.0316 9
0.0100 24
: Rxtio of the winb X-ray numinoaities ar a function of wind momentul eatio, $\eta$, for colliding wina systems with eqyal, wpatially iibariant wind aieeds, xqual abundancex, and near to the adiabadiz pimit.[]{data-label="tab:ad_lim"}
$L_{1}/L_{2}$ in-between ehese lomlts {... | 9 0.0316 9 0.0100 24 : Ratio wind luminosities as function of wind wind with equal, spatially wind speeds, equal and near to the adiabatic limit.[]{data-label="tab:ad_lim"} in-between these limits {#sec:inbetween} ------------------------------------- Estimates of the luminosity ratio for systems where shocked winds ar... | 9
0.0316 9
0.0100 24
: Ratio of the wind X-ray luminosIties as a fuNctioN of WinD mOmenTum rAtio, $\eta$, for collIDing Wind systems with equal, spAtialLy INvarIAnT wind Speeds, eQUaL ABunDaNcEs, aNd NEaR to thE adIabatic Limit.[]{data-lAbeL="tAb:ad_lim"}
$L_{1}/L_{2}$ in-bETwEen these liMitS {... | 9
0.0316 9
0 .0100 24
:Ra tioof t he wind X-rayl umin osities as a functionof wi nd mome n tu m rat io, $\e t a$ , for c ol lid in g w ind s yst ems wit h equal, s pat ia lly invarian t w ind speeds , e qual abundan ces , andne art o the ad iabat ic lim i t.[]{d ata-label =" t ab:ad_ l im"}
... | 9
_ 0.0316_ _ __ 9
__ 0.0100 _ _ 24
_ : Ratio_of_the wind X-ray luminosities as a function of wind momentum ratio, $\eta$, for colliding_wind_systems with_equal,_spatially_invariant wind speeds, equal abundances,_and near to the adiabatic_limit.[]{data-label="tab:ad_lim"}
$L_{1}/L_{2}$ in-between_these limits {... |
j,n})$ realizes the $A$-invariant type $p_1
\otimes \cdots \otimes p_n$ over $A'q_0 \subseteq
{\operatorname{dcl}}(q_0,a_{1,1},\ldots,a_{j-1,n}A)$. Applying Lemma \[shuffle-chaos\], we find a small model $M \supseteq A'$ and a non-constant global $M$-invariant type $r$ such that $$(a_{j,1},\ldots,a_{j,n},q_0) \mode... | j, n})$ realizes the $ A$ -invariant type $ p_1
\otimes \cdots \otimes p_n$ over $ A'q_0 \subseteq
{ \operatorname{dcl}}(q_0,a_{1,1},\ldots, a_{j-1,n}A)$. Applying Lemma \[shuffle - chaos\ ], we find a humble exemplar $ M \supseteq A'$ and a non - constant global $ M$-invariant character $ r$ such that $ $ (a... | j,n})$ gealizes the $A$-invariant uype $p_1
\otimes \cdots \otikes p_n$ over $A'q_0 \subseteq
{\operatorname{dcl}}(q_0,a_{1,1},\pdits,a_{j-1,b}A)$. Applying Lemma \[shuffue-chaos\], wv find a wmalo model $M \snlseteq A'$ and z non-eoistant global $M$-lnvariant tfpe $r$ such thad $$(x_{j,1},\pdots,a_{j,n},q_0) \mode... | j,n})$ realizes the $A$-invariant type $p_1 \otimes p_n$ $A'q_0 \subseteq Applying Lemma \[shuffle-chaos\], $M A'$ and a global $M$-invariant type such that $$(a_{j,1},\ldots,a_{j,n},q_0) \models p_1 \otimes \otimes p_n \otimes r|M.$$ The type $r$ extends ${\operatorname{tp}}(q_0/M)$ and therefore lives in $M$-definabl... | j,n})$ realizes the $A$-invariant tyPe $p_1
\otimes \cDots \oTimEs p_N$ oVer $A'Q_0 \subSeteq
{\operatornAMe{dcL}}(q_0,a_{1,1},\ldots,a_{j-1,n}A)$. Applying LeMma \[shUfFLe-chAOs\], We finD a small MOdEL $m \suPsEtEq A'$ AnD A nOn-conStaNt globaL $M$-invarianT tyPe $R$ such that $$(a_{j,1},\lDOtS,a_{j,n},q_0) \mode... | j,n})$ realizes the $A$-in variant ty pe $p _1 \ ot imes \cd ots \otimes p_ n $ ov er $A'q_0 \subseteq
{\ope ra t orna m e{ dcl}} (q_0,a_ { 1, 1 } ,\l do ts ,a_ {j - 1, n}A)$ . A pplying Lemma \[s huf fl e-chaos\], w e f ind a smal l m odel $M \sup set eq A'$ a nda non- con stant globa l $M$-i nvariantty p e $... | j,n})$ realizes_the $A$-invariant_type $p_1
\otimes_\cdots \otimes_p_n$_over $A'q_0_\subseteq
_ {\operatorname{dcl}}(q_0,a_{1,1},\ldots,a_{j-1,n}A)$. Applying_Lemma \[shuffle-chaos\], we find_a small model $M_\supseteq A'$ and_a_non-constant global $M$-invariant type $r$ such that $$(a_{j,1},\ldots,a_{j,n},q_0) \mode... |
(spin-orbit angle) is totally abnormal in (eclipsing) binary stars. The only exception I know of is that of DI Her [@Albrecht+09; @Albrecht+11], where the equatorial plane of both members of the binary are strongly tilted with respect to their orbital plane; though no definitive explanation has been given to such misa... | (spin - orbit angle) is totally abnormal in (overshadow) binary star. The only exception I know of is that of DI Her [ @Albrecht+09; @Albrecht+11 ], where the equatorial plane of both member of the binary are strongly tilt with regard to their orbital plane; though no authoritative explanation has been give to such mis... | (spln-orbit angle) is totally abnormal in (eclipsing) binarg stars. Ghe only exception I know of iw thau of DI Her [@Albrechg+09; @Albrechn+11], where tye ewyatorial plane of njth jcmberv of the binary are stroncly tilted witv fedpect to their orbital plane; though no defonltive explanatyon nws bsvn given to such misa... | (spin-orbit angle) is totally abnormal in (eclipsing) The exception I of is that where equatorial plane of members of the are strongly tilted with respect to orbital plane; though no definitive explanation has been given to such misalignment, it be the consequence of a relatively recent and strong orbital perturbation ... | (spin-orbit angle) is totally abNormal in (ecLipsiNg) bInaRy StarS. The Only exception I KNow oF is that of DI Her [@Albrecht+09; @albreChT+11], WherE ThE equaTorial pLAnE OF boTh MeMbeRs OF tHe binAry Are stroNgly tilted WitH rEspect to theiR OrBital plane; ThoUgh no definitIve ExplanAtIon HAs beeN giVen to Such miSA... | (spin-orbit angle) is tot ally abnor mal i n ( ecl ip sing ) bi nary stars. Th e onl y exception I know ofis th at of D I H er [@ Albrech t +0 9 ; @A lb re cht +1 1 ], wher e t he equa torial pla neof both member s o f the bina ryare strongly ti lted w it h r e spect to thei r orbi t al pla ne; thoug hn o def... | (spin-orbit_angle) is_totally abnormal in (eclipsing)_binary stars._The_only exception_I_know of is_that of DI_Her [@Albrecht+09; @Albrecht+11], where_the equatorial plane_of_both members of the binary are strongly tilted with respect to their orbital plane;_though_no definitive_explanation_has_been given to such misa... |
N$-body simulation ($f_{\rm pro}\sim0.55-0.60$). In the catalog of satellite systems extracted from the Sloan Digital Sky Survey (SDSS; @york00) Data Release 6 [@bai08], @hf08 determined the direction of rotation of some host galaxies by spectroscopic observations. They obtained the data for 78 satellites associated wi... | N$-body simulation ($ f_{\rm pro}\sim0.55 - 0.60 $). In the catalog of satellite systems extracted from the Sloan Digital Sky Survey (SDSS; @york00) Data Release 6 [ @bai08 ], @hf08 determined the steering of rotation of some server galaxies by spectroscopic observations. They obtained the datum for 78 satellites assoc... | N$-bofy simulation ($f_{\rm pro}\sim0.55-0.60$). In the catalog of satxllite aystems dxtracted from the Sloan Digmtal Sky Wurvey (SDSS; @york00) Data Felease 6 [@hai08], @hf08 dwternuned the dmdection of rofwtiou if some host gslaxies by spectroscopic ocszrvations. They obtained the data for 78 satelkihes associated wi... | N$-body simulation ($f_{\rm pro}\sim0.55-0.60$). In the catalog systems from the Digital Sky Survey [@bai08], determined the direction rotation of some galaxies by spectroscopic observations. They obtained data for 78 satellites associated with 63 hosts. Combining these data with those the satellites in @zar97, they fo... | N$-body simulation ($f_{\rm pro}\sim0.55-0.60$). IN the cataloG of saTelLitE sYsteMs exTracted from the sLoan digital Sky Survey (SDSS; @yoRk00) DatA RELeasE 6 [@BaI08], @hf08 deTermineD ThE DIreCtIoN of RoTAtIon of SomE host gaLaxies by spEctRoScopic observATiOns. They obtAinEd the data for 78 SatElliteS aSsoCIated Wi... | N$-body simulation ($f_{\r m pro}\sim 0.55- 0.6 0$) .In t he c atalog of sate l lite systems extracted fro m the S l oanD ig italSky Sur v ey ( SDS S; @ yor k0 0 )DataRel ease 6[@bai08],@hf 08 determinedt he direction of rotation of so me hos tgal a xiesbyspect roscop i c obse rvations. T h ey obt a ined th e ... | N$-body simulation_($f_{\rm pro}\sim0.55-0.60$)._In the catalog of_satellite systems_extracted_from the_Sloan_Digital Sky Survey_(SDSS; @york00) Data_Release 6 [@bai08], @hf08_determined the direction_of_rotation of some host galaxies by spectroscopic observations. They obtained the data for 78_satellites_associated wi... |
-------- --------------------------------------------------------
$\text{\ensuremath{\hat{\mathcal{U}}_{0,+}^{+}}},\text{\ensuremath{\hat{\mathcal{U}}_{T/2,+}^{+}}}$ AI $\mathcal{R}_{7-\delta}$ 0 ... | -------- --------------------------------------------------------
$ \text{\ensuremath{\hat{\mathcal{U}}_{0,+}^{+}}},\text{\ensuremath{\hat{\mathcal{U}}_{T/2,+}^{+}}}$ AI $ \mathcal{R}_{7-\delta}$ 0 ... | -------- --------------------------------------------------------
$\text{\ensuremath{\hat{\mxthcal{U}}_{0,+}^{+}}},\text{\ensutenath{\hav{\mathcam{U}}_{T/2,+}^{+}}}$ AI $\matical{E}_{7-\deltq}$ 0 ... | -------- -------------------------------------------------------- $\text{\ensuremath{\hat{\mathcal{U}}_{0,+}^{+}}},\text{\ensuremath{\hat{\mathcal{U}}_{T/2,+}^{+}}}$ AI $\mathcal{R}_{7-\delta}$ 0 0 0 $\mathbb{\mathbb{Z}}_{2}$ $\mathbb{Z}$ BDI $\mathcal{R}_{-\delta}$ $\mathbb{Z}$ $\mathbb{Z}_{2}$ $\ensuremath{\hat{\math... | -------- --------------------------------------------------------
$\text{\ensuremath{\hat{\mathcal{U}}_{0,+}^{+}}},\Text{\ensureMath{\hAt{\mAthCaL{U}}_{T/2,+}^{+}}}$ Ai $\matHcal{R}_{7-\delta}$ 0 ... | -------- ----------------- ---------- ----- --- --- -- ---- ---- --------
$\ text{\ensuremath{\hat{ \math ca l {U}} _ {0 ,+}^{ +}}},\t e xt { \ ens ur em ath {\ h at {\mat hca l{U}}_{ T/2,+}^{+} }}$ AI $\mathcal{R} _{7 -\delt a} $ 0 ... | -------- --------------------------------------------------------
_ _ _ $\text{\ensuremath{\hat{\mathcal{U}}_{0,+}^{+}}},\text{\ensuremath{\hat{\mathcal{U}}_{T/2,+}^{+}}}$__ __ _ AI _ _ __ $\mathcal{R}_{7-\delta}$ __ ___ _ _ _ __ _ _0 _ ___ _ _ _ _ _ _ _ ... |
. The data used in scaling analysis has been corrected for demagnetization factor.
The x-ray data has been analyzed by the Rietveld refinement method using FULLPROF software[@Carvajal] and the results show that the sample is single phase and crystallizes in rhombohedral structure with space group R-3c which is in agre... | . The data used in scaling analysis has been chastise for demagnetization agent.
The x - ray data has been analyze by the Rietveld refinement method use FULLPROF software[@Carvajal ] and the resultant role show that the sample distribution is individual phase and crystallizes in rhombohedral social organization with... | . Thf data used in scaling akalysis has been correcved for demagnegization factor.
The x-ray data hqs betu analyzed by the Ridtveld revinement metiod using FULLPROF software[@Carbwjal] end the results show that the sample is sknyle phase and crystallizes in rhomboredral xtgucture with skace bwoup G-3c which is in agre... | . The data used in scaling analysis corrected demagnetization factor. x-ray data has refinement using FULLPROF software[@Carvajal] the results show the sample is single phase and in rhombohedral structure with space group R-3c which is in agrement with the report.[@Fita] Figure \[fig: XRD\] displays the room temperatur... | . The data used in scaling analySis has been CorreCteD foR dEmagNetiZation factor.
ThE X-ray Data has been analyzed by tHe RieTvELd reFInEment Method uSInG fuLLpRoF SofTwARe[@carvaJal] And the rEsults show ThaT tHe sample is siNGlE phase and cRysTallizes in rhOmbOhedraL sTruCTure wIth Space Group R-3C Which iS in agre... | . The data used in scaling analysishas b een co rr ecte d fo r demagnetizat i on f actor.
The x-ray data hasbe e n an a ly zed b y the R i et v e ldre fi nem en t m ethod us ing FUL LPROF soft war e[ @Carvajal] a n dthe result s s how that the sa mple i ssin g le ph ase andcrysta l lizesin rhombo he d ral st r ... | . The_data used_in scaling analysis has_been corrected_for_demagnetization factor.
The_x-ray_data has been_analyzed by the_Rietveld refinement method using_FULLPROF software[@Carvajal] and_the_results show that the sample is single phase and crystallizes in rhombohedral structure with_space_group R-3c_which_is_in agre... |
^{m})
\bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}P(\pi';\theta^{m}) & (\pi\in[\tau_i]), \\
0 & (\pi\not\in[\tau_i]),
\end{cases}\label{optq}\\
q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1}: i=1,\ldots,n, \pi\in S_{r} \},\\
L(\theta;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}_{i,\pi}\log P(\p... | ^{m })
\bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}P(\pi';\theta^{m }) & (\pi\in[\tau_i ]), \\
0 & (\pi\not\in[\tau_i ]),
\end{cases}\label{optq}\\
q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1 }: i=1,\ldots, n, \pi\in S_{r } \},\\
L(\theta;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}... | ^{m})
\bihg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}K(\pi';\theta^{m}) & (\pi\in[\tao_i]), \\
0 & (\pi\nmt\in[\tah_i]),
\end{casds}\label{optq}\\
q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1}: i=1,\ldots,n, \'i\in S_{r} \},\\
L(\uketa;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pk\in[\tau_i]}q^{m+1}_{p,\pi}\log P(\p... | ^{m}) \bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}P(\pi';\theta^{m}) & (\pi\in[\tau_i]), \\ 0 & q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1}: \pi\in S_{r} L(\theta;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}_{i,\pi}\log P(\pi;\theta),\\ L_{\lambda}(\phi;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1... | ^{m})
\bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\taU_{i})}P(\pi';\theta^{M}) & (\pi\in[\Tau_I]), \\
0 & (\pi\NoT\in[\tAu_i]),
\eNd{cases}\label{opTQ}\\
q_{(n)}^{m+1}&:=\{Q_{i,\pi}^{m+1}: i=1,\ldots,n, \pi\in S_{r} \},\\
L(\theTa;\tau_{(N)},q^{M+1}_{(N)})&:=-\sum_{I=1}^N\sUm_{\pi\iN[\tau_i]}q^{m+1}_{I,\Pi}\LOG P(\p... | ^{m})
\bigg{/}\sum_{\pi'\i n[\tau_i]} \phi^ {m} _{\ pi ',t( \tau _{i})}P(\pi';\ t heta ^{m}) & (\pi\in[\tau_i ]), \ \0 & ( \ pi \not\ in[\tau _ i] ) ,
\e nd {c ase s} \ la bel{o ptq }\\
q_{ (n)}^{m+1} &:= \{ q_{i,\pi}^{m + 1} : i=1,\ldo ts, n, \pi\in S_ {r} \},\\
L (\t h eta;\ tau _{(n) },q^{m + 1}_{(n )})&:=-\s um... | ^{m})
\bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}P(\pi';\theta^{m}) &_(\pi\in[\tau_i]), \\
0_& (\pi\not\in[\tau_i]),
\end{cases}\label{optq}\\
q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1}: i=1,\ldots,n, \pi\in_S_{r} \},\\
L(\theta;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}_{i,\pi}\log_P(\p... |
\bigwedge_{i<|s|} a_{\theta(i)}^{s(i)} \in {\mathscr{A}}$, and let $$\begin{gathered}
\rho_{\theta,n} = \bigcup_{s\in 2^n} \pi_s^{b_{\theta,s}}(x),
\qquad
\rho_\theta = \bigcup_n \rho_{\theta,n}.
\end{gathered}$$ In other words, $\rho_\theta(c)$ holds if and only if, for every $s \in 2^{<\omega}$, $c$ sa... | \bigwedge_{i<|s| } a_{\theta(i)}^{s(i) } \in { \mathscr{A}}$, and let $ $ \begin{gathered }
\rho_{\theta, n } = \bigcup_{s\in 2^n } \pi_s^{b_{\theta, s}}(x),
\qquad
\rho_\theta = \bigcup_n \rho_{\theta, n }.
\end{gathered}$$ In other words, $ \rho_\theta(c)$ holds if and merely if, for every $ s \... | \bihwedge_{i<|s|} a_{\theta(i)}^{s(i)} \in {\mauhscr{A}}$, and let $$\beyun{gathxred}
\rho_{\thetx,n} = \bigcup_{s\in 2^n} \pi_s^{b_{\theta,s}}(x),
\qquae
\rho_\theta = \bigcup_n \rho_{\theta,j}.
\end{garhertd}$$ In other words, $\rho_\theta(c)$ holsd if end only if, for every $s \it 2^{<\omega}$, $c$ sa... | \bigwedge_{i<|s|} a_{\theta(i)}^{s(i)} \in {\mathscr{A}}$, and let $$\begin{gathered} \bigcup_{s\in \pi_s^{b_{\theta,s}}(x), \qquad = \bigcup_n \rho_{\theta,n}. holds and only if, every $s \in $c$ satisfies $\pi_s$ over $\bigwedge_{i<|s|} a_{\theta(i)}^{s(i)}$. is easy to check that $\rho_{\theta,n}$ is consistent and ... | \bigwedge_{i<|s|} a_{\theta(i)}^{s(i)} \in {\mathScr{A}}$, and let $$\Begin{GatHerEd}
\Rho_{\tHeta,N} = \bigcup_{s\in 2^n} \pi_s^{B_{\ThetA,s}}(x),
\qquad
\rho_\theta = \bigcup_N \rho_{\tHeTA,n}.
\enD{GaThereD}$$ In otheR WoRDS, $\rhO_\tHeTa(c)$ HoLDs If and OnlY if, for eVery $s \in 2^{<\omeGa}$, $c$ Sa... | \bigwedge_{i<|s|} a_{\the ta(i)}^{s( i)} \ in{\m at hscr {A}} $, and let $$\ b egin {gathered}
\rho_{\ theta ,n } = \ b ig cup_{ s\in 2^ n }\ p i_s ^{ b_ {\t he t a, s}}(x ), \qq uad
\r ho_ \t heta = \bigc u p_ n \rho_{\t het a,n}.
\end {ga thered }$ $ I n othe r w ords, $\rho _ \theta (c)$ hold si f a... | \bigwedge_{i<|s|}_a_{\theta(i)}^{s(i)} \in_{\mathscr{A}}$, and let $$\begin{gathered}
_ __\rho_{\theta,n} =_\bigcup_{s\in_2^n} \pi_s^{b_{\theta,s}}(x),
_ \qquad
_ \rho_\theta_= \bigcup_n \rho_{\theta,n}.
__\end{gathered}$$ In other words, $\rho_\theta(c)$ holds if and only if, for every $s \in_2^{<\omega}$,_$c$ sa... |
whether its values are statistically significant from those of the cell *to its left*. Statistical significance is computed using a two-tailed T-test with inequal variance.
----------------- -------------- ------------------------------ -----------------
Level Demo Length (Mean $\pm$ S... | whether its values are statistically significant from those of the cell * to its left *. Statistical meaning is calculate practice a two - tailed T - test with inequal division.
----------------- -------------- ------------------------------ -----------------
Level Demo Length (Mea... | whfther its values are stauistically signifnxant fcom thoae of thd cell *to its left*. Statisticel sugnifucance is computed usivg a two-twiled T-twst xith inequal varmznce.
----------------- -------------- ------------------------------ -----------------
Lebcl Demo Length (Maav $\'m$ S... | whether its values are statistically significant from the *to its Statistical significance is with variance. ----------------- -------------- ----------------- Level Demo (Mean $\pm$ Std) (lr)[2-3]{} BabyAI 1.0 1.1 GoToObj [**100**]{} [**100**]{} 5.18 $\pm$ 2.38 GoToRedBallGrey [**100**]{} [**100**]{} 5.81 $\pm$ 3.29 [... | whether its values are statisTically sigNificAnt FroM tHose Of thE cell *to its left*. sTatiStical significance is coMputeD uSIng a TWo-TaileD T-test wITh INEquAl VaRiaNcE.
----------------- -------------- ------------------------------ -----------------
leVel DeMo LEngth (MeAn $\pm$ S... | whether its values are st atisticall y sig nif ica nt fro m th ose of the cel l *to its left*. Statistica l sig ni f ican c eis co mputedu si n g atw o- tai le d T -test wi th ineq ual varian ce.
---------- - -- ---- ----- --- ------ ----- --- ------ -- --- - ----- --- -- -- ------ - ------ --
Leve l ... | whether_its values_are statistically significant from_those of_the_cell *to_its_left*. Statistical significance_is computed using_a two-tailed T-test with_inequal variance.
_-----------------_-------------- ------------------------------ -----------------
Level __ ___ _ _ _ Demo Length (Mean $\pm$ S... |
and $F$ acts trivially on $A(g_1)$. Let $\lambda$ be the non-trivial character of $A(g_1)$. As above, we obtain: $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl} q^2 &\quad\mbox{if
$g=g_1$},\\ -q^2 & \quad \mbox{if $g=g_1'$},\\ 0 & \quad \mbox{if
$g\not\in C^F$},
\end{array}\right.$$ where $g_1'\in C^F$ corresponds ... | and $ F$ acts trivially on $ A(g_1)$. Let $ \lambda$ be the non - trivial character of $ A(g_1)$. As above, we receive: $ $ \chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl } q^2 & \quad\mbox{if
$ deoxyguanosine monophosphate = g_1$},\\ -q^2 & \quad \mbox{if $ g = g_1'$},\\ 0 & \quad \mbox{if
$ g\not\in C^F$ },
\end... | anf $F$ acts trivially on $A(g_1)$. Let $\lambda$ be jhw non-tcivial dharactef of $A(g_1)$. As above, we obtain: $$\cii_{g_1,\lqmbda}(t)=\left\{\begin{array}{cl} q^2 &\quxd\mbox{if
$h=g_1$},\\ -q^2 & \quqd \muox{if $g=g_1'$},\\ 0 & \quad \mbox{if
$g\kjt\in G^F$},
\end{crcay}\right.$$ where $n_1'\in C^F$ corrasponds ... | and $F$ acts trivially on $A(g_1)$. Let the character of As above, we -q^2 \quad \mbox{if $g=g_1'$},\\ & \quad \mbox{if C^F$}, \end{array}\right.$$ where $g_1'\in C^F$ corresponds the non-trivial element of $A(g_1)$. Now we have $R_{x_0}=\zeta_{x_0}\chi_{g_1,\lambda}$. In order to show $\zeta_{x_0}=1$, we can use the k... | and $F$ acts trivially on $A(g_1)$. Let $\lAmbda$ be the Non-trIviAl cHaRactEr of $a(g_1)$. As above, we obtAIn: $$\chI_{g_1,\lambda}(g)=\left\{\begin{arraY}{cl} q^2 &\qUaD\Mbox{IF
$g=G_1$},\\ -q^2 & \quaD \mbox{if $G=G_1'$},\\ 0 & \qUAD \mbOx{If
$G\noT\iN c^F$},
\End{arRay}\Right.$$ whEre $g_1'\in C^F$ coRreSpOnds ... | and $F$ acts trivially on $A(g_1)$. Let$\l amb da $ be the non-trivial c h arac ter of $A(g_1)$. As ab ove,we obta i n: $$\c hi_{g_1 , \l a m bda }( g) =\l ef t \{ \begi n{a rray}{c l} q^2 &\q uad \m box{if
$g=g _ 1$ },\\ -q^2& \ quad \mbox{i f $ g=g_1' $} ,\\ 0 & \ qua d \mb ox{if$g\not \in C^F$} ,\ end{ar r ay... | and_$F$ acts_trivially on $A(g_1)$. Let_$\lambda$ be_the_non-trivial character_of_$A(g_1)$. As above,_we obtain: $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl}_q^2 &\quad\mbox{if
$g=g_1$},\\ -q^2_& \quad \mbox{if_$g=g_1'$},\\_0 & \quad \mbox{if
$g\not\in C^F$},
\end{array}\right.$$ where $g_1'\in C^F$ corresponds ... |
surface was advanced in [@Kudler-Flam:2018qjo]. Furthermore a [*proof*]{} for this proposal, based on the [*reflected entropy*]{} [@Dutta:2019gen] was established in another recent communication [@Kusuki:2019zsp]. The entanglement wedge was earlier shown to be the bulk subregion dual to the reduced density matrix of t... | surface was advanced in [ @Kudler - Flam:2018qjo ]. Furthermore a [ * proof * ] { } for this proposal, free-base on the [ * reflect entropy * ] { } [ @Dutta:2019gen ] was established in another recent communication [ @Kusuki:2019zsp ]. The entanglement hacek was earlier shown to be the bulk subregion double to the repr... | sugface was advanced in [@Kualer-Flam:2018qjo]. Furjhwrmore a [*prokf*]{} for tfis proposal, based on the [*rehlecred ebtropy*]{} [@Dutta:2019gen] was esgablished in anotyer cecent communicavjon [@Kusmhi:2019zsl]. The xntanglement wecge was easlier shown to bd che bulk subregion dual to the reducqd densotj matrix of t... | surface was advanced in [@Kudler-Flam:2018qjo]. Furthermore a this based on [*reflected entropy*]{} [@Dutta:2019gen] communication The entanglement wedge earlier shown to the bulk subregion dual to the density matrix of the dual $CFT$s in [@Czech:2012bh; @Wall:2012uf; @Headrick:2014cta; @Jafferis:2014lza; @Jafferis:201... | surface was advanced in [@KudleR-Flam:2018qjo]. FuRtherMorE a [*pRoOf*]{} foR thiS proposal, based ON the [*Reflected entropy*]{} [@Dutta:2019gEn] was EsTAbliSHeD in anOther reCEnT COmmUnIcAtiOn [@kUsUki:2019zsP]. ThE entangLement wedgE waS eArlier shown tO Be The bulk subRegIon dual to the RedUced deNsIty MAtrix Of t... | surface was advanced in [ @Kudler-Fl am:20 18q jo] .Furt herm ore a [*proof* ] {} f or this proposal, base d onth e [*r e fl ected entrop y *] { } [@ Du tt a:2 01 9 ge n] wa s e stablis hed in ano the rrecent commu n ic ation [@Ku suk i:2019zsp].The entan gl eme n t wed gewas e arlier shownto be the b u lk sub r ... | surface_was advanced_in [@Kudler-Flam:2018qjo]. Furthermore a_[*proof*]{} for_this_proposal, based_on_the [*reflected entropy*]{}_[@Dutta:2019gen] was established_in another recent communication_[@Kusuki:2019zsp]. The entanglement_wedge_was earlier shown to be the bulk subregion dual to the reduced density matrix_of_t... |
lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big]$$ (this is also valid when $s=0$, where the sum over $k$ is indeed multiplied by a vanishing coefficient). On the other hand, when $j>s+1$ we observe that $(j)_{s}-(j-1)_{s}=s(j-1)_{s-1}$ (once again this eq... | lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r - k}\big((\lambda^{>s})_{s}\big)\big]$$ (this is also valid when $ s=0 $, where the sum over $ k$ is indeed multiplied by a vanishing coefficient). On the other bridge player, when $ joule > s+1 $ we observe that $ (j)_{s}-(j-1)_{s}=s(j-1)_{s-... | lamhda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{y-1}e_{r-k}\big((\lambda^{>s})_{s}\yug)\big]$$ (vhis is also vauid when $s=0$, where the sum ovec $k$ us ineeed multiplied by a vxnishing boefficiebt). Oi the other hand, when $j>s+1$ we obacrve chet $(j)_{s}-(j-1)_{s}=s(j-1)_{s-1}$ (once again thiv eq... | lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big]$$ (this is also valid when $s=0$, sum $k$ is multiplied by a hand, $j>s+1$ we observe $(j)_{s}-(j-1)_{s}=s(j-1)_{s-1}$ (once again equality holds also when $s=0$, in form of $1-1=0$), so that we get $$jm_{j}... | lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\bIg)^{k-1}e_{r-k}\big((\lAmbda^{>S})_{s}\bIg)\bIg]$$ (This Is alSo valid when $s=0$, whERe thE sum over $k$ is indeed multiPlied By A VaniSHiNg coeFficienT). on THE otHeR hAnd, WhEN $j>S+1$ we obSerVe that $(j)_{S}-(j-1)_{s}=s(j-1)_{s-1}$ (once AgaIn This eq... | lambda^{>s})_{s}\big)-s(s) _{s-1}\sum _{k=1 }^{ r}\ bi g(-( s+1) _{s}\big)^{k-1 } e_{r -k}\big((\lambda^{>s}) _{s}\ bi g )\bi g ]$ $ (th is is a l so v ali dwh en$s = 0$ , whe rethe sum over $k$isin deed multipl i ed by a vani shi ng coefficie nt) . On t he ot h er ha nd, when $j>s+ 1 $ we o bserve th at $(j)_{ s ... | lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big]$$ (this_is also_valid when $s=0$, where_the sum_over_$k$ is_indeed_multiplied by a_vanishing coefficient). On_the other hand, when_$j>s+1$ we observe_that_$(j)_{s}-(j-1)_{s}=s(j-1)_{s-1}$ (once again this eq... |
(\[dNcp2\]) becomes $$\begin{aligned}
p_{1,t}=0,\quad
r_{1,t}=0,\quad
q_{1,t}=-\frac{1}{3}(p_1^2+r_1^2),
\label{dNcp21}\end{aligned}$$ which gives $$\begin{aligned}
p_{1}=c_1,\quad
r_{1}=c_2,\quad
q_{1}=-\frac{1}{3}(c_1^2+c_2^2)t,
\label{dNcp21s}\end{aligned}$$ where $c_{1}$ and $c_{2}$ are real valued integration con... | (\[dNcp2\ ]) becomes $ $ \begin{aligned }
p_{1,t}=0,\quad
r_{1,t}=0,\quad
q_{1,t}=-\frac{1}{3}(p_1 ^ 2+r_1 ^ 2),
\label{dNcp21}\end{aligned}$$ which gives $ $ \begin{aligned }
p_{1}=c_1,\quad
r_{1}=c_2,\quad
q_{1}=-\frac{1}{3}(c_1 ^ 2+c_2 ^ 2)t,
\label{dNcp21s}\end{aligned}$$ where $ c_{1}$ and $ c_{2}$... | (\[dNfp2\]) becomes $$\begin{aligned}
p_{1,u}=0,\quad
r_{1,t}=0,\quad
q_{1,t}=-\frac{1}{3}(k_1^2+r_1^2),
\oabel{dIcp21}\end{amigned}$$ wfich gives $$\begin{aligned}
p_{1}=c_1,\quav
r_{1}=c_2,\qyad
q_{1}=-\feac{1}{3}(c_1^2+c_2^2)t,
\label{dNcp21s}\end{aliened}$$ wherv $c_{1}$ and $c_{2}$ are eeal valuev integration dln... | (\[dNcp2\]) becomes $$\begin{aligned} p_{1,t}=0,\quad r_{1,t}=0,\quad q_{1,t}=-\frac{1}{3}(p_1^2+r_1^2), \label{dNcp21}\end{aligned}$$ $$\begin{aligned} r_{1}=c_2,\quad q_{1}=-\frac{1}{3}(c_1^2+c_2^2)t, where $c_{1}$ and constants. we arrive at single-peakon solution $$\begin{aligned} x+\frac{c_1^2+c_2^2}{3}t\mid}=ce^{... | (\[dNcp2\]) becomes $$\begin{aligned}
p_{1,t}=0,\Quad
r_{1,t}=0,\quad
Q_{1,t}=-\fraC{1}{3}(p_1^2+r_1^2),
\LabEl{DNcp21}\End{aLigned}$$ which givES $$\begIn{aligned}
p_{1}=c_1,\quad
r_{1}=c_2,\quad
q_{1}=-\Frac{1}{3}(c_1^2+C_2^2)t,
\LAbel{DncP21s}\end{Aligned}$$ WHeRE $C_{1}$ anD $c_{2}$ ArE reAl VAlUed inTegRation cOn... | (\[dNcp2\]) becomes $$\be gin{aligne d}
p_ {1, t}= 0, \qua d
r_ {1,t}=0,\quadq _{1, t}=-\frac{1}{3}(p_1^2+ r_1^2 ), \lab e l{ dNcp2 1}\end{ a li g n ed} $$ w hic hg iv es $$ \be gin{ali gned}
p_{1 }=c _1 ,\quad
r_{1} = c_ 2,\quad
q_ {1} =-\frac{1}{3 }(c _1^2+c _2 ^2) t ,
\la bel {dNcp 21s}\e n d{alig ned}$$ wh er e ... | (\[dNcp2\])_becomes $$\begin{aligned}
p_{1,t}=0,\quad
r_{1,t}=0,\quad
q_{1,t}=-\frac{1}{3}(p_1^2+r_1^2),
\label{dNcp21}\end{aligned}$$_which gives $$\begin{aligned}
p_{1}=c_1,\quad
r_{1}=c_2,\quad
q_{1}=-\frac{1}{3}(c_1^2+c_2^2)t,
\label{dNcp21s}\end{aligned}$$ where_$c_{1}$ and_$c_{2}$_are real_valued_integration con... |
$v^{th}$-order derivatives of the PGFs of the clutter cardinality distribution and the predicted cardinality distribution as $$\begin{aligned}
C^{(v)}_{j}(t) &= \frac{d^v C_{j}}{d t^v}(t) \,, \quad
\pgf^{(v)}(t) = \frac{d^v \pgf}{d t^v}(t).\end{aligned}$$
We use $\gamma$ to denote the probability, under the predictiv... | $ v^{th}$-order derivatives of the PGFs of the clutter cardinality distribution and the predicted cardinality distribution as $ $ \begin{aligned }
C^{(v)}_{j}(t) & = \frac{d^v C_{j}}{d t^v}(t) \, , \quad
\pgf^{(v)}(t) = \frac{d^v \pgf}{d t^v}(t).\end{aligned}$$
We use $ \gamma$ to denote the probability, under t... | $v^{tj}$-order derivatives of tht PGFs of the clujtwr carvinalitg districution and the predicted carvinaoity eistribution as $$\begin{auigned}
C^{(v)}_{j}(n) &= \frac{d^v C_{j}}{d r^v}(t) \,, \quad
\pjr^{(v)}(t) = \frac{d^v \pfn}{d t^v}(c).\eid{aligned}$$
We use $\gamma$ to genote the protacipity, under the predictiv... | $v^{th}$-order derivatives of the PGFs of the distribution the predicted distribution as $$\begin{aligned} \,, \pgf^{(v)}(t) = \frac{d^v t^v}(t).\end{aligned}$$ We use to denote the probability, under the PHD, that a target is detected by no sensor, and we thus have: {\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\t... | $v^{th}$-order derivatives of the PgFs of the clUtter CarDinAlIty dIstrIbution and the pREdicTed cardinality distribuTion aS $$\bEGin{aLIgNed}
C^{(v)}_{J}(t) &= \frac{d^V c_{j}}{D T^V}(t) \,, \qUaD
\pGf^{(v)}(T) = \fRAc{D^v \pgf}{D t^v}(T).\end{aliGned}$$
We use $\gAmmA$ tO denote the prOBaBility, undeR thE predictiv... | $v^{th}$-order derivative s of the P GFs o f t hecl utte r ca rdinality dist r ibut ion and the predictedcardi na l ityd is tribu tion as $$ \ b egi n{ al ign ed }
C ^{(v) }_{ j}(t) & = \frac{d^ v C _{ j}}{d t^v}(t ) \ ,, \quad
\ pgf ^{(v)}(t) =\fr ac{d^v \ pgf } {d t^ v}( t).\e nd{ali g ned}$$
We use$\ g amma$t ... | $v^{th}$-order_derivatives of_the PGFs of the_clutter cardinality_distribution_and the_predicted_cardinality distribution as_$$\begin{aligned}
C^{(v)}_{j}(t) &= \frac{d^v_C_{j}}{d t^v}(t) \,, \quad
\pgf^{(v)}(t)_= \frac{d^v \pgf}{d_t^v}(t).\end{aligned}$$
We_use $\gamma$ to denote the probability, under the predictiv... |
corresponds to the broken phase. In general the phase diagram depends on four parameters $\beta$, $\gamma$, $\nu$ and $\varphi$. In the $\mathcal{PT}$-symmetric phase only three of these parameters are independent and the phase diagram will contain a connected $3D$ region (symmetric phase) characterized by $F=0$. This... | corresponds to the broken phase. In general the phase diagram depends on four parameters $ \beta$, $ \gamma$, $ \nu$ and $ \varphi$. In the $ \mathcal{PT}$-symmetric phase merely three of these argument are independent and the phase diagram will contain a connected $ 3D$ area (symmetric phase) characterized by $ F=0$. ... | cogresponds to the broken khase. In general jhw phasx diagrzm depenas on four parameters $\beta$, $\gemma$, $\nu$ abd $\varphi$. In the $\mathcxl{PT}$-symmenric phasw onot three of these parametsvs arz mndependent and the phase diagram will wovtcin a connected $3D$ region (symmetric prase) chsrwcterized by $F=0$. Thix... | corresponds to the broken phase. In general diagram on four $\beta$, $\gamma$, $\nu$ phase three of these are independent and phase diagram will contain a connected region (symmetric phase) characterized by $F=0$. This is clearly a richer phase diagram that obtained by using exclusively eigenstates of the S-matrix wher... | corresponds to the broken phaSe. In generaL the pHasE diAgRam dEpenDs on four parameTErs $\bEta$, $\gamma$, $\nu$ and $\varphi$. In tHe $\matHcAL{PT}$-sYMmEtric Phase onLY tHREe oF tHeSe pArAMeTers aRe iNdependEnt and the pHasE dIagram will coNTaIn a connectEd $3D$ Region (symmetRic Phase) cHaRacTErizeD by $f=0$. This... | corresponds to the broken phase. In gene ral th ephas e di agram dependso n fo ur parameters $\beta$, $\ga mm a $, $ \ nu $ and $\varp h i$ . Inth e$\m at h ca l{PT} $-s ymmetri c phase on lyth ree of these pa rameters a reindependentand the p ha sed iagra m w ill c ontain a conn ected $3D $r egion( symmetr i c ... | corresponds_to the_broken phase. In general_the phase_diagram_depends on_four_parameters $\beta$, $\gamma$,_$\nu$ and $\varphi$._In the $\mathcal{PT}$-symmetric phase_only three of_these_parameters are independent and the phase diagram will contain a connected $3D$ region (symmetric_phase)_characterized by_$F=0$._This... |
interpolation between isochrones neither accounts for the nonlinear mapping of time onto the H-R diagram nor the non-uniform distribution of stellar masses observed in the galaxy. As a consequence, straightforward interpolation between isochrones results in an age distribution for field stars that is biased towards ol... | interpolation between isochrones neither accounts for the nonlinear mapping of meter onto the H - R diagram nor the non - consistent distribution of stellar masses observed in the galax. As a consequence, straightforward interpolation between isochrones consequence in an age distribution for field star that is biased t... | inherpolation between isocmrones neither aexounts for tge nonlivear mapping of time onto thx H-R diageam nor the non-uniform distribunion of sreller masses observxs in thc galzwy. As e consequence, sjraightforwasd interpolatimn bztween isochrones results in an age qistribitlon for field ftarx thaf is biased towards ol... | interpolation between isochrones neither accounts for the of onto the diagram nor the observed the galaxy. As consequence, straightforward interpolation isochrones results in an age distribution field stars that is biased towards older ages compared to the distribution predicted stellar evolutionary theory. Bayesian in... | interpolation between isochRones neithEr accOunTs fOr The nOnliNear mapping of tIMe onTo the H-R diagram nor the noN-unifOrM DistRIbUtion Of stellAR mASSes ObSeRveD iN ThE galaXy. AS a conseQuence, straIghTfOrward interpOLaTion betweeN isOchrones resuLts In an agE dIstRIbutiOn fOr fieLd starS That is Biased towArDS ol... | interpolation between iso chrones ne ither ac cou nt s fo r th e nonlinear ma p ping of time onto the H-Rdiagr am nort he non- uniform di s t rib ut io n o fs te llarmas ses obs erved in t hega laxy. As a c o ns equence, s tra ightforwardint erpola ti onb etwee n i sochr ones r e sultsin an age d i stribu t ion fo... | interpolation_between isochrones_neither accounts for the_nonlinear mapping_of_time onto_the_H-R diagram nor_the non-uniform distribution_of stellar masses observed_in the galaxy._As_a consequence, straightforward interpolation between isochrones results in an age distribution for field stars_that_is biased_towards_ol... |
of this section.
\[l:almost properness\] For each interval map $f : I \to I$ in ${\mathscr{A}}$ there is $\varepsilon > 0$ such that the following property holds. Let $J_0$ be an interval contained in $I$ satisfying $|J_0| \le \varepsilon$, let $n \ge 1$ be an integer, and let $J$ be a pull-back of $J_0$ by $f^n$ who... | of this section.
\[l: almost properness\ ] For each interval map $ farad: I \to I$ in $ { \mathscr{A}}$ there be $ \varepsilon > 0 $ such that the following property holds. get $ J_0 $ be an interval contained in $ I$ satisfy $ |J_0| \le \varepsilon$, let $ n \ge 1 $ be an integer, and let $ J$ be a ... | of this section.
\[l:almost prokerness\] For each nbtervan map $f : I \to I$ in ${\mathscr{A}}$ there is $\varepsilln > 0$ suxh that the following oroperty jolds. Ler $J_0$ bt an interval convzined ik $N$ satjdfyiug $|O_0| \le \varepsilon$, let $n \ge 1$ te an integer, dna pet $J$ be a pull-back of $J_0$ by $f^n$ who... | of this section. \[l:almost properness\] For each $f I \to in ${\mathscr{A}}$ there that following property holds. $J_0$ be an contained in $I$ satisfying $|J_0| \le let $n \ge 1$ be an integer, and let $J$ be a pull-back $J_0$ by $f^n$ whose closure is contained in the interior of $I$. Suppose addition for $j$ $\{1, \... | of this section.
\[l:almost propeRness\] For eaCh intErvAl mAp $F : I \to i$ in ${\mAthscr{A}}$ there is $\VArepSilon > 0$ such that the followIng prOpERty hOLdS. Let $J_0$ Be an intERvAL ConTaInEd iN $I$ SAtIsfyiNg $|J_0| \Le \varepSilon$, let $n \gE 1$ be An Integer, and leT $j$ bE a pull-back Of $J_0$ By $f^n$ who... | of this section.
\[l:alm ost proper ness\ ] F orea ch i nter val map $f : I \toI$ in ${\mathscr{A}}$there i s $\v a re psilo n > 0$s uc h tha tth e f ol l ow ing p rop erty ho lds. Let $ J_0 $be an interv a lcontainedin$I$ satisfyi ng$|J_0| \ le\ varep sil on$,let $n \ge 1$ be an in te g er, an d let $J $ be a ... | of_this section.
\[l:almost_properness\] For each interval_map $f :_I_\to I$_in ${\mathscr{A}}$_there is $\varepsilon >_0$ such that_the following property holds._Let $J_0$ be an_interval_contained in $I$ satisfying $|J_0| \le \varepsilon$, let $n \ge 1$ be an integer, and let $J$ be_a_pull-back of $J_0$_by $f^n$_who... |
the $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals of Mn while the remaining Mn orbitals are delocalized. Strong quantum fluctuations, possibly related to an electronic instability that forms the Mn moment or to the one-dimensional character of [Ti$_{4}$MnBi$_{2}$]{}, nearly overcome magnetic order.'
author:
- Abhishek Pande... | the $ d_{x^{2}-y^{2}}$ and $ d_{xy}$ orbitals of Mn while the remaining Mn orbitals are delocalized. Strong quantum fluctuations, possibly relate to an electronic imbalance that forms the Mn moment or to the one - dimensional fictional character of [ Ti$_{4}$MnBi$_{2}$ ] { }, closely overcome magnetic club.'
generato... | thf $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals on Mn while the rgmqining Mn orgitals afe delocalized. Strong quantul dluctyations, possibly relatdd to an vlectronix inwrability tizt forms the Jk momznv or to the one-cimensionan character of [Tk$_{4}$MuBi$_{2}$]{}, nearly overcome magnetic order.'
auehor:
- Abnidhek Pande... | the $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals of Mn remaining orbitals are Strong quantum fluctuations, instability forms the Mn or to the character of [Ti$_{4}$MnBi$_{2}$]{}, nearly overcome magnetic author: - Abhishek Pandey - Ping Miao - 'M. Klemm' - 'H. He' 'H. Wang' - 'X. Qian' - 'J. W. Lynn' - 'M. C. Aronson' 'Corr... | the $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals of Mn wHile the remAininG Mn OrbItAls aRe deLocalized. StronG QuanTum fluctuations, possiblY relaTeD To an ELeCtronIc instaBIlITY thAt FoRms ThE mn MomenT or To the onE-dimensionAl cHaRacter of [Ti$_{4}$MnbI$_{2}$]{}, nEarly overcOme Magnetic ordeR.'
auThor:
- AbHiSheK pande... | the $d_{x^{2}-y^{2}}$ and $d_{xy}$orbit als of M n wh ilethe remainingM n or bitals are delocalized . Str on g qua n tu m flu ctuatio n s, p oss ib ly re la t ed to a n e lectron ic instabi lit ythat forms t h eMn momentorto the one-d ime nsiona lcha r acter of [Ti$ _{4}$M n Bi$_{2 }$]{}, ne ar l y over c ome ... | the_$d_{x^{2}-y^{2}}$ and_$d_{xy}$ orbitals of Mn_while the_remaining_Mn orbitals_are_delocalized. Strong quantum_fluctuations, possibly related_to an electronic instability_that forms the_Mn_moment or to the one-dimensional character of [Ti$_{4}$MnBi$_{2}$]{}, nearly overcome magnetic order.'
author:
- Abhishek Pande... |
&\equiv& k^2+a^2(\eta)(m_{\chi}^2-2H^2) \nonumber \\
&\sim& k^2-2a^2(\eta)H^2<0.\end{aligned}$$ And, since these modes turn out to be the ones which are most important in order to calculate the relevant quantities [@li05; @v83], we would finally conclude that the diagonalization method is not the most appropriate one ... | & \equiv & k^2+a^2(\eta)(m_{\chi}^2 - 2H^2) \nonumber \\
& \sim & k^2 - 2a^2(\eta)H^2<0.\end{aligned}$$ And, since these modes turn out to be the ones which are most authoritative in decree to calculate the relevant quantities [ @li05; @v83 ], we would finally reason that the diagonalization method is not the most ... | &\equlv& k^2+a^2(\eta)(m_{\chi}^2-2H^2) \nonumber \\
&\rim& k^2-2a^2(\eta)H^2<0.\end{alntned}$$ Aid, sincs these oodes turn out to be the oned qhich are most important in order to calculare tie relevant quanvjties [@ll05; @v83], ws wounv finally conclode that the diagonalizatimn mzthod is not the most appropriate onq ... | &\equiv& k^2+a^2(\eta)(m_{\chi}^2-2H^2) \nonumber \\ &\sim& k^2-2a^2(\eta)H^2<0.\end{aligned}$$ And, modes out to the ones which to the relevant quantities @v83], we would conclude that the diagonalization method is the most appropriate one to be used in this case. What is more, order to compute the desired quantity, t... | &\equiv& k^2+a^2(\eta)(m_{\chi}^2-2H^2) \nonumber \\
&\siM& k^2-2a^2(\eta)H^2<0.\end{AlignEd}$$ ANd, sInCe thEse mOdes turn out to bE The oNes which are most importaNt in oRdER to cALcUlate The releVAnT QUanTiTiEs [@lI05; @v83], WE wOuld fInaLly concLude that thE diAgOnalization mEThOd is not the MosT appropriate One ... | &\equiv& k^2+a^2(\eta)(m_{ \chi}^2-2H ^2) \no num be r \\
&\s im& k^2-2a^2(\ e ta)H ^2<0.\end{aligned}$$ A nd, s in c e th e se mode s turno ut t o b eth e o ne s w hichare most i mportant i n o rd er to calcul a te the relev ant quantities[@l i05; @ v8 3], we wo uld fina lly co n cludethat thedi a gonali z atio... | &\equiv& k^2+a^2(\eta)(m_{\chi}^2-2H^2)_ \nonumber_\\
&\sim& k^2-2a^2(\eta)H^2<0.\end{aligned}$$ And, since_these modes_turn_out to_be_the ones which_are most important_in order to calculate_the relevant quantities_[@li05;_@v83], we would finally conclude that the diagonalization method is not the most appropriate_one_... |
is always smaller than the lowest energy gap, $p_{0,1}$. Once $eB$ becomes bigger than the squared system-size inverse (that is, $\alpha\gtrsim 10$ from Fig. \[fig:p01\]), however, the energy gap is significantly reduced and the anomalous coupling between the magnetic field and the rotation is then manifested [@Chen:2... | is always smaller than the lowest energy break, $ p_{0,1}$. Once $ eB$ become bigger than the squared system - size inverse (that is, $ \alpha\gtrsim 10 $ from Fig. \[fig: p01\ ]), however, the energy col is significantly reduced and the anomalous yoke between the magnetic field and the rotation is then manifested ... | is always smaller than the lowest energy yqp, $p_{0,1}$. Oice $eB$ gecomes cigger than the squared systxm-size incerse (that is, $\alpha\gtrrim 10$ from Fig. \[fig:p01\]), howtver, the energy gel is sinuificzktly xevuced and the akomalous cogpling between tfe magnetic field and the rotation is then msnlfested [@Chen:2... | is always smaller than the lowest energy Once becomes bigger the squared system-size from \[fig:p01\]), however, the gap is significantly and the anomalous coupling between the field and the rotation is then manifested [@Chen:2015hfc; @Hattori:2016njk]. Integration Measure and Reweighted functions {#sec:measure} ======... | is always smaller than the lowEst energy gAp, $p_{0,1}$. OnCe $eb$ beCoMes bIggeR than the squareD SystEm-size inverse (that is, $\alpHa\gtrSiM 10$ From fIg. \[Fig:p01\]), hOwever, tHE eNERgy GaP iS siGnIFiCantlY reDuced anD the anomalOus CoUpling betweeN ThE magnetic fIelD and the rotatIon Is then MaNifESted [@CHen:2... | is always smaller than th e lowest e nergy ga p,$p _{0, 1}$. Once $eB$ bec o mesbigger than the square d sys te m -siz e i nvers e (that is , $\a lp ha \gt rs i m10$ f rom Fig. \ [fig:p01\] ),ho wever, the e n er gy gap issig nificantly r edu ced an dthe anoma lou s cou plingb etween the magn et i c fiel d and t... | is_always smaller_than the lowest energy_gap, $p_{0,1}$._Once_$eB$ becomes_bigger_than the squared_system-size inverse (that_is, $\alpha\gtrsim 10$ from_Fig. \[fig:p01\]), however, the_energy_gap is significantly reduced and the anomalous coupling between the magnetic field and the_rotation_is then_manifested [@Chen:2... |
of squares. This gives that $$\sum_{i\ge 4} 2i \cdot 2q_i \ge 2\cdot 4\cdot 2q_2.$$ By Equation we then have $-3+3\sum_{i\ge 2}q_i\ge 4q_2+2q_2+3q_3$, that is, $q_2 < \sum_{i\ge 4}q_i$. But then in average each face has more than $6$ edges which contradicts Thm. \[sum\_rank2\]. Hence there is an object $a$ such that ... | of squares. This gives that $ $ \sum_{i\ge 4 } 2i \cdot 2q_i \ge 2\cdot 4\cdot 2q_2.$$ By Equation we then have $ -3 + 3\sum_{i\ge 2}q_i\ge 4q_2 + 2q_2 + 3q_3 $, that is, $ q_2 < \sum_{i\ge 4}q_i$. But then in modal each grimace has more than $ 6 $ edges which contradicts Thm. \[sum\_rank2\ ]. therefore there is a... | of squares. This gives that $$\sum_{i\ge 4} 2i \cdot 2q_i \ge 2\rdot 4\cdkt 2q_2.$$ By Dquation we then have $-3+3\sum_{i\ge 2}q_u\ge 4q_2+2w_2+3q_3$, that is, $q_2 < \sum_{i\ge 4}q_k$. But thej in aveeage wach face izs more than $6$ cdges xhich contradicjs Thm. \[sum\_rann2\]. Hence there hs au object $a$ such that ... | of squares. This gives that $$\sum_{i\ge 4} 2q_i 2\cdot 4\cdot By Equation we that $q_2 < \sum_{i\ge But then in each face has more than $6$ which contradicts Thm. \[sum\_rank2\]. Hence there is an object $a$ such that there $\alpha,\beta,\gamma\in R^a_+$ as above satisfying Equation . We have $\alpha+\gamma, \beta +\g... | of squares. This gives that $$\sum_{I\ge 4} 2i \cdot 2q_i \Ge 2\cdoT 4\cdOt 2q_2.$$ by equaTion We then have $-3+3\sum_{i\GE 2}q_i\gE 4q_2+2q_2+3q_3$, that is, $q_2 < \sum_{i\ge 4}q_i$. But Then iN aVEragE EaCh facE has morE ThAN $6$ EdgEs WhIch CoNTrAdictS ThM. \[sum\_ranK2\]. Hence therE is An Object $a$ such tHAt ... | of squares. This gives th at $$\sum_ {i\ge 4} 2i \ cdot 2q_ i \ge 2\cdot 4 \ cdot 2q_2.$$ By Equation we th en have $- 3+3\s um_{i\g e 2 } q _i\ ge 4 q_2 +2 q _2 +3q_3 $,that is , $q_2 < \ sum _{ i\ge 4}q_i$. Bu t then inave rage each fa cehas mo re th a n $6$ ed ges w hich c o ntradi cts Thm.\[ s um\_ra n k2... | of_squares. This_gives that $$\sum_{i\ge 4}_2i \cdot_2q_i_\ge 2\cdot_4\cdot_2q_2.$$ By Equation _we then have_$-3+3\sum_{i\ge 2}q_i\ge 4q_2+2q_2+3q_3$, that_is, $q_2 <_\sum_{i\ge_4}q_i$. But then in average each face has more than $6$ edges which contradicts_Thm. \[sum\_rank2\]._Hence there_is_an_object $a$ such that ... |
}^{N}
g(\tau_n,\boldsymbol{a}_n)\boldsymbol{a}_n\boldsymbol{a}_n^T\end{aligned}$$ where $g(\tau_n,\boldsymbol{a}_n)$ is defined as $$\begin{aligned}
g(\tau_n,\boldsymbol{a}_n) \triangleq \frac {f_{w}^2
(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)}
{F_{w}(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)(1-F_{w}(\boldsymbol{a}_... | } ^{N }
g(\tau_n,\boldsymbol{a}_n)\boldsymbol{a}_n\boldsymbol{a}_n^T\end{aligned}$$ where $ g(\tau_n,\boldsymbol{a}_n)$ is defined as $ $ \begin{aligned }
g(\tau_n,\boldsymbol{a}_n) \triangleq \frac { f_{w}^2
(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n) }
{ F_{w}(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)(1 - F_... | }^{N}
g(\twu_n,\boldsymbol{a}_n)\boldsymbou{a}_n\boldsymbol{a}_n^J\ebd{aligied}$$ whede $g(\tau_n,\coldsymbol{a}_n)$ is defined as $$\bxgin{qligntb}
g(\tau_n,\boldsymbol{a}_n) \tfiangleq \vrac {f_{w}^2
(\bildsbmbol{a}_n^{T}\boldsymbol{h}-\tau_n)}
{F_{w}(\boldsglbol{c}_n^{V}\boldsymbol{h}-\tau_k)(1-F_{w}(\boldsymbml{a}_... | }^{N} g(\tau_n,\boldsymbol{a}_n)\boldsymbol{a}_n\boldsymbol{a}_n^T\end{aligned}$$ where $g(\tau_n,\boldsymbol{a}_n)$ is defined as \triangleq {f_{w}^2 (\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)} \label{g-function}\end{aligned}$$ in which function of $w_n$. Accordingly, CRB matrix for estimation problem (\[quantized-da... | }^{N}
g(\tau_n,\boldsymbol{a}_n)\boldsymBol{a}_n\boldsYmbol{A}_n^T\End{AlIgneD}$$ wheRe $g(\tau_n,\boldsymBOl{a}_n)$ Is defined as $$\begin{aligneD}
g(\tau_N,\bOLdsyMBoL{a}_n) \trIangleq \FRaC {F_{W}^2
(\boLdSyMboL{a}_N^{t}\bOldsyMboL{h}-\tau_n)}
{F_{W}(\boldsymboL{a}_n^{t}\bOldsymbol{h}-\taU_N)(1-F_{W}(\boldsymboL{a}_... | }^{N}
g(\tau_n,\boldsymbol {a}_n)\bol dsymb ol{ a}_ n\ bold symb ol{a}_n^T\end{ a lign ed}$$ where $g(\tau_n, \bold sy m bol{ a }_ n)$ i s defin e da s $$ \b eg in{ al i gn ed}
g (\t au_n,\b oldsymbol{ a}_ n) \triangleq\ fr ac {f_{w}^ 2
( \boldsymbol{ a}_ n^{T}\ bo lds y mbol{ h}- \tau_ n)}
{F _ {w}(\b oldsymbol {a }... | }^{N}
g(\tau_n,\boldsymbol{a}_n)\boldsymbol{a}_n\boldsymbol{a}_n^T\end{aligned}$$ where_$g(\tau_n,\boldsymbol{a}_n)$ is_defined as $$\begin{aligned}
g(\tau_n,\boldsymbol{a}_n) \triangleq_\frac {f_{w}^2
(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)}
{F_{w}(\boldsymbol{a}_n^{T}\boldsymbol{h}-\tau_n)(1-F_{w}(\boldsymbol{a}_... |
),
$E(2,0,0)=2 \epsilon_{\alpha}+U_{\alpha \alpha}$ (2e singlet),
$E(1,1,1)=\epsilon_{\alpha}+ \epsilon_{\beta}+U_{\alpha \beta}-J_{\alpha \beta}$ (2e triplet),
$E(1,1,0)=\epsilon_{\alpha}+ \epsilon_{\beta}+U_{\alpha \beta}+J_{\alpha \beta}$ (2e singlet),
$E(2,1,\frac{1}{2})=2\epsilon_{\alpha}+ \epsilon_{\beta}+2U_... | ),
$ E(2,0,0)=2 \epsilon_{\alpha}+U_{\alpha \alpha}$ (2e singlet),
$ E(1,1,1)=\epsilon_{\alpha}+ \epsilon_{\beta}+U_{\alpha \beta}-J_{\alpha \beta}$ (2e triplet),
$ E(1,1,0)=\epsilon_{\alpha}+ \epsilon_{\beta}+U_{\alpha \beta}+J_{\alpha \beta}$ (2e singlet),
$ E(2,1,\frac{1}{2})=2\epsilon_{\alpha}+ \epsilon... | ),
$E(2,0,0)=2 \eosilon_{\alpha}+U_{\alpha \alpha}$ (2t singlet),
$E(1,1,1)=\epsilon_{\copha}+ \e'silon_{\bsta}+U_{\alphx \beta}-J_{\alpha \beta}$ (2e triplet),
$E(1,1,0)=\xpsioon_{\alkka}+ \epsilon_{\beta}+U_{\alpha \beta}+J_{\alpja \beta}$ (2w siiglet),
$E(2,1,\frac{1}{2})=2\epsiloi_{\zlpha}+ \eifiloh_{\neta}+2U_... | ), $E(2,0,0)=2 \epsilon_{\alpha}+U_{\alpha \alpha}$ (2e singlet), $E(1,1,1)=\epsilon_{\alpha}+ \beta}$ triplet), $E(1,1,0)=\epsilon_{\alpha}+ \beta}+J_{\alpha \beta}$ (2e \beta}$ $E(2,2,0)=2\epsilon_{\alpha}+ 2\epsilon_{\beta}+4U_{\alpha \beta}+U_{\alpha \beta}-2J_{\alpha \beta}$ (4e). addition energies measured in exp... | ),
$E(2,0,0)=2 \epsilon_{\alpha}+U_{\alpha \alpha}$ (2e Singlet),
$E(1,1,1)=\epSilon_{\AlpHa}+ \ePsIlon_{\Beta}+u_{\alpha \beta}-J_{\alpHA \betA}$ (2e triplet),
$E(1,1,0)=\epsilon_{\alpha}+ \EpsilOn_{\BEta}+U_{\ALpHa \betA}+J_{\alpha \BEtA}$ (2E SinGlEt),
$e(2,1,\frAc{1}{2})=2\EPsIlon_{\aLphA}+ \epsiloN_{\beta}+2U_... | ),
$E(2,0,0)=2 \epsilon_{ \alpha}+U_ {\alp ha\al ph a}$(2esinglet),
$E( 1 ,1,1 )=\epsilon_{\alpha}+ \ epsil on _ {\be t a} +U_{\ alpha \ b et a } -J_ {\ al pha \ b et a}$ ( 2etriplet ),
$E(1,1 ,0) =\ epsilon_{\al p ha }+ \epsilo n_{ \beta}+U_{\a lph a \bet a} +J_ { \alph a \ beta} $ (2es inglet ),
$E(2, 1, \ frac{... | ),
$E(2,0,0)=2 \epsilon_{\alpha}+U_{\alpha_\alpha}$ (2e_singlet),
$E(1,1,1)=\epsilon_{\alpha}+ \epsilon_{\beta}+U_{\alpha \beta}-J_{\alpha \beta}$_(2e triplet),
$E(1,1,0)=\epsilon_{\alpha}+_\epsilon_{\beta}+U_{\alpha_\beta}+J_{\alpha \beta}$_(2e_singlet),
$E(2,1,\frac{1}{2})=2\epsilon_{\alpha}+ \epsilon_{\beta}+2U_... |
\delta_{ij} - n^{-1} \big)^2 \cdot \mathbf{E}_{ii} } \right\Vert}
= {\left\Vert {\sum_{i=1}^d \sum_{j=1}^n n^{-1}\big(1 - n^{-1}\big) \cdot \mathbf{E}_{ii} } \right\Vert}
\approx 1.$$ The large deviation parameter is $$L^2 = {\operatorname{\mathbb{E}}}\max\nolimits_{i,j} {{\left\Vert { \big(\delta_{ij} - n^{-1... | \delta_{ij } - n^{-1 } \big)^2 \cdot \mathbf{E}_{ii } } \right\Vert }
= { \left\Vert { \sum_{i=1}^d \sum_{j=1}^n n^{-1}\big(1 - n^{-1}\big) \cdot \mathbf{E}_{ii } } \right\Vert }
\approx 1.$$ The large deviation parameter is $ $ L^2 = { \operatorname{\mathbb{E}}}\max\nolimits_{i, j } { { \left\Vert { \big(\... | \depta_{ij} - n^{-1} \big)^2 \cdot \mathbf{T}_{ii} } \right\Vert}
= {\left\Vxrt {\sum_{j=1}^d \sum_{j=1}^n n^{-1}\big(1 - n^{-1}\big) \cdot \mathbf{E}_{ii} } \cighr\Vert}
\approx 1.$$ The large aeviation parametwr iw $$L^2 = {\operatorname{\mabkbb{E}}}\mzw\nolikmts_{i,j} {{\left\Vert { \big(\delta_{ib} - n^{-1... | \delta_{ij} - n^{-1} \big)^2 \cdot \mathbf{E}_{ii} } {\left\Vert \sum_{j=1}^n n^{-1}\big(1 n^{-1}\big) \cdot \mathbf{E}_{ii} large parameter is $$L^2 {\operatorname{\mathbb{E}}}\max\nolimits_{i,j} {{\left\Vert { - n^{-1} \big)\cdot \mathbf{E}_{ii} } \right\Vert}^2} 1.$$ Therefore, the large-deviation term drives the up... | \delta_{ij} - n^{-1} \big)^2 \cdot \mathbf{E}_{ii} } \rIght\Vert}
= {\leFt\VerT {\suM_{i=1}^d \SuM_{j=1}^n n^{-1}\Big(1 - n^{-1}\Big) \cdot \mathbf{E}_{II} } \rigHt\Vert}
\approx 1.$$ The large deViatiOn PAramETeR is $$L^2 = {\oPeratorNAmE{\MAthBb{e}}}\mAx\nOlIMiTs_{i,j} {{\lEft\vert { \big(\Delta_{ij} - n^{-1... | \delta_{ij} - n^{-1} \big )^2 \cdot\math bf{ E}_ {i i} } \ri ght\Vert}
= {\l eft\Vert {\sum_{i=1}^d \sum _{ j =1}^ n n ^{-1} \big(1- n ^ { -1} \b ig ) \ cd o t\math bf{ E}_{ii} } \right\ Ver t}
\approx 1. $$ The lar gedeviation pa ram eter i s$$L ^ 2 = { \op erato rname{ \ mathbb {E}}}\max \n o limits _ ... | \delta_{ij}_- n^{-1}_\big)^2 \cdot \mathbf{E}_{ii} }_\right\Vert}
__ =_{\left\Vert_{\sum_{i=1}^d \sum_{j=1}^n n^{-1}\big(1_- n^{-1}\big) \cdot_\mathbf{E}_{ii} } \right\Vert}
_ \approx_1.$$_The large deviation parameter is $$L^2 = {\operatorname{\mathbb{E}}}\max\nolimits_{i,j} {{\left\Vert { \big(\delta_{ij} - n^{-1... |
the effect of dropout can be found in Appendix D.4.
Discussion {#sec:discussion}
==========
The results show that the system has an acceptable TRE of $3.3\%\pm1.4\%$. The MVPM outperforms DeepLab V3+ and Dense 3D CRF for semantic segmentation, with statistically significant improvement in less frequently appearing c... | the effect of dropout can be found in Appendix D.4.
Discussion { # sec: discussion }
= = = = = = = = = =
The results show that the arrangement have an acceptable TRE of $ 3.3\%\pm1.4\%$. The MVPM outperforms DeepLab V3 + and Dense 3D CRF for semantic division, with statistically significant improvement in less ... | thf effect of dropout can ne found in Appeueix D.4.
Dmscussikn {#sec:dircussion}
==========
The results show thav thw sysuvm has an acceptable GRE of $3.3\%\pm1.4\%$. The MVPN ouuperforms DeepLab V3+ and Dcuse 3D GRF fmc semantic segmgntation, witv statisticallf riynificant improvement in less frequegtly aplewring c... | the effect of dropout can be found D.4. {#sec:discussion} ========== results show that TRE $3.3\%\pm1.4\%$. The MVPM DeepLab V3+ and 3D CRF for semantic segmentation, with significant improvement in less frequently appearing classes such as Human, Mayo Stand, Sterile and Anesthesia Cart. Ablation studies have also show... | the effect of dropout can be foUnd in AppenDix D.4.
DIscUssIoN {#sec:DiscUssion}
==========
The resulTS shoW that the system has an accEptabLe trE of $3.3\%\PM1.4\%$. THe MVPm outperFOrMS deePLAb v3+ anD DENsE 3D CRF For SemantiC segmentatIon, WiTh statisticaLLy SignificanT imProvement in lEss FrequeNtLy aPPeariNg c... | the effect of dropout can be foundin Ap pen dix D .4.
Dis cussion {#sec: d iscu ssion}
==========
The resu lt s sho w t hat t he syst e mh a s a nac cep ta b le TREof$3.3\%\ pm1.4\%$.The M VPM outperfo r ms DeepLab V 3+and Dense 3D CR F forse man t ic se gme ntati on, wi t h stat istically s i gnific a nt i... | the_effect of_dropout can be found_in Appendix_D.4.
Discussion_{#sec:discussion}
==========
The results_show_that the system_has an acceptable_TRE of $3.3\%\pm1.4\%$. The_MVPM outperforms DeepLab_V3+_and Dense 3D CRF for semantic segmentation, with statistically significant improvement in less frequently_appearing_c... |
in $K$ will be done using iterative methods. In this case, the computational effort to computing (\[score\]) or (\[ascore\]) is roughly linear in the number of solves required (although see Section \[sec4\] for methods that make $N$ solves for a common matrix $K$ somewhat less than $N$ times the effort of one solve), ... | in $ K$ will be done using iterative methods. In this case, the computational campaign to calculate (\[score\ ]) or (\[ascore\ ]) is roughly linear in the numeral of solves ask (although see Section \[sec4\ ] for method acting that have $ N$ solves for a common matrix $ K$ slightly less than $ N$ times the effort of on... | in $K$ will be done using ittrative methods. Iu this rase, ths computxtional effort to computing (\[dcire\]) oe (\[ascore\]) is roughly livear in tje numbee of wolves reqnjred (albkough dee Vxction \[sec4\] for kethods thdt make $N$ solvas flr a common matrix $K$ somewhat less ehan $N$ yiles the effort of pge sklve), ... | in $K$ will be done using iterative this the computational to computing (\[score\]) in number of solves (although see Section for methods that make $N$ solves a common matrix $K$ somewhat less than $N$ times the effort of one so that (\[ascore\]) is much easier to compute than (\[score\]) when $N/n$ is An feature the (... | in $K$ will be done using iteratiVe methods. IN this CasE, thE cOmpuTatiOnal effort to coMPutiNg (\[score\]) or (\[ascore\]) is roughLy linEaR In thE NuMber oF solves REqUIRed (AlThOugH sEE SEctioN \[seC4\] for metHods that maKe $N$ SoLves for a commON mAtrix $K$ someWhaT less than $N$ tiMes The effOrT of ONe solVe), ... | in $K$ will be done using iterative meth ods . I nthis cas e, the computa t iona l effort to computing(\[sc or e \])o r(\[as core\]) is r oug hl ylin ea r i n the nu mber of solves re qui re d (althoughs ee Section \ [se c4\] for met hod s that m ake $N$ s olv es fo r a co m mon ma trix $K$so m ewhatl ess tha ... | in_$K$ will_be done using iterative_methods. In_this_case, the_computational_effort to computing_(\[score\]) or (\[ascore\])_is roughly linear in_the number of_solves_required (although see Section \[sec4\] for methods that make $N$ solves for a common_matrix_$K$ somewhat_less_than_$N$ times the effort of_one solve), ... |
dR_{k+1}}F_{_{k+1}}^T=F_{2_{k}}^T
{\ensuremath{\left( F_{_{k}}J_{dR_{k}}-J_{dR_{k}}F_{_{k}}^T \right)}}F_{2_{k}}-h^2S(M_{k+1}),\label{eqn:findFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T
=F_{2_{k}}^T{\ensuremath{\left( F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T \right)}}F_{2_{k}}
+h^2X_{_{k+1}}\times{\ensuremath{\frac{\partia... | dR_{k+1}}F_{_{k+1}}^T = F_{2_{k}}^T
{ \ensuremath{\left (F_{_{k}}J_{dR_{k}}-J_{dR_{k}}F_{_{k}}^T \right)}}F_{2_{k}}-h^2S(M_{k+1}),\label{eqn: findFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T
= F_{2_{k}}^T{\ensuremath{\left (F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T \right)}}F_{2_{k } }
+ h^2X_{_{k+1}}\times{\ensuremath... | dR_{k+1}}V_{_{k+1}}^T=F_{2_{k}}^T
{\ensuremath{\left( F_{_{k}}J_{aR_{k}}-J_{dR_{k}}F_{_{k}}^T \righj)}}F_{2_{j}}-h^2S(M_{k+1}),\lebel{eqn:rindFl}\\F_{2_{k+1}}G_{d_2}-J_{d_2}F_{2_{k+1}}^T
=F_{2_{k}}^T{\ensuremath{\left( F_{2_{k}}J_{v_2}-J_{d_2}F_{2_{j}}^T \ritht)}}F_{2_{k}}
+h^2X_{_{k+1}}\times{\ensurematf{\frac{\partpa... | dR_{k+1}}F_{_{k+1}}^T=F_{2_{k}}^T {\ensuremath{\left( F_{_{k}}J_{dR_{k}}-J_{dR_{k}}F_{_{k}}^T \right)}}F_{2_{k}}-h^2S(M_{k+1}),\label{eqn:findFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T =F_{2_{k}}^T{\ensuremath{\left( F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T \right)}}F_{2_{k}} X_{_{k+1}}}}}+h^2S(M_{k+1}) R_{_{k+1}}=F_{2_{k... | dR_{k+1}}F_{_{k+1}}^T=F_{2_{k}}^T
{\ensuremath{\left( F_{_{k}}j_{dR_{k}}-J_{dR_{k}}F_{_{k}}^t \righT)}}F_{2_{k}}-H^2S(M_{K+1}),\lAbel{Eqn:fIndFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T
=f_{2_{K}}^T{\enSuremath{\left( F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T \rIght)}}F_{2_{K}}
+h^2x_{_{K+1}}\timES{\eNsureMath{\fraC{\PaRTIa... | dR_{k+1}}F_{_{k+1}}^T=F_{2 _{k}}^T
{\ ensur ema th{ \l eft( F_{ _{k}}J_{dR_{k} } -J_{ dR_{k}}F_{_{k}}^T \rig ht)}} F_ { 2_{k } }- h^2S( M_{k+1} ) ,\ l a bel {e qn :fi nd F l} \\F_{ 2_{ k+1}}J_ {d_2}-J_{d _2} F_ {2_{k+1}}^T= F_ {2_{k}}^T{ \en suremath{\le ft( F_{2_ {k }}J _ {d_2} -J_ {d_2} F_{2_{ k }}^T \ right)}}F _{... | dR_{k+1}}F_{_{k+1}}^T=F_{2_{k}}^T
{\ensuremath{\left( F_{_{k}}J_{dR_{k}}-J_{dR_{k}}F_{_{k}}^T_\right)}}F_{2_{k}}-h^2S(M_{k+1}),\label{eqn:findFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T
=F_{2_{k}}^T{\ensuremath{\left( F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T_\right)}}F_{2_{k}}
+h^2X_{_{k+1}}\times{\ensuremath{\frac{\partia... |
}}{
\varepsilon}_k = 0$ such that ${
\lim _{k \to +\infty }}W_1 ({
f ^{\varepsilon _k}}(t), f(t)) = 0$ uniformly for $t \in [0,T], T>0$ if ${
\mathrm{supp\;}}{
f ^{\mathrm{in}}}\subset \{ (x,v) \;:\;|x|\leq L_0, |v| = r\}$ and ${
\lim _{k \to +\infty }}W_1 ({
f ^{\varepsilon _k}}(t), f(t)) = 0$ uniformly for $t \in [\d... | } } {
\varepsilon}_k = 0 $ such that $ {
\lim _ { k \to + \infty } } W_1 ({
f ^{\varepsilon _ k}}(t), f(t) ) = 0 $ uniformly for $ t \in [ 0,T ], T>0 $ if $ {
\mathrm{supp\; } } {
f ^{\mathrm{in}}}\subset \ { (x, v) \;:\;|x|\leq L_0, |v| = r\}$ and $ {
\lim _ { k \to + \infty } } W_1 ({
f ^{\varepsilon _ ... | }}{
\varfpsilon}_k = 0$ such that ${
\lim _{k \to +\infty }}W_1 ({
f ^{\varepsmlon _k}}(t), f(t)) = 0$ unkformly for $t \in [0,T], T>0$ if ${
\mathcm{supp\;}}{
f ^{\mqthrm{in}}}\subset \{ (x,v) \;:\;|x|\leq L_0, |v| = r\}$ ajd ${
\lim _{k \to +\mnfty }}W_1 ({
f ^{\varepsmmon _k}}(t), n(c)) = 0$ uhlformnb for $t \in [\d... | }}{ \varepsilon}_k = 0$ such that ${ \to }}W_1 ({ ^{\varepsilon _k}}(t), f(t)) \in T>0$ if ${ f ^{\mathrm{in}}}\subset \{ \;:\;|x|\leq L_0, |v| = r\}$ and \lim _{k \to +\infty }}W_1 ({ f ^{\varepsilon _k}}(t), f(t)) = 0$ uniformly $t \in [\delta,T], T>\delta >0$ if ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v)... | }}{
\varepsilon}_k = 0$ such that ${
\lim _{k \to +\Infty }}W_1 ({
f ^{\varEpsilOn _k}}(T), f(t)) = 0$ UnIforMly fOr $t \in [0,T], T>0$ if ${
\mathrM{Supp\;}}{
F ^{\mathrm{in}}}\subset \{ (x,v) \;:\;|x|\leq L_0, |V| = r\}$ and ${
\LiM _{K \to +\iNFtY }}W_1 ({
f ^{\vaRepsiloN _K}}(t), F(T)) = 0$ UniFoRmLy fOr $T \In [\D... | }}{
\varepsilon}_k = 0$ su ch that ${
\lim _{ k \ to +\i nfty }}W_1 ({
f ^{ \ vare psilon _k}}(t), f(t))= 0$un i form l yfor $ t \in [ 0 ,T ] , T> 0$ i f $ {\ ma thrm{ sup p\;}}{f ^{\mathr m{i n} }}\subset \{ (x ,v) \;:\;| x|\ leq L_0, |v| =r\}$ a nd ${ \lim_{k \to+\inft y }}W_1 ({
f ^{\ va r epsilo n _k}}(... | }}{
\varepsilon}_k =_0$ such_that ${
\lim _{k \to_+\infty }}W_1_({
f_^{\varepsilon _k}}(t),_f(t))_= 0$ uniformly_for $t \in_[0,T], T>0$ if ${
\mathrm{supp\;}}{
f_^{\mathrm{in}}}\subset \{ (x,v)_\;:\;|x|\leq_L_0, |v| = r\}$ and ${
\lim _{k \to +\infty }}W_1 ({
f ^{\varepsilon _k}}(t), f(t))_=_0$ uniformly_for_$t_\in [\d... |
m)=2^a-1=l\#m$ whenever $2^{a-1}{\leqslant}l,m<2^a$.
Let $r,s,t\in\N$. From Proposition \[backdiag\] follows that $\tau(r,s){\geqslant}t$ if, and only if, there exist $\rho{\leqslant}r$ and $\s{\leqslant}s$ with $\rho+\s=t$ such that $Q_t^{(\rho,\s)}\equiv1$. Now suppose that $r,s{\leqslant}2^a$, and consider $\rho{\l... | m)=2^a-1 = l\#m$ whenever $ 2^{a-1}{\leqslant}l, m<2^a$.
Let $ r, s, t\in\N$. From Proposition \[backdiag\ ] follows that $ \tau(r, s){\geqslant}t$ if, and only if, there exist $ \rho{\leqslant}r$ and $ \s{\leqslant}s$ with $ \rho+\s = t$ such that $ Q_t^{(\rho,\s)}\equiv1$. nowadays presuppose that $ r, s{\leqsla... | m)=2^a-1=l\#l$ whenever $2^{a-1}{\leqslant}l,m<2^a$.
Ltt $r,s,t\in\N$. From Propositimn \[backsiag\] foluows that $\tau(r,s){\geqslant}t$ if, end inly uf, there exist $\rho{\leqsuant}r$ and $\s{\leqslabt}s$ xith $\rho+\s=t$ such vgat $Q_t^{(\rmj,\s)}\eqhlv1$. Noc wuppose that $r,x{\leqslant}2^a$, and consider $\shu{\l... | m)=2^a-1=l\#m$ whenever $2^{a-1}{\leqslant}l,m<2^a$. Let $r,s,t\in\N$. From Proposition that if, and if, there exist such $Q_t^{(\rho,\s)}\equiv1$. Now suppose $r,s{\leqslant}2^a$, and consider and $\s{\leqslant}s$. If $\rho+\s<2^a$ then $$Q_{\rho+\s}^{(\rho,\s)}\equiv\binom{\rho+\s}{\rho}= \binom{\rho+\s+2^a}{\rho+2^a... | m)=2^a-1=l\#m$ whenever $2^{a-1}{\leqslant}l,m<2^a$.
LEt $r,s,t\in\N$. FrOm ProPosItiOn \[BackDiag\] Follows that $\tau(R,S){\geqSlant}t$ if, and only if, there Exist $\RhO{\LeqsLAnT}r$ and $\S{\leqslaNT}s$ WITh $\rHo+\S=t$ SucH tHAt $q_t^{(\rho,\S)}\eqUiv1$. Now sUppose that $R,s{\lEqSlant}2^a$, and conSIdEr $\rho{\l... | m)=2^a-1=l\#m$ whenever $2 ^{a-1}{\le qslan t}l ,m< 2^ a$.
Let $r,s,t\in\N$. From Proposition \[backdia g\] f ol l owst ha t $\t au(r,s) { \g e q sla nt }t $ i f, an d onl y i f, ther e exist $\ rho {\ leqslant}r$a nd $\s{\leqs lan t}s$ with $\ rho +\s=t$ s uch that$Q_ t^{(\ rho,\s ) }\equi v1$. Nowsu p pose t h at... | m)=2^a-1=l\#m$ whenever_$2^{a-1}{\leqslant}l,m<2^a$.
Let $r,s,t\in\N$._From Proposition \[backdiag\] follows that_$\tau(r,s){\geqslant}t$ if,_and_only if,_there_exist $\rho{\leqslant}r$ and_$\s{\leqslant}s$ with $\rho+\s=t$_such that $Q_t^{(\rho,\s)}\equiv1$. Now_suppose that $r,s{\leqslant}2^a$,_and_consider $\rho{\l... |
\max_{R^d, R^{ud}} \left\{ R^d + R^{ud} \right\},\end{aligned}$$ where $$\begin{aligned}
\begin{aligned}
\left( R^d, R^d, R^{ud}, R^d \right) &\in
\col{}{\\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\Cmac{\bH_d, \bI + P \bH_{ud} \bH_{ud}^T + P \bHinter \bHinter^T}.
\end{aligned}\end{aligned}$$
Time Sh... | \max_{R^d, R^{ud } } \left\ { R^d + R^{ud } \right\},\end{aligned}$$ where $ $ \begin{aligned }
\begin{aligned }
\left (R^d, R^d, R^{ud }, R^d \right) & \in
\col{}{\\ & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! }
\Cmac{\bH_d, \bI + P \bH_{ud } \bH_{ud}^T + P \bHinter \bHinter^T }.
\end{aligned}... | \mad_{R^d, R^{ud}} \left\{ R^d + R^{ud} \rigmt\},\end{aligned}$$ whete $$\begin{eligned}
\gegin{aliened}
\left( R^d, R^d, R^{ud}, R^d \rmght) &\in
\col{}{\\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\Cmac{\bH_d, \cI + P \bH_{uf} \bH_{ud}^T + P \bIinter \bHinter^T}.
\eis{aligned}\end{aljnned}$$
Tnmx Sh... | \max_{R^d, R^{ud}} \left\{ R^d + R^{ud} \right\},\end{aligned}$$ \begin{aligned} R^d, R^d, R^d \right) &\in P \bH_{ud}^T + P \bHinter^T}. \end{aligned}\end{aligned}$$ Time {#s:time-sharing} ============ The simplest scheme that scheduling is that of time sharing. A two-phase time-sharing scheme alternates between trans... | \max_{R^d, R^{ud}} \left\{ R^d + R^{ud} \right\},\end{Aligned}$$ wheRe $$\begIn{aLigNeD}
\begIn{alIgned}
\left( R^d, R^d, R^{UD}, R^d \rIght) &\in
\col{}{\\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\Cmac{\bH_d, \bI + P \bH_{uD} \bH_{ud}^t + P \BhintER \bhinteR^T}.
\end{alIGnED}\End{AlIgNed}$$
tiME SH... | \max_{R^d, R^{ud}} \left\ { R^d + R^ {ud}\ri ght \} ,\en d{al igned}$$ where $$\b egin{aligned}
\begin{a ligne d} \ le ft( R ^d, R^d , R ^ { ud} ,R^ d \r i gh t) &\ in \co l{}{\\ &\! \!\ !\ !\!\!\!\!\!\ ! \! \!\!\!\!\! \!\ !\!}
\C mac{\b H_ d,\ bI +P \ bH_{u d} \bH _ {ud}^T + P \bHi nt e r \bHi n ter^T}... | \max_{R^d,_R^{ud}} \left\{_R^d + R^{ud} \right\},\end{aligned}$$_where $$\begin{aligned}
\begin{aligned}
__ _\left(_R^d, R^d, R^{ud},_R^d \right)_&\in
_\col{}{\\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
__ \Cmac{\bH_d, \bI + P \bH_{ud} \bH_{ud}^T + P \bHinter_\bHinter^T}.
\end{aligned}\end{aligned}$$
Time_Sh... |
it can query the value of the function at any point with one computation step, although it does not have the full description of the function. (See [@goldreichFound] for a detailed description.)
We now define *pseudorandom functions* (see [@goldreich1986construct]). Intuitively, this is a family of functions indexed ... | it can query the value of the function at any point with one computation gradation, although it does not take the full description of the function. (See [ @goldreichFound ] for a detailed description .)
We immediately define * pseudorandom functions * (experience [ @goldreich1986construct ]). Intuitively, this is a ... | it can query the value of uhe function at aut poinv with kne compjtation step, although it doed bot hqve the full descriptiun of the functiob. (Set [@goldreichFound] hkr a debciled fescxi'tion.)
We now deflne *pseudordndom functionv* (rez [@goldreich1986construct]). Intuitively, thif is a galily of functijns pnqexes ... | it can query the value of the any with one step, although it description the function. (See for a detailed We now define *pseudorandom functions* (see Intuitively, this is a family of functions indexed by a seed, such that is hard to distinguish a random member of the family from a truly selected A function (PRF)* is a... | it can query the value of the fuNction at anY poinT wiTh oNe CompUtatIon step, althougH It doEs not have the full descriPtion Of THe fuNCtIon. (SeE [@goldreIChfOUnd] FoR a DetAiLEd DescrIptIon.)
We noW define *pseUdoRaNdom functionS* (SeE [@goldreich1986ConStruct]). IntuitIveLy, this Is A faMIly of FunCtionS indexED ... | it can query the value of the funct ion a t a nypo intwith one computati o n st ep, although it does n ot ha ve thef ul l des criptio n o f the f un cti on . ( See [ @go ldreich Found] for ade tailed descr i pt ion.)
Wenow define *pse udo random f unc t ions* (s ee [@ goldre i ch1986 construct ]) . Intui t ... | it_can query_the value of the_function at_any_point with_one_computation step, although_it does not_have the full description_of the function._(See_[@goldreichFound] for a detailed description.)
We now define *pseudorandom functions* (see [@goldreich1986construct]). Intuitively, this is_a_family of_functions_indexed_... |
such that $X(\sigma)=0$ or, equivalently, $X_t(\sigma)=\sigma$ for all $t \in {{\mathbb R}}$. The set formed by singularities is the *singular set of $X$* denoted ${\mathrm{Sing}}(X)$ and ${\operatorname{Per}}(X)$ is the set of periodic points of $X$. We say that a *singularity is hyperbolic* if the eigenvalues of the... | such that $ X(\sigma)=0 $ or, equivalently, $ X_t(\sigma)=\sigma$ for all $ t \in { { \mathbb R}}$. The set formed by singularities is the * remarkable hardening of $ X$ * denoted $ { \mathrm{Sing}}(X)$ and $ { \operatorname{Per}}(X)$ is the set of periodic points of $ X$. We allege that a * singularity is hyperbolic *... | sufh that $X(\sigma)=0$ or, equivauently, $X_t(\sigma)=\sntma$ foc all $t \in {{\mathcb R}}$. The set formed by singupaeitiew is the *singular set uf $X$* denoned ${\mathrn{Sinj}}(X)$ and ${\operatorneje{Per}}(X)$ lf ths set if periodic polnts of $X$. Wa say that a *shneuparity is hyperbolic* if the eigenvajues of tje... | such that $X(\sigma)=0$ or, equivalently, $X_t(\sigma)=\sigma$ for \in R}}$. The formed by singularities $X$* ${\mathrm{Sing}}(X)$ and ${\operatorname{Per}}(X)$ the set of points of $X$. We say that *singularity is hyperbolic* if the eigenvalues of the derivative $DX(\sigma)$ of the vector at the singularity $\sigma$ h... | such that $X(\sigma)=0$ or, equivalenTly, $X_t(\sigma)=\Sigma$ For All $T \iN {{\matHbb R}}$. the set formed by SInguLarities is the *singular sEt of $X$* DeNOted ${\MAtHrm{SiNg}}(X)$ and ${\oPErATOrnAmE{PEr}}(X)$ Is THe Set of PerIodic poInts of $X$. We sAy tHaT a *singularitY Is Hyperbolic* If tHe eigenvalueS of The... | such that $X(\sigma)=0$ o r, equival ently , $ X_t (\ sigm a)=\ sigma$ for all $t \ in {{\mathbb R}}$. The setfo r medb ysingu laritie s i s the * si ngu la r s et of $X $* deno ted ${\mat hrm {S ing}}(X)$ an d $ {\operator nam e{Per}}(X)$isthe se tofp eriod icpoint s of $ X $. Wesay thata* singul a rity is h yp... | such_that $X(\sigma)=0$_or, equivalently, $X_t(\sigma)=\sigma$ for_all $t_\in_{{\mathbb R}}$._The_set formed by_singularities is the_*singular set of $X$*_denoted ${\mathrm{Sing}}(X)$ and_${\operatorname{Per}}(X)$_is the set of periodic points of $X$. We say that a *singularity is_hyperbolic*_if the_eigenvalues_of_the... |
$C$ has no first generation ancestors, i.e. $\mathcal{A}^\Lambda_1(C)=\emptyset$, then the value of its flag $u$ alone will determine whether $C$ is kept or not. Otherwise, the value of $u$ will decide if $C$ is kept once we determine the fate of all the first generation ancestors of $C$.
2. To decide whether any gi... | $ C$ has no first generation ancestors, i.e. $ \mathcal{A}^\Lambda_1(C)=\emptyset$, then the value of its pin $ u$ entirely will determine whether $ C$ is kept or not. differently, the value of $ u$ will decide if $ C$ is kept once we decide the fate of all the inaugural generation ancestors of $ C$.
2. To decide ... | $C$ jas no first generation xncestors, i.e. $\majhxal{A}^\Lakbda_1(C)=\ejptyset$, ghen the value of its flag $u$ aoone qill determine whether $C$ is kepn or not. Ithecwise, the value of $u$ will decisc if $E$ ms kept once we determine the fate of anl tke first generation ancestors of $C$.
2. Eo decice whether any gy... | $C$ has no first generation ancestors, i.e. the of its $u$ alone will or Otherwise, the value $u$ will decide $C$ is kept once we determine fate of all the first generation ancestors of $C$. 2. To decide whether given first generation ancestor $\tilde{C} \in \mathcal{A}^\Lambda_1(C)$ is kept, one must repeat step for i... | $C$ has no first generation anceStors, i.e. $\matHcal{A}^\lamBda_1(c)=\eMptySet$, tHen the value of iTS flaG $u$ alone will determine whEther $c$ iS Kept OR nOt. OthErwise, tHE vALUe oF $u$ WiLl dEcIDe If $C$ is KepT once we Determine tHe fAtE of all the firST gEneration aNceStors of $C$.
2. To deCidE whethEr Any GI... | $C$ has no first generati on ancesto rs, i .e. $\ ma thca l{A} ^\Lambda_1(C)= \ empt yset$, then the valueof it sf lag$ u$ alon e willd et e r min ewh eth er $C $ iskep t or no t. Otherwi se, t he value of$ u$ will deci deif $C$ is ke ptonce w edet e rmine th e fat e of a l l thefirst gen er a tion a n cestors o ... | $C$_has no_first generation ancestors, i.e._$\mathcal{A}^\Lambda_1(C)=\emptyset$, then_the_value of_its_flag $u$ alone_will determine whether_$C$ is kept or_not. Otherwise, the_value_of $u$ will decide if $C$ is kept once we determine the fate of_all_the first_generation_ancestors_of $C$.
2. To decide_whether any gi... |
bottom) panel corresponds to the 2+2 (3+1) configuration.
The first orbital sampling method (lognormal orbital period and Rayleigh eccentricity distributions; see [Fig.]{}\[fig:IC\_comp\_per1\]) agrees reasonably with the MSC for periods $\gtrsim 10^3\,{\mathrm{d}}$. There is a clear excess of systems with inner perio... | bottom) panel corresponds to the 2 + 2 (3 + 1) configuration.
The first orbital sample distribution method acting (lognormal orbital menstruation and Rayleigh eccentricity distributions; visualize [ Fig.]{}\[fig: IC\_comp\_per1\ ]) agree reasonably with the MSC for periods $ \gtrsim 10 ^ 3\,{\mathrm{d}}$. There be a... | bothom) panel corresponds to the 2+2 (3+1) configurcrion.
Thx first orbital sampling method (lognormal ocbitql peeiod and Rayleigh eccevtricity fistriburionw; see [Fig.]{}\[fij:JC\_comp\_pcx1\]) agrscs recsinably with thg MSC for pesiods $\gtrsim 10^3\,{\mdtfrl{d}}$. There is a clear excess of systeis with ijner perio... | bottom) panel corresponds to the 2+2 (3+1) first sampling method orbital period and agrees with the MSC periods $\gtrsim 10^3\,{\mathrm{d}}$. is a clear excess of systems inner periods between $\sim 1$ and $\sim 10^2\,{\mathrm{d}}$, as noted before by @2008MNRAS.389..925T. our aim in §\[sect:pop\_syn\] is to establish ... | bottom) panel corresponds to tHe 2+2 (3+1) configurAtion.
the FirSt OrbiTal sAmpling method (lOGnorMal orbital period and RayLeigh EcCEntrICiTy disTributiONs; SEE [FiG.]{}\[fIg:iC\_cOmP\_PeR1\]) agreEs rEasonabLy with the MsC fOr Periods $\gtrsiM 10^3\,{\MaThrm{d}}$. There Is a Clear excess oF syStems wItH inNEr perIo... | bottom) panel correspondsto the 2+2 (3+1 ) c onf ig urat ion.
The first or b ital sampling method (logn ormal o r bita l p eriod and Ra y le i g h e cc en tri ci t ydistr ibu tions;see [Fig.] {}\ [f ig:IC\_comp\ _ pe r1\]) agre esreasonably w ith the M SC fo r peri ods $\gt rsim 1 0 ^3\,{\ mathrm{d} }$ . Th... | bottom) panel_corresponds to_the 2+2 (3+1) configuration.
The_first orbital_sampling_method (lognormal_orbital_period and Rayleigh_eccentricity distributions; see_[Fig.]{}\[fig:IC\_comp\_per1\]) agrees reasonably with_the MSC for_periods_$\gtrsim 10^3\,{\mathrm{d}}$. There is a clear excess of systems with inner perio... |
:tran1\]) and (\[eq:chan\_k\]), the received signal can be expressed as follows: $$\begin{aligned}
\label{eq:rec_MP}
r(t)=\sum_{k=1}^{N_u}\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\,
b^{(k)}_{\lfloor
j/N_f\rfloor}u^{(k)}(t-jT_f-c^{(k)}_jT_c-\tau_k)+\sigma_nn(t),\end{aligned}$$ where $n(t)$ is a white Gaus... | : tran1\ ]) and (\[eq: chan\_k\ ]), the received signal can be expressed as follows: $ $ \begin{aligned }
\label{eq: rec_MP }
r(t)=\sum_{k=1}^{N_u}\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\,
b^{(k)}_{\lfloor
j / N_f\rfloor}u^{(k)}(t - jT_f - c^{(k)}_jT_c-\tau_k)+\sigma_nn(t),\end{aligned}$$ where... | :traj1\]) and (\[eq:chan\_k\]), the receivtd signal can be gxpressev as fomlows: $$\beein{aligned}
\label{eq:rec_MP}
r(t)=\sum_{k=1}^{I_u}\sqet{\frax{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\,
c^{(k)}_{\lfloor
j/J_f\rfloor}y^{(k)}(t-jU_f-c^{(k)}_jT_c-\tau_k)+\sigma_ni(f),\end{alinued}$$ wgcre $n(c)$ ms a white Gaus... | :tran1\]) and (\[eq:chan\_k\]), the received signal can as $$\begin{aligned} \label{eq:rec_MP} b^{(k)}_{\lfloor j/N_f\rfloor}u^{(k)}(t-jT_f-c^{(k)}_jT_c-\tau_k)+\sigma_nn(t),\end{aligned}$$ where noise zero mean and spectral density, and \label{eq:u_k} u^{(k)}(t)=\sum_{l=1}^{L}\alpha_l^{(k)}w_{rx}\left(t-(l-1)T_c\right... | :tran1\]) and (\[eq:chan\_k\]), the received Signal can bE exprEssEd aS fOlloWs: $$\beGin{aligned}
\labeL{Eq:reC_MP}
r(t)=\sum_{k=1}^{N_u}\sqrt{\frac{E_k}{N_F}}\sum_{j=-\InFTy}^{\inFTy}D^{(k)}_j\,
b^{(k)}_{\Lfloor
j/n_F\rFLOor}U^{(k)}(T-jt_f-c^{(K)}_jt_C-\tAu_k)+\siGma_Nn(t),\end{aLigned}$$ wherE $n(t)$ Is A white Gaus... | :tran1\]) and (\[eq:chan\_ k\]), therecei ved si gn al c an b e expressed as foll ows: $$\begin{aligned}
\lab el { eq:r e c_ MP}
r (t)=\su m _{ k = 1}^ {N _u }\s qr t {\ frac{ E_k }{N_f}} \sum_{j=-\ inf ty }^{\infty}d^ { (k )}_j\,
b^{ (k) }_{\lfloor
j /N_ f\rflo or }u^ { (k)}( t-j T_f-c ^{(k)} _ jT_c-\ tau_k)+\s ig m... | :tran1\]) and_(\[eq:chan\_k\]), the_received signal can be_expressed as_follows:_$$\begin{aligned}
\label{eq:rec_MP}
r(t)=\sum_{k=1}^{N_u}\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\,
b^{(k)}_{\lfloor
j/N_f\rfloor}u^{(k)}(t-jT_f-c^{(k)}_jT_c-\tau_k)+\sigma_nn(t),\end{aligned}$$ where_$n(t)$_is a white_Gaus... |
cosmic variance term. For example experiments currently being considered for measuring the B-modes of CMB polarization would be cosmic variance limited for E polarization over a wide range of $ls$ (e.g. [@bmodes]). Such an optimal experiment, exploring structures into a multipole range of $l=4000$, represents the limi... | cosmic variance term. For example experiments presently being view for measuring the boron - mode of CMB polarization would be cosmic variance limited for vitamin e polarization over a wide range of $ ls$ (for example [ @bmodes ]). Such an optimum experiment, exploring structures into a multipole stove of $ l=4000 $, r... | codmic variance term. For ewample experimenjs curreitly bejng conskdered for measuring the B-moves if CMV polarization would bd cosmic nariance oimiued for E polarizefion ovcx a wjfe rcnje of $ls$ (e.g. [@bmoces]). Such at optimal expesioeut, exploring structures into a multi[ole ramgf of $l=4000$, represegts uhe limj... | cosmic variance term. For example experiments currently for the B-modes CMB polarization would E over a wide of $ls$ (e.g. Such an optimal experiment, exploring structures a multipole range of $l=4000$, represents the limit of how much information on one could extract from the CMB in principle. We found an expected err... | cosmic variance term. For examPle experimEnts cUrrEntLy BeinG conSidered for measURing The B-modes of CMB polarizaTion wOuLD be cOSmIc varIance liMItED For e pOlAriZaTIoN over A wiDe range Of $ls$ (e.g. [@bmodEs]). SUcH an optimal exPErIment, exploRinG structures iNto A multiPoLe rANge of $L=4000$, rePreseNts the LImi... | cosmic variance term. For example e xperi men tscu rren tlybeing consider e d fo r measuring the B-mode s ofCM B pol a ri zatio n would be c osm ic v ari an c elimit edfor E p olarizatio n o ve r a wide ran g eof $ls$ (e .g. [@bmodes]). Su ch anop tim a l exp eri ment, explo r ing st ructuresin t o a mu l tipole... | cosmic_variance term._For example experiments currently_being considered_for_measuring the_B-modes_of CMB polarization_would be cosmic_variance limited for E_polarization over a_wide_range of $ls$ (e.g. [@bmodes]). Such an optimal experiment, exploring structures into a multipole_range_of $l=4000$,_represents_the_limi... |
ach, J. V. 1983,, 202, 113 Gunn, J. E., & Gott, J. R. I. 1972,, 176, 1 Jørgensen, I., Franx, M., & Kjærgard, P. 1995,, 273, 1097 Jørgensen, I., Franx, M., & Kjærgard, P. 1996,, 280, 167 Kormendy, J., & Freeman, K. C. 2004, in IAU Symp. 220, Dark Matter in Galaxies, ed. S. D. Ryder et al. (San Francisco: ASP), 377 Krona... | ach, J. V. 1983, , 202, 113 Gunn, J. E., & Gott, J. R. I. 1972, , 176, 1 Jørgensen, I., Franx, M., & Kjærgard, P. 1995, , 273, 1097 Jørgensen, I., Franx, M., & Kjærgard, P. 1996, , 280, 167 Kormendy, J., & Freeman, K. C. 2004, in IAU Symp. 220, Dark Matter in Galaxies, ed. S. D. Ryder et al. (San Francisc... | ach, J. V. 1983,, 202, 113 Gunn, J. E., & Gott, J. R. I. 1972,, 176, 1 Jørgensen, I., Ftabx, M., & Njærgars, P. 1995,, 273, 1097 Jøfgensen, I., Franx, M., & Kjærgard, P. 1996,, 280, 167 Kornendy, J., & Freeman, K. C. 2004, iv IAU Symi. 220, Dark Mqttec in Galaxies, ed. S. D. Rydev et zp. (Sau Hrancisco: ASP), 377 Lrona... | ach, J. V. 1983,, 202, 113 Gunn, & J. R. 1972,, 176, 1 Kjærgard, 1995,, 273, 1097 I., Franx, M., Kjærgard, P. 1996,, 280, 167 Kormendy, & Freeman, K. C. 2004, in IAU Symp. 220, Dark Matter in Galaxies, S. D. Ryder et al. (San Francisco: ASP), 377 Kronawitter, A., Saglia, R. Gerhard, E., R. 144, 53 Mehlert, D., Saglia, ... | ach, J. V. 1983,, 202, 113 Gunn, J. E., & Gott, J. R. I. 1972,, 176, 1 JørgensEn, I., Franx, M., & KJærgaRd, P. 1995,, 273, 1097 jørGeNsen, i., FraNx, M., & Kjærgard, P. 1996,, 280, 167 KoRMendY, J., & Freeman, K. C. 2004, in IAU Symp. 220, DaRk MatTeR In GaLAxIes, ed. s. D. Ryder ET aL. (sAn FRaNcIscO: Asp), 377 KRona... | ach, J. V. 1983,, 202, 113 Gunn, J.E., & Go tt, J . R. I.1972,, 176, 1J ørge nsen, I., Franx, M., & Kjær ga r d, P . 1 995,, 273, 1 0 97 J ørg en se n,I. , F ranx, M. , & Kjæ rgard, P.199 6, , 280, 167 K o rm endy, J.,& F reeman, K. C . 2 004, i nIAU Symp. 22 0, Da rk Mat t er inGalaxies, e d . S. D . Ryder... | ach, J. V._1983,, 202,_113 Gunn, J. E., &_Gott, J. R. I. 1972,,_176,_1 Jørgensen,_I.,_Franx, M., &_Kjærgard, P. 1995,,_273, 1097 Jørgensen, I.,_Franx, M., &_Kjærgard,_P. 1996,, 280, 167 Kormendy, J., & Freeman, K. C. 2004, in IAU Symp. 220,_Dark_Matter in_Galaxies,_ed._S. D. Ryder et al. (San_Francisco: ASP), 377 Krona... |
^*S\to E$ is an isomorphism of DG ${{\mathcal A}}^0$-modules such that the following holds: there exists an isomorphism of graded ${{\mathcal A}}^0 _{{{\mathcal R}}}$-modules $\eta :(E\otimes
{{\mathcal R}})^{\gr} \to S^{\gr}$ so that the composition $$E= i^*(E\otimes {{\mathcal R}})
\stackrel{i^*(\eta)}{\to} i^*S\stac... | ^*S\to E$ is an isomorphism of DG $ { { \mathcal A}}^0$-modules such that the following holds: there exists an isomorphism of graded $ { { \mathcal A}}^0 _ { { { \mathcal R}}}$-modules $ \eta :( E\otimes
{ { \mathcal R}})^{\gr } \to S^{\gr}$ so that the composing $ $ E= i^*(E\otimes { { \mathcal radius } })
\stackr... | ^*S\to E$ is an isomorphism of AG ${{\mathcal A}}^0$-modolws suci that fhe folluwing holds: there exists an msomirphiwm of graded ${{\mathcal A}}^0 _{{{\mathcal G}}}$-modules $\wta :(T\otimes
{{\mathcal R}})^{\gc} \to S^{\gr}$ so thzb the romposition $$E= i^*(G\otimes {{\mathwal R}})
\stackrel{i^*(\atx)}{\tl} i^*S\stac... | ^*S\to E$ is an isomorphism of DG such the following there exists an _{{{\mathcal $\eta :(E\otimes {{\mathcal \to S^{\gr}$ so the composition $$E= i^*(E\otimes {{\mathcal R}}) i^*S\stackrel{\sigma}{\to}E$$ is the identity. Given objects $(S,\sigma),(S^\prime,\sigma ^\prime)\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}... | ^*S\to E$ is an isomorphism of DG ${{\maThcal A}}^0$-moduLes suCh tHat ThE folLowiNg holds: there exISts aN isomorphism of graded ${{\maThcal a}}^0 _{{{\mAThcaL r}}}$-mOduleS $\eta :(E\otIMeS
{{\MAthCaL R}})^{\Gr} \tO S^{\GR}$ sO that The ComposiTion $$E= i^*(E\otiMes {{\MaThcal R}})
\stackrEL{i^*(\Eta)}{\to} i^*S\staC... | ^*S\to E$ is an isomorphis m of DG ${ {\mat hca l A }} ^0$- modu les such thatt he f ollowing holds: thereexist sa n is o mo rphis m of gr a de d ${{ \m at hca lA }} ^0 _{ {{\ mathcal R}}}$-mod ule s$\eta :(E\ot i me s
{{\mathc alR}})^{\gr} \ toS^{\gr }$ so thatthe comp ositio n $$E=i^*(E\oti me s {{\ma t hcal R} ... | ^*S\to E$_is an_isomorphism of DG ${{\mathcal_A}}^0$-modules such_that_the following_holds:_there exists an_isomorphism of graded_${{\mathcal A}}^0 _{{{\mathcal R}}}$-modules_$\eta :(E\otimes
{{\mathcal R}})^{\gr}_\to_S^{\gr}$ so that the composition $$E= i^*(E\otimes {{\mathcal R}})
\stackrel{i^*(\eta)}{\to} i^*S\stac... |
@{-->}[rr]^-{\displaystyle{{A}{}^{{\circlearrowleft}}}}
&& \operatorname{W}^*_Z({Y},L_2)\,.
}$$
By Corollary \[coro:desc\] for $\bar A=\bar A_2\circ\bar A_1{^{-1}}$, there exists $A:L_1{\leadsto}L_2$ such that $f^!A\circ\bar A_1\simeq
\bar A_2$. By Proposition \[prop:gen-func\], ${{A}{}^{{\circlearrowleft}}}\circ{\ope... | @{-->}[rr]^-{\displaystyle{{A}{}^{{\circlearrowleft } } } }
& & \operatorname{W}^*_Z({Y},L_2)\, .
} $ $
By Corollary \[coro: desc\ ] for $ \bar A=\bar A_2\circ\bar A_1{^{-1}}$, there exists $ A: L_1{\leadsto}L_2 $ such that $ f^!A\circ\bar A_1\simeq
\bar A_2$. By Proposition \[prop: gen - func\ ], $ { { A}... | @{-->}[rr]^-{\dlsplaystyle{{A}{}^{{\circlearrowltft}}}}
&& \operatorname{W}^*_E({Y},O_2)\,.
}$$
By Cocollary \[doro:desc\] for $\bar A=\bar A_2\circ\bar A_1{^{-1}}$, thece ezists $A:L_1{\leadsto}L_2$ such that $w^!A\circ\bar A_1\simeq
\bqr A_2$. Vy Proposivjon \[prop:nzn-fund\], ${{A}{}^{{\cixcoearrowleft}}}\cirg{\ope... | @{-->}[rr]^-{\displaystyle{{A}{}^{{\circlearrowleft}}}} && \operatorname{W}^*_Z({Y},L_2)\,. }$$ By Corollary \[coro:desc\] A=\bar A_1{^{-1}}$, there $A:L_1{\leadsto}L_2$ such that Proposition ${{A}{}^{{\circlearrowleft}}}\circ{\operatorname{Push}_{f,\bar A_1}}={\operatorname{Push}_{f,f^!A\circ \bar A_2}}$. \[rema:feel-... | @{-->}[rr]^-{\displaystyle{{A}{}^{{\circlearroWleft}}}}
&& \operaTornaMe{W}^*_z({Y},L_2)\,.
}$$
by coroLlarY \[coro:desc\] for $\baR a=\bar a_2\circ\bar A_1{^{-1}}$, there exists $A:L_1{\LeadsTo}l_2$ Such THaT $f^!A\ciRc\bar A_1\sIMeQ
\BAr A_2$. by prOpoSiTIoN \[prop:Gen-Func\], ${{A}{}^{{\ciRclearrowlEft}}}\CiRc{\ope... | @{-->}[rr]^-{\displaystyle {{A}{}^{{\ circl ear row le ft}} }}
& & \operatornam e {W}^ *_Z({Y},L_2)\,.
}$$
B y Cor ol l ary\ [c oro:d esc\] f o r$ \ bar A =\ bar A _ 2\ circ\ bar A_1{^{ -1}}$, the reex ists $A:L_1{ \ le adsto}L_2$ su ch that $f^! A\c irc\ba rA_1 \ simeq
\b ar A_ 2$. By Propos ition \[p ro p :gen-... | @{-->}[rr]^-{\displaystyle{{A}{}^{{\circlearrowleft}}}}
&& \operatorname{W}^*_Z({Y},L_2)\,.
}$$
By_Corollary \[coro:desc\] for_$\bar A=\bar A_2\circ\bar A_1{^{-1}}$,_there exists_$A:L_1{\leadsto}L_2$_such that_$f^!A\circ\bar_A_1\simeq
\bar A_2$. By_Proposition \[prop:gen-func\], ${{A}{}^{{\circlearrowleft}}}\circ{\ope... |
I \rangle^{{ 2k }}+ \langle v_*, I_* \rangle^{{ 2k }}
\right)
\mathcal{B}(v,v_*,I,I_*,r,R,\sigma) \\ \times \varphi_\alpha(r) \, (1-R) R^{1/2} \psi_\alpha(R) \, \mathrm{d} \sigma\, \mathrm{d}r \, \mathrm{d}R \, \mathrm{d}I_* \mathrm{d} v_* \mathrm{d}I \mathrm{d}v,
\end{gathered}$$ so that $$\label{W}
\ma... | I \rangle^ { { 2k } } + \langle v _ *, I _ * \rangle^ { { 2k } }
\right)
\mathcal{B}(v, v_*,I, I_*,r, R,\sigma) \\ \times \varphi_\alpha(r) \, (1 - R) R^{1/2 } \psi_\alpha(R) \, \mathrm{d } \sigma\, \mathrm{d}r \, \mathrm{d}R \, \mathrm{d}I _ * \mathrm{d } v _ * \mathrm{d}I \mathrm{d}v,
\end{gathered}$$ s... | I \gangle^{{ 2k }}+ \langle v_*, I_* \rannle^{{ 2k }}
\right)
\mathcan{B}(v,v_*,I,I_*,d,R,\sigma) \\ \times \varphi_\alpha(r) \, (1-R) R^{1/2} \psi_\elphq(R) \, \mqthrm{d} \sigma\, \mathrm{d}r \, \mathrm{d}R \, \mathrm{e}I_* \methrm{d} v_* \mathrm{d}M \mathrm{d}v,
\skd{gatkeced}$$ so that $$\labgl{W}
\ma... | I \rangle^{{ 2k }}+ \langle v_*, I_* }} \mathcal{B}(v,v_*,I,I_*,r,R,\sigma) \\ \varphi_\alpha(r) \, (1-R) \mathrm{d}r \mathrm{d}R \, \mathrm{d}I_* v_* \mathrm{d}I \mathrm{d}v, so that $$\label{W} \mathcal{W}=\mathcal{W}^+-\mathcal{W}^-.$$ We treat term separately. For the gain part, we use the bound from above stated $... | I \rangle^{{ 2k }}+ \langle v_*, I_* \rangle^{{ 2k }}
\riGht)
\mathcal{b}(v,v_*,I,I_*,R,R,\sIgmA) \\ \tImes \VarpHi_\alpha(r) \, (1-R) R^{1/2} \psi_\aLPha(R) \, \Mathrm{d} \sigma\, \mathrm{d}r \, \maThrm{d}r \, \mAThrm{D}i_* \mAthrm{D} v_* \mathrM{D}I \MAThrM{d}V,
\eNd{gAtHErEd}$$ so tHat $$\Label{W}
\mA... | I \rangle^{{ 2k }}+ \lang le v_*, I_ * \ra ngl e^{ {2k } }
\right)
\m a thca l{B}(v,v_*,I,I_*,r,R,\ sigma )\ \ \t i me s \va rphi_\a l ph a ( r)\, ( 1-R )R ^{ 1/2}\ps i_\alph a(R) \, \m ath rm {d} \sigma\, \m athrm{d}r\,\mathrm{d}R\,\mathr m{ d}I _ * \ma thr m{d}v_* \m a thrm{d }I \mathr m{ d }v,
\end{g a t he... | I_\rangle^{{ 2k_}}+ \langle v_*, I_*_\rangle^{{ 2k_}}_
__ \right)
\mathcal{B}(v,v_*,I,I_*,r,R,\sigma)_\\ \times \varphi_\alpha(r)_\, (1-R) R^{1/2} \psi_\alpha(R)_\, \mathrm{d} \sigma\,_\mathrm{d}r_\, \mathrm{d}R \, \mathrm{d}I_* \mathrm{d} v_* \mathrm{d}I \mathrm{d}v,
\end{gathered}$$ so that_$$\label{W}
_ __\ma... |
an example, for $N = 2$, repeating the above procedure will yield us $y_1 = 0.5, y_2 = 0.75, y_3 = 0.25, y_4 = 0.875, \cdots$. Note that the above values are in fact the thresholds reported in Table \[tab:osa\_two\_users\].
3. In Remark \[rem:osa\_2\], for the $N = 2$ case, we noticed that the feedback from the base... | an example, for $ N = 2 $, repeating the above procedure will yield us $ y_1 = 0.5, y_2 = 0.75, y_3 = 0.25, y_4 = 0.875, \cdots$. notice that the above value are in fact the thresholds reported in Table \[tab: osa\_two\_users\ ].
3. In Remark \[rem: osa\_2\ ], for the $ N = 2 $ subject, we notice that the feed... | an example, for $N = 2$, repeatikg the above proewdure xill yisld us $y_1 = 0.5, y_2 = 0.75, y_3 = 0.25, y_4 = 0.875, \cdots$. Note thet tye abive values are in fact the thredholds rwporued in Table \[tab:ose\_fwo\_users\].
3. In Dcmark \[xen:osa\_2\], for the $N = 2$ case, we noticed that dhd yeedback from the base... | an example, for $N = 2$, repeating procedure yield us = 0.5, y_2 y_4 0.875, \cdots$. Note the above values in fact the thresholds reported in \[tab:osa\_two\_users\]. 3. In Remark \[rem:osa\_2\], for the $N = 2$ case, we noticed the feedback from the base station corresponding to a threshold can be viewed the represent... | an example, for $N = 2$, repeating the Above proceDure wIll YieLd Us $y_1 = 0.5, y_2 = 0.75, Y_3 = 0.25, y_4 = 0.875, \cdOts$. Note that the ABove Values are in fact the threSholdS rEPortED iN TablE \[tab:osa\_TWo\_USErs\].
3. in reMarK \[rEM:oSa\_2\], for The $n = 2$ case, we Noticed thaT thE fEedback from tHE bAse... | an example, for $N = 2$,repeatingthe a bov e p ro cedu re w ill yield us $ y _1 = 0.5, y_2 = 0.75, y_3= 0.2 5, y_4= 0 .875, \cdots $ .N o teth at th ea bo ve va lue s are i n fact the th re sholds repor t ed in Table\[t ab:osa\_two\ _us ers\].
3.In Re mar k \[r em:osa \ _2\],for the $ N= 2$ ca s e, we n o t ic... | an_example, for_$N = 2$, repeating_the above_procedure_will yield_us_$y_1 = 0.5,_y_2 = 0.75,_y_3 = 0.25, y_4_= 0.875, \cdots$._Note_that the above values are in fact the thresholds reported in Table \[tab:osa\_two\_users\].
3. In_Remark \[rem:osa\_2\],_for the_$N_=_2$ case, we noticed that_the feedback from the base... |
uniformly along the surface [@Smoluchowski1941], thus giving weak corrugation. The crude proposal (\[eq:4\]) might be viewed as the precursor to the effective-medium theory [@EMT].
{width="85.00000%"}
The form (\[eq:4\]) provides an interpretation of the mechanism by which the increas... | uniformly along the surface [ @Smoluchowski1941 ], thus giving weak corrugation. The unrefined marriage proposal (\[eq:4\ ]) might be viewed as the precursor to the effective - culture medium hypothesis [ @EMT ].
! [ image](fig1CuContourPlots.png){width="85.00000% " }
The form (\[eq:4\ ]) provides an interpretatio... | unlformly along the surfact [@Smoluchowski1941], thos givinj weak dorrugatkon. The crude proposal (\[eq:4\]) mijht ve vitced as the precursor to the evfective-nedinm theory [@EMT].
{width="85.00000%"}
The form (\[gq:4\]) provides dn interpretathov lf the mechanism by which the increws... | uniformly along the surface [@Smoluchowski1941], thus giving The proposal (\[eq:4\]) be viewed as theory {width="85.00000%"} The form provides an interpretation the mechanism by which the increasing corrugation for (111) $<$ (100) $<$ (110) causes increasing amplitudes of modulation $V_1... | uniformly along the surface [@SMoluchowskI1941], thus GivIng WeAk coRrugAtion. The crude pROposAl (\[eq:4\]) might be viewed as the PrecuRsOR to tHE eFfectIve-mediUM tHEOry [@eMt].
{width="85.00000%"}
The fOrm (\[Eq:4\]) Provides an inTErPretation oF thE mechanism by WhiCh the iNcReaS... | uniformly along the surfa ce [@Smolu chows ki1 941 ], thu s gi ving weak corr u gati on. The crude proposal (\[e q: 4 \])m ig ht be viewed as t hepr ec urs or to theeff ective- medium the ory [ @EMT].
 {wi dth="8 5. 000 0 0%"}
Th e for m (\[e q :4\])providesan interp r etatio... | uniformly_along the_surface [@Smoluchowski1941], thus giving_weak corrugation._The_crude proposal_(\[eq:4\])_might be viewed_as the precursor_to the effective-medium theory_[@EMT].
{width="85.00000%"}
The form (\[eq:4\])_provides_an interpretation of the mechanism by which the increas... |
shells, hence line emission from such shells is strongly enhanced compared to collisional thermal radiations. For instance, the Ly$\rm \delta$ line at 14.8 $\rm \AA$ is stronger than the Ly$\rm \gamma$ line at 15.2 $\rm \AA$. Similar conditions can be found in many other transitions, e.g., the He$\rm \delta$ line at 1... | shells, hence line emission from such carapace is powerfully enhanced compared to collisional thermal radiation sickness. For example, the Ly$\rm \delta$ line at 14.8 $ \rm \AA$ is stronger than the Ly$\rm \gamma$ tune at 15.2 $ \rm \AA$. like conditions can be found in many other transitions, for example, the He$\rm \... | shflls, hence line emission from such shells is svrongly enhancea compared to collisional thxrmao raduations. For instance, tfe Ly$\rm \dvlta$ line at 14.8 $\em \AA$ is svdonger bkan tgc Ly$\rk \gamma$ line at 15.2 $\rm \AA$. Sikilar conditiots ccn be found in many other transitionf, e.g., thr Je$\rm \delta$ ling at 1... | shells, hence line emission from such shells enhanced to collisional radiations. For instance, 14.8 \AA$ is stronger the Ly$\rm \gamma$ at 15.2 $\rm \AA$. Similar conditions be found in many other transitions, e.g., the He$\rm \delta$ line at 17.4 \AA$, the Ly$\rm \delta$ line at 19.4 $\rm \AA$, and Ly$\rm \delta$ line... | shells, hence line emission frOm such shelLs is sTroNglY eNhanCed cOmpared to colliSIonaL thermal radiations. For iNstanCe, THe Ly$\RM \dElta$ lIne at 14.8 $\rm \aa$ iS STroNgEr ThaN tHE LY$\rm \gaMma$ Line at 15.2 $\rM \AA$. Similar ConDiTions can be foUNd In many otheR trAnsitions, e.g., tHe HE$\rm \delTa$ LinE At 1... | shells, hence line emissi on from su ch sh ell s i sstro ngly enhanced comp a redto collisional thermal radi at i ons. Fo r ins tance,t he L y$\ rm \ del ta $ l ine a t 1 4.8 $\r m \AA$ isstr on ger than the Ly $\rm \gamm a$line at 15.2 $\ rm \AA $. Si m ilarcon ditio ns can be fou nd in man yo ther t r ansitio ... | shells,_hence line_emission from such shells_is strongly_enhanced_compared to_collisional_thermal radiations. For_instance, the Ly$\rm_\delta$ line at 14.8_$\rm \AA$ is_stronger_than the Ly$\rm \gamma$ line at 15.2 $\rm \AA$. Similar conditions can be found_in_many other_transitions,_e.g.,_the He$\rm \delta$ line at_1... |
distance keeps on attaining linearity. At infinite time, the steady state values are linear with position as given by Eq. \[infinite-time\]. Similarly, the material energy density initially exhibits slight non-linear variation and finally attains the linearity (Fig. \[vxt-planar\]). The non-linear variation at early s... | distance keeps on attaining linearity. At infinite meter, the regular state values are linear with status as given by Eq. \[infinite - time\ ]. Similarly, the material department of energy density initially exhibits slight non - analogue mutant and finally reach the linearity (Fig. \[vxt - planar\ ]). The non - linear ... | didtance keeps on attaininn linearity. At iudinite time, fhe steaay state values are linear wmth posituon as given by Eq. \[infknite-time\]. Similaroy, tie material enerjg densibv iniflally xxhibits slight non-linear variation and fknclly attains the linearity (Fig. \[vxt-plwnar\]). Thr jon-linear variwtiom at sarly s... | distance keeps on attaining linearity. At infinite steady values are with position as the energy density initially slight non-linear variation finally attains the linearity (Fig. \[vxt-planar\]). non-linear variation at early stages occurs due to net absorption of energy by initially cold material (as u(x,0)=v(x,0)=0).... | distance keeps on attaining lInearity. At InfinIte TimE, tHe stEady State values are LIneaR with position as given by eq. \[infInITe-tiME\]. SImilaRly, the mATeRIAl eNeRgY deNsITy InitiAllY exhibiTs slight noN-liNeAr variation aND fInally attaIns The linearity (fig. \[Vxt-plaNaR\]). ThE Non-liNeaR variAtion aT Early s... | distance keeps on attaini ng lineari ty. A t i nfi ni te t ime, the steady st a te v alues are linear withposit io n asg iv en by Eq. \[ i nf i n ite -t im e\] .S im ilarl y,the mat erial ener gyde nsity initia l ly exhibitssli ght non-line arvariat io n a n d fin all y att ains t h e line arity (Fi g. \[vxt- p la... | distance_keeps on_attaining linearity. At infinite_time, the_steady_state values_are_linear with position_as given by_Eq. \[infinite-time\]. Similarly, the_material energy density_initially_exhibits slight non-linear variation and finally attains the linearity (Fig. \[vxt-planar\]). The non-linear variation_at_early s... |
a contradiction. Hence, the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ is non decreasing, and equality cannot occur.
Let us prove that it suffices to choose $\iota_k$ as the smallest in $\eta(n(j_{k-1})\setminus \eta(n(j_k))$. If at step $k$, a smallest $\iota_k$ is not chosen in $\eta(n(j_{k-1})\setminus \eta(n(j... | a contradiction. Hence, the sequence $ \iota_n,\iota_{n-1},\ldots,\iota_1 $ is non decreasing, and equality cannot occur.
lease us rise that it suffices to choose $ \iota_k$ as the smallest in $ \eta(n(j_{k-1})\setminus \eta(n(j_k))$. If at step $ k$, a minor $ \iota_k$ is not chosen in $ \eta(n(j_{k-1})\setminus \e... | a fontradiction. Hence, the requence $\iota_n,\iota_{n-1},\ldovs,\iota_1$ js non ddcreasing, and equality cannov ocxur.
Leu us prove that it ruffices no choose $\iote_k$ as the smallest in $\eta(n(j_{k-1})\seflinuv \eta(n(j_k))$. If at xtep $k$, a skallest $\iota_k$ hs nlt chosen in $\eta(n(j_{k-1})\setminus \eta(n(j... | a contradiction. Hence, the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ is and cannot occur. us prove that as smallest in $\eta(n(j_{k-1})\setminus If at step a smallest $\iota_k$ is not chosen $\eta(n(j_{k-1})\setminus \eta(n(j_k))$, it will be taken after, and the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ will ... | a contradiction. Hence, the seqUence $\iota_n,\Iota_{n-1},\LdoTs,\iOtA_1$ is nOn deCreasing, and equALity Cannot occur.
Let us prove tHat it SuFFiceS To ChoosE $\iota_k$ aS ThE SMalLeSt In $\eTa(N(J_{k-1})\SetmiNus \Eta(n(j_k))$. IF at step $k$, a sMalLeSt $\iota_k$ is not CHoSen in $\eta(n(j_{K-1})\seTminus \eta(n(j... | a contradiction. Hence, t he sequenc e $\i ota _n, \i ota_ {n-1 },\ldots,\iota _ 1$ i s non decreasing, andequal it y can n ot occu r.
Let us p rov eth atit su ffice s t o choos e $\iota_k $ a sthe smallest in $\eta(n(j _{k -1})\setminu s \ eta(n( j_ k)) $ . Ifatstep$k$, a smalle st $\iota _k $ is no t chosen i ... | a_contradiction. Hence,_the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ is_non decreasing,_and_equality cannot_occur.
Let_us prove that_it suffices to_choose $\iota_k$ as the_smallest in $\eta(n(j_{k-1})\setminus_\eta(n(j_k))$._If at step $k$, a smallest $\iota_k$ is not chosen in $\eta(n(j_{k-1})\setminus \eta(n(j... |
a finite model for ${\underline{E}G}$ implies that the $G$-sphere spectrum is small in the triangulated homotopy category $\operatorname{Ho}(\operatorname{Sp}_G)$. The example below illustrates that this is not true in general, and that even a finite-dimensional model for ${\underline{E}G}$ does not imply smallness of... | a finite model for $ { \underline{E}G}$ implies that the $ G$-sphere spectrum is small in the triangulated homotopy class $ \operatorname{Ho}(\operatorname{Sp}_G)$. The case below illustrates that this is not true in cosmopolitan, and that even a finite - dimensional model for $ { \underline{E}G}$ does not incriminat... | a vinite model for ${\underlike{E}G}$ implies thaj rhe $G$-s'here slectrum ks small in the triangulated himotokj category $\operatornamd{Ho}(\operatlrname{Sp}_T)$. Tht example below illustratcf thzb thiv is not true ik general, atd that even a fknnte-dimensional model for ${\underline{E}G}$ does npt imply smallnefs og... | a finite model for ${\underline{E}G}$ implies that spectrum small in triangulated homotopy category that is not true general, and that a finite-dimensional model for ${\underline{E}G}$ does imply smallness of ${\mathbb S}_G$. In fact, [@barcenas-degrijse-patchkoria:stable_finiteness Thm.5.4] shows that for a discrete g... | a finite model for ${\underline{E}g}$ implies thAt the $g$-spHerE sPectRum iS small in the triANgulAted homotopy category $\opEratoRnAMe{Ho}(\OPeRatorName{Sp}_G)$. tHe EXAmpLe BeLow IlLUsTrateS thAt this iS not true in GenErAl, and that eveN A fInite-dimenSioNal model for ${\uNdeRline{E}g}$ dOes NOt impLy sMallnEss of... | a finite model for ${\und erline{E}G }$ im pli esth at t he $ G$-sphere spec t rumis small in the triang ulate dh omot o py cate gory $\ o pe r a tor na me {Ho }( \ op erato rna me{Sp}_ G)$. The e xam pl e below illu s tr ates thatthi s is not tru e i n gene ra l,a nd th atevena fini t e-dime nsional m od e l for$ {\... | a_finite model_for ${\underline{E}G}$ implies that_the $G$-sphere_spectrum_is small_in_the triangulated homotopy_category $\operatorname{Ho}(\operatorname{Sp}_G)$. The example_below illustrates that this_is not true_in_general, and that even a finite-dimensional model for ${\underline{E}G}$ does not imply smallness of... |
\] below, using the characterization of $N$-${\mathcal{F}}$-amenability from Remark \[rem:covers-not-maps\] by the existence of $S$-wide covers of $G {{\times}}\Delta$.
To outline the construction of these covers and to prepare for the mapping class group we introduce some notation. Pick a $G$-invariant proper metric ... | \ ] below, using the characterization of $ N$-${\mathcal{F}}$-amenability from Remark \[rem: covers - not - maps\ ] by the being of $ S$-wide cover charge of $ G { { \times}}\Delta$.
To outline the structure of these covers and to train for the mapping course group we introduce some notation. Pick a $ G$-invariant... | \] bepow, using the characterieation of $N$-${\mathcal{F}}$-amenauility rrom Remxrk \[rem:covers-not-maps\] by the eeistwnce if $S$-wide covers of $G {{\tkmes}}\Delta$.
No outlinw tht construction of these covers zkd to 'repare for the mapping cnass group we hngrlduce some notation. Pick a $G$-invariagt proprr metric ... | \] below, using the characterization of $N$-${\mathcal{F}}$-amenability \[rem:covers-not-maps\] the existence $S$-wide covers of construction these covers and prepare for the class group we introduce some notation. a $G$-invariant proper metric on the set $E$ of edges of $\Gamma$; this possible as $G$ and $E$ are count... | \] below, using the characterizaTion of $N$-${\matHcal{F}}$-AmeNabIlIty fRom REmark \[rem:covers-NOt-maPs\] by the existence of $S$-widE coveRs OF $G {{\tiMEs}}\delta$.
to outliNE tHE ConStRuCtiOn OF tHese cOveRs and to Prepare for The MaPping class grOUp We introducE soMe notation. PiCk a $g$-invarIaNt pROper mEtrIc ... | \] below, using the charac terization of $ N$- ${\ ma thca l{F} }$-amenability from Remark \[rem:covers-n ot-ma ps \ ] by th e exi stenceo f$ S $-w id ecov er s o f $G{{\ times}} \Delta$.
Toou tline the co n st ruction of th ese covers a ndto pre pa ref or th e m appin g clas s group we intro du c e some notatio ... | \] below,_using the_characterization of $N$-${\mathcal{F}}$-amenability from_Remark \[rem:covers-not-maps\] by_the_existence of_$S$-wide_covers of $G_{{\times}}\Delta$.
To outline the_construction of these covers_and to prepare_for_the mapping class group we introduce some notation. Pick a $G$-invariant proper metric ... |
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