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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m V/X,Y),\circ) \to (KK(X,Y),\circ)$$ to the “$KK$-theory via correspondences" of Emerson-Meyer [@EM2; @EM3] (and more generally to their counterpart based on a complex oriented cohomology theory).
Let $\bB$ be a bivariant theory on $\m V$ such that a smooth morphism $f: X\to Y$ has a [*stable orientation*]{} $\theta(... | m V / X, Y),\circ) \to (KK(X, Y),\circ)$$ to the “ $ KK$-theory via correspondences " of Emerson - Meyer [ @EM2; @EM3 ] (and more generally to their counterpart based on a complex oriented cohomology hypothesis).
lease $ \bB$ be a bivariant theory on $ \m V$ such that a smooth morphism $ f: X\to Y$ experience a [ * ... | m V/D,Y),\circ) \to (KK(X,Y),\circ)$$ to tht “$KK$-theory via cotrwspondxnces" or Emersov-Meyer [@EM2; @EM3] (and more generaplt to ukeir counterpart basdd on a clmplex oeienued cohomology thxkry).
Let $\nY$ be z bivcrmant theory on $\k V$ such tvat a smooth mmrohnsm $f: X\to Y$ has a [*stable orientation*]{} $\theta(... | m V/X,Y),\circ) \to (KK(X,Y),\circ)$$ to the “$KK$-theory of [@EM2; @EM3] more generally to complex cohomology theory). Let be a bivariant on $\m V$ such that a morphism $f: X\to Y$ has a [*stable orientation*]{} $\theta(f)\in \bB(f)$, like $\bM(\m V/-)$, $\theta(f):=[X \xrightarrow {\op {id}_X} X]$ (these notions will... | m V/X,Y),\circ) \to (KK(X,Y),\circ)$$ to the “$KK$-Theory via cOrresPonDenCeS" of EMersOn-Meyer [@EM2; @EM3] (and MOre gEnerally to their counterPart bAsED on a COmPlex oRiented COhOMOloGy ThEorY).
LET $\bb$ be a bIvaRiant thEory on $\m V$ suCh tHaT a smooth morpHIsM $f: X\to Y$ has a [*StaBle orientatiOn*]{} $\tHeta(... | m V/X,Y),\circ) \to (KK(X, Y),\circ)$ $ tothe “$ KK $-th eory via correspon d ence s" of Emerson-Meyer [@ EM2;@E M 3] ( a nd more genera l ly t o t he ir co un t er partbas ed on a complex o rie nt ed cohomolog y t heory).
L et$\bB$ be a b iva riantth eor y on $ \mV$ su ch tha t a smo oth morph is m $f: X \ ... | m V/X,Y),\circ)_\to (KK(X,Y),\circ)$$_to the “$KK$-theory via_correspondences" of_Emerson-Meyer_[@EM2; @EM3]_(and_more generally to_their counterpart based_on a complex oriented_cohomology theory).
Let $\bB$_be_a bivariant theory on $\m V$ such that a smooth morphism $f: X\to Y$_has_a [*stable_orientation*]{}_$\theta(... |
I^{\otimes n-1})V|0^n\rangle,\\
f(0^{n+1},U_2)&=&\langle0^n|V^\dagger(|0\rangle\langle0|^{\otimes 2}
\otimes I^{\otimes n-2})V|0^n\rangle.\end{aligned}$$ Now we show that calculating $f(0^{n+1},U_1)$ and $f(0^{n+1},U_2)$ with a constant multiplicative error $0\le \epsilon<1$ is postBQP-hard. Since ${\rm postBQP}={\r... | I^{\otimes n-1})V|0^n\rangle,\\
f(0^{n+1},U_2)&=&\langle0^n|V^\dagger(|0\rangle\langle0|^{\otimes 2 }
\otimes I^{\otimes n-2})V|0^n\rangle.\end{aligned}$$ Now we show that calculating $ f(0^{n+1},U_1)$ and $ f(0^{n+1},U_2)$ with a constant multiplicative mistake $ 0\le \epsilon<1 $ is postBQP - unvoiced. Since $ {... | I^{\ohimes n-1})V|0^n\rangle,\\
f(0^{n+1},U_2)&=&\langue0^n|V^\dagger(|0\ranglg\lqngle0|^{\ovimes 2}
\ofimes I^{\ogimes n-2})V|0^n\rangle.\end{aligned}$$ Nox we show that calculating $f(0^{n+1},U_1)$ xnd $f(0^{n+1},U_2)$ wpth a conwtanu multiplicative xdror $0\le \epsilkk<1$ is 'owtBQP-hard. Sincg ${\rm postBQP}={\s... | I^{\otimes n-1})V|0^n\rangle,\\ f(0^{n+1},U_2)&=&\langle0^n|V^\dagger(|0\rangle\langle0|^{\otimes 2} \otimes I^{\otimes n-2})V|0^n\rangle.\end{aligned}$$ show calculating $f(0^{n+1},U_1)$ $f(0^{n+1},U_2)$ with a is Since ${\rm postBQP}={\rm [@postBQP] and ${\rm PP}={\rm P}^{\#{\rm P}}$, it means that calculation is $\#... | I^{\otimes n-1})V|0^n\rangle,\\
f(0^{n+1},U_2)&=&\langle0^N|V^\dagger(|0\raNgle\lAngLe0|^{\oTiMes 2}
\oTimeS I^{\otimes n-2})V|0^n\ranGLe.\enD{aligned}$$ Now we show that cAlculAtINg $f(0^{n+1},u_1)$ AnD $f(0^{n+1},U_2)$ wIth a conSTaNT MulTiPlIcaTiVE eRror $0\lE \epSilon<1$ is PostBQP-harD. SiNcE ${\rm postBQP}={\r... | I^{\otimes n-1})V|0^n\ran gle,\\
f (0^{n +1} ,U_ 2) &=&\ lang le0^n|V^\dagge r (|0\ rangle\langle0|^{\otim es 2}
\ o time s I ^{\ot imes n- 2 }) V | 0^n \r an gle .\ e nd {alig ned }$$ Now we show t hat c alculating $ f (0 ^{n+1},U_1 )$and $f(0^{n+ 1}, U_2)$wi tha cons tan t mul tiplic a tive e rror $0\l e\ eps... | I^{\otimes_n-1})V|0^n\rangle,\\ _
f(0^{n+1},U_2)&=&\langle0^n|V^\dagger(|0\rangle\langle0|^{\otimes 2}
\otimes I^{\otimes n-2})V|0^n\rangle.\end{aligned}$$_Now we_show_that calculating_$f(0^{n+1},U_1)$_and $f(0^{n+1},U_2)$ with_a constant multiplicative_error $0\le \epsilon<1$ is_postBQP-hard. Since ${\rm_postBQP}={\r... |
replica theory of spin glasses [@parisi:83].
Measurement of the overlap function {#sect:q_measurement}
===================================
\[sec:meas\]
In this section we outline the key process of this work, i.e., the basic experimental procedure by which physical replicas can be prepared and used to measure the o... | replica theory of spin glasses [ @parisi:83 ].
Measurement of the overlap affair { # faction: q_measurement }
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
\[sec: meas\ ]
In this section we outline the cardinal process of this oeuvre, i.e., the basic experimental procedure by which ph... | reolica theory of spin glarses [@parisi:83].
Measorwment mf the overlap function {#sect:q_measurement}
===================================
\[ser:meaw\]
In tyis section we outline the key irocess od thms work, i.e., the beaic expcximenfwl pxoredure by which physical seplicas can ba orzpared and used to measure the o... | replica theory of spin glasses [@parisi:83]. Measurement overlap {#sect:q_measurement} =================================== In this section of work, i.e., the experimental procedure by physical replicas can be prepared and to measure the overlap function $q$ defined in (\[eq:q\]). This consists of three steps: (1) initi... | replica theory of spin glasseS [@parisi:83].
MeaSuremEnt Of tHe OverLap fUnction {#sect:q_meASureMent}
===================================
\[sec:meas\]
In this sectiOn we oUtLIne tHE kEy proCess of tHIs WORk, i.E., tHe BasIc EXpErimeNtaL procedUre by which PhySiCal replicas cAN bE prepared aNd uSed to measure The O... | replica theory of spin gl asses [@pa risi: 83] .
Me asur emen t of the overl a p fu nction {#sect:q_measur ement }= ==== = == ===== ======= = == = = === == == ===
\ [s ec:me as\ ]
In t his sectio n w eoutline thek ey process o f t his work, i. e., the b as ice xperi men tal p rocedu r e by w hich phys ic a l repl ... | replica_theory of_spin glasses [@parisi:83].
Measurement of_the overlap_function_{#sect:q_measurement}
===================================
\[sec:meas\]
In this_section_we outline the_key process of_this work, i.e., the_basic experimental procedure_by_which physical replicas can be prepared and used to measure the o... |
s$ [@ref:POWHEG2009; @ref:singletop-mcgen].
Kinematic properties of background events associated with the $W$+jets and $Z$+jets processes are simulated using a leading-order MC generator [@ref:alpgen], and those of diboson processes ($WW$, $WZ$ and $ZZ$) and are modeled using MC [@ref:pythia]. In all cases is used to ... | s$ [ @ref: POWHEG2009; @ref: singletop - mcgen ].
Kinematic properties of background events associated with the $ W$+jets and $ Z$+jets summons are simulate using a leading - order MC generator [ @ref: alpgen ], and those of diboson process ($ WW$, $ WZ$ and $ ZZ$) and are modeled using MC [ @ref: pythia ]. In... | s$ [@rev:POWHEG2009; @ref:singletop-mcgek].
Kinematic propettues of backgdound evdnts associated with the $W$+jevs abd $Z$+jtns processes are simuuated usijg a leaeing-ieder MC geisrator [@rcy:alpgsk], and vhose of dibosok processes ($WW$, $WZ$ and $ZZ$) dna cre modeled using MC [@ref:pythia]. In all cases os used to ... | s$ [@ref:POWHEG2009; @ref:singletop-mcgen]. Kinematic properties of background with $W$+jets and processes are simulated [@ref:alpgen], those of diboson ($WW$, $WZ$ and and are modeled using MC [@ref:pythia]. all cases is used to model proton remnants and simulate the hadronization of generated partons. All MC events a... | s$ [@ref:POWHEG2009; @ref:singletop-mcgEn].
KinematiC propErtIes Of BackGrouNd events associATed wIth the $W$+jets and $Z$+jets proCesseS aRE simULaTed usIng a leaDInG-ORdeR Mc gEneRaTOr [@Ref:alPgeN], and thoSe of dibosoN prOcEsses ($WW$, $WZ$ and $zz$) aNd are modelEd uSing MC [@ref:pytHia]. in all cAsEs iS Used tO ... | s$ [@ref:POWHEG2009; @ref: singletop- mcgen ].
Ki ne mati c pr operties of ba c kgro und events associatedwithth e $W$ + je ts an d $Z$+j e ts p roc es se s a re si mulat edusing a leading-o rde rMC generator [@ ref:alpgen ],and those of di bosonpr oce s ses ( $WW $, $W Z$ and $ZZ$)and are m od e led us i ng MC [ @ ... | s$ [@ref:POWHEG2009; @ref:singletop-mcgen].
Kinematic_properties of_background events associated with_the $W$+jets_and_$Z$+jets processes_are_simulated using a_leading-order MC generator [@ref:alpgen],_and those of diboson_processes ($WW$, $WZ$_and_$ZZ$) and are modeled using MC [@ref:pythia]. In all cases is used to ... |
a_1}{\rm ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \eqno(4.36)$$ In terms of the angular variable $\gamma_1$ and $\gamma_2$ defined by (4.6) and (4.7), this expression can be rewritten in the form $$\Delta x_1={1\over a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over \sin{1\over 2}\left(\gamma_1-\... | a_1}{\rm ln}\left|{p_1+p_2^*\over p_1 - p_2}\right|. \eqno(4.36)$$ In terms of the angular variable $ \gamma_1 $ and $ \gamma_2 $ defined by (4.6) and (4.7), this expression can be rewrite in the human body $ $ \Delta x_1={1\over a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over \sin{1\over 2}\left... | a_1}{\rl ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \edno(4.36)$$ In terms of the anjular vzriable $\eamma_1$ and $\gamma_2$ defined by (4.6) end (4.7), this expression can be rewfitten in the forn $$\Deora x_1={1\over a_1}\,{\cj ln}\lefb|{\fin{1\obcr 2}\leyt(\jamma_1+\gamma_2\right)\pver \sin{1\ovar 2}\left(\gamma_1-\... | a_1}{\rm ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \eqno(4.36)$$ In terms of variable and $\gamma_2$ by (4.6) and rewritten the form $$\Delta a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over 2}\left(\gamma_1-\gamma_2\right)}\right|, \quad a_1=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\,\sin\,\gamma_1,\... | a_1}{\rm ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \eqnO(4.36)$$ In terms of The anGulAr vArIablE $\gamMa_1$ and $\gamma_2$ defiNEd by (4.6) And (4.7), this expression can be RewriTtEN in tHE fOrm $$\DeLta x_1={1\oveR A_1}\,{\rM LN}\leFt|{\SiN{1\ovEr 2}\LEfT(\gammA_1+\gaMma_2\righT)\over \sin{1\ovEr 2}\lEfT(\gamma_1-\... | a_1}{\rm ln}\left|{p_1+p_ 2^*\over p _1-p_ 2}\ rig ht |. \ eqno (4.36)$$ In te r ms o f the angular variable $\ga mm a _1$a nd $\ga mma_2$d ef i n edby ( 4.6 )a nd (4.7 ),this ex pression c anbe rewritten i n t he form $$ \De lta x_1={1\o ver a_1}\ ,{ \rm ln}\l eft |{\si n{1\ov e r 2}\l eft(\gamm a_ 1 +\gamm a ... | a_1}{\rm_ln}\left|{p_1+p_2^*\over p_1-p_2}\right|._\eqno(4.36)$$ In terms of_the angular_variable_$\gamma_1$ and_$\gamma_2$_defined by (4.6)_and (4.7), this_expression can be rewritten_in the form_$$\Delta_x_1={1\over a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over \sin{1\over 2}\left(\gamma_1-\... |
_1+\dotsm+ t_n= t$ such that $f( x, t)= x+ r_1 t_1+ p_1+\dotsm+ r_n t_n+ p_n$ and $x+\dotsm+ r_j t_j\ge b_j$ for all $j= 1,\dotsc, n$. Let $t_k'= t_k+ t_{ k+ 1}$, $t_{ k+ 1}'= 0$, and $t_j'= t_j$ for all $j\notin\{ k, k+ 1\}$. By $r_k\ge r_{ k+ 1}$, we know that $x+\dotsm+ r_k t_k'\ge b_k$ and $x+\dotsm+ r_{ k+ 1} t_{ ... | _ 1+\dotsm+ t_n= t$ such that $ f (x, t)= x+ r_1 t_1 + p_1+\dotsm+ r_n t_n+ p_n$ and $ x+\dotsm+ r_j t_j\ge b_j$ for all $ j= 1,\dotsc, n$. Let $ t_k'= t_k+ t _ { k+ 1}$, $ t _ { k+ 1}'= 0 $, and $ t_j'= t_j$ for all $ j\notin\ { k, k+ 1\}$. By $ r_k\ge radius _ { k+ 1}$, we acknowledge that $ x+\dotsm+ r_k t_k'\ge b_k... | _1+\dotdm+ t_n= t$ such that $f( x, t)= w+ r_1 t_1+ p_1+\dotsm+ r_n j_n+ p_n$ anv $x+\dotsj+ r_j t_j\gd b_j$ for all $j= 1,\dotsc, n$. Let $t_n'= r_k+ t_{ j+ 1}$, $t_{ k+ 1}'= 0$, and $t_j'= t_j$ for all $j\notpn\{ k, k+ 1\}$. Bt $r_k\je r_{ k+ 1}$, we know vgat $x+\dobfm+ r_i t_k'\gz u_k$ and $x+\dotsm+ r_{ k+ 1} t_{ ... | _1+\dotsm+ t_n= t$ such that $f( x, r_1 p_1+\dotsm+ r_n p_n$ and $x+\dotsm+ $j= n$. Let $t_k'= t_{ k+ 1}$, k+ 1}'= 0$, and $t_j'= t_j$ all $j\notin\{ k, k+ 1\}$. By $r_k\ge r_{ k+ 1}$, we know that r_k t_k'\ge b_k$ and $x+\dotsm+ r_{ k+ 1} t_{ k+ 1}'\ge b_{ k+ hence r_j b_j$ all $j= 1,\dotsc, n$. Hence this new run is ... | _1+\dotsm+ t_n= t$ such that $f( x, t)= x+ r_1 t_1+ p_1+\doTsm+ r_n t_n+ p_n$ aNd $x+\doTsm+ R_j t_J\gE b_j$ fOr alL $j= 1,\dotsc, n$. Let $t_k'= t_K+ T_{ k+ 1}$, $t_{ k+ 1}'= 0$, And $t_j'= t_j$ for all $j\notin\{ k, k+ 1\}$. BY $r_k\ge R_{ k+ 1}$, WE knoW ThAt $x+\doTsm+ r_k t_k'\GE b_K$ ANd $x+\DoTsM+ r_{ k+ 1} T_{ ... | _1+\dotsm+ t_n= t$ such th at $f( x,t)= x + r _1t_ 1+ p _1+\ dotsm+ r_n t_n + p_n $ and $x+\dotsm+ r_j t _j\ge b _ j$ f o rall $ j= 1,\d o ts c , n$ .Le t $ t_ k '= t_k+ t_ { k+ 1} $, $t_{ k+ 1} '= 0$, and $t_ j '= t_j$ forall $j\notin\{k,k+ 1\} $. By $r_k\ ger_{ k + 1}$, we kno w that $x +\ d otsm+r _k t_k' ... | _1+\dotsm+ t_n=_t$ such_that $f( x, t)=_x+ r_1_t_1+_p_1+\dotsm+ r_n_t_n+_p_n$ and $x+\dotsm+_r_j t_j\ge b_j$_for all $j= 1,\dotsc,_n$. Let $t_k'=_t_k+_t_{ k+ 1}$, $t_{ k+ 1}'= 0$, and $t_j'= t_j$ for all $j\notin\{ k,_k+_1\}$. By_$r_k\ge_r_{_k+ 1}$, we know that_$x+\dotsm+ r_k t_k'\ge b_k$ and_$x+\dotsm+ r_{_k+ 1} t_{ ... |
$(\pi,0)$ (i.e. in the second zone) are such, that one is ‘looking into the node’ of the radiation characteristics of the ZRS, whereas for smaller $\bbox{k}_{\|}$ one is looking at the maximum (see the inset). Hence there is a strong asymmetry around $(\pi,0)$. Increasing the energy to $\ge 30 eV$ the angles $\Theta$ ... | $ (\pi,0)$ (i.e. in the second zone) are such, that one is ‘ looking into the node ’ of the radiation characteristics of the ZRS, whereas for little $ \bbox{k}_{\|}$ one is expect at the maximum (visualize the gusset). Hence there is a solid asymmetry around $ (\pi,0)$. increase the energy to $ \ge 30 eV$ the angle $ \... | $(\pi,0)$ (i.e. in the second zone) ave such, that one is ‘looning ihto the vode’ of the radiation characveriwtics of the ZRS, whereas fof smaller $\bbox{k}_{\|}$ obe iw looking av the mawnmum (ace thz mnset). Hence thete is a strotg asymmetry asojnb $(\pi,0)$. Increasing the energy to $\ge 30 eV$ the anblfs $\Theta$ ... | $(\pi,0)$ (i.e. in the second zone) are one ‘looking into node’ of the whereas smaller $\bbox{k}_{\|}$ one looking at the (see the inset). Hence there is strong asymmetry around $(\pi,0)$. Increasing the energy to $\ge 30 eV$ the angles becomes smaller, one is no longer sampling the node and the the intensity much symm... | $(\pi,0)$ (i.e. in the second zone) are sucH, that one is ‘LookiNg iNto ThE nodE’ of tHe radiation chaRActeRistics of the ZRS, whereas For smAlLEr $\bbOX{k}_{\|}$ One is Looking AT tHE MaxImUm (See ThE InSet). HeNce There is A strong asyMmeTrY around $(\pi,0)$. IncREaSing the eneRgy To $\ge 30 eV$ the angLes $\theta$ ... | $(\pi,0)$ (i.e. in the se cond zone) aresuc h,th at o ne i s ‘looking int o the node’ of the radiatio n cha ra c teri s ti cs of the ZR S ,w h ere as f orsm a ll er $\ bbo x{k}_{\ |}$ one is lo ok ing at the m a xi mum (see t heinset). Henc e t here i sa s t rongasy mmetr y arou n d $(\p i,0)$. In cr e asingt he... | $(\pi,0)$_(i.e. in_the second zone) are_such, that_one_is ‘looking_into_the node’ of_the radiation characteristics_of the ZRS, whereas_for smaller $\bbox{k}_{\|}$_one_is looking at the maximum (see the inset). Hence there is a strong asymmetry_around_$(\pi,0)$. Increasing_the_energy_to $\ge 30 eV$ the_angles $\Theta$ ... |
A first result shown by Barthélemy shows that capacities collapse to belief functions when $L$ is linear [@bar00].
Any capacity on $L$ is a belief function if and only if $L$ is a linear lattice.
In the sequel, we address the combination of belief functions and their decomposition in terms of simple support functions... | A first result shown by Barthélemy shows that capacity crack up to belief functions when $ L$ is linear [ @bar00 ].
Any capacitance on $ L$ is a belief function if and entirely if $ L$ is a analogue lattice.
In the sequel, we address the combination of belief functions and their decomposition in term of simple su... | A flrst result shown by Baruhélemy shows that capacivies comlapse tu belief functions when $L$ is lunear [@bar00].
Any capacity on $L$ ks a belivf functiin ih and only if $L$ ma a linccr lafbice.
Iu vhe sequel, we acdress the combination ox cepief functions and their decomposityon in yegms of simple fupppwt fhnctions... | A first result shown by Barthélemy shows collapse belief functions $L$ is linear is belief function if only if $L$ a linear lattice. In the sequel, address the combination of belief functions and their decomposition in terms of simple functions. We will see that classical results generalize. \[def:drc\] Let $\bel_1,\be... | A first result shown by BarthéLemy shows tHat caPacItiEs CollApse To belief functiONs whEn $L$ is linear [@bar00].
Any capacIty on $l$ iS A belIEf FunctIon if anD OnLY If $L$ Is A lIneAr LAtTice.
IN thE sequel, We address tHe cOmBination of beLIeF functions And Their decompoSitIon in tErMs oF SimplE suPport FunctiONs... | A first result shown by Ba rthélemy s howstha t c ap acit iescollapse to be l ieffunctions when $L$ islinea r[ @bar 0 0] .
An y capac i ty o n $ L$ i s a b e li ef fu nct ion ifand only i f $ L$ is a linear la ttice.
In th e sequel, we ad dressth e c o mbina tio n ofbelief functi ons and t he i r deco m positio ... | A first_result shown_by Barthélemy shows that_capacities collapse_to_belief functions_when_$L$ is linear_[@bar00].
Any capacity on_$L$ is a belief_function if and_only_if $L$ is a linear lattice.
In the sequel, we address the combination of belief_functions_and their_decomposition_in_terms of simple support functions... |
}})$. Let $L: R^k\rightarrow R^k$ be defined by $$L_j({\mbox{\bf w}})=m^{-1}\sum_{i=1}^m b_{ij}sgn(K_i-{\mbox{\bf b}}_i^T{\mbox{\bf w}})$$ where $sgn(x)$ is the sign function of $x$ and equals zero when $x=0$. Then we want to prove that there is a root ${\widehat {\mbox{\bf w}}}$ of the equation $L({\mbox{\bf w}})=0$ s... | } }) $. Let $ L: R^k\rightarrow R^k$ be defined by $ $ L_j({\mbox{\bf w}})=m^{-1}\sum_{i=1}^m b_{ij}sgn(K_i-{\mbox{\bf b}}_i^T{\mbox{\bf w}})$$ where $ sgn(x)$ is the sign affair of $ x$ and equal zero when $ x=0$. Then we want to prove that there is a solution $ { \widehat { \mbox{\bf w}}}$ of the equation $ L({\mbox{... | }})$. Leh $L: R^k\rightarrow R^k$ be dtfined by $$L_j({\mbox{\by w}})=m^{-1}\sum_{m=1}^m b_{ij}sfn(K_i-{\mbox{\cf b}}_i^T{\mbox{\bf w}})$$ where $sgn(x)$ is tye sitn function of $x$ and eduals zerl when $x=0$. Thei we want to prots that bkere jd a xoit ${\widehat {\mbow{\bf w}}}$ of tha equation $L({\mbmx{\cf w}})=0$ s... | }})$. Let $L: R^k\rightarrow R^k$ be defined w}})=m^{-1}\sum_{i=1}^m b}}_i^T{\mbox{\bf w}})$$ $sgn(x)$ is the equals when $x=0$. Then want to prove there is a root ${\widehat {\mbox{\bf of the equation $L({\mbox{\bf w}})=0$ satisfying $\|{\widehat {\mbox{\bf w}}}\|_2^2=O_p(k/m)$. By classical convexity argument, suffic... | }})$. Let $L: R^k\rightarrow R^k$ be definEd by $$L_j({\mbox{\Bf w}})=m^{-1}\sUm_{i=1}^M b_{iJ}sGn(K_i-{\Mbox{\Bf b}}_i^T{\mbox{\bf w}})$$ whERe $sgN(x)$ is the sign function of $x$ And eqUaLS zerO WhEn $x=0$. ThEn we wanT To PROve ThAt TheRe IS a Root ${\wIdeHat {\mbox{\Bf w}}}$ of the eqUatIoN $L({\mbox{\bf w}})=0$ s... | }})$. Let $L: R^k\rightarr ow R^k$ be defi ned by $ $L_j ({\m box{\bf w}})=m ^ {-1} \sum_{i=1}^m b_{ij}sgn (K_i- {\ m box{ \ bf b}}_ i^T{\mb o x{ \ b f w }} )$ $ w he r e$sgn( x)$ is the sign func tio nof $x$ and e q ua ls zero wh en$x=0$. Thenwewant t opro v e tha t t hereis a r o ot ${\ widehat { \m b ox{\bf w}... | }})$. Let_$L: R^k\rightarrow_R^k$ be defined by_$$L_j({\mbox{\bf w}})=m^{-1}\sum_{i=1}^m_b_{ij}sgn(K_i-{\mbox{\bf_b}}_i^T{\mbox{\bf w}})$$_where_$sgn(x)$ is the_sign function of_$x$ and equals zero_when $x=0$. Then_we_want to prove that there is a root ${\widehat {\mbox{\bf w}}}$ of the equation_$L({\mbox{\bf_w}})=0$ s... |
of $A$ is the mapping $J_A=(A+{\ensuremath{\operatorname{Id}}})^{-1}$. We say that $A$ is a linear relation if ${\ensuremath{\operatorname{gra}}}A$ is a linear subspace of ${\ensuremath{\mathcal H}}\times {\ensuremath{\mathcal H}}$. $A$ is said to be a maximally monotone linear relation if $A$ is both maximally monoto... | of $ A$ is the mapping $ J_A=(A+{\ensuremath{\operatorname{Id}}})^{-1}$. We say that $ A$ is a linear relation if $ { \ensuremath{\operatorname{gra}}}A$ is a analogue subspace of $ { \ensuremath{\mathcal H}}\times { \ensuremath{\mathcal H}}$. $ A$ is say to be a maximally monotone linear relation back if $ A$ is both m... | of $A$ is the mapping $J_A=(A+{\ensmremath{\operatorncne{Id}}})^{-1}$. Wx say tgat $A$ is a linear relation if ${\ensurelarh{\opeeatorname{gra}}}A$ is a lindar subspwce of ${\ebsurtmath{\mathcal H}}\timxa {\ensurciath{\jwthccl H}}$. $A$ is said tp be a maxhmally monotona uiuear relation if $A$ is both maximally monoto... | of $A$ is the mapping $J_A=(A+{\ensuremath{\operatorname{Id}}})^{-1}$. We $A$ a linear if ${\ensuremath{\operatorname{gra}}}A$ is H}}\times H}}$. $A$ is to be a monotone linear relation if $A$ is maximally monotone and a linear relation. The mapping $T: {\ensuremath{\mathcal H}}\to {\ensuremath{\mathcal H}}$ said to be... | of $A$ is the mapping $J_A=(A+{\ensuremAth{\operatoRname{id}}})^{-1}$. WE saY tHat $A$ Is a lInear relation iF ${\EnsuRemath{\operatorname{gra}}}A$ Is a liNeAR subSPaCe of ${\eNsuremaTH{\mATHcaL H}}\TiMes {\EnSUrEmath{\MatHcal H}}$. $A$ iS said to be a MaxImAlly monotone LInEar relatioN if $a$ is both maximAllY monotO... | of $A$ is the mapping $J_ A=(A+{\ens urema th{ \op er ator name {Id}}})^{-1}$. We s ay that $A$ is a linea r rel at i on i f $ {\ens uremath { \o p e rat or na me{ gr a }} }A$ i s a linear subspaceof${ \ensuremath{ \ ma thcal H}}\ tim es {\ensurem ath {\math ca l H } }$. $ A$is sa id tob e a ma ximally m on o tone ... | of_$A$ is_the mapping $J_A=(A+{\ensuremath{\operatorname{Id}}})^{-1}$. We_say that_$A$_is a_linear_relation if ${\ensuremath{\operatorname{gra}}}A$_is a linear_subspace of ${\ensuremath{\mathcal H}}\times_{\ensuremath{\mathcal H}}$. $A$_is_said to be a maximally monotone linear relation if $A$ is both maximally monoto... |
band represents the systematic uncertainty.[]{data-label="step"}](step.png){width="10cm"}
Moreover, sharp peaks in the energy dependence of the ratios $K^{+}/\pi^{+}$ and $\Lambda/\pi$ were found for Pb+Pb collisions by the NA49 collaboration. Figure \[horn\] shows a comparison of the new measurements by NA61/SHINE f... | band represents the systematic uncertainty.[]{data - label="step"}](step.png){width="10 cm " }
Moreover, acute acme in the energy dependence of the ratios $ K^{+}/\pi^{+}$ and $ \Lambda/\pi$ were establish for Pb+Pb collisions by the NA49 collaboration. Figure \[horn\ ] show a comparison of the modern measurements... | bajd represents the systemxtic uncertainti.[]{dqta-labxl="step"}](sfep.png){wiath="10cm"}
Moreover, sharp peaks in tye entggy dependence of the ratios $K^{+}/\ii^{+}$ and $\Lanbda/\ki$ were found for Pb+Pb collisiohd by vhe NA49 collaborstion. Figuse \[horn\] shows a cum'arison of the new measurements by NW61/SHINE g... | band represents the systematic uncertainty.[]{data-label="step"}](step.png){width="10cm"} Moreover, sharp the dependence of ratios $K^{+}/\pi^{+}$ and collisions the NA49 collaboration. \[horn\] shows a of the new measurements by NA61/SHINE inelastic p+p interactions with the world data. Candidates of charged decays of... | band represents the systematIc uncertaiNty.[]{daTa-lAbeL="sTep"}](sTep.pNg){width="10cm"}
MoreoVEr, shArp peaks in the energy depEndenCe OF the RAtIos $K^{+}/\pI^{+}$ and $\LamBDa/\PI$ WerE fOuNd fOr pB+PB collIsiOns by thE NA49 collaboRatIoN. Figure \[horn\] sHOwS a comparisOn oF the new measuRemEnts by nA61/sHIne f... | band represents the syste matic unce rtain ty. []{ da ta-l abel ="step"}](step . png) {width="10cm"}
Moreov er, s ha r p pe a ks in t he ener g yd e pen de nc e o ft he rati os$K^{+}/ \pi^{+}$ a nd$\ Lambda/\pi$w er e found fo r P b+Pb collisi ons by th eNA4 9 coll abo ratio n. Fig u re \[h orn\] sho ws a comp a ... | band_represents the_systematic uncertainty.[]{data-label="step"}](step.png){width="10cm"}
Moreover, sharp peaks_in the_energy_dependence of_the_ratios $K^{+}/\pi^{+}$ and_$\Lambda/\pi$ were found_for Pb+Pb collisions by_the NA49 collaboration._Figure \[horn\]_shows a comparison of the new measurements by NA61/SHINE f... |
,7$ and $\theta(x)\neq x^{-1}$, which is the fourth row;
- $n=4$, $q=3,5, 7,9$ and $x\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$. This case is considered in the last row, as, up to rack isomorphism, this class is represented by an element in an $F$-stable maximal torus with associated partition ${\boldsy... | , 7 $ and $ \theta(x)\neq x^{-1}$, which is the fourth row;
- $ n=4 $, $ q=3,5, 7,9 $ and $ x\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$. This case is considered in the last rowing, as, up to scud isomorphism, this class is represented by an element in an $ F$-stable maximal torus with consort partiti... | ,7$ anf $\theta(x)\neq x^{-1}$, which is tme fourth row;
- $u=4$, $q=3,5, 7,9$ anv $x\in{{\mafhcal O}}_1^{\tfeta,{\operatorname{PGL}}_n(q)}$. This cese us cobsidered in the last ruw, as, up no rack iwomocphism, this class is repvzsentsf by en element in ak $F$-stable mdximal torus whtf cssociated partition ${\boldsy... | ,7$ and $\theta(x)\neq x^{-1}$, which is the - $q=3,5, 7,9$ $x\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$. This last as, up to isomorphism, this class represented by an element in an maximal torus with associated partition ${\boldsymbol{1}}$. - $n$ twice an odd number, $x\in{{\mathcal which is the fifth row... | ,7$ and $\theta(x)\neq x^{-1}$, which is the foUrth row;
- $n=4$, $q=3,5, 7,9$ aNd $x\in{{\MatHcaL O}}_1^{\ThetA,{\opeRatorname{PGL}}_n(q)}$. tHis cAse is considered in the laSt row, As, UP to rACk IsomoRphism, tHIs CLAss Is RePreSeNTeD by an EleMent in aN $F$-stable maXimAl Torus with assOCiAted partitIon ${\Boldsy... | ,7$ and $\theta(x)\neq x^{ -1}$, whic h isthe fo ur th r ow;
- $n=4$, $q = 3,5, 7,9$ and $x\in{{\math cal O }} _ 1^{\ t he ta,{\ operato r na m e {PG L} }_ n(q )} $ .Thiscas e is co nsidered i n t he last row, a s ,up to rack is omorphism, t his class i s r e prese nte d byan ele m ent in an $F$-s ta b le max ... | ,7$ and_$\theta(x)\neq x^{-1}$,_which is the fourth_row;
- __$n=4$, $q=3,5,_7,9$_and $x\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$._This case is_considered in the last_row, as, up_to_rack isomorphism, this class is represented by an element in an $F$-stable maximal torus_with_associated partition_${\boldsy... |
a minimalist phenomenological model, finding a good agreement.'
address:
- 'Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900, Maringá, PR, Brazil'
- 'National Institute of Science and Technology for Complex Systems, CNPq, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil... | a minimalist phenomenological model, finding a good agreement.'
address:
-' Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020 - 900, Maringá, PR, Brazil'
-' National Institute of Science and Technology for Complex Systems, CNPq, Rua Xavier Sigaud 150, 22290 - 180, Rio de Janeiro, R... | a linimalist phenomenologigal model, findiny a goov agreejent.'
addrdss:
- 'Departamento de Física, Unmverwidadt Estadual de Marineá, Av. Cololbo 5790, 87020-900, Maeingá, PR, Brazil'
- 'Iztional Instifmte oy Wcience and Teghnology fos Complex Systamr, ENPq, Rua Xavier Sigaud 150, 22290-180, Rio de Janeyro, RJ, Nrwzil... | a minimalist phenomenological model, finding a good - de Física, Estadual de Maringá, PR, - 'National Institute Science and Technology Complex Systems, CNPq, Rua Xavier Sigaud 22290-180, Rio de Janeiro, RJ, Brazil' author: - 'H.V. Ribeiro' - 'R.S. Mendes' 'E.K. Lenzi' - 'M.P. Belancon' - 'L.C. Malacarne' title: On the ... | a minimalist phenomenologicAl model, finDing a GooD agReEmenT.'
addRess:
- 'DepartamenTO de FÍsica, Universidade EstadUal de maRIngá, aV. COlombO 5790, 87020-900, MaringÁ, pR, bRAziL'
- 'NAtIonAl iNsTitutE of science And TechnolOgy FoR Complex SystEMs, cNPq, Rua XavIer sigaud 150, 22290-180, Rio de JAneIro, RJ, BRaZil... | a minimalist phenomenolog ical model , fin din g a g oodagre ement.'
addres s :
-'Departamento de Físic a, Un iv e rsid a de Esta dual de Ma r i ngá ,Av . C ol o mb o 579 0,87020-9 00, Maring á,PR , Brazil'
-' Na tional Ins tit ute of Scien ceand Te ch nol o gy fo r C omple x Syst e ms, CN Pq, Rua X av i er Sig a ud... | a_minimalist phenomenological_model, finding a good_agreement.'
address:
- 'Departamento_de_Física, Universidade_Estadual_de Maringá, Av._Colombo 5790, 87020-900,_Maringá, PR, Brazil'
- 'National_Institute of Science_and_Technology for Complex Systems, CNPq, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ,_Brazil... |
{I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k}\big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2}} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t} \nonumber\\
& \quad\qquad+\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{H}{2}} \... | { I}_{N_t } - \frac{\bar{\lambda}_k}{\gamma_k}\big(\mathbf{I}_{N_t } + \sum_{j\neq k } \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2 } } \mathbf{h}_{Pk } \mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t } \nonumber\\
& \quad\qquad+\sum_{j\neq k } \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\... | {I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k}\bin(\mathbf{I}_{N_t} +\sum_{j\ngq k} \bar{\nambda}_n \mathbf{f}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2}} \mathbf{i}_{Pk} \nathbd{h}_{Pk}^H\big(\mathbf{I}_{N_t} \nonuober\\
& \quwd\qquad+\sym_{j\ntq k} \bar{\lambda}_j \mefhbf{h}_{Pj}\mathbf{g}_{Ij}^H\biy)^{-\fcac{H}{2}} \... | {I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k}\big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2}} \nonumber\\ \quad\qquad+\sum_{j\neq k} \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{H}{2}} \succeq \mathbf{0}_{N_t}, constraint (\[E:semi-definite\]) means that minimal... | {I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k}\biG(\mathbf{I}_{N_t} +\Sum_{j\nEq k} \Bar{\LaMbda}_J \matHbf{h}_{Pj}\mathbf{h}_{PJ}^h\big)^{-\Frac{1}{2}} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^h\big(\mAtHBf{I}_{N_T} \NoNumbeR\\
& \quad\qqUAd+\SUM_{j\nEq K} \bAr{\lAmBDa}_J \mathBf{h}_{pj}\mathbF{h}_{Pj}^H\big)^{-\frAc{H}{2}} \... | {I}_{N_t} - \frac{\bar{\la mbda}_k}{\ gamma _k} \bi g( \mat hbf{ I}_{N_t} +\sum _ {j\n eq k} \bar{\lambda}_j\math bf { h}_{ P j} \math bf{h}_{ P j} ^ H \bi g) ^{ -\f ra c {1 }{2}} \ mathbf{ h}_{Pk} \m ath bf {h}_{Pk}^H\b i g( \mathbf{I} _{N _t} \nonumbe r\\
& \ qu ad\ q quad+ \su m_{j\ neq k} \bar{\ lambda}_j \ m... | {I}_{N_t} -_\frac{\bar{\lambda}_k}{\gamma_k}\big(\mathbf{I}_{N_t} +\sum_{j\neq_k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2}} _\mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t}_\nonumber\\
_ &_\quad\qquad+\sum_{j\neq_k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{H}{2}}_\... |
has $\left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1$ terms, where $\left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is the size of the orbit of $\w_j$.
An efficient way to find the decompositions is to work with products of Weyl group orbits, rather than with orbit functions. Their decompositi... | has $ \left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1 $ terms, where $ \left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is the size of the orbit of $ \w_j$.
An efficient way to witness the decomposition is to work with products of Weyl group orbits, quite than with orbit functions. Their decay... | had $\left(\begin{smallmatrix}n+1\\ m\\\end{smallmatrix}\rntht)+1$ tecms, whede $\left(\bdgin{smallmatrix}n+1\\j\\\end{smallmatcix}\rught)$ us the size of the orbkt of $\w_j$.
Aj efficiwnt xay to find the vscomposlcions ls to xork with produgts of Weyl group orbits, saghzr than with orbit functions. Their dqcomposotl... | has $\left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1$ terms, where $\left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is of orbit of An efficient way to with products of group orbits, rather with orbit functions. Their decomposition has studied, and many examples have been described in [@HLP]. It ... | has $\left(\begin{smallmatrix}n+1\\ j\\\End{smallmaTrix}\rIghT)+1$ teRmS, wheRe $\leFt(\begin{smallmaTRix}n+1\\J\\\end{smallmatrix}\right)$ is The siZe OF the ORbIt of $\w_J$.
An effiCIeNT Way To FiNd tHe DEcOmposItiOns is to Work with prOduCtS of Weyl group ORbIts, rather tHan With orbit funCtiOns. TheIr DecOMposiTi... | has $\left(\begin{smallma trix}n+1\\ j\\\ end {sm al lmat rix} \right)+1$ ter m s, w here $\left(\begin{sma llmat ri x }n+1 \ \j \\\en d{small m at r i x}\ ri gh t)$ i s t he si zeof theorbit of $ \w_ j$ .
An effici e nt way to fi ndthe decompos iti ons is t o w o rk wi thprodu cts of Weyl g roup orbi ts , rath... | has_$\left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1$_terms, where $\left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is_the size_of_the orbit_of_$\w_j$.
An efficient way_to find the_decompositions is to work_with products of_Weyl_group orbits, rather than with orbit functions. Their decompositi... |
have $$\overline{G}^j(\overline{F}^i(x)) =
G^j(F^i(x) + i\ell) =
G^j(F^i(x)) + i\ell.$$ This gives (b) for $i=0,1,\dots,q-1$. The fact that $\overline{P}_q(x) = P_q(x) + q\ell$ follows from (b) for $i=0$ and from the definition of these two sets.
Suppose that $\ell$ satisfies the assumption of (c). From (b) and the c... | have $ $ \overline{G}^j(\overline{F}^i(x) ) =
G^j(F^i(x) + i\ell) =
G^j(F^i(x) ) + i\ell.$$ This gives (b) for $ i=0,1,\dots, q-1$. The fact that $ \overline{P}_q(x) = P_q(x) + q\ell$ follows from (boron) for $ i=0 $ and from the definition of these two set.
Suppose that $ \ell$ satisfies the assumption of (spee... | hage $$\overline{G}^j(\overline{F}^i(x)) =
G^j(F^i(x) + i\ell) =
G^j(Y^u(x)) + i\enl.$$ Thia gives (c) for $i=0,1,\dots,q-1$. The fact that $\oterlune{P}_q(z) = P_q(x) + q\ell$ follows ffom (b) for $i=0$ and feom uhe definition of these two seta.
Duppmwe that $\ell$ sajisfies the dssumption of (w). Wrlm (b) and the c... | have $$\overline{G}^j(\overline{F}^i(x)) = G^j(F^i(x) + i\ell) = i\ell.$$ gives (b) $i=0,1,\dots,q-1$. The fact q\ell$ from (b) for and from the of these two sets. Suppose that satisfies the assumption of (c). From (b) and the choice of $\ell$, we $$\min \Re(\overline{P}_{i+1}(x)) - \max \Re(\overline{P}_i(x)) = \min \... | have $$\overline{G}^j(\overline{F}^i(x)) =
g^j(F^i(x) + i\ell) =
G^J(F^i(x)) + i\Ell.$$ thiS gIves (B) for $I=0,1,\dots,q-1$. The fact tHAt $\ovErline{P}_q(x) = P_q(x) + q\ell$ followS from (B) fOR $i=0$ anD FrOm the DefinitIOn OF TheSe TwO seTs.
sUpPose tHat $\Ell$ satiSfies the asSumPtIon of (c). From (b) aND tHe c... | have $$\overline{G}^j(\ov erline{F}^ i(x)) =G^j (F ^i(x ) +i\ell) =
G^j(F ^ i(x) ) + i\ell.$$ This give s (b) f o r $i = 0, 1,\do ts,q-1$ . T h e fa ct t hat $ \ ov erlin e{P }_q(x)= P_q(x) + q\ el l$ follows f r om (b) for $ i=0 $ and from t hedefini ti ono f the setwo s ets.
S uppose that $\e ll $ satis f ... | have_$$\overline{G}^j(\overline{F}^i(x)) =
G^j(F^i(x)_+ i\ell) =
G^j(F^i(x)) +_i\ell.$$ This_gives_(b) for_$i=0,1,\dots,q-1$._The fact that_$\overline{P}_q(x) = P_q(x)_+ q\ell$ follows from_(b) for $i=0$_and_from the definition of these two sets.
Suppose that $\ell$ satisfies the assumption of (c)._From_(b) and_the_c... |
that $|A' \cap Z| > m/2$ and $|Z| {\leqslant}|(B+B) \setminus [\lambda k]|$. It follows that, given $B$ such that $B = A \setminus [\lambda k / 2]$ for some $A \in {\mathcal{D}}(b,\mu)$, the number of choices for $A'$ is at most $$\sum_{\ell > m/2} \binom{\mu b}{\ell}\binom{\lambda k / 2}{k - b - \ell} {\leqslant}\sum... | that $ |A' \cap Z| > m/2 $ and $ |Z| { \leqslant}|(B+B) \setminus [ \lambda k]|$. It follows that, given $ B$ such that $ B = A \setminus [ \lambda k / 2]$ for some $ A \in { \mathcal{D}}(b,\mu)$, the number of option for $ A'$ is at about $ $ \sum_{\ell > m/2 } \binom{\mu b}{\ell}\binom{\lambda k / 2}{k - b - \ell } {... | thwt $|A' \cap Z| > m/2$ and $|Z| {\leqsuant}|(B+B) \setminus [\lambda k]|$. It rollows ghat, given $B$ such that $B = A \derminuw [\lambda k / 2]$ for some $X \in {\mathbal{D}}(b,\mu)$, tye nnmber of choices for $A'$ is at mkdt $$\sbm_{\xll > m/2} \binom{\mu n}{\ell}\binom{\lakbda k / 2}{k - b - \alu} {\peqslant}\sum... | that $|A' \cap Z| > m/2$ and \setminus k]|$. It that, given $B$ \setminus k / 2]$ some $A \in the number of choices for $A'$ at most $$\sum_{\ell > m/2} \binom{\mu b}{\ell}\binom{\lambda k / 2}{k - b - {\leqslant}\sum_{\ell > m/2} \left( \frac{e\mu b}{\ell} \cdot \frac{2}{\lambda-2} \right)^\ell {\lambda k / 2 k b} \le... | that $|A' \cap Z| > m/2$ and $|Z| {\leqslant}|(B+B) \sEtminus [\lamBda k]|$. IT foLloWs That, GiveN $B$ such that $B = A \seTMinuS [\lambda k / 2]$ for some $A \in {\mathCal{D}}(b,\Mu)$, THe nuMBeR of chOices foR $a'$ iS AT moSt $$\SuM_{\elL > m/2} \BInOm{\mu b}{\Ell}\Binom{\laMbda k / 2}{k - b - \ell} {\LeqSlAnt}\sum... | that $|A' \cap Z| > m/2$and $|Z| { \leqs lan t}| (B +B)\set minus [\lambda k]|$ . It follows that, giv en $B $s ucht ha t $B= A \se t mi n u s [ \l am bda k /2]$ f orsome $A \in {\mat hca l{ D}}(b,\mu)$, th e number o f c hoices for $ A'$ is at m ost $$\su m_{ \ell> m/2} \binom {\mu b}{\ el l }\bino m {\lambd a k/... | that_$|A' \cap_Z| > m/2$ and_$|Z| {\leqslant}|(B+B)_\setminus_[\lambda k]|$._It_follows that, given_$B$ such that_$B = A \setminus_[\lambda k /_2]$_for some $A \in {\mathcal{D}}(b,\mu)$, the number of choices for $A'$ is at most_$$\sum_{\ell_> m/2}_\binom{\mu_b}{\ell}\binom{\lambda_k / 2}{k - b_- \ell} {\leqslant}\sum... |
2$. We establish the closure properties of all classes of quantitative languages with respect to these four operations.
author:
- Krishnendu Chatterjee
- Laurent Doyen
- 'Thomas A. Henzinger'
bibliography:
- 'biblio.bib'
title: Expressiveness and Closure Properties for Quantitative Languages
---
Introduction
=========... | 2$. We establish the closure properties of all classes of quantitative lyric with obedience to these four operations.
author:
- Krishnendu Chatterjee
- Laurent Doyen
-' Thomas A. Henzinger'
bibliography:
-' biblio.bib'
title: Expressiveness and Closure Properties for Quantitative Languages
---
insert... | 2$. We establish the closure pvoperties of all classev of qhantitatkve languages with respect tl rhese four operations.
author:
- Krishnenfu Chattwrjet
- Laurent Doyen
- 'Tikmas A. Mznzinfcr'
bibnmography:
- 'biblio.nib'
title: Ex[ressiveness atd Cposure Properties for Quantitative Janguagrs
---
Lntroduction
=========... | 2$. We establish the closure properties of of languages with to these four - Doyen - 'Thomas Henzinger' bibliography: - title: Expressiveness and Closure Properties for Languages --- Introduction ============ A boolean language $L$ can be viewed as a that assigns to each word $w$ a boolean value, namely, $L(w) = 1$ the... | 2$. We establish the closure propErties of alL clasSes Of qUaNtitAtivE languages with REspeCt to these four operationS.
authOr:
- kRishNEnDu ChaTterjee
- lAuRENt DOyEn
- 'thoMaS a. HEnzinGer'
BibliogRaphy:
- 'bibliO.biB'
tItle: ExpressiVEnEss and ClosUre properties foR QuAntitaTiVe LANguagEs
---
INtrodUction
=========... | 2$. We establish the closu re propert ies o f a llcl asse s of quantitativel angu ages with respect to t hesefo u r op e ra tions .
autho r :- Kri sh ne ndu C h at terje e
- Lauren t Doyen
-'Th om as A. Henzin g er '
bibliogr aph y:
- 'biblio .bi b'
tit le : E x press ive nessand Cl o sure P roperties f o r Quan t ... | 2$. We_establish the_closure properties of all_classes of_quantitative_languages with_respect_to these four_operations.
author:
- Krishnendu Chatterjee
-_Laurent Doyen
- 'Thomas A._Henzinger'
bibliography:
- 'biblio.bib'
title: Expressiveness_and_Closure Properties for Quantitative Languages
---
Introduction
=========... |
" points on $C$, instead choose $Z$ to consist of 192 general points on $C$ and one general point of $\lambda$. The first difference of the Hilbert function of $Z$ is
deg 0 1 2 3 4 5 6 7 8 9 10 11
-------------- --- --- -... | " points on $ C$, instead choose $ Z$ to consist of 192 general period on $ C$ and one cosmopolitan point of $ \lambda$. The first remainder of the Hilbert routine of $ Z$ is
deg 0 1 2 3 4 5 6 7 8 9 10 11
-... | " polnts on $C$, instead choose $Z$ to consist oy 192 genecal poihts on $C$ and one general point of $\lalbea$. Tht first difference uf the Hipbert fubctiib of $Z$ is
deg 0 1 2 3 4 5 6 7 8 9 10 11
-------------- --- --- -... | " points on $C$, instead choose $Z$ of general points $C$ and one first of the Hilbert of $Z$ is 0 1 2 3 4 5 7 8 9 10 11 -------------- --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h_Z$ 1 3 6 10 15 20 24 27 29 29 29 0 that this is exactly the same as what would h... | " points on $C$, instead choose $Z$ to Consist of 192 gEneraL poIntS oN $C$ anD one General point of $\LAmbdA$. The first difference of tHe HilBeRT funCTiOn of $Z$ Is
deg 0 1 2 3 4 5 6 7 8 9 10 11
-------------- --- --- -... | " points on $C$, instead c hoose $Z$to co nsi stof 192 gen eral points on $C$and one general pointof $\ la m bda$ . T he fi rst dif f er e n ceof t heHi l be rt fu nct ion of$Z$ is
deg 0 1 2 3 4 5 6 7 8 9 10 11
----------- - -- --- --- -... | " points_on $C$,_instead choose $Z$ to_consist of_192_general points_on_$C$ and one_general point of_$\lambda$. The first difference_of the Hilbert_function_of $Z$ is
deg __ _0__ 1 2_ 3 _ 4_ 5 6__ 7_ 8 9 _ 10 11_ ___ _ _ _ _ _ _ _ _ _ _
_-------------- --- --- -... |
training set, to evaluate the attribute generation performance, whose average precision on CelebA testing set is $91.7\%$, close to state-of-the-art performance. F1 score is reported as precision and recall may vary due to threshold setting.
We apply 3DA-GAN on the negative samples (without the target attribute) to g... | training set, to evaluate the attribute coevals operation, whose average precision on CelebA examination set is $ 91.7\%$, airless to state - of - the - artwork operation. F1 score is reported as precision and recall may vary due to threshold mount.
We apply 3DA - GAN on the negative sample distribution (without the... | trwining set, to evaluate tme attribute gengrqtion 'erformznce, whore average precision on CeleuA twstint set is $91.7\%$, close to stage-of-the-arn performqnce. D1 score is reported as pdccisimi and recall mai vary due tm threshold sedtkny.
We apply 3DA-GAN on the negative sampjes (witnokt the target wttrpbtte) fo g... | training set, to evaluate the attribute generation average on CelebA set is $91.7\%$, score reported as precision recall may vary to threshold setting. We apply 3DA-GAN the negative samples (without the target attribute) to generate the images with target serving as positive samples. Further, FID [@fid2017] is computed... | training set, to evaluate the aTtribute geNeratIon PerFoRmanCe, whOse average precISion On CelebA testing set is $91.7\%$, clOse to StATe-of-THe-Art peRformanCE. F1 SCOre Is RePorTeD As PreciSioN and recAll may vary Due To Threshold setTInG.
We apply 3DA-gAN On the negativE saMples (wItHouT The taRgeT attrIbute) tO G... | training set, to evaluate the attri butegen era ti on p erfo rmance, whosea vera ge precision on CelebA test in g set is $91. 7\%$, c l os e tost at e-o f- t he -artper formanc e. F1 scor e i sreported asp re cision and re call may var y d ue toth res h old s ett ing.
We ap p ly 3DA -GAN on t he negati v e sampl ... | training_set, to_evaluate the attribute generation_performance, whose_average_precision on_CelebA_testing set is_$91.7\%$, close to_state-of-the-art performance. F1 score_is reported as_precision_and recall may vary due to threshold setting.
We apply 3DA-GAN on the negative samples_(without_the target_attribute)_to_g... |
the Elba network with networks obtained for tourism regions in other continents, such as America and Asia, in order to search for similar and distinct properties. The reported methodology can also be applied as a means to obtain a comparative analysis between tourism destinations in the first and third world.
*Simula... | the Elba network with networks obtained for tourism regions in early continent, such as America and Asia, in order to search for exchangeable and clear-cut properties. The reported methodology can also be lend oneself as a means to obtain a relative psychoanalysis between tourism destinations in the first and third wor... | thf Elba network with netwurks obtained for tourmsm regjons in uther continents, such as Amecica and Qsia, in order to searcf for simplar and eistmnct properties. Vge repovced msbhodonigy can also bg applied as a means to obdakn a comparative analysis between touwism dextlnations in thg firxe ans third world.
*Simula... | the Elba network with networks obtained for in continents, such America and Asia, similar distinct properties. The methodology can also applied as a means to obtain comparative analysis between tourism destinations in the first and third world. *Simulations:* Given network such as that analyzed in this work and its res... | the Elba network with networkS obtained fOr touRisM reGiOns iN othEr continents, suCH as AMerica and Asia, in order to SearcH fOR simILaR and dIstinct PRoPERtiEs. thE rePoRTeD methOdoLogy can Also be applIed As A means to obtaIN a ComparativE anAlysis betweeN toUrism dEsTinATions In tHe firSt and tHIrd worLd.
*Simula... | the Elba network with net works obta inedfor to ur ismregi ons in other c o ntin ents, such as Americaand A si a , in or der t o searc h f o r si mi la r a nd di stinc t p roperti es. The re por te d methodolog y c an also be ap plied as a m ean s to o bt ain a com par ative analy s is bet ween tour is m desti ... | the_Elba network_with networks obtained for_tourism regions_in_other continents,_such_as America and_Asia, in order_to search for similar_and distinct properties._The_reported methodology can also be applied as a means to obtain a comparative analysis_between_tourism destinations_in_the_first and third world.
*Simula... |
This percentage rises to 80 if core growth is very rapid. These results indicate that if pebble accretion driven core growth is vigourous, it may be able to destroy many of these otherwise bound protoplanets within their first few thousand years of life.
In the bottom panel of Figure \[fig:Cass\] we plot a histogram ... | This percentage rises to 80 if core growth is very rapid. These results bespeak that if pebble accretion drive core growth is vigourous, it may be able to destroy many of these differently bound protoplanets within their first few thousand years of animation.
In the bottom panel of Figure \[fig: Cass\ ] we plot a hi... | Thls percentage rises to 80 lf core growth is very capid. Tgese resjlts indicate that if pebble axcretuon driven core growth is vigougous, it mqy bt able to destroy many of these lthexwmse bound protoklanets withhn their first fdw thousand years of life.
In the bottoi panel ov Figure \[fig:Cafs\] wt pjot z histogram ... | This percentage rises to 80 if core very These results that if pebble vigourous, may be able destroy many of otherwise bound protoplanets within their first thousand years of life. In the bottom panel of Figure \[fig:Cass\] we plot histogram of core masses that are able to unbind H17 protoplanets in less 1500 Using \[e... | This percentage rises to 80 if coRe growth is Very rApiD. ThEsE resUlts Indicate that if PEbblE accretion driven core grOwth iS vIGourOUs, It may Be able tO DeSTRoy MaNy Of tHeSE oTherwIse Bound prOtoplanets WitHiN their first fEW tHousand yeaRs oF life.
In the boTtoM panel Of figURe \[fig:casS\] we plOt a hisTOgram ... | This percentage rises to80 if core grow thisve ry r apid . These result s ind icate that if pebble a ccret io n dri v en core growth is v igo ur ou s,it ma y beabl e to de stroy many of t hese otherwi s ebound prot opl anets within th eir fi rs t f e w tho usa nd ye ars of life.
In the b ot t om pan e l of Fi ... | This_percentage rises_to 80 if core_growth is_very_rapid. These_results_indicate that if_pebble accretion driven_core growth is vigourous,_it may be_able_to destroy many of these otherwise bound protoplanets within their first few thousand years_of_life.
In the_bottom_panel_of Figure \[fig:Cass\] we plot_a histogram ... |
steps, when $(H,K)$ is a generalized strong Shoda pair of $G$. Let us elaborate on how to proceed by steps. If $(H,K)$ is a generalized strong Shoda pair of $G$, $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ is a strong inductive chain from $H$ to $G$ and $\mathcal{C}\in \mathcal{C}_{q}(H/K)$, set $$\varepsilon_{\mathca... | steps, when $ (H, K)$ is a generalized strong Shoda pair of $ G$. Let us complicate on how to continue by steps. If $ (H, K)$ is a generalized strong Shoda couple of $ G$, $ H = H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ is a strong inductive range from $ H$ to $ G$ and $ \mathcal{C}\in \mathcal{C}_{q}(H / K)$, set $ $ \... | stfps, when $(H,K)$ is a generallzed strong Shodc pair mf $G$. Lst us elxborate on how to proceed by sreps. Uf $(H,K)$ is a generalized strong Sjoda paie of $T$, $H=H_{0}\leq H_{1}\lxs \cdots \leq H_{h}=N$ is c wtrong inductiye chain frmm $H$ to $G$ and $\kagheal{C}\in \mathcal{C}_{q}(H/K)$, set $$\varepsilon_{\matrca... | steps, when $(H,K)$ is a generalized strong of Let us on how to is generalized strong Shoda of $G$, $H=H_{0}\leq \cdots \leq H_{n}=G$ is a strong chain from $H$ to $G$ and $\mathcal{C}\in \mathcal{C}_{q}(H/K)$, set $$\varepsilon_{\mathcal{C}}^{(0)}(H,K) = \varepsilon_{\mathcal{C}}(H,K),$$ $$\varepsilon_{\mathcal{C}}^{(... | steps, when $(H,K)$ is a generalized Strong ShodA pair Of $G$. let Us ElabOratE on how to proceeD By stEps. If $(H,K)$ is a generalized sTrong shODa paIR oF $G$, $H=H_{0}\lEq H_{1}\leq \cDOtS \LEq H_{N}=G$ Is A stRoNG iNductIve Chain frOm $H$ to $G$ and $\mAthCaL{C}\in \mathcal{C}_{Q}(h/K)$, Set $$\varepsiLon_{\Mathca... | steps, when $(H,K)$ is ageneralize d str ong Sh od a pa ir o f $G$. Let use labo rate on how to proceed by s te p s. I f $ (H,K) $ is ag en e r ali ze dstr on g S hodapai r of $G $, $H=H_{0 }\l eq H_{1}\leq \ c do ts \leq H_ {n} =G$ is a str ong induc ti vec hainfro m $H$ to $G $ and $ \mathcal{ C} \ in \ma t ... | steps,_when $(H,K)$_is a generalized strong_Shoda pair_of_$G$. Let_us_elaborate on how_to proceed by_steps. If $(H,K)$ is_a generalized strong_Shoda_pair of $G$, $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ is a strong inductive chain from_$H$_to $G$_and_$\mathcal{C}\in_\mathcal{C}_{q}(H/K)$, set $$\varepsilon_{\mathca... |
to incorporate the special conditions then is solved with orthodox methods or the task is considered as it is and the solving methods are generalized. The most straightforward example for the first approach is the complex scaling method which has a long history [@cs]. In a recent nuclear three-body application [@atk] ... | to incorporate the special conditions then is solved with orthodox method or the undertaking is considered as it is and the solving methods are generalize. The most straightforward exercise for the beginning approach is the complex scaling method acting which have a farseeing history [ @cs ]. In a recent nuclear three ... | to incorporate the special conditions theu is sonved wjth orthudox methods or the task is ronsuderee as it is and the solxing methlds are tenecalized. The most straighbyorwadf excm'le for the firxt approacv is the complax sealing method which has a long histowy [@cs]. Im w recent nuclewr tnwee-bkdy application [@atk] ... | to incorporate the special conditions then is orthodox or the is considered as methods generalized. The most example for the approach is the complex scaling method has a long history [@cs]. In a recent nuclear three-body application [@atk] the problem of this method is solved. The usage of the complex scaling method CC... | to incorporate the special coNditions thEn is sOlvEd wItH ortHodoX methods or the tASk is Considered as it is and the SolviNg MEthoDS aRe genEralizeD. thE MOst StRaIghTfORwArd exAmpLe for thE first apprOacH iS the complex sCAlIng method wHicH has a long hisTorY [@cs]. In a ReCenT NucleAr tHree-bOdy appLIcatioN [@atk] ... | to incorporate the specia l conditio ns th enisso lved wit h orthodox met h odsor the task is conside red a si t is an d the solvin g m e t hod sar e g en e ra lized . T he most straightf orw ar d example fo r t he first a ppr oach is thecom plex s ca lin g meth odwhich has a long h istory [@ cs ] . In a recent... | to_incorporate the_special conditions then is_solved with_orthodox_methods or_the_task is considered_as it is_and the solving methods_are generalized. The_most_straightforward example for the first approach is the complex scaling method which has a_long_history [@cs]._In_a_recent nuclear three-body application [@atk]_... |
As one can check easily, one cannot obtain an exact closed set of dynamical equations, since the time derivatives of two-operator products involve three-operator products, whose time derivatives contain terms being the products of four operators, etc. Because of this hierarchy, we have to restrict ourselves to expecta... | As one can check easily, one cannot obtain an exact shut hardening of dynamical equations, since the meter derivative of two - operator products necessitate three - hustler products, whose time derivatives contain terms being the product of four operators, etc. Because of this hierarchy, we have to qualify ourselves to... | As one can check easily, ont cannot obtain au exact closes set of dynamical equations, since tie tume dtgivatives of two-operagor produbts involce tiree-operator provhcts, whose tijc dernvetives contain jerms being dhe products ox wobr operators, etc. Because of this hiewarchy, ee have to restryct ptrsemnew to expecta... | As one can check easily, one cannot exact set of equations, since the involve products, whose time contain terms being products of four operators, etc. Because this hierarchy, we have to restrict ourselves to expectation values and – at point – introduce a factorization that is based on physical considerations. The sim... | As one can check easily, one canNot obtain aN exacT clOseD sEt of DynaMical equations, SInce The time derivatives of twO-operAtOR proDUcTs invOlve thrEE-oPERatOr PrOduCtS, WhOse tiMe dErivatiVes contain TerMs Being the prodUCtS of four opeRatOrs, etc. BecausE of This hiErArcHY, we haVe tO restRict ouRSelves To expecta... | As one can check easily,one cannot obta inanex actclos ed set of dyna m ical equations, since thetimede r ivat i ve s oftwo-ope r at o r pr od uc tsin v ol ve th ree -operat or product s,wh ose time der i va tives cont ain terms being th e prod uc tso f fou r o perat ors, e t c. Bec ause of t hi s hiera r chy, w... | As_one can_check easily, one cannot_obtain an_exact_closed set_of_dynamical equations, since_the time derivatives_of two-operator products involve_three-operator products, whose_time_derivatives contain terms being the products of four operators, etc. Because of this hierarchy,_we_have to_restrict_ourselves_to expecta... |
0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... | 0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... | 0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... | 0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... | 0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... | 0.9800568,0.7951193)(0.996 3817,0.792 2913) (1. 029 03 15,0 .786 6843)(1.094331 2 ,0.7 756631)(1.2249305,0.75 43668 )( 1 .486 1 29 2,0.7 145679) ( 1. 5 0 245 41 ,0 .71 21 9 7) (1.51 877 9,0.709 8393)(1.55 142 89 ,0.7051629)( 1 .6 167285,0.6 959 638)(1.74732 79, 0.6781 60 9)( 2 .0085 266 ,0.64 47885) ( 2.0237 696,0.642 ... | 0.9800568,0.7951193)(0.9963817,0.7922913)(1.0290315,0.7866843)(1.0943312,0.7756631)(1.2249305,0.7543668)(1.4861292,0.7145679)(1.5024541,0.712197)(1.518779,0.7098393)(1.5514289,0.7051629)(1.6167285,0.6959638)(1.7473279,0.6781609)(2.0085266,0.6447885)(2.0237696,0.6429278)(2.0390127,0.6410762)(2.0694989,0.6374005)(2.13047... |
$ omits the partitions obtained from $S$ by amalgamating adjacent parts of odd size, so that these edges may be regarded as removed or “clipped”. This is illustrated in figure \[clippedcube.fig\], which should be compared with figure 2 of [@graphs03 p. 885]. We note that ${\partial^\lambda}$ and ${\partial^\lambda_{\en... | $ omits the partitions obtained from $ S$ by amalgamating adjacent part of curious size, so that these edges may be regarded as removed or “ trot ”. This is illustrated in figure \[clippedcube.fig\ ], which should be compare with number 2 of [ @graphs03 p. 885 ]. We note that $ { \partial^\lambda}$ and $ { \parti... | $ omlts the partitions obtaiked from $S$ by amcogamatmng adjzcent pafts of odd size, so that thesx edtes mqy be regarded as remoxed or “clppped”. Thiw is ullustratev in figmxe \[cliliedcuye.hig\], which shoulc be compased with figura 2 uf [@yraphs03 p. 885]. We note that ${\partial^\lambda}$ and ${\pattlal^\lambda_{\en... | $ omits the partitions obtained from $S$ adjacent of odd so that these removed “clipped”. This is in figure \[clippedcube.fig\], should be compared with figure 2 [@graphs03 p. 885]. We note that ${\partial^\lambda}$ and ${\partial^\lambda_{\ensuremath{\mathbf{Q}}}}$ co-incide if each entry $S$ and $\mu_a(S)$ have size ... | $ omits the partitions obtaineD from $S$ by amAlgamAtiNg aDjAcenT parTs of odd size, so tHAt thEse edges may be regarded aS remoVeD Or “clIPpEd”. ThiS is illuSTrATEd iN fIgUre \[ClIPpEdcubE.fiG\], which sHould be comParEd With figure 2 of [@GRaPhs03 p. 885]. We note ThaT ${\partial^\lambDa}$ aNd ${\partIaL^\laMBda_{\en... | $ omits the partitions obt ained from $S$byama lg amat ingadjacent parts of o dd size, so that these edge sm ay b e r egard ed as r e mo v e d o r“c lip pe d ”. This is illust rated in f igu re \[clippedcu b e. fig\], whi chshould be co mpa red wi th fi g ure 2 of [@gr aphs03 p. 885 ]. We not et hat ${ \ part... | $ omits_the partitions_obtained from $S$ by_amalgamating adjacent_parts_of odd_size,_so that these_edges may be_regarded as removed or_“clipped”. This is_illustrated_in figure \[clippedcube.fig\], which should be compared with figure 2 of [@graphs03 p. 885]. We note that_${\partial^\lambda}$_and ${\partial^\lambda_{\en... |
states. In a recent work [@Olmos09; @Olmos09-3] this was shown for a system in which the atoms are confined to a deep ring lattice. Here the collective excitations of the atomic ensemble were calculated analytically and it was shown that - due to the special geometry of the ring - these excitations possess a particula... | states. In a recent work [ @Olmos09; @Olmos09 - 3 ] this was shown for a arrangement in which the atom are confined to a deep ring wicket. Here the collective excitations of the atomic corps de ballet were forecast analytically and it was shown that - ascribable to the limited geometry of the ring - these excitation po... | stwtes. In a recent work [@Oloos09; @Olmos09-3] this cqs shoxn for z system in which the atoms are confmned to a deep ring lattice. Herd the colpective wxciuations of the atomic enscible aere ralculated analitically and it was shown dhxt - due to the special geometry of thq ring - tjese excitatiogs ppfsesa a particula... | states. In a recent work [@Olmos09; @Olmos09-3] shown a system which the atoms ring Here the collective of the atomic were calculated analytically and it was that - due to the special geometry of the ring - these excitations a particularly symmetric structure. In this paper we show how these symmetric states be to sing... | states. In a recent work [@Olmos09; @OLmos09-3] this waS showN foR a sYsTem iN whiCh the atoms are cONfinEd to a deep ring lattice. HeRe the CoLLectIVe ExcitAtions oF ThE ATomIc EnSemBlE WeRe calCulAted anaLytically aNd iT wAs shown that - dUE tO the speciaL geOmetry of the rIng - These eXcItaTIons pOssEss a pArticuLA... | states. In a recent work[@Olmos09; @Olm os0 9-3 ]this was shown for a s y stem in which the atoms ar e con fi n ed t o a deep ring l a tt i c e.He re th ec ol lecti veexcitat ions of th e a to mic ensemble we re calcula ted analyticall y a nd itwa s s h own t hat - du e to t h e spec ial geome tr y of th e ... | states._In a_recent work [@Olmos09; @Olmos09-3]_this was_shown_for a_system_in which the_atoms are confined_to a deep ring_lattice. Here the_collective_excitations of the atomic ensemble were calculated analytically and it was shown that -_due_to the_special_geometry_of the ring - these_excitations possess a particula... |
right-handed slepton and $\langle v^2 \rangle$ represents the average of the squared velocity of the LSP, which could be determined by a close inspection of the scattering of the LSP off the thermal radiations. For the case of $E\gg m_{LSP}$, we estimate the elastic scattering cross section off an electron in the ther... | right - handed slepton and $ \langle v^2 \rangle$ represents the average of the squared velocity of the LSP, which could be determined by a cheeseparing inspection of the dispersion of the LSP off the thermal radiations. For the case of $ E\gg m_{LSP}$, we estimate the elastic disperse cross section off an electron in ... | rihht-handed slepton and $\lakgle v^2 \rangle$ rekrwsents the aberage ow the squared velocity of thx LSP, whixh could be determined by a clode inspextioi of the scattermhg of tmz LSP lff chx thermal radiajions. For tha case of $E\gg k_{LRP}$, we estimate the elastic scattering cross xeftion off an ejectgog in nht ther... | right-handed slepton and $\langle v^2 \rangle$ represents of squared velocity the LSP, which close of the scattering the LSP off thermal radiations. For the case of m_{LSP}$, we estimate the elastic scattering cross section off an electron in the bath to be of the order of $$\langle \sigma v_{rel} \rangle \sim \alpha^{... | right-handed slepton and $\langLe v^2 \rangle$ rEpresEntS thE aVeraGe of The squared veloCIty oF the LSP, which could be detErminEd BY a clOSe InspeCtion of THe SCAttErInG of ThE lSp off tHe tHermal rAdiations. FOr tHe Case of $E\gg m_{LSp}$, We Estimate thE elAstic scatterIng Cross sEcTioN Off an EleCtron In the tHEr... | right-handed slepton and$\langle v ^2 \r ang le$ r epre sent s the averageo f th e squared velocity ofthe L SP , whi c hcould be det e rm i n edby a cl os e i nspec tio n of th e scatteri ngof the LSP off th e thermalrad iations. For th e case o f $ E \gg m _{L SP}$, we es t imatethe elast ic scatte r ing cr... | right-handed_slepton and_$\langle v^2 \rangle$ represents_the average_of_the squared_velocity_of the LSP,_which could be_determined by a close_inspection of the_scattering_of the LSP off the thermal radiations. For the case of $E\gg m_{LSP}$, we_estimate_the elastic_scattering_cross_section off an electron in_the ther... |
JAS, VHCh, LKE were partially supported by the research grants No. 97-02-17168 and 1.2.2.2 from the Russian Foundation for Basic Research and from the State Programm “Astronomy” respectively.
Afanasiev, V.L., Lipovetsky, V.A., Mikhailov, V.P, Nazarov, E.A., & Shapovalova, A.I. 1991, Astrofiz. Issled. (Izv. SAO.), 31,... | JAS, VHCh, LKE were partially supported by the research grants No. 97 - 02 - 17168 and 1.2.2.2 from the Russian Foundation for Basic Research and from the State Programm “ Astronomy ” respectively.
Afanasiev, V.L., Lipovetsky, V.A., Mikhailov, V.P, Nazarov, E.A., & Shapovalova, A.I. 1991, Astrofiz. Issled. (Izv. SAO... | JAD, VHCh, LKE were partialln supported by tkw reseerch grznts No. 97-02-17168 and 1.2.2.2 from the Russian Foundetiob for Basic Research and frum the Stwte Progeamm “Qstronomy” csspectiyzly.
Afzkasier, T.L., Lipovetsky, V.S., Mikhailoe, V.P, Nazarov, E.D., & Skapovalova, A.I. 1991, Astrofiz. Issled. (Izv. SWO.), 31,... | JAS, VHCh, LKE were partially supported by grants 97-02-17168 and from the Russian from State Programm “Astronomy” Afanasiev, V.L., Lipovetsky, Mikhailov, V.P, Nazarov, E.A., & Shapovalova, 1991, Astrofiz. Issled. (Izv. SAO.), 31, 121 Afanasiev, V.L., Burenkov, A.N., Vlasyuk, V.V., Drabek, S.V. 1995, SAO technical repo... | JAS, VHCh, LKE were partially suPported by tHe resEarCh gRaNts NO. 97-02-17168 and 1.2.2.2 From the Russian fOundAtion for Basic Research aNd froM tHE StaTE PRograMm “AstroNOmY” REspEcTiVelY.
AFAnAsiev, v.L., LIpovetsKy, V.A., MikhaiLov, v.P, nazarov, E.A., & ShaPOvAlova, A.I. 1991, AstRofIz. Issled. (Izv. SaO.), 31,... | JAS, VHCh, LKE were parti ally suppo rtedbythe r esea rchgrants No. 97- 0 2-17 168 and 1.2.2.2 from t he Ru ss i an F o un datio n for B a si c Res ea rc h a nd fr om th e S tate Pr ogramm “As tro no my” respecti v el y.
Afanas iev , V.L., Lipo vet sky, V .A .,M ikhai lov , V.P , Naza r ov, E. A., & Sha po v alova, ... | JAS,_VHCh, LKE_were partially supported by_the research_grants_No. 97-02-17168_and_1.2.2.2 from the_Russian Foundation for_Basic Research and from_the State Programm_“Astronomy”_respectively.
Afanasiev, V.L., Lipovetsky, V.A., Mikhailov, V.P, Nazarov, E.A., & Shapovalova, A.I. 1991, Astrofiz. Issled._(Izv._SAO.), 31,... |
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\,,$$ where overdot denotes differentiation with respect to time. Bellow we shall represent the Hamiltonian corresponding to Lagrangian (\[eq:bilag\]) in terms of a special parameterization, adapted to the action of ${SO(n, \mathbb{R})}$ symmetry group of the system. We shall demonstrate that on the Principal orbit str... | \,,$$ where overdot denotes differentiation with respect to time. Bellow we shall constitute the Hamiltonian correspond to Lagrangian (\[eq: bilag\ ]) in terms of a special parameterization, adapted to the natural process of $ { SO(n, \mathbb{R})}$ symmetry group of the system. We shall show that on the Principal orbit... | \,,$$ whfre overdot denotes difftrentiation with tewpect vo time. Bellow de shall represent the Hamilvoniqn coeresponding to Lagrangkan (\[eq:bilwg\]) in teems id a special paramebzrizaflon, aba'ted to the actlon of ${SO(n, \kathbb{R})}$ symmetsy gxoup of the system. We shall demonstrwte thay ln the Principwl ogbyt sfg... | \,,$$ where overdot denotes differentiation with respect Bellow shall represent Hamiltonian corresponding to a parameterization, adapted to action of ${SO(n, symmetry group of the system. We demonstrate that on the Principal orbit stratum of ${SO(n, \mathbb{R})}$ group action the Hamiltonian defines a new generalizatio... | \,,$$ where overdot denotes differEntiation wIth reSpeCt tO tIme. BElloW we shall represENt thE Hamiltonian correspondIng to laGRangIAn (\[Eq:bilAg\]) in terMS oF A SpeCiAl ParAmETeRizatIon, Adapted To the actioN of ${sO(N, \mathbb{R})}$ symmETrY group of thE syStem. We shall dEmoNstratE tHat ON the PRinCipal Orbit sTR... | \,,$$ where overdot denote s differen tiati onwit hresp ectto time. Bello w weshall represent the Ha milto ni a n co r re spond ing toL ag r a ngi an ( \[e q: b il ag\]) in termsof a speci alpa rameterizati o n, adapted t o t he action of ${ SO(n,\m ath b b{R}) }$symme try gr o up ofthe syste m. We sha l l demon ... | \,,$$ where_overdot denotes_differentiation with respect to_time. Bellow_we_shall represent_the_Hamiltonian corresponding to_Lagrangian (\[eq:bilag\]) in_terms of a special_parameterization, adapted to_the_action of ${SO(n, \mathbb{R})}$ symmetry group of the system. We shall demonstrate that on_the_Principal orbit_str... |
f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is uniformly smooth.
A remark regarding the sharpness of our results is now in order. Under the generality of the hypotheses of Theorem \[uniform monotonicity with 1-coercive modulus is recessive\], ev... | f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is uniformly smooth.
A remark regarding the sharpness of our results is nowadays in decree. Under the generality of the hypotheses of Theorem \[uniform monotonicity with 1 - coercive modulus is rece... | f},{\blldsymbol \lambda})=p_\mu({\bf f},{\buldsymbol \lambdc})^{**}=p_{\mu^{-1}}({\bf h^*},{\boldsyjbol \lamcda})^*$ is uniformly smooth.
A remerk eegareing the sharpness of uur resulns is now in ieder. Under the gencxalitg of chx hypotheses of Theorem \[unhform monotoniwigy with 1-coercive modulus is recessive\], ev... | f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is A regarding the of our results the of the hypotheses Theorem \[uniform monotonicity 1-coercive modulus is recessive\], even when that for each $i\in I$, ${\ensuremath{\varphi}}_i$ is the largest possi... | f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boLdsymbol \laMbda})^{**}=p_{\Mu^{-1}}({\bF f^*},{\bOlDsymBol \lAmbda})^*$ is uniformLY smoOth.
A remark regarding the SharpNeSS of oUR rEsultS is now iN OrDER. UnDeR tHe gEnERaLity oF thE hypothEses of TheoRem \[UnIform monotonICiTy with 1-coerCivE modulus is reCesSive\], ev... | f},{\boldsymbol \lambda}) =p_\mu({\b f f}, {\b old sy mbol \la mbda})^{**}=p_ { \mu^ {-1}}({\bf f^*},{\bold symbo l\ lamb d a} )^*$is unif o rm l y sm oo th .
Ar em ark r ega rding t he sharpne ssof our results is now in or der . Under thegen eralit yoft he hy pot heses of Th e orem \ [uniformmo n otonic i ty wit... | f},{\boldsymbol_\lambda})=p_\mu({\bf f},{\boldsymbol_\lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is_uniformly smooth.
A_remark_regarding the_sharpness_of our results_is now in_order. Under the generality_of the hypotheses_of_Theorem \[uniform monotonicity with 1-coercive modulus is recessive\], ev... |
mathcal{Y}}(W_s) - \gamma \right) \dd s} \right] \le
K_1 N_{\cC}^{5/2}\left(\frac{r}{a} \right)^{\tfrac{d}2} \left(1 + \frac{\gamma + (1+\theta) r^{-2}}{\gamma - \lambda_{\cC}} \right)$$ and $$\label{e:semigroupbb}
\sup_{ t \ge 0}\E_x \left[ {\end{equation}}^{\int_0^t \left(\theta V^{(a)}_{\mathcal{Y}}(W_s) - \gamma \... | mathcal{Y}}(W_s) - \gamma \right) \dd s } \right ] \le
K_1 N_{\cC}^{5/2}\left(\frac{r}{a } \right)^{\tfrac{d}2 } \left(1 + \frac{\gamma + (1+\theta) r^{-2}}{\gamma - \lambda_{\cC } } \right)$$ and $ $ \label{e: semigroupbb }
\sup _ { t \ge 0}\E_x \left [ { \end{equation}}^{\int_0^t \left(\theta V^{(a)}_{\mathcal{Y}... | matjcal{Y}}(W_s) - \gamma \right) \dd r} \right] \le
K_1 N_{\cE}^{5/2}\oeft(\frec{r}{a} \rifht)^{\tfrac{a}2} \left(1 + \frac{\gamma + (1+\theta) r^{-2}}{\galmq - \lanbda_{\cC}} \right)$$ and $$\label{d:semigrouibb}
\sup_{ t \te 0}\E_e \left[ {\end{equation}}^{\int_0^t \lcyt(\thefw V^{(a)}_{\kethcal{Y}}(W_s) - \gamms \... | mathcal{Y}}(W_s) - \gamma \right) \dd s} \right] N_{\cC}^{5/2}\left(\frac{r}{a} \left(1 + + (1+\theta) r^{-2}}{\gamma \sup_{ \ge 0}\E_x \left[ \left(\theta V^{(a)}_{\mathcal{Y}}(W_s) - \right) \dd s} \mathbbm{1}_{\{\tau_{\cC^\cc} > t \le K_1 N_{\cC}^{5/2} \left(\frac{r}{a} \right)^{\tfrac{d}2} \left( 1 + \frac{\gamma +... | mathcal{Y}}(W_s) - \gamma \right) \dd s} \riGht] \le
K_1 N_{\cC}^{5/2}\lEft(\frAc{r}{A} \riGhT)^{\tfrAc{d}2} \lEft(1 + \frac{\gamma + (1+\thETa) r^{-2}}{\gAmma - \lambda_{\cC}} \right)$$ and $$\laBel{e:sEmIGrouPBb}
\Sup_{ t \gE 0}\E_x \left[ {\ENd{EQUatIoN}}^{\iNt_0^t \LeFT(\tHeta V^{(A)}_{\maThcal{Y}}(W_S) - \gamma \... | mathcal{Y}}(W_s) - \gamma\right) \d d s}\ri ght ]\le
K_1 N_{\cC}^{5/2} \ left (\frac{r}{a} \right)^{ \tfra c{ d }2}\ le ft(1+ \frac { \g a m ma+(1 +\t he t a) r^{- 2}} {\gamma - \lambda _{\ cC }} \right)$$ an d $$\label {e: semigroupbb}
\s up_{ t \ ge0 }\E_x \l eft[{\end{ e quatio n}}^{\int _0 ^ t \lef t (\theta ... | mathcal{Y}}(W_s) -_\gamma \right)_\dd s} \right] \le_
K_1 N_{\cC}^{5/2}\left(\frac{r}{a}_\right)^{\tfrac{d}2}_\left(1 +_\frac{\gamma_+ (1+\theta) r^{-2}}{\gamma_- \lambda_{\cC}} \right)$$_and $$\label{e:semigroupbb}
\sup_{ t \ge_0}\E_x \left[ {\end{equation}}^{\int_0^t_\left(\theta_V^{(a)}_{\mathcal{Y}}(W_s) - \gamma \... |
_x\\#4&s_3&j_y\\#7&\sigma&J_3 \end{array}\right\}}
\cdot {A \left\{\begin{array}{ccc} l_x'&\sigma_x'&j_x'\\#4&s_4&j_y'\\#7&\sigma'&J_3' \end{array}\right\}}\\
&\cdot&\displaystyle {A \left\{\begin{array}{ccc} l_{xy}&\sigma&J_3\\#4&s_4&j_z\\#7&S&J \end{array}\right\}}
\cdot {A \left\{\begin{array}{ccc} l_{xy}'&\sigm... | _ x\\#4&s_3&j_y\\#7&\sigma&J_3 \end{array}\right\ } }
\cdot { A \left\{\begin{array}{ccc } l_x'&\sigma_x'&j_x'\\#4&s_4&j_y'\\#7&\sigma'&J_3' \end{array}\right\}}\\
& \cdot&\displaystyle { A \left\{\begin{array}{ccc } l_{xy}&\sigma&J_3\\#4&s_4&j_z\\#7&S&J \end{array}\right\ } }
\cdot { A \left\{\begin{array}{c... | _x\\#4&s_3&j_j\\#7&\sigma&J_3 \end{array}\right\}}
\cdut {A \left\{\begin{atrqy}{ccc} n_x'&\sigmz_x'&j_x'\\#4&s_4&j_y'\\#7&\skgma'&J_3' \end{array}\right\}}\\
&\cdot&\disppatstylt {A \left\{\begin{array}{czc} l_{xy}&\sigla&J_3\\#4&s_4&j_z\\#7&S&J \end{erray}\right\}}
\cdot {A \left\{\bcyin{ardwy}{cce} o_{xy}'&\sigm... | _x\\#4&s_3&j_y\\#7&\sigma&J_3 \end{array}\right\}} \cdot {A \left\{\begin{array}{ccc} l_x'&\sigma_x'&j_x'\\#4&s_4&j_y'\\#7&\sigma'&J_3' \end{array}\right\}}\\ \left\{\begin{array}{ccc} \end{array}\right\}} \cdot \left\{\begin{array}{ccc} l_{xy}'&\sigma'&J_3'\\#4&s_2&j_z'\\#7&S&J \end{array}\right\}} &\cdot&\displaystyl... | _x\\#4&s_3&j_y\\#7&\sigma&J_3 \end{array}\right\}}
\cdOt {A \left\{\begIn{arrAy}{cCc} l_X'&\sIgma_X'&j_x'\\#4&s_4&J_y'\\#7&\sigma'&J_3' \end{arrAY}\rigHt\}}\\
&\cdot&\displaystyle {A \lefT\{\begiN{aRRay}{cCC} l_{Xy}&\sigMa&J_3\\#4&s_4&j_z\\#7&S&j \EnD{ARraY}\rIgHt\}}
\cDoT {a \lEft\{\beGin{Array}{ccC} l_{xy}'&\sigm... | _x\\#4&s_3&j_y\\#7&\sigma& J_3 \end{a rray} \ri ght \} }
\ cdot {A \left\{\be g in{a rray}{ccc} l_x'&\sigma _x'&j _x ' \\#4 & s_ 4&j_y '\\#7&\ s ig m a '&J _3 '\en d{ a rr ay}\r igh t\}}\\
&\cdot&\d isp la ystyle {A \l e ft \{\begin{a rra y}{ccc} l_{x y}& \sigma &J _3\ \ #4&s_ 4&j _z\\# 7&S&J\ end{ar ray}\righ t\ } }... | _x\\#4&s_3&j_y\\#7&\sigma&J_3 \end{array}\right\}}
_\cdot {A_\left\{\begin{array}{ccc} l_x'&\sigma_x'&j_x'\\#4&s_4&j_y'\\#7&\sigma'&J_3' \end{array}\right\}}\\
&\cdot&\displaystyle_{A \left\{\begin{array}{ccc}_l_{xy}&\sigma&J_3\\#4&s_4&j_z\\#7&S&J_\end{array}\right\}}
\cdot__{A \left\{\begin{array}{ccc} l_{xy}'&\sigm... |
space: $$\notag
{\rm tr}[\,(Q\Psi)\,\phi\,]+{\rm tr}[\,\Psi\Psi\,\phi\,]=0.$$
The Ellwood invariant {#el}
---------------------
In [@Ellwood:2008jh], Ellwood conjectured that there exists a relation between the gauge-invariant observables of open string field theory which were discovered in [@HI; @GRSZ], and the clo... | space: $ $ \notag
{ \rm tr}[\,(Q\Psi)\,\phi\,]+{\rm tr}[\,\Psi\Psi\,\phi\,]=0.$$
The Ellwood invariant { # el }
---------------------
In [ @Ellwood:2008jh ], Ellwood conjectured that there exist a relation back between the gauge - invariant observables of open drawstring field theory which were discover in [ ... | spwce: $$\notag
{\rm tr}[\,(Q\Psi)\,\phi\,]+{\rm ur}[\,\Psi\Psi\,\phi\,]=0.$$
The Ellwood iivarianf {#el}
---------------------
In [@Eulwood:2008jh], Ellwood conjectured tyat tyere exists a relation between nhe gauge-unvaciant observables of opek strjkg fizlv theory which eere discoeered in [@HI; @GRVZ], aud the clo... | space: $$\notag {\rm tr}[\,(Q\Psi)\,\phi\,]+{\rm tr}[\,\Psi\Psi\,\phi\,]=0.$$ The Ellwood --------------------- [@Ellwood:2008jh], Ellwood that there exists observables open string field which were discovered [@HI; @GRSZ], and the closed string on a disk.[^12] In this paper, we call these gauge-invariant observables th... | space: $$\notag
{\rm tr}[\,(Q\Psi)\,\phi\,]+{\rm tr}[\,\psi\Psi\,\phi\,]=0.$$
THe EllWooD inVaRianT {#el}
---------------------
IN [@Ellwood:2008jh], EllwOOd coNjectured that there exisTs a reLaTIon bETwEen thE gauge-iNVaRIAnt ObSeRvaBlES oF open StrIng fielD theory whiCh wErE discovered iN [@hI; @gRSZ], and the Clo... | space: $$\notag
{\rm tr}[ \,(Q\Psi)\ ,\phi \,] +{\ rm tr} [\,\ Psi\Psi\,\phi\ , ]=0. $$
The Ellwood invari ant { #e l }
-- - -- ----- ------- - -- -
In [ @E llw oo d :2 008jh ],Ellwood conjectur edth at there exi s ts a relatio n b etween the g aug e-inva ri ant obser vab les o f open string field th eo r y whi... | space:_$$\notag
{\rm tr}[\,(Q\Psi)\,\phi\,]+{\rm_tr}[\,\Psi\Psi\,\phi\,]=0.$$
The Ellwood invariant {#el}
---------------------
In_[@Ellwood:2008jh], Ellwood_conjectured_that there_exists_a relation between_the gauge-invariant observables_of open string field_theory which were_discovered_in [@HI; @GRSZ], and the clo... |
and for all objects $X_{1},\dots,X_{n},Y\in \underline{\mathcal{O}}$, the map of spaces $$\mul_{\mathcal{O}}\left(\left\{ X_{1},\dots,X_{n}\right\} ;Y\right)\to\mul_{h_{d}\mathcal{O}}\left(\left\{ \theta_{d}\left(X_{1}\right),\dots,\theta_{d}\left(X_{n}\right)\right\} ;\theta_{d}\left(Y\right)\right)$$ is the $\left(d... | and for all objects $ X_{1},\dots, X_{n},Y\in \underline{\mathcal{O}}$, the map of spaces $ $ \mul_{\mathcal{O}}\left(\left\ { X_{1},\dots, X_{n}\right\ }; Y\right)\to\mul_{h_{d}\mathcal{O}}\left(\left\ { \theta_{d}\left(X_{1}\right),\dots,\theta_{d}\left(X_{n}\right)\right\ }; \theta_{d}\left(Y\right)\right)$$ is the ... | anf for all objects $X_{1},\dots,X_{k},Y\in \underline{\majhxal{O}}$, tie map kf spacer $$\mul_{\mathcal{O}}\left(\left\{ X_{1},\dots,X_{i}\rigyt\} ;Y\rught)\to\mul_{h_{d}\mathcal{O}}\lefg(\left\{ \thena_{d}\left(X_{1}\rught),\vots,\theta_{d}\left(X_{n}\cjght)\rigmc\} ;\thefw_{d}\leyt(B\right)\right)$$ is jhe $\left(d... | and for all objects $X_{1},\dots,X_{n},Y\in \underline{\mathcal{O}}$, the spaces X_{1},\dots,X_{n}\right\} ;Y\right)\to\mul_{h_{d}\mathcal{O}}\left(\left\{ ;\theta_{d}\left(Y\right)\right)$$ is the universal of $\theta_{d}$) and to the proof $d$-categories. We conclude with a simple of the theory of $d$-operads, that s... | and for all objects $X_{1},\dots,X_{n},Y\iN \underline{\MathcAl{O}}$, The MaP of sPaceS $$\mul_{\mathcal{O}}\leFT(\lefT\{ X_{1},\dots,X_{n}\right\} ;Y\right)\to\mUl_{h_{d}\mAtHCal{O}}\LEfT(\left\{ \Theta_{d}\lEFt(x_{1}\RIghT),\dOtS,\thEtA_{D}\lEft(X_{n}\RigHt)\right\} ;\Theta_{d}\left(y\riGhT)\right)$$ is the $\lEFt(D... | and for all objects $X_{1 },\dots,X_ {n},Y \in \u nd erli ne{\ mathcal{O}}$,t he m ap of spaces $$\mul_{\ mathc al { O}}\ l ef t(\le ft\{ X_ { 1} , \ dot s, X_ {n} \r i gh t\} ; Y\r ight)\t o\mul_{h_{ d}\ ma thcal{O}}\le f t( \left\{ \t het a_{d}\left(X _{1 }\righ t) ,\d o ts,\t het a_{d} \left( X _{n}\r ight)\rig ht... | and_for all_objects $X_{1},\dots,X_{n},Y\in \underline{\mathcal{O}}$, the_map of_spaces_$$\mul_{\mathcal{O}}\left(\left\{ X_{1},\dots,X_{n}\right\}_;Y\right)\to\mul_{h_{d}\mathcal{O}}\left(\left\{_\theta_{d}\left(X_{1}\right),\dots,\theta_{d}\left(X_{n}\right)\right\} ;\theta_{d}\left(Y\right)\right)$$ is_the $\left(d... |
Then $N_1$ is a line bundle of degree $-l$, say. One clearly has $l\leq d_1$. Define the stratification of $S_{k-2}$ given by the subsets $$T_t=\{ (L_2,V_2) \in S_{k-2} \; |\; h^0(N^*_1{\otimes}L_2)=t\}.$$ Clearly $\dim T_t \leq \dim G(1,d_2+l,t)$. Also we stratify $G(1,d_1,2)$ by the subsets $W_l$ of those coherent s... | Then $ N_1 $ is a line bundle of degree $ -l$, say. One intelligibly have $ l\leq d_1$. Define the stratification of $ S_{k-2}$ given by the subsets $ $ T_t=\ { (L_2,V_2) \in S_{k-2 } \; |\; h^0(N^*_1{\otimes}L_2)=t\}.$$ distinctly $ \dim T_t \leq \dim G(1,d_2+l, t)$. Also we stratify $ G(1,d_1,2)$ by the subsets $ W_l... | Thfn $N_1$ is a line bundle of degree $-l$, say. Ouw cleacly has $l\leq d_1$. Aefine the stratification of $S_{j-2}$ givtu by the subsets $$T_t=\{ (U_2,V_2) \in S_{k-2} \; |\; h^0(N^*_1{\otimws}L_2)=t\}.$$ Xlearly $\dim T_t \leq \dim G(1,s_2+p,t)$. Anwo we stratify $G(1,d_1,2)$ by the subsets $W_l$ of tfode coherent s... | Then $N_1$ is a line bundle of say. clearly has d_1$. Define the the $$T_t=\{ (L_2,V_2) \in \; |\; h^0(N^*_1{\otimes}L_2)=t\}.$$ $\dim T_t \leq \dim G(1,d_2+l,t)$. Also stratify $G(1,d_1,2)$ by the subsets $W_l$ of those coherent systems $(L_1,V_1)$ such that image of the map ${{{\mathcal O}}}^2\to L_1$ is a line bundl... | Then $N_1$ is a line bundle of degreE $-l$, say. One clEarly Has $L\leQ d_1$. defiNe thE stratificatioN Of $S_{k-2}$ Given by the subsets $$T_t=\{ (L_2,V_2) \iN S_{k-2} \; |\; h^0(N^*_1{\OtIMes}L_2)=T\}.$$ clEarly $\Dim T_t \leQ \DiM g(1,D_2+l,t)$. alSo We sTrATiFy $G(1,d_1,2)$ bY thE subsetS $W_l$ of those CohErEnt s... | Then $N_1$ is a line bund le of degr ee $- l$, sa y. One cle arly has $l\le q d_1 $. Define the stratifi catio no f $S _ {k -2}$given b y t h e su bs et s $ $T _ t= \{ (L _2, V_2) \i n S_{k-2}\;|\ ; h^0(N^*_1{ \ ot imes}L_2)= t\} .$$ Clearly$\d im T_t \ leq \dimG(1 ,d_2+ l,t)$. Also w e stratif y$ G(1,d_ 1 ,2)$... | Then_$N_1$ is_a line bundle of_degree $-l$,_say._One clearly_has_$l\leq d_1$. Define_the stratification of_$S_{k-2}$ given by the_subsets $$T_t=\{ (L_2,V_2)_\in_S_{k-2} \; |\; h^0(N^*_1{\otimes}L_2)=t\}.$$ Clearly $\dim T_t \leq \dim G(1,d_2+l,t)$. Also we stratify $G(1,d_1,2)$_by_the subsets_$W_l$_of_those coherent s... |
(\epsilon_{\alpha'})^2 \sin(\vartheta_{\beta'})
\sin(\varphi_{\gamma})(\Delta \epsilon)_{\alpha'} (\Delta\vartheta)_{\beta'}.\end{aligned}$$ The bin center values $\epsilon_{\alpha'}$ and $\vartheta_{\beta'}$ are given just as $r_{i'}$ and $\theta_{j'}$ in (\[rcenter\]) and (\[thetacenter\]). Also, the momentum bin ... | (\epsilon_{\alpha'})^2 \sin(\vartheta_{\beta' })
\sin(\varphi_{\gamma})(\Delta \epsilon)_{\alpha' } (\Delta\vartheta)_{\beta'}.\end{aligned}$$ The bin center values $ \epsilon_{\alpha'}$ and $ \vartheta_{\beta'}$ are given just as $ r_{i'}$ and $ \theta_{j'}$ in (\[rcenter\ ]) and (\[thetacenter\ ]). besides, the m... | (\epdilon_{\alpha'})^2 \sin(\vartheta_{\beua'})
\sin(\varphi_{\gamma})(\Bwlta \e'silon)_{\ampha'} (\Deuta\vartheta)_{\beta'}.\end{aligned}$$ Thx bib cenuvr values $\epsilon_{\alphx'}$ and $\varnheta_{\beta'}$ are tiven just as $r_{i'}$ akb $\thefw_{j'}$ iu (\[ccenter\]) and (\[thejacenter\]). Alsm, the momentum bkn ... | (\epsilon_{\alpha'})^2 \sin(\vartheta_{\beta'}) \sin(\varphi_{\gamma})(\Delta \epsilon)_{\alpha'} (\Delta\vartheta)_{\beta'}.\end{aligned}$$ The bin $\epsilon_{\alpha'}$ $\vartheta_{\beta'}$ are just as $r_{i'}$ (\[thetacenter\]). the momentum bin are given by difference of the bounding edge values, as in the case of t... | (\epsilon_{\alpha'})^2 \sin(\vartheta_{\beTa'})
\sin(\varphI_{\gammA})(\DeLta \EpSiloN)_{\alpHa'} (\Delta\varthetA)_{\Beta'}.\End{aligned}$$ The bin center ValueS $\ePSiloN_{\AlPha'}$ anD $\vartheTA_{\bETA'}$ arE gIvEn jUsT As $R_{i'}$ and $\TheTa_{j'}$ in (\[rcEnter\]) and (\[thEtaCeNter\]). Also, the mOMeNtum bin ... | (\epsilon_{\alpha'})^2 \s in(\varthe ta_{\ bet a'} )
\si n(\v arphi_{\gamma} ) (\De lta \epsilon)_{\alpha' } (\ De l ta\v a rt heta) _{\beta ' }. \ e nd{ al ig ned }$ $ T he bi n c enter v alues $\ep sil on _{\alpha'}$a nd $\varthet a_{ \beta'}$ are gi ven ju st as $r_{i '}$ and$\thet a _{j'}$ in (\[rc en t er\... | (\epsilon_{\alpha'})^2_\sin(\vartheta_{\beta'})
\sin(\varphi_{\gamma})(\Delta_\epsilon)_{\alpha'} (\Delta\vartheta)_{\beta'}.\end{aligned}$$ The_bin center_values_$\epsilon_{\alpha'}$ and_$\vartheta_{\beta'}$_are given just_as $r_{i'}$ and_$\theta_{j'}$ in (\[rcenter\]) and_(\[thetacenter\]). Also, the_momentum_bin ... |
H_{V} = g \nu$. Then conclusion (i) of Theorem \[thm:mainreg\_g0\_version1\] holds. Furthermore, if for all odd $j$, the hypersurfaces $M_j$ as in conclusion (ii) of Theorem \[thm:mainreg\_g0\_version1\] are two-sided, then there is an $n$-manifold $S_{V}$ and a proper $C^2$ immersion $\iota:S_V \to N$ such that $V=\io... | H_{V } = g \nu$. Then conclusion (i) of Theorem \[thm: mainreg\_g0\_version1\ ] holds. Furthermore, if for all curious $ j$, the hypersurfaces $ M_j$ as in ending (ii) of Theorem \[thm: mainreg\_g0\_version1\ ] are two - sided, then there is an $ n$-manifold $ S_{V}$ and a proper $ C^2 $ immersion $ \iota: S_V \to ... | H_{V} = g \nu$. Then conclusion (i) uf Theorem \[thm:manbreg\_g0\_vxrsion1\] golds. Fufthermore, if for all odd $j$, tie htpersyrfaces $M_j$ as in concljsion (ii) lf Theorwm \[thn:nainreg\_g0\_vecaion1\] arc two-alded, chxn there is an $k$-manifold $S_{E}$ and a proper $C^2$ ilmersion $\iota:S_V \to N$ such that $V=\io... | H_{V} = g \nu$. Then conclusion (i) \[thm:mainreg\_g0\_version1\] Furthermore, if all odd $j$, conclusion of Theorem \[thm:mainreg\_g0\_version1\] two-sided, then there an $n$-manifold $S_{V}$ and a proper immersion $\iota:S_V \to N$ such that $V=\iota_\#(|S_V|),$ and there is a continuous choice unit normal $\nu$ on $... | H_{V} = g \nu$. Then conclusion (i) of TheOrem \[thm:maiNreg\_g0\_VerSioN1\] hOlds. furtHermore, if for alL Odd $j$, The hypersurfaces $M_j$ as in ConclUsIOn (ii) OF THeoreM \[thm:maiNReG\_G0\_VerSiOn1\] Are TwO-SiDed, thEn tHere is aN $n$-manifold $s_{V}$ aNd A proper $C^2$ immeRSiOn $\iota:S_V \to n$ suCh that $V=\io... | H_{V} = g \nu$. Then concl usion (i)of Th eor em\[ thm: main reg\_g0\_versi o n1\] holds. Furthermore, i f for a l l od d $ j$, t he hype r su r f ace s$M _j$ a s i n con clu sion (i i) of Theo rem \ [thm:mainreg \ _g 0\_version 1\] are two-sid ed, thenth ere is an $n $-man ifold$ S_{V}$ and a pr op e r $C^2 ... | H_{V} =_g \nu$._Then conclusion (i) of_Theorem \[thm:mainreg\_g0\_version1\] holds._Furthermore,_if for_all_odd $j$, the_hypersurfaces $M_j$ as_in conclusion (ii) of_Theorem \[thm:mainreg\_g0\_version1\] are two-sided,_then_there is an $n$-manifold $S_{V}$ and a proper $C^2$ immersion $\iota:S_V \to N$ such_that_$V=\io... |
,, 126, 2291
Osterbrock, D. E. & Ferland, G. J. 2006, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (Sausalito, CA: Univ. Sci.)
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 2002,, 124, 266.
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 2010,, 139, 2097.
Robinson, A., et al. 1994,, 291, 351... | ,, 126, 2291
Osterbrock, D. E. & Ferland, G. J. 2006, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (Sausalito, CA: Univ. Sci .)
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 2002, , 124, 266.
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 2010, , 139, 2097.
Robinso... | ,, 126, 2291
Odterbrock, D. E. & Ferland, G. J. 2006, Astrophysics of Gasxous Negulae ana Active Galactic Nuclei (Saudaoito, XA: Univ. Sci.)
Peng, C. Y., Ho, U. C., Impey, B. D., & Rix, H. -Q. 2002,, 124, 266.
Keng, C. Y., Ho, L. C., Impxg, C. D., & Rlr, H. -W. 2010,, 139, 2097.
Robnnwon, A., et al. 1994,, 291, 351... | ,, 126, 2291 Osterbrock, D. E. & J. Astrophysics of Nebulae and Active Sci.) C. Y., Ho, C., Impey, C. & Rix, H. -W. 2002,, 124, Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 139, 2097. Robinson, A., et al. 1994,, 291, 351 Rupke, D. S., Veilleux, & D. 2005,, 87 Sanders, D. B. & Mirabel, I. F. 1996, ARA&A, 34, 749 ... | ,, 126, 2291
Osterbrock, D. E. & Ferland, G. J. 2006, AstrOphysics of gaseoUs NEbuLaE and actiVe Galactic NuclEI (SauSalito, CA: Univ. Sci.)
Peng, C. Y., HO, L. C., ImPeY, c. D., & RiX, h. -W. 2002,, 124, 266.
peng, C. y., Ho, L. C., ImPEy, c. d., & rix, h. -W. 2010,, 139, 2097.
roBinSoN, a., eT al. 1994,, 291, 351... | ,, 126, 2291
Osterbrock,D. E. & Fe rland , G . J .2006 , As trophysics ofG aseo us Nebulae and ActiveGalac ti c Nuc l ei (Sau salito, CA : Uni v. S ci. )Pe ng, C . Y ., Ho,L. C., Imp ey, C . D., & Rix, H. -W. 2002, , 1 24, 266.
Pe ng, C. Y. ,Ho, L. C. , I mpey, C. D. , & Rix , H. -W.20 1 0,, 13 9 , 2097. Ro b... | ,, 126,_2291
Osterbrock, D._E. & Ferland, G._J. 2006, Astrophysics_of_Gaseous Nebulae_and_Active Galactic Nuclei_(Sausalito, CA: Univ._Sci.)
Peng, C. Y., Ho, L. C.,_Impey, C. D., &_Rix,_H. -W. 2002,, 124, 266.
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H. -W. 2010,, 139,_2097.
Robinson,_A., et_al. 1994,,_291,_351... |
different energy, and non degenerate. Notice that one of them corresponds to a center of mass excitation and that merges with other levels at $E=9$ when $g\rightarrow \infty$. Using these type of arguments one can assign the spin to the eigenstates of the Hamiltonian even if the total wave function could be complicate... | different energy, and non degenerate. Notice that one of them correspond to a plaza of mass excitation and that merges with early levels at $ E=9 $ when $ g\rightarrow \infty$. Using these type of argument one can assign the spin to the eigenstates of the Hamiltonian even if the entire wave function could be complicate... | divferent energy, and non dtgenerate. Notice jhqt one of thsm correrponds to a center of mass eecitqtion and that merges with uther levvls at $E=9$ qhen $t\rightarrox \infty$. Mfing bhese vype of argumenjs one can avsign the spin tu che eigenstates of the Hamiltonian eden if yhf total wave fonctipg cohld be complicate... | different energy, and non degenerate. Notice that them to a of mass excitation levels $E=9$ when $g\rightarrow Using these type arguments one can assign the spin the eigenstates of the Hamiltonian even if the total wave function could be and most of the times non-factorizable in a spin and a space part. excitation ====... | different energy, and non degeNerate. NotiCe thaT onE of ThEm coRresPonds to a center OF masS excitation and that mergEs witH oTHer lEVeLs at $E=9$ When $g\riGHtARRow \InFtY$. UsInG ThEse tyPe oF argumeNts one can aSsiGn The spin to the EIgEnstates of The hamiltonian eVen If the tOtAl wAVe funCtiOn couLd be coMPlicatE... | different energy, and non degenerat e. No tic e t ha t on e of them correspo n ds t o a center of mass exc itati on andt ha t mer ges wit h o t h erle ve lsat $E =9$ w hen $g\rig htarrow \i nft y$ . Using thes e t ype of arg ume nts one canass ign th espi n to t heeigen states of the Hamilton ia n eveni f the ... | different_energy, and_non degenerate. Notice that_one of_them_corresponds to_a_center of mass_excitation and that_merges with other levels_at $E=9$ when_$g\rightarrow_\infty$. Using these type of arguments one can assign the spin to the eigenstates_of_the Hamiltonian_even_if_the total wave function could_be complicate... |
left\langle z^jz^k \right\rangle, \left\langle y^iy^lz^k \right\rangle, \left\langle y^my^ny^jy^l\right\rangle, \left\langle z^k \right\rangle$ form a closed system of evolution equations given the flow field $\mathbf{u}$, and are also sufficient to describe the stress tensor $\sigma^{jk}$ in (\[stress2final\]) complet... | left\langle z^jz^k \right\rangle, \left\langle y^iy^lz^k \right\rangle, \left\langle y^my^ny^jy^l\right\rangle, \left\langle z^k \right\rangle$ form a closed system of evolution equation render the flow field $ \mathbf{u}$, and are also sufficient to trace the stress tensor $ \sigma^{jk}$ in (\[stress2final\ ]) complet... | lefh\langle z^jz^k \right\rangle, \left\langle y^iy^lz^k \rigit\rangls, \left\lavgle y^my^ny^jy^l\right\rangle, \lefv\lantle z^j \right\rangle$ form a cuosed sysnem of evilutmon equations gitsn the njow rleld $\kethbf{u}$, and are slso suffiwient to descrhbd che stress tensor $\sigma^{jk}$ in (\[stress2fynal\]) cokppet... | left\langle z^jz^k \right\rangle, \left\langle y^iy^lz^k \right\rangle, \left\langle z^k form a system of evolution $\mathbf{u}$, are also sufficient describe the stress $\sigma^{jk}$ in (\[stress2final\]) completely. It is to interpret these moments as tensor fields – see appendix **\[app:vect\]** for more This is pot... | left\langle z^jz^k \right\rangle, \Left\langle Y^iy^lz^K \riGht\RaNgle, \Left\Langle y^my^ny^jy^l\RIght\Rangle, \left\langle z^k \righT\rangLe$ FOrm a CLoSed syStem of eVOlUTIon EqUaTioNs GIvEn the FloW field $\mAthbf{u}$, and aRe aLsO sufficient tO DeScribe the sTreSs tensor $\sigmA^{jk}$ In (\[streSs2FinAL\]) compLet... | left\langle z^jz^k \right\ rangle, \l eft\l ang ley^ iy^l z^k\right\rangle, \lef t\langle y^my^ny^jy^l\ right \r a ngle , \ left\ langlez ^k \ rig ht \r ang le $ f orm a cl osed sy stem of ev olu ti on equations gi ven the fl owfield $\math bf{ u}$, a nd ar e also su ffici ent to descri be the st re s s tens o r ... | left\langle z^jz^k_\right\rangle, \left\langle_y^iy^lz^k \right\rangle, \left\langle y^my^ny^jy^l\right\rangle,_\left\langle z^k_\right\rangle$_form a_closed_system of evolution_equations given the_flow field $\mathbf{u}$, and_are also sufficient_to_describe the stress tensor $\sigma^{jk}$ in (\[stress2final\]) complet... |
beta,\gamma,\lambda)$ for $2\leq
j\leq J.$ As in the proof of Proposition 4 of Newey (1994a), for any $w_{t}$ we have$$\frac{\partial}{\partial\tau}E[w_{t}|x_{t},y_{tj}=1,\tau]=E[\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}\{w_{t}-E[w_{t}|x_{t},y_{tj}=1]\}S(z_{t})|x_{t}].$$ It follows that$$\begin{aligned}
\frac{\partial E[m(z_... | beta,\gamma,\lambda)$ for $ 2\leq
j\leq J.$ As in the proof of Proposition 4 of Newey (1994a), for any $ w_{t}$ we have$$\frac{\partial}{\partial\tau}E[w_{t}|x_{t},y_{tj}=1,\tau]=E[\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}\{w_{t}-E[w_{t}|x_{t},y_{tj}=1]\}S(z_{t})|x_{t}].$$ It follows that$$\begin{aligned }
\frac{\partial... | betw,\gamma,\lambda)$ for $2\leq
j\leq J.$ As in the proof of 'roposifion 4 of Newey (1994a), for any $w_{t}$ we have$$\fcac{\pqrtiao}{\partial\tau}E[w_{t}|x_{t},y_{tj}=1,\tau]=D[\frac{y_{tj}}{P_{u}(\tilde{v}_{t})}\{q_{t}-E[w_{u}|x_{t},y_{tj}=1]\}S(z_{t})|x_{t}].$$ It follows that$$\begih{wliguev}
\frac{\partial E[m(e_... | beta,\gamma,\lambda)$ for $2\leq j\leq J.$ As in of 4 of (1994a), for any that$$\begin{aligned} E[m(z_{t},\beta_{0},\gamma_{j}(\tau),\gamma_{-j,0})]}{\partial\tau} & =-\delta E[u_{1,t+1}+H_{t+1}|x_{t},y_{tj}=1,\tau]}{\partial\tau}]\\ & =-\delta\frac{\partial}{\partial\tau}E[E[A(x_{t})P_{vj}(\tilde{v}_{t})\{u_{1,t+1}+H_... | beta,\gamma,\lambda)$ for $2\leq
j\leq j.$ As in the prOof of proPosItIon 4 oF NewEy (1994a), for any $w_{t}$ we hAVe$$\frAc{\partial}{\partial\tau}E[w_{t}|X_{t},y_{tj}=1,\TaU]=e[\fraC{Y_{tJ}}{P_{j}(\tiLde{v}_{t})}\{w_{t}-e[W_{t}|X_{T},Y_{tj}=1]\}s(z_{T})|x_{T}].$$ It FoLLoWs thaT$$\beGin{aligNed}
\frac{\parTiaL E[M(z_... | beta,\gamma,\lambda)$ for$2\leq
j\l eq J. $ A s i ntheproo f of Propositi o n 4of Newey (1994a), forany $ w_ { t}$w ehave$ $\frac{ \ pa r t ial }{ \p art ia l \t au}E[ w_{ t}|x_{t },y_{tj}=1 ,\t au ]=E[\frac{y_ { tj }}{P_{j}(\ til de{v}_{t})}\ {w_ {t}-E[ w_ {t} | x_{t} ,y_ {tj}= 1]\}S( z _{t})| x_{t}].$$ I t follo w ... | beta,\gamma,\lambda)$ for_$2\leq
j\leq J.$_As in the proof_of Proposition_4_of Newey_(1994a),_for any $w_{t}$_we have$$\frac{\partial}{\partial\tau}E[w_{t}|x_{t},y_{tj}=1,\tau]=E[\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}\{w_{t}-E[w_{t}|x_{t},y_{tj}=1]\}S(z_{t})|x_{t}].$$ It_follows that$$\begin{aligned}
\frac{\partial E[m(z_... |
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M. Kriener, C._Zobel, A. Reichl,_J._Baier, M. Cwik, K. Berggold, H. Kierspel, O. Zabara, A. Freimuth, and T. Lorenz,_Phys._Rev. B_**69**,_094417_(2004).
J. Wu and C. Leighton,_Phys... |
a first numerical experiment we have performed the robust stability test as described in [@Alcubierre03d]. For this test one takes Minkowski initial data and adds random noise with a small amplitude to all dynamical variables. For the evolution we have used both harmonic slicing with $f=1$ and standard 1+log slicing w... | a first numerical experiment we have performed the robust constancy trial as described in [ @Alcubierre03d ]. For this test one lead Minkowski initial datum and adds random noise with a small amplitude to all dynamic variable. For the evolution we have use both harmonic slicing with $ f=1 $ and standard 1+log slice w... | a virst numerical experimekt we have perfotmwd the robusf stabilkty test as described in [@Alcuuieree03d]. Fir this test one takes Minkowskp initial date and adds random noise with a dmaln amplitude to sll dynamiwal variables. Xof che evolution we have used both harmjnic sloclng with $f=1$ and stamqard 1+log slicing w... | a first numerical experiment we have performed stability as described [@Alcubierre03d]. For this data adds random noise a small amplitude all dynamical variables. For the evolution have used both harmonic slicing with $f=1$ and standard 1+log slicing with $f=2/\alpha$. the simulations discussed below we used 1,000 grid... | a first numerical experiment We have perfOrmed The RobUsT staBiliTy test as descriBEd in [@alcubierre03d]. For this test One taKeS minkOWsKi iniTial datA AnD ADds RaNdOm nOiSE wIth a sMalL amplitUde to all dyNamIcAl variables. FOR tHe evolutioN we Have used both HarMonic sLiCinG With $f=1$ And StandArd 1+log SLicing W... | a first numerical experim ent we hav e per for med t he r obus t stability te s t as described in [@Alcubi erre0 3d ] . Fo r t his t est one ta k e s M in ko wsk ii ni tialdat a and a dds random no is e with a sma l lamplitudetoall dynamica l v ariabl es . F o r the ev oluti on weh ave us ed both h ar m onic s l ... | a_first numerical_experiment we have performed_the robust_stability_test as_described_in [@Alcubierre03d]. For this_test one takes_Minkowski initial data and_adds random noise_with_a small amplitude to all dynamical variables. For the evolution we have used both_harmonic_slicing with_$f=1$_and_standard 1+log slicing w... |
${\mathbf{GrVect}}$ is the category of ${\mathbb{N}}_0$-graded vector spaces over ${\mathbbm{k}}$. Most of the content of this section generalises to the graded case in a completely straightforward way, so to avoid too much duplication, we only state it for the ungraded case.
To be able to extract any sort of useful ... | $ { \mathbf{GrVect}}$ is the category of $ { \mathbb{N}}_0$-graded vector spaces over $ { \mathbbm{k}}$. Most of the content of this section popularize to the grade case in a completely straightforward means, so to avoid besides much duplicate, we only state it for the ungraded case.
To be able to extract any kind o... | ${\mahhbf{GrVect}}$ is the categovy of ${\mathbb{N}}_0$-grabwd vecvor spades over ${\mathbbm{k}}$. Most of the contenv of this section generalises tu the grafed case in e completely strejghtforward wzn, so co avoid too mucm duplicatimn, we only stade ic for the ungraded case.
To be able to extracy wny sort of usgful ... | ${\mathbf{GrVect}}$ is the category of ${\mathbb{N}}_0$-graded vector ${\mathbbm{k}}$. of the of this section in completely straightforward way, to avoid too duplication, we only state it for ungraded case. To be able to extract any sort of useful information from modules, we need to understand their structure. One way... | ${\mathbf{GrVect}}$ is the category Of ${\mathbb{N}}_0$-gRaded VecTor SpAces Over ${\Mathbbm{k}}$. Most of THe coNtent of this section geneRalisEs TO the GRaDed caSe in a coMPlETEly StRaIghTfORwArd waY, so To avoid Too much dupLicAtIon, we only staTE iT for the ungRadEd case.
To be abLe tO extraCt Any SOrt of UseFul ... | ${\mathbf{GrVect}}$ is th e category of $ {\m ath bb {N}} _0$- graded vectors pace s over ${\mathbbm{k}}$ . Mos to f th e c onten t of th i ss e cti on g ene ra l is es to th e grade d case ina c om pletely stra i gh tforward w ay, so to avoid to o much d upl i catio n,we on ly sta t e it f or the un gr a ded c... | ${\mathbf{GrVect}}$_is the_category of ${\mathbb{N}}_0$-graded vector_spaces over_${\mathbbm{k}}$._Most of_the_content of this_section generalises to_the graded case in_a completely straightforward_way,_so to avoid too much duplication, we only state it for the ungraded case.
To_be_able to_extract_any_sort of useful ... |
mathbf{C}}_{11}^{-1} = \left( \breve{\mathbf{C}}_{12} \breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T},\end{aligned}$$ holds by considering the orthogonality of the eigenvector matrix $\breve{\mathbf{C}}$, $$\begin{aligned}
\mathbf{I} = \breve{\mathbf{C}}^\mathsf{T} \breve{\mathbf{C}}.\end{aligned}$$ This implies that s... | mathbf{C}}_{11}^{-1 } = \left (\breve{\mathbf{C}}_{12 } \breve{\mathbf{C}}_{22}^{-1 } \right)^\mathsf{T},\end{aligned}$$ holds by considering the orthogonality of the eigenvector matrix $ \breve{\mathbf{C}}$, $ $ \begin{aligned }
\mathbf{I } = \breve{\mathbf{C}}^\mathsf{T } \breve{\mathbf{C}}.\end{aligned}$$ This imp... | matjbf{C}}_{11}^{-1} = \left( \breve{\mathbf{C}}_{12} \nreve{\mathbf{C}}_{22}^{-1} \rigkr)^\mathsh{T},\end{aljgned}$$ houds by considering the orthojonaoity if the eigenvector matfix $\breve{\lathbf{C}}$, $$\vegii{aligned}
\mathbf{I} = \breve{\mabkbf{C}}^\mzbhsf{T} \ureve{\mathbf{C}}.\end{sligned}$$ Thhs implies thad r... | mathbf{C}}_{11}^{-1} = \left( \breve{\mathbf{C}}_{12} \breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T},\end{aligned}$$ holds the of the matrix $\breve{\mathbf{C}}$, $$\begin{aligned} implies subsystem and environment are orthogonal, leading the expression, $$\begin{aligned} \mathbf{0} &= \breve{\mathbf{C}}_\mathcal{E}^\... | mathbf{C}}_{11}^{-1} = \left( \breve{\mathbf{C}}_{12} \brEve{\mathbf{C}}_{22}^{-1} \Right)^\MatHsf{t},\eNd{alIgneD}$$ holds by considERing The orthogonality of the eIgenvEcTOr maTRiX $\brevE{\mathbf{c}}$, $$\BeGIN{alIgNeD}
\maThBF{I} = \Breve{\MatHbf{C}}^\matHsf{T} \breve{\mAthBf{c}}.\end{aligned}$$ THIs Implies thaT s... | mathbf{C}}_{11}^{-1} = \le ft( \breve {\mat hbf {C} }_ {12} \br eve{\mathbf{C} } _{22 }^{-1} \right)^\mathsf {T},\ en d {ali g ne d}$$holds b y c o n sid er in g t he or thogo nal ity ofthe eigenv ect or matrix $\br e ve {\mathbf{C }}$ , $$\begin{a lig ned}
\ ma thb f {I} = \b reve{ \mathb f {C}}^\ mathsf{T} \ b r... | mathbf{C}}_{11}^{-1} =_\left( \breve{\mathbf{C}}_{12}_\breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T},\end{aligned}$$ holds by_considering the_orthogonality_of the_eigenvector_matrix $\breve{\mathbf{C}}$, $$\begin{aligned}
\mathbf{I}_= \breve{\mathbf{C}}^\mathsf{T} \breve{\mathbf{C}}.\end{aligned}$$_This implies that s... |
nu}{eV}})$ in the differential conductance when the edge was coupled to one vortex (Eq. ). This can be generalized if the chiral Majoranas have single-point couplings to several vortices. In that case $\phi$ is replaced by $\sum_i \phi_i - \sum_j \phi_j$, where $\phi_i = \arctan({\frac{\nu_i}{eV}})$ comes from vortex-e... | nu}{eV}})$ in the differential conductance when the edge was coupled to one vortex (Eq .). This can be generalized if the chiral Majoranas get individual - point couplings to several whirlpool. In that case $ \phi$ is replaced by $ \sum_i \phi_i - \sum_j \phi_j$, where $ \phi_i = \arctan({\frac{\nu_i}{eV}})$ come from ... | nu}{eG}})$ in the differential cokductance when tkw edge was ckupled tu one vortex (Eq. ). This can be gwneraoized if the chiral Magoranas hwve singoe-pomnt couplings to several vortidcs. In vhat case $\phi$ ix replaced by $\sum_i \phi_i - \sjm_l \phi_j$, where $\phi_i = \arctan({\frac{\nu_i}{eV}})$ cjmes frpm vortex-e... | nu}{eV}})$ in the differential conductance when the coupled one vortex ). This can Majoranas single-point couplings to vortices. In that $\phi$ is replaced by $\sum_i \phi_i \sum_j \phi_j$, where $\phi_i = \arctan({\frac{\nu_i}{eV}})$ comes from vortex-edge-coupling with the upper edge $\phi_j$ from coupling with the l... | nu}{eV}})$ in the differential condUctance wheN the eDge Was CoUpleD to oNe vortex (Eq. ). This CAn be Generalized if the chiral majorAnAS havE SiNgle-pOint couPLiNGS to SeVeRal VoRTiCes. In ThaT case $\phI$ is replaceD by $\SuM_i \phi_i - \sum_j \phI_J$, wHere $\phi_i = \arCtaN({\frac{\nu_i}{eV}})$ coMes From voRtEx-e... | nu}{eV}})$ in the differen tial condu ctanc e w hen t he e dgewas coupled to onevortex (Eq. ). This ca n bege n eral i ze d ifthe chi r al M ajo ra na s h av e s ingle -po int cou plings tosev er al vortices. In that case $\ phi$ is repl ace d by $ \s um_ i \phi _i- \su m_j \p h i_j$,where $\p hi _ i = \a r ctan({... | nu}{eV}})$ in_the differential_conductance when the edge_was coupled_to_one vortex_(Eq._). This can_be generalized if_the chiral Majoranas have_single-point couplings to_several_vortices. In that case $\phi$ is replaced by $\sum_i \phi_i - \sum_j \phi_j$, where_$\phi_i_= \arctan({\frac{\nu_i}{eV}})$_comes_from_vortex-e... |
_\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\ell}^{\alpha_{\gamma_\ell}}, c_\gamma \in {K}, \alpha_{\gamma_i} \in {K}, \sum_{i=1}^\ell \alpha_{\gamma_i} \leq \alpha,\end{aligned}$$ where $\Gamma$ ranges over distinct monomials in the free algebra and all but finitely many of the $c_\gamma$ are zero. The homogenizat... | _ \gamma x_{i_1}^{\alpha_{\gamma_1 } } \cdots x_{i_\ell}^{\alpha_{\gamma_\ell } }, c_\gamma \in { K }, \alpha_{\gamma_i } \in { K }, \sum_{i=1}^\ell \alpha_{\gamma_i } \leq \alpha,\end{aligned}$$ where $ \Gamma$ ranges over distinct monomials in the free algebra and all but finitely many of the $ c_\gamma$ are zero. Th... | _\gamla x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\tll}^{\alpha_{\gamma_\ell}}, e_\tamma \mn {K}, \allha_{\gamma_k} \in {K}, \sum_{i=1}^\ell \alpha_{\gamma_i} \lxq \aopha,\ebd{aligned}$$ where $\Gamma$ fanges ovvr distinxt mibomials in the frec algsnra aud all but finitgly many of dhe $c_\gamma$ are zdrl. The homogenizat... | _\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\ell}^{\alpha_{\gamma_\ell}}, c_\gamma \in {K}, {K}, \alpha_{\gamma_i} \leq where $\Gamma$ ranges free and all but many of the are zero. The homogenization of $f$ then $$\begin{aligned} \hat{f} = \sum_{\gamma \in \Gamma} c_\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\e... | _\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\eLl}^{\alpha_{\gamMa_\ell}}, C_\gaMma \In {k}, \alpHa_{\gaMma_i} \in {K}, \sum_{i=1}^\ell \ALpha_{\Gamma_i} \leq \alpha,\end{alignEd}$$ wheRe $\gAmma$ RAnGes ovEr distiNCt MONomIaLs In tHe FReE algeBra And all bUt finitely ManY oF the $c_\gamma$ arE ZeRo. The homogEniZat... | _\gamma x_{i_1}^{\alpha_{\ gamma_1}}\cdot s x _{i _\ ell} ^{\a lpha_{\gamma_\ e ll}} , c_\gamma \in {K}, \a lpha_ {\ g amma _ i} \in{K}, \s u m_ { i =1} ^\ el l \ al p ha _{\ga mma _i} \le q \alpha,\ end {a ligned}$$ wh e re $\Gamma$ran ges over dis tin ct mon om ial s in t hefreealgebr a and a ll but fi ni t ely m... | _\gamma x_{i_1}^{\alpha_{\gamma_1}}_\cdots x_{i_\ell}^{\alpha_{\gamma_\ell}},_c_\gamma \in {K}, \alpha_{\gamma_i}_\in {K},_\sum_{i=1}^\ell_\alpha_{\gamma_i} \leq_\alpha,\end{aligned}$$_where $\Gamma$ ranges_over distinct monomials_in the free algebra_and all but_finitely_many of the $c_\gamma$ are zero. The homogenizat... |
:P^n =
P(V) \rightarrow ]L_i[)$ is equivalent to the choice of $m-n-1$ surjective linear operators $a_i: W^*\rightarrow V^*$. Namely, given such $a_i$, we associate to any hyperplane in $V^*$ i.e. to any point of $P(V)$ its inverse image under $a_i$. Thus a point $\pi\in G^n(W)$ corresponding to $x\in P(V)$ is $\bigca... | : P^n =
P(V) \rightarrow ] L_i[)$ is equivalent to the choice of $ m - n-1 $ surjective linear operators $ a_i: W^*\rightarrow V^*$. Namely, given such $ a_i$, we consort to any hyperplane in $ V^*$ i.e. to any item of $ P(V)$ its inverse image under $ a_i$. Thus a point $ \pi\in G^n(W)$ equate to $ x\in P(V)$ is $ ... | :P^n =
P(V) \rightarrow ]L_i[)$ is equlvalent to the ckiice oh $m-n-1$ sudjective linear operators $a_i: W^*\rightacrow V^*$. Nanely, given such $a_i$, we xssociate to any yypecplane in $V^*$ i.e. to any polut of $I(V)$ itv inverse image under $a_i$. Dhus a point $\ph\iv Y^n(W)$ corresponding to $x\in P(V)$ is $\bigca... | :P^n = P(V) \rightarrow ]L_i[)$ is equivalent choice $m-n-1$ surjective operators $a_i: W^*\rightarrow we to any hyperplane $V^*$ i.e. to point of $P(V)$ its inverse image $a_i$. Thus a point $\pi\in G^n(W)$ corresponding to $x\in P(V)$ is $\bigcap {\rm (a_i(x))$. Define a linear map $A: {\bf C}^{m-n-1} \rightarrow {\r... | :P^n =
P(V) \rightarrow ]L_i[)$ is equivalEnt to the chOice oF $m-n-1$ SurJeCtivE linEar operators $a_i: w^*\RighTarrow V^*$. Namely, given such $A_i$, we aSsOCiatE To Any hyPerplanE In $v^*$ I.E. to AnY pOinT oF $p(V)$ Its inVerSe image Under $a_i$. ThuS a pOiNt $\pi\in G^n(W)$ corREsPonding to $x\In P(v)$ is $\bigca... | :P^n =
P(V) \rightarrow ] L_i[)$ isequiv ale ntto the cho ice of $m-n-1$ surj ective linear operator s $a_ i: W^*\ r ig htarr ow V^*$ . N a m ely ,gi ven s u ch $a_i $,we asso ciate to a nyhy perplane in$ V^ *$ i.e. to an y point of $ P(V )$ its i nve r se im age unde r $a_i $ . Thus a point$\ p i\in G ^ n(W)... | :P^n =
_P(V) \rightarrow_]L_i[)$ is equivalent to_the choice_of_$m-n-1$ surjective_linear_operators $a_i: W^*\rightarrow_V^*$. Namely, given_such $a_i$, we associate_to any hyperplane_in_$V^*$ i.e. to any point of $P(V)$ its inverse image under $a_i$. Thus a_point_$\pi\in G^n(W)$_corresponding_to_$x\in P(V)$ is $\bigca... |
to construct a Lie contact structure on $M$ from the later data. In general, the Levi bracket is a section of the bundle ${\Lambda}^2 H^*\otimes TM/H$, which decomposes according to the isomorphism $H\cong L^*\otimes R$ as $({\Lambda}^2L\otimes S^2R^*\otimes TM/H)\oplus(S^2L\otimes{\Lambda}^2R^*\otimes TM/H)$. Since w... | to construct a Lie contact structure on $ M$ from the later datum. In cosmopolitan, the Levi bracket is a part of the bundle $ { \Lambda}^2 H^*\otimes TM / H$, which decomposes accord to the isomorphism $ H\cong L^*\otimes R$ as $ ({ \Lambda}^2L\otimes S^2R^*\otimes TM / H)\oplus(S^2L\otimes{\Lambda}^2R^*\otimes TM / H... | to construct a Lie contact structure on $M$ from tie lated data. Iv general, the Levi bracket id q secupon of the bundle ${\Lamcda}^2 H^*\otimvs TM/H$, whuch vecomposes accorvjng to bke isklorpkiwm $H\cong L^*\otimgs R$ as $({\Lambga}^2L\otimes S^2R^*\othmds TM/H)\oplus(S^2L\otimes{\Lambda}^2R^*\otimes TM/H)$. Fince w... | to construct a Lie contact structure on the data. In the Levi bracket bundle H^*\otimes TM/H$, which according to the $H\cong L^*\otimes R$ as $({\Lambda}^2L\otimes S^2R^*\otimes TM/H)$. Since we assume ${\varphi}\otimes R$ is isotropic with respect to $\L$, for ${\varphi}$, the Levi bracket $\L$ factorizes through the... | to construct a Lie contact strUcture on $M$ fRom thE laTer DaTa. In GeneRal, the Levi bracKEt is A section of the bundle ${\LamBda}^2 H^*\oTiMEs TM/h$, WhIch deComposeS AcCORdiNg To The IsOMoRphisM $H\cOng L^*\otiMes R$ as $({\LambDa}^2L\OtImes S^2R^*\otimes tm/H)\Oplus(S^2L\otiMes{\lambda}^2R^*\otimeS TM/h)$. Since W... | to construct a Lie contac t structur e on$M$ fr om the lat er data. In ge n eral , the Levi bracket isa sec ti o n of th e bun dle ${\ L am b d a}^ 2H^ *\o ti m es TM/H $,which d ecomposesacc or ding to thei so morphism $ H\c ong L^*\otim esR$ as$( {\L a mbda} ^2L \otim es S^2 R ^*\oti mes TM/H) \o p lus(S^ 2 L\ot... | to_construct a_Lie contact structure on_$M$ from_the_later data._In_general, the Levi_bracket is a_section of the bundle_${\Lambda}^2 H^*\otimes TM/H$,_which_decomposes according to the isomorphism $H\cong L^*\otimes R$ as $({\Lambda}^2L\otimes S^2R^*\otimes TM/H)\oplus(S^2L\otimes{\Lambda}^2R^*\otimes TM/H)$. Since_w... |
n \leq X \\ P^-(n) > Y^2}} 1 \ll X \prod_{q \leq Y^2} \left(1 - \frac{1}{q}\right),$$ where the final inequality follows from applying Brun’s Sieve (cf. [@hr Theorem 2.2]). By Mertens’ Theorem (cf. [@pollack Theorem 3.15]), we have $$\label{brunexcept} X \prod_{q \leq Y^2} \left(1 - \frac{1}{q}\right) \ll \frac{X}{\log... | n \leq X \\ P^-(n) > Y^2 } } 1 \ll X \prod_{q \leq Y^2 } \left(1 - \frac{1}{q}\right),$$ where the final inequality follows from applying Brun ’s Sieve (cf. [ @hr Theorem 2.2 ]). By Mertens ’ Theorem (californium. [ @pollack Theorem 3.15 ]), we take $ $ \label{brunexcept } X \prod_{q \leq Y^2 } \left(1 - \frac{1}{q}\ri... | n \lfq X \\ P^-(n) > Y^2}} 1 \ll X \prod_{q \ueq Y^2} \left(1 - \frae{1}{w}\right),$$ where the finxl inequality follows from a'plyung Beun’s Sieve (cf. [@hr Theordm 2.2]). By Megtens’ Theirem (xf. [@pollack Theorem 3.15]), we hzye $$\layeo{brunexcept} X \krod_{q \leq Y^2} \neft(1 - \frac{1}{q}\rigvt) \lp \frac{X}{\log... | n \leq X \\ P^-(n) > Y^2}} X \leq Y^2} - \frac{1}{q}\right),$$ where applying Sieve (cf. [@hr 2.2]). By Mertens’ (cf. [@pollack Theorem 3.15]), we have X \prod_{q \leq Y^2} \left(1 - \frac{1}{q}\right) \ll \frac{X}{\log Y}.$$ Now, suppose that > 1$. On one hand, for all $k > 1$, we have $$\label{lpeq1} + \leq \ell^*_p(... | n \leq X \\ P^-(n) > Y^2}} 1 \ll X \prod_{q \leq Y^2} \left(1 - \fRac{1}{q}\right),$$ wHere tHe fInaL iNequAlitY follows from apPLyinG Brun’s Sieve (cf. [@hr Theorem 2.2]). by MerTeNS’ TheOReM (cf. [@poLlack ThEOrEM 3.15]), We hAvE $$\lAbeL{bRUnExcepT} X \pRod_{q \leq y^2} \left(1 - \frac{1}{q}\RigHt) \Ll \frac{X}{\log... | n \leq X \\ P^-(n) > Y^2}} 1 \ll X \ prod_ {q\le qY^2} \le ft(1 - \frac{1 } {q}\ right),$$ where the fi nal i ne q uali t yfollo ws from ap p l yin gBr un’ sS ie ve (c f.[@hr Th eorem 2.2] ).By Mertens’ Th e or em (cf. [@ pol lack Theorem 3. 15]),we ha v e $$\ lab el{br unexce p t} X \ prod_{q \ le q Y^2}\ left(1... | n \leq_X \\_P^-(n) > Y^2}} 1_\ll X_\prod_{q_\leq Y^2}_\left(1_- \frac{1}{q}\right),$$ where_the final inequality_follows from applying Brun’s_Sieve (cf. [@hr_Theorem_2.2]). By Mertens’ Theorem (cf. [@pollack Theorem 3.15]), we have $$\label{brunexcept} X \prod_{q \leq_Y^2}_\left(1 -_\frac{1}{q}\right)_\ll_\frac{X}{\log... |
{\mathsf{Mat}}_{n \times n}({\mathbb{C}})$ such that for all $a, b \in \operatorname{\mathfrak{g}}$ we have: $
\omega(a, b) = \omega_K(a, b) := {\mathsf{tr}}(K^t \cdot \bigl([a, b]\bigr).$ Let $1 \le e \le n$ be such that $\gcd(n, e) = 1$. Then the parabolic subalgebra $\operatorname{\mathfrak{p}}_e$ is Frobenius. If ... | { \mathsf{Mat}}_{n \times n}({\mathbb{C}})$ such that for all $ a, b \in \operatorname{\mathfrak{g}}$ we have: $
\omega(a, b) = \omega_K(a, b): = { \mathsf{tr}}(K^t \cdot \bigl([a, b]\bigr).$ Let $ 1 \le e \le n$ be such that $ \gcd(n, e) = 1$. Then the parabolic subalgebra $ \operatorname{\mathfrak{p}}_e$ is Froben... | {\matjsf{Mat}}_{n \times n}({\mathbb{C}})$ smch that for all $a, b \in \operaforname{\mxthfrak{g}}$ we have: $
\omega(a, b) = \lmwga_K(a, b) := {\mathsf{tr}}(K^t \cdot \biel([a, b]\bigr).$ Let $1 \le e \lt n$ be such that $\jdd(n, e) = 1$. Then fme paxauolic subalgebrs $\operatortame{\mathfrak{p}}_e$ ir Yrobenius. If ... | {\mathsf{Mat}}_{n \times n}({\mathbb{C}})$ such that for all \in we have: \omega(a, b) = \bigl([a, Let $1 \le \le n$ be that $\gcd(n, e) = 1$. Then parabolic subalgebra $\operatorname{\mathfrak{p}}_e$ is Frobenius. If $(\operatorname{\mathfrak{g}}, e, \omega)$ is a Stolin triple $\omega_K$ has to define a Frobenius pai... | {\mathsf{Mat}}_{n \times n}({\mathbb{C}})$ suCh that for aLl $a, b \iN \opEraToRnamE{\matHfrak{g}}$ we have: $
\omEGa(a, b) = \Omega_K(a, b) := {\mathsf{tr}}(K^t \cdot \Bigl([a, B]\bIGr).$ LeT $1 \Le E \le n$ bE such thAT $\gCD(N, e) = 1$. THeN tHe pArABoLic suBalGebra $\opEratorname{\MatHfRak{p}}_e$ is FrobeNIuS. If ... | {\mathsf{Mat}}_{n \times n }({\mathbb {C}}) $ s uch t hatforall $a, b \in\ oper atorname{\mathfrak{g}} $ weha v e: $ \ omega (a, b)= \ o m ega _K (a , b ): ={\mat hsf {tr}}(K ^t \cdot \ big l( [a, b]\bigr) . $Let $1 \le e\le n$ be su chthat $ \g cd( n , e)= 1 $. Th en the parabo lic subal ge b ra $\o p eratorn a m ... | {\mathsf{Mat}}_{n \times_n}({\mathbb{C}})$ such_that for all $a,_b \in_\operatorname{\mathfrak{g}}$_we have:_$
_\omega(a, b) =_\omega_K(a, b) :=_{\mathsf{tr}}(K^t \cdot \bigl([a, b]\bigr).$_Let $1 \le_e_\le n$ be such that $\gcd(n, e) = 1$. Then the parabolic subalgebra $\operatorname{\mathfrak{p}}_e$_is_Frobenius. If_... |
$ does not have fixed components, we get $K_{W} \cdot \Sigma \leqslant 0$ and hence $m \leqslant 2 + Z^{2}$.
Moreover, if $Z$ is the tautological section on $V \simeq \mathbb{F}_{n}$, $n \geqslant 0$, then $Z^{2} = n$ and $$K_{W} \cdot \Sigma = g^{*}(K_{V} + c_{1}(\mathcal{E})) \cdot \Sigma = (K_{V} + c_{1}(\mathcal{E... | $ does not have fixed components, we get $ K_{W } \cdot \Sigma \leqslant 0 $ and therefore $ m \leqslant 2 + Z^{2}$.
furthermore, if $ Z$ is the pleonastic section on $ V \simeq \mathbb{F}_{n}$, $ n \geqslant 0 $, then $ Z^{2 } = n$ and $ $ K_{W } \cdot \Sigma = g^{*}(K_{V } + c_{1}(\mathcal{E }) ) \cdot \Sigma = (K... | $ dofs not have fixed compontnts, we get $K_{W} \cdot \Sigme \leqslznt 0$ and hence $m \leqslant 2 + Z^{2}$.
Moreovec, if $Z$ is the tautological sectkon on $V \dimeq \marhbb{H}_{n}$, $n \geqslant 0$, tisn $Z^{2} = n$ and $$K_{S} \cdoc \Wigma = g^{*}(K_{V} + c_{1}(\msthcal{E})) \cdmt \Sigma = (K_{V} + w_{1}(\mxtkcal{E... | $ does not have fixed components, we \cdot \leqslant 0$ hence $m \leqslant $Z$ the tautological section $V \simeq \mathbb{F}_{n}$, \geqslant 0$, then $Z^{2} = n$ $$K_{W} \cdot \Sigma = g^{*}(K_{V} + c_{1}(\mathcal{E})) \cdot \Sigma = (K_{V} + c_{1}(\mathcal{E})) Z = -n - 2 + c_{1}(\mathcal{E}) \cdot Z.$$ From we obtain... | $ does not have fixed componentS, we get $K_{W} \cdOt \SigMa \lEqsLaNt 0$ anD henCe $m \leqslant 2 + Z^{2}$.
MoREoveR, if $Z$ is the tautological sEctioN oN $v \simEQ \mAthbb{f}_{n}$, $n \geqsLAnT 0$, THen $z^{2} = n$ AnD $$K_{W} \CdOT \SIgma = g^{*}(k_{V} + c_{1}(\Mathcal{e})) \cdot \Sigma = (k_{V} + c_{1}(\MaThcal{E... | $ does not have fixed comp onents, we get$K_ {W} \ cdot \Si gma \leqslant0 $ an d hence $m \leqslant 2 + Z^ {2 } $.
M or eover , if $Z $ i s the t au tol og i ca l sec tio n on $V \simeq \m ath bb {F}_{n}$, $n \g eqslant 0$ , t hen $Z^{2} = n$ and $ $K _{W } \cdo t \ Sigma = g^{ * }(K_{V } + c_{1} (\ m athca... | $ does_not have_fixed components, we get_$K_{W} \cdot_\Sigma_\leqslant 0$_and_hence $m \leqslant_2 + Z^{2}$.
Moreover,_if $Z$ is the_tautological section on_$V_\simeq \mathbb{F}_{n}$, $n \geqslant 0$, then $Z^{2} = n$ and $$K_{W} \cdot \Sigma =_g^{*}(K_{V}_+ c_{1}(\mathcal{E}))_\cdot_\Sigma_= (K_{V} + c_{1}(\mathcal{E... |
)^2 \mid \mathcal{F}_{t-1} ]] \\
= (N-1){{\mathbb{E}}_{x_{0:t},\tilde x_{0:t}}}[{\mathds{1}}\!\left(1\in I_{t-1} \right) {{\mathbb{E}}_{x_{0:t},\tilde x_{0:t}}}[ {\mathds{1}}\!\left(a_{t-1}^1 = \tilde a_{t-1}^1 = 1 \right) \psi_t(x_{0:t}^1)^2 \mid \mathcal{F}_{t-1} ]].\end{gathered}$$ The inner conditional expecta... | ) ^2 \mid \mathcal{F}_{t-1 } ] ] \\
= (N-1){{\mathbb{E}}_{x_{0: t},\tilde x_{0: t}}}[{\mathds{1}}\!\left(1\in I_{t-1 } \right) { { \mathbb{E}}_{x_{0: t},\tilde x_{0: t } } } [ { \mathds{1}}\!\left(a_{t-1}^1 = \tilde a_{t-1}^1 = 1 \right) \psi_t(x_{0: t}^1)^2 \mid \mathcal{F}_{t-1 } ] ] .\end{gathered}$$ The inn... | )^2 \mif \mathcal{F}_{t-1} ]] \\
= (N-1){{\mathbn{E}}_{x_{0:t},\tilde x_{0:t}}}[{\mathbw{1}}\!\left(1\ii I_{t-1} \rifht) {{\mathcb{E}}_{x_{0:t},\tilde x_{0:t}}}[ {\mathds{1}}\!\left(a_{t-1}^1 = \vildw a_{t-1}^1 = 1 \right) \psi_t(x_{0:t}^1)^2 \mid \matfcal{F}_{t-1} ]].\ejd{gatherwd}$$ Tie inner conditional expceta... | )^2 \mid \mathcal{F}_{t-1} ]] \\ = (N-1){{\mathbb{E}}_{x_{0:t},\tilde \right) x_{0:t}}}[ {\mathds{1}}\!\left(a_{t-1}^1 \tilde a_{t-1}^1 = ]].\end{gathered}$$ inner conditional expectation be computed as \label{eq:cruce:h2-inner} {{\mathbb{E}}_{x_{0:t},\tilde x_{0:t}}}[ {\mathds{1}}\!\left(a_{t-1}^1 = \tilde = 1 \right)... | )^2 \mid \mathcal{F}_{t-1} ]] \\
= (N-1){{\mathbb{E}}_{x_{0:t},\tilDe x_{0:t}}}[{\mathds{1}}\!\Left(1\iN I_{t-1} \RigHt) {{\MathBb{E}}_{x_{0:T},\tilde x_{0:t}}}[ {\mathds{1}}\!\LEft(a_{T-1}^1 = \tilde a_{t-1}^1 = 1 \right) \psi_t(x_{0:t}^1)^2 \mid \MathcAl{f}_{T-1} ]].\end{GAtHered}$$ the inneR CoNDItiOnAl ExpEcTA... | )^2 \mid \mathcal{F}_{t-1} ]] \\
= (N -1) {{\ ma thbb {E}} _{x_{0:t},\til d e x_ {0:t}}}[{\mathds{1}}\! \left (1 \ in I _ {t -1} \ right){ {\ m a thb b{ E} }_{ x_ { 0: t},\t ild e x_{0: t}}}[ {\ma thd s{ 1}}\!\left(a _ {t -1}^1 = \t ild e a_{t-1}^1= 1 \righ t) \p s i_t(x _{0 :t}^1 )^2 \m i d \mat hcal{F}_{ t- 1 }... | )^2 \mid_\mathcal{F}_{t-1} ]]_\\
_= (N-1){{\mathbb{E}}_{x_{0:t},\tilde_x_{0:t}}}[{\mathds{1}}\!\left(1\in_I_{t-1} \right)_{{\mathbb{E}}_{x_{0:t},\tilde_x_{0:t}}}[ {\mathds{1}}\!\left(a_{t-1}^1 =_\tilde a_{t-1}^1 =_1 \right) \psi_t(x_{0:t}^1)^2 \mid_\mathcal{F}_{t-1} ]].\end{gathered}$$_The_inner conditional expecta... |
(-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1);
(2,-4) to (2,-4+2); (2,-4+2) to (2+1,-4+2) to (2+1,-4+2+1); (2+1,-4+2+1) to (2+1,-4+2+1+3) to (2+1+1,-4+2+1+3+1);
at (-3.5,-3.5) [$\alpha + \beta$]{}; at (4.3,-3.5) [$\alpha + \beta$]{}; at (-4.5,2.5) [$\alpha$]{}; at (2.0,2.5) [$-\alpha$]{};
Note that using det... | (-1,-2) to (-1,-2 + 1) to (-1 + 1,-2 + 1) to (-1 + 1,-2 + 1 + 1);
(2,-4) to (2,-4 + 2); (2,-4 + 2) to (2 + 1,-4 + 2) to (2 + 1,-4 + 2 + 1); (2 + 1,-4 + 2 + 1) to (2 + 1,-4 + 2 + 1 + 3) to (2 + 1 + 1,-4 + 2 + 1 + 3 + 1);
at (-3.5,-3.5) [ $ \alpha + \beta$ ] { }; at (4.3,-3.5) [ $ \alpha + \beta$ ] { }; at (-4.5,2.... |
(-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1);
(2,-4) to (2,-4+2); (2,-4+2) to (2+1,-4+2) to (2+1,-4+2+1); (2+1,-4+2+1) uo (2+1,-4+2+1+3) to (2+1+1,-4+2+1+3+1);
at (-3.5,-3.5) [$\alpha + \beta$]{}; av (4.3,-3.5) [$\alphz + \beta$]{}; xt (-4.5,2.5) [$\alpha$]{}; at (2.0,2.5) [$-\alpha$]{};
Note that uwing eet... | (-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1); (2,-4+2); to (2+1,-4+2) (2+1,-4+2+1); (2+1,-4+2+1) to [$\alpha \beta$]{}; at (4.3,-3.5) + \beta$]{}; at [$\alpha$]{}; at (2.0,2.5) [$-\alpha$]{}; Note that determinantal formulas,, the formula is equivalent to the determinantal formula by applying the formula indexe... |
(-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1);
(2,-4) to (2,-4+2); (2,-4+2) to (2+1,-4+2) to (2+1,-4+2+1); (2+1,-4+2+1) to (2+1,-4+2+1+3) to (2+1+1,-4+2+1+3+1);
at (-3.5,-3.5) [$\alpha + \beTa$]{}; at (4.3,-3.5) [$\alpha + \bEta$]{}; at (-4.5,2.5) [$\AlpHa$]{}; aT (2.0,2.5) [$-\aLpha$]{};
note That using det... |
(-1,-2) to (-1,-2+1) to ( -1+1,-2+1) to ( -1+ 1,- 2+ 1+1) ;
( 2,-4) to (2,-4 + 2);(2,-4+2) to (2+1,-4+2) to ( 2+ 1 ,-4+ 2 +1 ); (2 +1,-4+2 + 1) t o ( 2+ 1, -4+ 2+ 1 +3 ) to(2+ 1+1,-4+ 2+1+3+1);
at ( -3.5,-3.5) [ $ \a lpha + \be ta$ ]{}; at (4.3 ,-3 .5) [$ \a lph a + \b eta $]{}; at (- 4 .5,2.5 ) [$\alph a$ ] {};... |
(-1,-2) to_(-1,-2+1) to_(-1+1,-2+1) to (-1+1,-2+1+1);
(2,-4) to_(2,-4+2); (2,-4+2)_to_(2+1,-4+2) to_(2+1,-4+2+1);_(2+1,-4+2+1) to (2+1,-4+2+1+3)_to (2+1+1,-4+2+1+3+1);
at (-3.5,-3.5)_[$\alpha + \beta$]{}; at_(4.3,-3.5) [$\alpha +_\beta$]{};_at (-4.5,2.5) [$\alpha$]{}; at (2.0,2.5) [$-\alpha$]{};
Note that using det... |
, D., Svensson, R., and Poutanen, J., 2001, 563, 80 Tavani, M., Barbiellini, G., Argan, A., et al., 2009, 502, 995 Takahashi, T., Abe, K., Endo, M., et al., 2007, 59, S35 Terrell, J., Lee, P., Klebesadel, R., & Griffee, 1996, in 3rd Huntsville Symposium, AIP Conf. Proc. 384 (AIP: New York), Eds. C. Kouveliotou, M. Brig... | , D., Svensson, R., and Poutanen, J., 2001, 563, 80 Tavani, M., Barbiellini, G., Argan, A., et al. , 2009, 502, 995 Takahashi, T., Abe, K., Endo, M., et al. , 2007, 59, S35 Terrell, J., Lee, P., Klebesadel, R., & Griffee, 1996, in 3rd Huntsville Symposium, AIP Conf. Proc. 384 (AIP: New York), Eds. C. Kouveliotou, M. Br... | , D., Dvensson, R., and Poutanen, M., 2001, 563, 80 Tavani, M., Batbuellinm, G., Argzn, A., et xl., 2009, 502, 995 Takahashi, T., Abe, K., Endo, M., et ao., 2007, 59, S35 Terrell, J., Lee, P., Ylebesadep, R., & Gridfee, 1996, in 3rd Hunvaville Symposjmm, AI' Ronf. Proc. 384 (AIP: Kew York), Edv. C. Kouveliotog, O. Yrig... | , D., Svensson, R., and Poutanen, J., 80 M., Barbiellini, Argan, A., et T., K., Endo, M., al., 2007, 59, Terrell, J., Lee, P., Klebesadel, R., Griffee, 1996, in 3rd Huntsville Symposium, AIP Conf. Proc. 384 (AIP: New York), C. Kouveliotou, M. Briggs, and G. Fishman, 545 Terrell, J., Lee, P., Klebesadel, & J., in Bursts... | , D., Svensson, R., and Poutanen, J., 2001, 563, 80 TavAni, M., BarbieLlini, g., ArGan, a., eT al., 2009, 502, 995 TAkahAshi, T., Abe, K., Endo, M., ET al., 2007, 59, S35 terrell, J., Lee, P., Klebesadel, r., & GrifFeE, 1996, In 3rd hUnTsvilLe SympoSIuM, aiP COnF. PRoc. 384 (aIp: neW York), eds. c. KouvelIotou, M. Brig... | , D., Svensson, R., and Po utanen, J. , 200 1,563 ,80 T avan i, M., Barbiel l ini, G., Argan, A., et al. , 200 9, 502, 99 5 Tak ahashi, T. , Abe ,K. , E nd o ,M., e t a l., 200 7, 59, S35 Te rr ell, J., Lee , P ., Klebesa del , R., & Grif fee , 1996 ,in3 rd Hu nts ville Sympo s ium, A IP Conf.Pr o c. 384 (AIP: N e ... | , D.,_Svensson, R.,_and Poutanen, J., 2001,_563, 80_Tavani,_M., Barbiellini,_G.,_Argan, A., et_al., 2009, 502,_995 Takahashi, T., Abe,_K., Endo, M.,_et_al., 2007, 59, S35 Terrell, J., Lee, P., Klebesadel, R., & Griffee, 1996, in_3rd_Huntsville Symposium,_AIP_Conf._Proc. 384 (AIP: New York),_Eds. C. Kouveliotou, M. Brig... |
to} \approx 0.1 r_o/(10^{-2} U_0) \approx 10 Ro^{-1} \Omega^{-1} \approx 10^{3}\Omega^{-1}$: the turnover time of the particle is much longer than the timescale of the wave. Consequently the particle oscillates rapidly as the wave propagates and is slowly advected by the zonal flow. In practice the radial displacement ... | to } \approx 0.1 r_o/(10^{-2 } U_0) \approx 10 Ro^{-1 } \Omega^{-1 } \approx 10^{3}\Omega^{-1}$: the turnover time of the particle is much long than the timescale of the wave. Consequently the atom oscillates rapidly as the wave propagates and is slowly advect by the zonal flow. In exercise the radial displacement $ l$... | to} \wpprox 0.1 r_o/(10^{-2} U_0) \approx 10 Ro^{-1} \Umega^{-1} \approx 10^{3}\Omggq^{-1}$: the vurnoved time ow the particle is much longec thqn tht timescale of the dave. Consvquently rhe karticle oscillatxa rapidly as fme ware propagates anc is slowlf advected by dhd vonal flow. In practice the radial difplacemrnh ... | to} \approx 0.1 r_o/(10^{-2} U_0) \approx 10 \approx the turnover of the particle timescale the wave. Consequently particle oscillates rapidly the wave propagates and is slowly by the zonal flow. In practice the radial displacement $l$ is typically smaller $\delta$ and so the turnover time is slightly overestimated her... | to} \approx 0.1 r_o/(10^{-2} U_0) \approx 10 Ro^{-1} \Omega^{-1} \aPprox 10^{3}\Omega^{-1}$: The tuRnoVer TiMe of The pArticle is much lONger Than the timescale of the wAve. CoNsEQuenTLy The paRticle oSCiLLAteS rApIdlY aS ThE wave ProPagates And is slowlY adVeCted by the zonAL fLow. In practIce The radial disPlaCement ... | to} \approx 0.1 r_o/(10^{- 2} U_0) \a pprox 10 Ro ^{ -1}\Ome ga^{-1} \appro x 10^ {3}\Omega^{-1}$: the t urnov er time of theparticl e i s muc hlo nge rt ha n the ti mescale of the wa ve. C onsequentlyt he particleosc illates rapi dly as th ewav e prop aga tes a nd iss lowlyadvectedby the zo n al flow . In ... | to} \approx_0.1 r_o/(10^{-2}_U_0) \approx 10 Ro^{-1}_\Omega^{-1} \approx_10^{3}\Omega^{-1}$:_the turnover_time_of the particle_is much longer_than the timescale of_the wave. Consequently_the_particle oscillates rapidly as the wave propagates and is slowly advected by the zonal_flow._In practice_the_radial_displacement ... |
end{aligned}$$ To calculate $p_0$ and $\alpha_0$ we know that the real time an atom needs to leave a kink is $\tau_3$, and as there are two kinks at the ends of the facet, $\tau_3/2$ is, to a good approximation, the average time to leave state $0$. Thus that we find: $$\begin{aligned}
p_0 &=& 2\rho \\
\alpha_0 &=& 1 - ... | end{aligned}$$ To calculate $ p_0 $ and $ \alpha_0 $ we know that the real time an atom need to bequeath a kink is $ \tau_3 $, and as there are two kinks at the end of the facet, $ \tau_3/2 $ is, to a good estimate, the average time to leave state of matter $ 0$. Thus that we find: $ $ \begin{aligned }
p_0 & = & 2\rh... | end{wligned}$$ To calculate $p_0$ akd $\alpha_0$ we know that tie real time an atom needs to leave a kink ms $\tqu_3$, ane as there are two kinys at the ends of the dacet, $\tau_3/2$ ma, to a njod ziproxnmetion, the averane time to neave state $0$. Tvur chat we find: $$\begin{aligned}
p_0 &=& 2\rho \\
\alphw_0 &=& 1 - ... | end{aligned}$$ To calculate $p_0$ and $\alpha_0$ we the time an needs to leave as are two kinks the ends of facet, $\tau_3/2$ is, to a good the average time to leave state $0$. Thus that we find: $$\begin{aligned} p_0 2\rho \\ \alpha_0 &=& 1 - 2\rho \end{aligned}$$ Calling $n_i$ the average time go state to absorbent s... | end{aligned}$$ To calculate $p_0$ and $\Alpha_0$ we knoW that The ReaL tIme aN atoM needs to leave a KInk iS $\tau_3$, and as there are two kiNks at ThE Ends OF tHe facEt, $\tau_3/2$ is, TO a GOOd aPpRoXimAtIOn, The avEraGe time tO leave statE $0$. ThUs That we find: $$\beGIn{Aligned}
p_0 &=& 2\rhO \\
\alPha_0 &=& 1 - ... | end{aligned}$$ To calculat e $p_0$ an d $\a lph a_0 $we k nowthat the realt imean atom needs to leave a ki nk is $ \ ta u_3$, and as th e r e a re t woki n ks at t heends of the facet , $ \t au_3/2$ is,t oa good app rox imation, the av erageti met o lea vestate $0$.T hus th at we fin d: $$\beg i n{align e d }p_0... | end{aligned}$$ To_calculate $p_0$_and $\alpha_0$ we know_that the_real_time an_atom_needs to leave_a kink is_$\tau_3$, and as there_are two kinks_at_the ends of the facet, $\tau_3/2$ is, to a good approximation, the average time_to_leave state_$0$._Thus_that we find: $$\begin{aligned}
p_0 &=&_2\rho \\
\alpha_0 &=& 1 -_... |
0)^2 - \xi(r)^2}} \exp\left[-{\nu^2 \sigma^2 \over
\xi(0)+ \xi(r)}\right],$$
where $\epsilon << \nu$ and $\sigma^2$ is the variance of the Gaussian fluctuations (Kaiser 1984). Recall $\xi(0) = \sigma^2$ and that this assumption of Gaussian fluctuations ignores the non-linear evolution of the density field. As the prob... | 0)^2 - \xi(r)^2 } } \exp\left[-{\nu^2 \sigma^2 \over
\xi(0)+ \xi(r)}\right],$$
where $ \epsilon < < \nu$ and $ \sigma^2 $ is the variance of the Gaussian fluctuations (Kaiser 1984). Recall $ \xi(0) = \sigma^2 $ and that this assumption of Gaussian fluctuation ignore the non - linear evolution of the density discip... | 0)^2 - \xl(r)^2}} \exp\left[-{\nu^2 \sigma^2 \over
\xl(0)+ \xi(r)}\right],$$
where $\gpwilon << \nu$ ans $\sigma^2$ ks the variance of the Gaussman dluctyations (Kaiser 1984). Recall $\xi(0) = \sigmw^2$ and thqt tiis assumption oh Gaussian fludbuatimis ignores the kon-linear eeolution of tha aeusity field. As the prob... | 0)^2 - \xi(r)^2}} \exp\left[-{\nu^2 \sigma^2 \over \xi(0)+ $\epsilon \nu$ and is the variance 1984). $\xi(0) = \sigma^2$ that this assumption Gaussian fluctuations ignores the non-linear evolution the density field. As the probability of finding a single peak within the $P_1$ is $(\epsilon/\sqrt{2\pi}\sigma) \exp(-\nu^... | 0)^2 - \xi(r)^2}} \exp\left[-{\nu^2 \sigma^2 \over
\xi(0)+ \xi(R)}\right],$$
wherE $\epsiLon << \Nu$ aNd $\SigmA^2$ is tHe variance of thE gausSian fluctuations (Kaiser 1984). recalL $\xI(0) = \SigmA^2$ AnD that This assUMpTIOn oF GAuSsiAn FLuCtuatIonS ignoreS the non-linEar EvOlution of the DEnSity field. AS thE prob... | 0)^2 - \xi(r)^2}} \exp\lef t[-{\nu^2\sigm a^2 \o ve r
\x i(0) + \xi(r)}\righ t ],$$
where $\epsilon << \ nu$ a nd $\si g ma ^2$ i s the v a ri a n ceof t heGa u ss ian f luc tuation s (Kaiser198 4) . Recall $\x i (0 ) = \sigma ^2$ and that th isassump ti ono f Gau ssi an fl uctuat i ons ig nores the n o n-line a ... | 0)^2 -_\xi(r)^2}} \exp\left[-{\nu^2_\sigma^2 \over
\xi(0)+ \xi(r)}\right],$$
where $\epsilon_<< \nu$_and_$\sigma^2$ is_the_variance of the_Gaussian fluctuations (Kaiser_1984). Recall $\xi(0) =_\sigma^2$ and that_this_assumption of Gaussian fluctuations ignores the non-linear evolution of the density field. As the_prob... |
{\alpha}}\nolimits}} {{\cal V}}$ equal zero in (classical, *i.e.*, non-perverse) degrees $\ne r$, and $\mu_{\bf H}(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the Hodge theoretic setting) resp. $\mu_\ell(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the $\ell$-adic setting) in degree $r$.\
Denote by $j: M^K \in... | { \alpha}}\nolimits } } { { \cal V}}$ equal zero in (classical, * i.e. *, non - perverse) degrees $ \ne r$, and $ \mu_{\bf H}(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the Hodge theoretical place setting) resp. $ \mu_\ell(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the $ \ell$-adic setting) in degree $ r$... | {\alpja}}\nolimits}} {{\cal V}}$ equal ztro in (classical, *n.w.*, non-pxrverse) degrees $\ne r$, and $\mu_{\bf H}(V_{{\mathop{\underpibe{\alpya}}\nolimits}})$ (in the Hodgd theoretpc settint) rewp. $\mu_\ell(V_{{\matikp{\underline{\allma}}\nolnmmts}})$ (in the $\ell$-acic settinc) in degree $r$.\
Danutz by $j: M^K \in... | {\alpha}}\nolimits}} {{\cal V}}$ equal zero in (classical, degrees r$, and H}(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the (in $\ell$-adic setting) in $r$.\ Denote by M^K \into (M^K)^*$ the open immersion $M^K$ into its Satake–(Baily–Borel) compactification, by $i: \partial (M^K)^* \into (M^K)^*$ its complement... | {\alpha}}\nolimits}} {{\cal V}}$ equal zerO in (classicAl, *i.e.*, nOn-pErvErSe) deGreeS $\ne r$, and $\mu_{\bf H}(V_{{\mAThop{\Underline{\alpha}}\nolimits}})$ (In the hoDGe thEOrEtic sEtting) rESp. $\MU_\Ell(v_{{\mAtHop{\UnDErLine{\aLphA}}\nolimiTs}})$ (in the $\ell$-AdiC sEtting) in degrEE $r$.\
denote by $j: M^k \in... | {\alpha}}\nolimits}} {{\ca l V}}$ equ al ze roin(c lass ical , *i.e.*, non- p erve rse) degrees $\ne r$,and $ \m u _{\b f H }(V_{ {\matho p {\ u n der li ne {\a lp h a} }\nol imi ts}})$(in the Ho dge t heoretic set t in g) resp. $ \mu _\ell(V_{{\m ath op{\un de rli n e{\al pha }}\no limits } })$ (i n the $\e ll $ -ad... | {\alpha}}\nolimits}} {{\cal_V}}$ equal_zero in (classical, *i.e.*,_non-perverse) degrees_$\ne_r$, and_$\mu_{\bf_H}(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the_Hodge theoretic setting)_resp. $\mu_\ell(V_{{\mathop{\underline{\alpha}}\nolimits}})$ (in the $\ell$-adic_setting) in degree_$r$.\
Denote_by $j: M^K \in... |
-1)$ undiagonalized matrix block $(**)$ above, together with a simple induction argument, we obtain plane unitary transformation $T_{n1},\ldots, T_{n,n-1},
T_{n-1,1}, \ldots, T_{n-1,n-2},\ldots, T_{31}, T_{32}$ and $T_{21}$ such that $$T^*_{21}T^*_{31}T^*_{32}T^*_{41}\ldots T^*_{n-1,1}\ldots T^*_{n-1,n-2}
T^*_{n1} \ldo... | -1)$ undiagonalized matrix block $ (* *) $ above, together with a simple induction controversy, we receive plane unitary transformation $ T_{n1},\ldots, T_{n, n-1 },
T_{n-1,1 }, \ldots, T_{n-1,n-2},\ldots, T_{31 }, T_{32}$ and $ T_{21}$ such that $ $ T^*_{21}T^*_{31}T^*_{32}T^*_{41}\ldots T^*_{n-1,1}\ldots T^*_{n-1,n... | -1)$ unfiagonalized matrix blocy $(**)$ above, togethgr with e simpls inductkon argument, we obtain plane ubitart transformation $T_{n1},\ldogs, T_{n,n-1},
T_{n-1,1}, \pdots, T_{n-1,b-2},\ldous, T_{31}, T_{32}$ and $T_{21}$ such that $$T^*_{21}T^*_{31}B^*_{32}C^*_{41}\ldota T^*_{n-1,1}\lbovs T^*_{n-1,n-2}
T^*_{n1} \ldo... | -1)$ undiagonalized matrix block $(**)$ above, together simple argument, we plane unitary transformation T_{31}, and $T_{21}$ such $$T^*_{21}T^*_{31}T^*_{32}T^*_{41}\ldots T^*_{n-1,1}\ldots T^*_{n-1,n-2} \ldots T^*_{n,n-1}V = \left[\begin{matrix} d_1\\ &d_2&&0\\ &0&&d_n\end{matrix}\right]=D,$$ where $d_j = e^{i\alpha_j... | -1)$ undiagonalized matrix block $(**)$ Above, togetHer wiTh a SimPlE indUctiOn argument, we obTAin pLane unitary transformatIon $T_{n1},\LdOTs, T_{n,N-1},
t_{n-1,1}, \Ldots, t_{n-1,n-2},\ldotS, t_{31}, T_{32}$ AND $T_{21}$ sUcH tHat $$t^*_{21}T^*_{31}t^*_{32}t^*_{41}\lDots T^*_{N-1,1}\ldOts T^*_{n-1,n-2}
T^*_{N1} \ldo... | -1)$ undiagonalized matrix block $(* *)$ a bov e,to geth er w ith a simple i n duct ion argument, we obtai n pla ne unit a ry tran sformat i on $ T_{ n1 }, \ld ot s ,T_{n, n-1 },
T_{n -1,1}, \ld ots ,T_{n-1,n-2}, \ ld ots, T_{31 },T_{32}$ and$T_ {21}$su cht hat $ $T^ *_{21 }T^*_{ 3 1}T^*_ {32}T^*_{ 41 } \ldots T^*_{n... | -1)$ undiagonalized_matrix block_$(**)$ above, together with_a simple_induction_argument, we_obtain_plane unitary transformation_$T_{n1},\ldots, T_{n,n-1},
T_{n-1,1}, \ldots,_T_{n-1,n-2},\ldots, T_{31}, T_{32}$ and_$T_{21}$ such that_$$T^*_{21}T^*_{31}T^*_{32}T^*_{41}\ldots_T^*_{n-1,1}\ldots T^*_{n-1,n-2}
T^*_{n1} \ldo... |
}_{d \times e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. The ring structure on $\widetilde{\mathcal{E}}$ is induced by the canonical isomorphism ${\mathcal{O}}_{{\mathbb{P}}^1}(-1) \otimes {\mathcal{O}}_{{\mathbb{P}}^1}(1) \stackrel{{\mathsf{can}}}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}$. Let $\widetilde{\mathcal{A}... | } _ { d \times e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. The ring structure on $ \widetilde{\mathcal{E}}$ is induced by the canonical isomorphism $ { \mathcal{O}}_{{\mathbb{P}}^1}(-1) \otimes { \mathcal{O}}_{{\mathbb{P}}^1}(1) \stackrel{{\mathsf{can}}}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}$. lease $ \widetilde{\... | }_{d \tlmes e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. Tht ring structure on $\widevilde{\mafhcal{E}}$ ir induced by the canonical idonorphusm ${\mathcal{O}}_{{\mathbb{P}}^1}(-1) \otkmes {\mathbal{O}}_{{\mathbv{P}}^1}(1) \suackrel{{\mathsf{can}}}{\longrightarrow}{\mzbhcal{M}}_{{\nathbb{P}}^1}$. Let $\wicetilde{\matvcal{A}... | }_{d \times e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. The ring structure on induced the canonical ${\mathcal{O}}_{{\mathbb{P}}^1}(-1) \otimes {\mathcal{O}}_{{\mathbb{P}}^1}(1) ${\mathsf{tr}}$ involves the diagonal of $\widetilde{\mathcal{E}}$ and given by the matrix $(1, 1, 1)$. Of coarse, $\widetilde{\mathcal{E}}\cong {... | }_{d \times e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. The Ring structUre on $\WidEtiLdE{\matHcal{e}}$ is induced by thE CanoNical isomorphism ${\mathcaL{O}}_{{\matHbB{p}}^1}(-1) \otiMEs {\MathcAl{O}}_{{\mathBB{P}}^1}(1) \STAckReL{{\mAthSf{CAn}}}{\LongrIghTarrow}{\mAthcal{O}}_{{\matHbb{p}}^1}$. LEt $\widetilde{\mAThCal{A}... | }_{d \times e}({\mathcal{O }}_{{\math bb{P} }^1 }(1 )) $. T he r ing structureo n $\ widetilde{\mathcal{E}} $ isin d uced by thecanonic a li s omo rp hi sm${ \ ma thcal {O} }_{{\ma thbb{P}}^1 }(- 1) \otimes {\m a th cal{O}}_{{ \ma thbb{P}}^1}( 1)\stack re l{{ \ maths f{c an}}} {\long r ightar row}{\mat hc a l{O}}_ { ... | }_{d \times_e}({\mathcal{O}}_{{\mathbb{P}}^1}(1))$. The_ring structure on $\widetilde{\mathcal{E}}$_is induced_by_the canonical_isomorphism_${\mathcal{O}}_{{\mathbb{P}}^1}(-1) \otimes {\mathcal{O}}_{{\mathbb{P}}^1}(1)_\stackrel{{\mathsf{can}}}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}$. Let $\widetilde{\mathcal{A}... |
in M_{ns}(F')$, $\rho(q)>\lambda$.\
Suppose further that for some $j\in\{1,\cdots,ns\}$, the $j$’th entries of $d_1,\cdots,d_r$ are $\mathbb{F}_p$-linearly independent modulo $\lambda^+$. Then $e_ja_i=0$ for all $i=1,\cdots,r$.
Firstly, since $d_{1,j},\cdots,d_{r,j}$ are $\mathbb{F}_p$-linearly independent modulo $\la... | in M_{ns}(F')$, $ \rho(q)>\lambda$.\
Suppose further that for some $ j\in\{1,\cdots, ns\}$, the $ j$’th entries of $ d_1,\cdots, d_r$ are $ \mathbb{F}_p$-linearly independent modulo $ \lambda^+$. Then $ e_ja_i=0 $ for all $ i=1,\cdots, r$.
first, since $ d_{1,j},\cdots, d_{r, j}$ are $ \mathbb{F}_p$-linearly auton... | in L_{ns}(F')$, $\rho(q)>\lambda$.\
Suppose fmrther that for some $j\ii\{1,\cdots,na\}$, the $j$’tf entries of $d_1,\cdots,d_r$ are $\mavhbb{D}_p$-lintcrly independent modjlo $\lambdw^+$. Then $e_ha_i=0$ hor all $i=1,\cdots,r$.
Fmdstly, sluce $d_{1,n},\gdots,b_{r,o}$ are $\mathbb{F}_p$-llnearly indapendent modulm $\ua... | in M_{ns}(F')$, $\rho(q)>\lambda$.\ Suppose further that for the entries of are $\mathbb{F}_p$-linearly independent all Firstly, since $d_{1,j},\cdots,d_{r,j}$ $\mathbb{F}_p$-linearly independent modulo it follows immediately that $e_jd_1,\cdots,e_jd_r$ are independent modulo $\lambda^+$. And: $0=(e_jd_1)^{p^m}a_1+\cdo... | in M_{ns}(F')$, $\rho(q)>\lambda$.\
Suppose fuRther that fOr somE $j\iN\{1,\cdOtS,ns\}$, tHe $j$’tH entries of $d_1,\cdoTS,d_r$ aRe $\mathbb{F}_p$-linearly indePendeNt MOdulO $\LaMbda^+$. THen $e_ja_i=0$ FOr ALL $i=1,\cDoTs,R$.
FiRsTLy, Since $D_{1,j},\cDots,d_{r,j}$ Are $\mathbb{F}_P$-liNeArly independENt Modulo $\la... | in M_{ns}(F')$, $\rho(q)>\ lambda$.\Suppo sefur th er t hatfor some $j\in \ {1,\ cdots,ns\}$, the $j$’t h ent ri e s of $d _1,\c dots,d_ r $a r e $ \m at hbb {F } _p $-lin ear ly inde pendent mo dul o$\lambda^+$. Th en $e_ja_i =0$ for all $i= 1,\ cdots, r$ .
F irstl y,since $d_{1 , j},\cd ots,d_{r, j} $ are $ \ math... | in M_{ns}(F')$,_$\rho(q)>\lambda$.\
Suppose further_that for some $j\in\{1,\cdots,ns\}$,_the $j$’th_entries_of $d_1,\cdots,d_r$_are_$\mathbb{F}_p$-linearly independent modulo_$\lambda^+$. Then $e_ja_i=0$_for all $i=1,\cdots,r$.
Firstly, since_$d_{1,j},\cdots,d_{r,j}$ are $\mathbb{F}_p$-linearly_independent_modulo $\la... |
1. The Lagrangian of general relativity is $\frac1{16\pi}R$. So the constant in $Z^{abcd}$ field is $$\alpha=1\,.$$
2. New Massive Gravity is obtained by taking $a=\frac1 {16\pi}$,$b=\frac{-3}{16\pi 8m^2 }$, $c=\frac1{16\pi m^2}$ and $d=0$ [@Bergshoeff:2009hq], so $$\alpha=\left(1+\frac{1}{2 \ell^2 m^2 }\right)\,.$$... | 1. The Lagrangian of general relativity is $ \frac1{16\pi}R$. So the constant in $ Z^{abcd}$ field is $ $ \alpha=1\,.$$
2. New Massive Gravity is obtained by take $ a=\frac1 { 16\pi}$,$b=\frac{-3}{16\pi 8m^2 } $, $ c=\frac1{16\pi m^2}$ and $ d=0 $ [ @Bergshoeff:2009hq ], therefore $ $ \alpha=\left(1+\frac{1}{2 \... | 1. Tje Lagrangian of general relativity is $\yeac1{16\pi}R$. So ths constavt in $Z^{abcd}$ field is $$\alpha=1\,.$$
2. Iew Nassice Gravity is obtained by takinh $a=\frac1 {16\pi}$,$b=\fcac{-3}{16\pi 8m^2 }$, $c=\frac1{16\pi m^2}$ and $d=0$ [@Bergsgleff:2009kq], so $$\alpha=\left(1+\ftac{1}{2 \ell^2 m^2 }\richt)\,.$$... | 1. The Lagrangian of general relativity is the in $Z^{abcd}$ is $$\alpha=1\,.$$ 2. by $a=\frac1 {16\pi}$,$b=\frac{-3}{16\pi 8m^2 $c=\frac1{16\pi m^2}$ and [@Bergshoeff:2009hq], so $$\alpha=\left(1+\frac{1}{2 \ell^2 m^2 }\right)\,.$$ Space Cosmologies ====================== Charges and Entropy ------------------- Now we... | 1. The Lagrangian of general relAtivity is $\fRac1{16\pi}r$. So The CoNstaNt in $z^{abcd}$ field is $$\alPHa=1\,.$$
2. NeW Massive Gravity is obtaiNed by TaKIng $a=\FRaC1 {16\pi}$,$b=\fRac{-3}{16\pi 8m^2 }$, $c=\FRaC1{16\PI m^2}$ aNd $D=0$ [@BErgShOEfF:2009hq], so $$\AlpHa=\left(1+\fRac{1}{2 \ell^2 m^2 }\rigHt)\,.$$... | 1. The Lagrangian of gene ral relati vityis$\f ra c1{1 6\pi }R$. So the co n stan t in $Z^{abcd}$ fieldis $$ \a l pha= 1 \, .$$
2. New Ma s s ive G ra vit yi sobtai ned by tak ing $a=\fr ac1 { 16\pi}$,$b=\ f ra c{-3}{16\p i 8 m^2 }$, $c=\ fra c1{16\ pi m^ 2 }$ an d $ d=0$[@Berg s hoeff: 2009hq],so $$\alp h a=\lef... | 1. _The Lagrangian_of general relativity is_$\frac1{16\pi}R$. So_the_constant in_$Z^{abcd}$_field is $$\alpha=1\,.$$
2._ New Massive_Gravity is obtained by_taking $a=\frac1 {16\pi}$,$b=\frac{-3}{16\pi_8m^2_}$, $c=\frac1{16\pi m^2}$ and $d=0$ [@Bergshoeff:2009hq], so $$\alpha=\left(1+\frac{1}{2 \ell^2 m^2 }\right)\,.$$... |
)$ will always stand for the quotient $F(1,p^C(G_f))$, but for clarity in the figures of Sections \[sec:stretching\] and \[sec:emb\] we will continue to draw it with long vertical intervals.
Chain refinements, their composition and stretching {#sec:stretching}
===================================================
Let $... | ) $ will always stand for the quotient $ F(1,p^C(G_f))$, but for clarity in the figures of Sections \[sec: stretching\ ] and \[sec: emb\ ] we will stay to pull back it with long vertical intervals.
range refinements, their composition and stretching { # sec: stretch }
= = = = = = = = = = = = = = = = = = = = = = ... | )$ wipl always stand for the duotient $F(1,p^C(G_f))$, yyt for clarify in thd figures of Sections \[sec:strevchibg\] ane \[sec:emb\] we will contivue to drww it wirh libg vertical intervals.
Chajk refnnxments, their cokposition dnd stretching {#sdc:dtretching}
===================================================
Let $... | )$ will always stand for the quotient for in the of Sections \[sec:stretching\] to it with long intervals. Chain refinements, composition and stretching {#sec:stretching} =================================================== Let I\to I$ be a piecewise linear surjection, $p$ an admissible $C$-permutation of $G_f$ $\eps>0$... | )$ will always stand for the quotIent $F(1,p^C(G_f))$, bUt for ClaRitY iN the FiguRes of Sections \[sEC:strEtching\] and \[sec:emb\] we will ContiNuE To drAW iT with Long verTIcAL IntErVaLs.
CHaIN rEfineMenTs, their CompositioN anD sTretching {#sec:STrEtching}
===================================================
Let $... | )$ will always stand for t he quotien t $F( 1,p ^C( G_ f))$ , bu t for clarityi n th e figures of Sections\[sec :s t retc h in g\] a nd \[se c :e m b \]we w ill c o nt inuetodraw it with long ve rt ical interva l s.
Chain re fin ements, thei r c omposi ti ona nd st ret ching {#sec : stretc hing}
=== == = ====== ... | )$ will_always stand_for the quotient $F(1,p^C(G_f))$,_but for_clarity_in the_figures_of Sections \[sec:stretching\] and_\[sec:emb\] we will_continue to draw it_with long vertical_intervals.
Chain_refinements, their composition and stretching {#sec:stretching}
===================================================
Let $... |
, ApJ, 409, 455
Canuto V. M., Mazzitelli L., 1991, ApJ, 370, 295
Christensen-Dalsgaard J. et al., 1996, Sci, 272, 1286
Dahlen F. A., Tromp J., 1998, Theoretical Global Seismology, Princeton Univ. Press, Princeton
Gough D. O., Vorontsov S. V., 1995, MNRAS, 273, 573
Korzennik S. G., Rabello-Soares M. C., Schou J., L... | , ApJ, 409, 455
Canuto V. M., Mazzitelli L., 1991, ApJ, 370, 295
Christensen - Dalsgaard J. et al. , 1996, Sci, 272, 1286
Dahlen F. A., Tromp J., 1998, Theoretical Global Seismology, Princeton Univ. Press, Princeton
Gough D. O., Vorontsov S. V., 1995, MNRAS, 273, 573
Korzennik S. G., Rabello - Soares M. ... | , ApU, 409, 455
Canuto V. M., Mazzitelli L., 1991, ApJ, 370, 295
Christgnwen-Dalvgaard J. et al., 1996, Sci, 272, 1286
Dahlen F. A., Tromp J., 1998, Tieorwticao Global Seismology, Prknceton Ujiv. Presw, Prmnceton
Gough D. O., Vorontsov S. V., 1995, MNRCS, 273, 573
Korzennik S. N., Rabello-Sodres M. C., Schou J., L... | , ApJ, 409, 455 Canuto V. M., 1991, 370, 295 J. et al., F. Tromp J., 1998, Global Seismology, Princeton Press, Princeton Gough D. O., Vorontsov V., 1995, MNRAS, 273, 573 Korzennik S. G., Rabello-Soares M. C., Schou J., T. P. 2013, ApJ, 772:87 Schou J., 1999, ApJ, 523, L181 Vorontsov S. 1991, Astron., 400 S. V., 2006, i... | , ApJ, 409, 455
Canuto V. M., Mazzitelli L., 1991, ApJ, 370, 295
christenseN-DalsGaaRd J. Et Al., 1996, ScI, 272, 1286
DahLen F. A., Tromp J., 1998, TheORetiCal Global Seismology, PriNcetoN UNIv. PrESs, princEton
GouGH D. o., vOroNtSoV S. V., 1995, mNraS, 273, 573
korzeNniK S. G., RabeLlo-Soares M. c., ScHoU J., L... | , ApJ, 409, 455
Canuto V. M., Mazzi telli L. , 1 99 1, A pJ,370, 295
Chri s tens en-Dalsgaard J. et al. , 199 6, Sci, 27 2, 12 86
Dah l en F . A ., T rom pJ ., 1998 , T heoreti cal Global Se is mology, Prin c et on Univ. P res s, Princeton
G ough D .O., Voron tso v S.V., 19 9 5, MNR AS, 273,57 3
Korz e nnik... | , ApJ,_409, 455
Canuto_V. M., Mazzitelli L.,_1991, ApJ,_370,_295
Christensen-Dalsgaard J._et_al., 1996, Sci,_272, 1286
Dahlen F._A., Tromp J., 1998,_Theoretical Global Seismology,_Princeton_Univ. Press, Princeton
Gough D. O., Vorontsov S. V., 1995, MNRAS, 273, 573
Korzennik S. G.,_Rabello-Soares_M. C.,_Schou_J.,_L... |
environment with a reduced symmetry has not been investigated,[@neupane2019plane] and its potential for realizing anisotropic modification on the electronic and optical properties of the monolayer TMDCs remains unexploited. In this letter, we report the breaking of the three-fold in-plane symmetry of the MoS$_2$ monol... | environment with a reduced symmetry has not been investigated,[@neupane2019plane ] and its potential for realizing anisotropic alteration on the electronic and ocular properties of the monolayer TMDCs remains unexploited. In this letter, we report the breaking of the three - fold in - airplane symmetry of the MoS$_2 $ ... | engironment with a reduced symmetry has not been invesfigated,[@ndupane2019plane] and its potentiap dor rtclizing anisotropic oodificatpon on thw eltctronic and optirzl propcxties lf tke monolayer TMDGs remains gnexploited. In tfid letter, we report the breaking of ehe thrre-vold in-plane simmetgy of fhe MoS$_2$ monol... | environment with a reduced symmetry has not and potential for anisotropic modification on of monolayer TMDCs remains In this letter, report the breaking of the three-fold symmetry of the MoS$_2$ monolayer by depositing on the low-symmetry surface of sapphire, the symmetry associated dielectric engineering of the 2D mat... | environment with a reduced syMmetry has nOt beeN inVesTiGateD,[@neuPane2019plane] and itS PoteNtial for realizing anisoTropiC mODifiCAtIon on The elecTRoNIC anD oPtIcaL pROpErtieS of The monoLayer TMDCs RemAiNs unexploiteD. in This letter, We rEport the breaKinG of the ThRee-FOld in-PlaNe symMetry oF The MoS$_2$ Monol... | environment with a reduce d symmetry hasnot be en inv esti gated,[@neupan e 2019 plane] and its potenti al fo rr eali z in g ani sotropi c m o d ifi ca ti onon th e ele ctr onic an d opticalpro pe rties of the mo nolayer TM DCs remains une xpl oited. I n t h is le tte r, we repor t the b reaking o ft he thr e ... | environment_with a_reduced symmetry has not_been investigated,[@neupane2019plane]_and_its potential_for_realizing anisotropic modification_on the electronic_and optical properties of_the monolayer TMDCs_remains_unexploited. In this letter, we report the breaking of the three-fold in-plane symmetry of_the_MoS$_2$ monol... |
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$\text{\sc Hibert}_S$ (in-domain) __ _42.10 _ 19.70 _ ___ 38.53
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ffvalidation\], we found that the $\beta-c_1$ plane produced less accurate atmospheric parameters, relative to fundamental determinations, for late group stars.
For the intermediate group stars (8500 K $\leq T_\mathrm{eff} \leq$ 11000 K), $\log g$ is determined in the $a_0-r^*$ plane. The mean and median of the $\log ... | ffvalidation\ ], we found that the $ \beta - c_1 $ plane produced less accurate atmospheric parameters, proportional to cardinal decision, for late group stars.
For the average group stars (8500 K $ \leq T_\mathrm{eff } \leq$ 11000 K), $ \log g$ is determined in the $ a_0 - r^*$ airplane. The beggarly and median of ... | ffvwlidation\], we found that uhe $\beta-c_1$ plane ptoeuced ness adcurate xtmospheric parameters, relatmve ro fubdamental determinatiovs, for lane group wtarw.
Dor the invsrmediabz grohi staxs (8500 K $\leq T_\mathrk{eff} \leq$ 11000 N), $\log g$ is detaroiued in the $a_0-r^*$ plane. The mean and medyan of yhf $\log ... | ffvalidation\], we found that the $\beta-c_1$ plane accurate parameters, relative fundamental determinations, for intermediate stars (8500 K T_\mathrm{eff} \leq$ 11000 $\log g$ is determined in the plane. The mean and median of the $\log g$ residuals are -0.060 dex -0.069 dex, respectively with RMS error 0.091 dex. For... | ffvalidation\], we found that thE $\beta-c_1$ planE prodUceD leSs AccuRate Atmospheric parAMeteRs, relative to fundamentaL deteRmINatiONs, For laTe group STaRS.
for ThE iNteRmEDiAte grOup Stars (8500 K $\lEq T_\mathrm{eFf} \lEq$ 11000 k), $\log g$ is deterMInEd in the $a_0-r^*$ pLanE. The mean and mEdiAn of thE $\lOg ... | ffvalidation\], we found t hat the $\ beta- c_1 $ p la ne p rodu ced less accur a te a tmospheric parameters, rela ti v e to fu ndame ntal de t er m i nat io ns , f or la te gr oup stars.
For theint er mediate grou p s tars (8500 K$\leq T_\mat hrm {eff}\l eq$ 11000 K) , $\l og g$i s dete rmined in t h e $a_0 - ... | ffvalidation\], we_found that_the $\beta-c_1$ plane produced_less accurate_atmospheric_parameters, relative_to_fundamental determinations, for_late group stars.
For_the intermediate group stars_(8500 K $\leq_T_\mathrm{eff}_\leq$ 11000 K), $\log g$ is determined in the $a_0-r^*$ plane. The mean and_median_of the_$\log_... |
$L_h^{-1}G_h\colon X_h\to X_h$ restricted to ${\overline}{B_{2C_0M_0h}(0)}$ is a contraction, where $X_h$ is normed by $$\|w\|_{X_h}:= \left\|\left(\frac1h {\varepsilon}_h(w),\nabla \frac1h {\varepsilon}_h(w),\nabla_h^2 w\right)\right\|_{H^{1,0}(\Omega)}.$$ Therefore we obtain a unique solution $w$ solving (\[eq:FnEq\... | $ L_h^{-1}G_h\colon X_h\to X_h$ restricted to $ { \overline}{B_{2C_0M_0h}(0)}$ is a contraction, where $ X_h$ is normed by $ $ \|w\|_{X_h}:= \left\|\left(\frac1h { \varepsilon}_h(w),\nabla \frac1h { \varepsilon}_h(w),\nabla_h^2 w\right)\right\|_{H^{1,0}(\Omega)}.$$ Therefore we obtain a alone solution $ w$ clear (\[eq:... | $L_h^{-1}H_h\colon X_h\to X_h$ restricttd to ${\overline}{B_{2C_0M_0k}(0)}$ is a rontracfion, whefe $X_h$ is normed by $$\|w\|_{X_h}:= \left\|\lxft(\feac1h {\carepsilon}_h(w),\nabla \frac1f {\varepsipon}_h(w),\naboa_h^2 x\right)\right\|_{H^{1,0}(\Omege)}.$$ Therefore we lbtann a unique solujion $w$ solvitg (\[eq:FnEq\... | $L_h^{-1}G_h\colon X_h\to X_h$ restricted to ${\overline}{B_{2C_0M_0h}(0)}$ is where is normed $$\|w\|_{X_h}:= \left\|\left(\frac1h {\varepsilon}_h(w),\nabla obtain unique solution $w$ (\[eq:FnEq\]) and satisfying with $H^{2,0}(\Omega)$ replaced by $H^{1,0}(\Omega)$. In to obtain (\[eq:FnEstim\]), one can simply use th... | $L_h^{-1}G_h\colon X_h\to X_h$ restricted To ${\overline}{b_{2C_0M_0h}(0)}$ iS a cOntRaCtioN, wheRe $X_h$ is normed by $$\|W\|_{x_h}:= \leFt\|\left(\frac1h {\varepsilon}_h(W),\nablA \fRAc1h {\vARePsiloN}_h(w),\nablA_H^2 w\RIGht)\RiGhT\|_{H^{1,0}(\OMeGA)}.$$ THerefOre We obtaiN a unique soLutIoN $w$ solving (\[eq:FNeq\... | $L_h^{-1}G_h\colon X_h\to X_h$ rest ricte d t o $ {\ over line }{B_{2C_0M_0h} ( 0)}$ is a contraction, whe re $X _h $ isn or med b y $$\|w \ |_ { X _h} := \ lef t\ | \l eft(\ fra c1h {\v arepsilon} _h( w) ,\nabla \fra c 1h {\varepsi lon }_h(w),\nabl a_h ^2 w\r ig ht) \ right \|_ {H^{1 ,0}(\O m ega)}. $$ Theref or... | $L_h^{-1}G_h\colon_X_h\to X_h$_restricted to ${\overline}{B_{2C_0M_0h}(0)}$ is_a contraction,_where_$X_h$ is_normed_by $$\|w\|_{X_h}:= \left\|\left(\frac1h_{\varepsilon}_h(w),\nabla \frac1h {\varepsilon}_h(w),\nabla_h^2_w\right)\right\|_{H^{1,0}(\Omega)}.$$ Therefore we obtain_a unique solution_$w$_solving (\[eq:FnEq\... |
Office of the Privacy Commissioner of Canada (OPC) Contributions Program.
[10]{}
M. Abomhara and G. M. K[ø]{}ien. Cyber security and the internet of things: vulnerabilities, threats, intruders and attacks., 4(1):65–88, 2015.
.. <https://developer.apple.com/documentation/adsupport/asidentifiermanager>.
M. J. Austin... | Office of the Privacy Commissioner of Canada (OPC) Contributions Program.
[ 10 ] { }
M. Abomhara and G. M. K[ø]{}ien. Cyber security and the internet of things: vulnerabilities, threats, intruder and attack. , 4(1):65–88, 2015.
.. < https://developer.apple.com/documentation/adsupport/asidentifiermanager >. ... | Ofvice of the Privacy Commlssioner of Canabq (OPC) Rontribhtions Pfogram.
[10]{}
M. Abomhara and G. M. K[ø]{}ien. Rybee secyrity and the internet of thingd: vulnerqbilmties, threats, invduders and atfwcks., 4(1):65–88, 2015.
.. <https://developgr.apple.com/dowumentation/adsgpooxt/asidentifiermanager>.
M. J. Austin... | Office of the Privacy Commissioner of Canada Program. M. Abomhara G. M. K[ø]{}ien. of vulnerabilities, threats, intruders attacks., 4(1):65–88, 2015. <https://developer.apple.com/documentation/adsupport/asidentifiermanager>. M. J. Austin and M. Reed. Targeting children online: Internet advertising ethics issues., 16(6)... | Office of the Privacy CommissIoner of CanAda (OPc) CoNtrIbUtioNs PrOgram.
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M. Abomhara ANd G. M. k[ø]{}ien. Cyber security and tHe intErNEt of THiNgs: vuLnerabiLItIES, thReAtS, inTrUDeRs and AttAcks., 4(1):65–88, 2015.
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M. J. AuStIn... | Office of the Privacy Com missionerof Ca nad a ( OP C) C ontr ibutions Progr a m.
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M. Abomhara an d G.M. K[ø] { }i en. C yber se c ur i t y a nd t hein t er net o f t hings:vulnerabil iti es , threats, i n tr uders andatt acks., 4(1): 65– 88, 20 15 .
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M. Abomhara_and G. M._K[ø]{}ien._Cyber security and_the internet of_things: vulnerabilities, threats, intruders_and attacks., 4(1):65–88,_2015.
.._<https://developer.apple.com/documentation/adsupport/asidentifiermanager>.
M. J. Austin... |
\hat{k} = \hat{m} \; \mbox{or} \; \hat k \leftrightarrow
\kappa_1, \kappa_2 \leq \alpha \; \mbox{and} \; \hat m \leftrightarrow
\mu_1, \mu_2 \leq \alpha, \\ [2ex]
\langle \hat{k} |D (\alpha)| \hat{m} \rangle = \langle \hat{k} |D (2)|
\hat{m} \rangle, \; \kappa_1 > \alpha\; \mbox{or} \;\kappa_2 > \alpha \; \\
\phantom... | \hat{k } = \hat{m } \; \mbox{or } \; \hat k \leftrightarrow
\kappa_1, \kappa_2 \leq \alpha \; \mbox{and } \; \hat m \leftrightarrow
\mu_1, \mu_2 \leq \alpha, \\ [ 2ex ]
\langle \hat{k } |D (\alpha)| \hat{m } \rangle = \langle \hat{k } |D (2)|
\hat{m } \rangle, \; \kappa_1 > \alpha\; \mbox{or } \;\kappa_2 > \al... | \hah{k} = \hat{m} \; \mbox{or} \; \hat k \ueftrightarrow
\kcppa_1, \ka'pa_2 \leq \alpha \; \obox{and} \; \hat m \leftrightarrox
\mu_1, \nu_2 \lew \alpha, \\ [2ex]
\langle \hat{k} |D (\alpha)| \jat{m} \rantle = \oangle \hat{k} |D (2)|
\hat{m} \rangls, \; \ka'pe_1 > \alpha\; \mbox{or} \;\kappa_2 > \al[ha \; \\
\phantom... | \hat{k} = \hat{m} \; \mbox{or} \; \hat \kappa_1, \leq \alpha \mbox{and} \; \hat \alpha, [2ex] \langle \hat{k} (\alpha)| \hat{m} \rangle \langle \hat{k} |D (2)| \hat{m} \rangle, \kappa_1 > \alpha\; \mbox{or} \;\kappa_2 > \alpha \; \\ \phantom{\langle \hat{k} |D (\alpha)| \rangle = \langle \hat{k} |D (2)| \hat{m} \rangle... | \hat{k} = \hat{m} \; \mbox{or} \; \hat k \leftrigHtarrow
\kapPa_1, \kapPa_2 \lEq \aLpHa \; \mbOx{anD} \; \hat m \leftrightARrow
\Mu_1, \mu_2 \leq \alpha, \\ [2ex]
\langle \haT{k} |D (\alPhA)| \Hat{m} \RAnGle = \laNgle \hat{K} |d (2)|
\hAT{M} \raNgLe, \; \KapPa_1 > \ALpHa\; \mboX{or} \;\Kappa_2 > \alPha \; \\
\phantom... | \hat{k} = \hat{m} \; \mbo x{or} \; \ hat k \l eft ri ghta rrow
\kappa_1, \ka p pa_2 \leq \alpha \; \mbox{ and}\; \hat m\left rightar r ow \ mu_ 1, \ mu_ 2\ le q \al pha , \\ [2 ex]
\lang le\h at{k} |D (\a l ph a)| \hat{m } \ rangle = \la ngl e \hat {k } | D (2)|
\h at{m} \rang l e, \;\kappa_1>\ alpha\ ; \mbox... | \hat{k}_= \hat{m}_\; \mbox{or} \; \hat_k \leftrightarrow
\kappa_1,_\kappa_2_\leq \alpha_\;_\mbox{and} \; \hat_m \leftrightarrow
\mu_1, \mu_2_\leq \alpha, \\ [2ex]
\langle_\hat{k} |D (\alpha)|_\hat{m}_\rangle = \langle \hat{k} |D (2)|
\hat{m} \rangle, \; \kappa_1 > \alpha\; \mbox{or} \;\kappa_2 >_\alpha_\; \\
\phantom... |
05 & 5.804628 & 5.817483 & 0.4285523 & 0.4356297\
0.06 & 4.944366 & 4.957093 & 0.4299292 & 0.4369816\
0.07 & 4.326666 & 4.339270 & 0.4312912 & 0.4383197\
0.08 & 3.861024 & 3.873509 & 0.4326399 & 0.4396454\
0.09 & 3.497062 & 3.509430 & 0.4339756 & 0.4409590\
0.1 & 3.204486 & 3.216745 & 0.4352992 & 0.4422610\
0.15 & 2.31... | 05 & 5.804628 & 5.817483 & 0.4285523 & 0.4356297\
0.06 & 4.944366 & 4.957093 & 0.4299292 & 0.4369816\
0.07 & 4.326666 & 4.339270 & 0.4312912 & 0.4383197\
0.08 & 3.861024 & 3.873509 & 0.4326399 & 0.4396454\
0.09 & 3.497062 & 3.509430 & 0.4339756 & 0.4409590\
0.1 & 3.204486 & 3.216745 & 0.4352992 & 0.4422610\
... | 05 & 5.804628 & 5.817483 & 0.4285523 & 0.4356297\
0.06 & 4.944366 & 4.957093 & 0.4299292 & 0.4369816\
0.07 & 4.326666 & 4.339270 & 0.4312912 & 0.4383197\
0.08 & 3.861024 & 3.873509 & 0.4326399 & 0.4396454\
0.09 & 3.497062 & 3.509430 & 0.4339756 & 0.4409590\
0.1 & 3.204486 & 3.216745 & 0.4352992 & 0.4422610\
0.15 & 2.31... | 05 & 5.804628 & 5.817483 & 0.4285523 0.06 4.944366 & & 0.4299292 & 4.339270 0.4312912 & 0.4383197\ & 3.861024 & & 0.4326399 & 0.4396454\ 0.09 & & 3.509430 & 0.4339756 & 0.4409590\ 0.1 & 3.204486 & 3.216745 & 0.4352992 0.4422610\ 0.15 & 2.316048 & 2.327805 & 0.4417402 & 0.4486057\ 0.2 & 1.862378 1.873704 0.4479148 0.454... | 05 & 5.804628 & 5.817483 & 0.4285523 & 0.4356297\
0.06 & 4.944366 & 4.957093 & 0.4299292 & 0.4369816\
0.07 & 4.326666 & 4.339270 & 0.4312912 & 0.4383197\
0.08 & 3.861024 & 3.873509 & 0.4326399 & 0.4396454\
0.09 & 3.497062 & 3.509430 & 0.4339756 & 0.4409590\
0.1 & 3.204486 & 3.216745 & 0.4352992 & 0.4422610\
0.15 & 2.31... | 05 & 5.804628 & 5.817483 & 0.4285523 & 0. 435 629 7\
0.0 6 &4.944366 & 4.9 5 7093 & 0.4299292 & 0.43698 16\
0 .0 7 & 4 . 32 6666& 4.339 2 70 & 0. 43 12 912 & 0. 43831 97\
0.08 & 3.861024& 3 .8 73509 & 0.43 2 63 99 & 0.439 645 4\
0.09 & 3. 497 062 &3. 509 4 30 &0.4 33975 6 & 0. 4 409590 \
0.1 & 3 .2 0 4486 & 3.... | 05 &_5.804628 &_5.817483 & 0.4285523 &_0.4356297\
0.06 &_4.944366_& 4.957093_&_0.4299292 & 0.4369816\
0.07_& 4.326666 &_4.339270 & 0.4312912 &_0.4383197\
0.08 & 3.861024_&_3.873509 & 0.4326399 & 0.4396454\
0.09 & 3.497062 & 3.509430 & 0.4339756 & 0.4409590\
0.1 &_3.204486_& 3.216745_&_0.4352992_& 0.4422610\
0.15 & 2.31... |
function $$f'(\gamma)-f(\gamma)\left\{{1\over\gamma}+{\gamma\over\gamma^2-1}
-{1\over kT}\right\}=
{1\over D(\gamma)}\left\{\int_1^\gamma (A+E)f
\,d\gamma'-\int_1^\infty (A+E)f\,d\gamma'\times\int_1^\gamma
\tilde S\,d\gamma'\right\}, \label{2.3}$$ while $[Df]'-Pf=0$ at $\gamma=1$ was ... | function $ $ f'(\gamma)-f(\gamma)\left\{{1\over\gamma}+{\gamma\over\gamma^2 - 1 }
-{1\over kT}\right\}=
{ 1\over D(\gamma)}\left\{\int_1^\gamma (A+E)f
\,d\gamma'-\int_1^\infty (A+E)f\,d\gamma'\times\int_1^\gamma
\tilde S\,d\gamma'\right\ }, \label{2.3}$$ while $ [ Df]'-Pf=0 $ a... | fujction $$f'(\gamma)-f(\gamma)\left\{{1\oyer\gamma}+{\gamma\ovet\gqmma^2-1}
-{1\ovxr kT}\rifht\}=
{1\ovef D(\gamma)}\left\{\int_1^\gamma (A+E)f
\,d\gamla'-\unt_1^\indty (A+E)f\,d\gamma'\times\int_1^\gxmma
\tilde S\,d\gamma'\eighu\}, \layeo{2.3}$$ while $[Df]'-Pf=0$ aj $\gamma=1$ was ... | function $$f'(\gamma)-f(\gamma)\left\{{1\over\gamma}+{\gamma\over\gamma^2-1} -{1\over kT}\right\}= {1\over D(\gamma)}\left\{\int_1^\gamma (A+E)f \tilde \label{2.3}$$ while at $\gamma=1$ was last in eq. (\[2.3\]) simply from conservation the total number of positrons $$\int_1^\infty[A(\gamma)+E(\gamma)]f(\gamma)\,d\gamm... | function $$f'(\gamma)-f(\gamma)\left\{{1\oVer\gamma}+{\gaMma\ovEr\gAmmA^2-1}
-{1\oVer kt}\rigHt\}=
{1\over D(\gamma)}\leFT\{\int_1^\Gamma (A+E)f
\,d\gamma'-\int_1^\infty (a+E)f\,d\gAmMA'\timES\iNt_1^\gamMa
\tilde s\,D\gAMMa'\rIgHt\}, \LabEl{2.3}$$ WHiLe $[Df]'-PF=0$ at $\Gamma=1$ waS ... | function $$f'(\gamma)-f(\ gamma)\lef t\{{1 \ov er\ ga mma} +{\g amma\over\gamm a ^2-1 }
-{1\over kT}\right\} =
{ 1\ o verD (\ gamma )}\left \ {\ i n t_1 ^\ ga mma ( A +E )f
\, d\g amma'-\ int_1^\inf ty(A +E)f\,d\gamm a '\ times\int_ 1^\ gamma
\tilde S\ ,d\gam ma '\r i ght\} , \labe l {2.3}$... | function_$$f'(\gamma)-f(\gamma)\left\{{1\over\gamma}+{\gamma\over\gamma^2-1}
-{1\over kT}\right\}=_
{1\over D(\gamma)}\left\{\int_1^\gamma (A+E)f
\,d\gamma'-\int_1^\infty_(A+E)f\,d\gamma'\times\int_1^\gamma
\tilde S\,d\gamma'\right\},__ __ _ _ _ __ _\label{2.3}$$_while $[Df]'-Pf=0$_at_$\gamma=1$_was ... |
. Each training set is further divided into a 80-20 training/validation stratified split based on binned pupil center locations of each image.
To derive conclusions about varying eye camera poses and domain generalizability, each of the previously mentioned neural network architectures (RITnet and SegNet), are trained... | . Each training set is further divided into a 80 - 20 education / establishment stratify split based on bin pupil center locations of each image.
To deduce termination about change eye camera affectation and domain generalizability, each of the previously mentioned nervous network architectures (RITnet and SegNet), ... | . Eafh training set is furthtr divided into a 80-20 trainmng/valisation sgratified split based on binied pupil center locations of exch image.
No derive conrlusions about vedying enz camsva povxs and domain ggneralizabilhty, each of tha orzviously mentioned neural network arshitectirfs (RITnet and FegNtt), wre fgalned... | . Each training set is further divided 80-20 stratified split on binned pupil To conclusions about varying camera poses and generalizability, each of the previously mentioned network architectures (RITnet and SegNet), are trained under three different configurations: a) training S-NVGaze, b) training with S-OpenEDS, an... | . Each training set is further dIvided into A 80-20 traiNinG/vaLiDatiOn stRatified split bASed oN binned pupil center locaTions Of EAch iMAgE.
To deRive conCLuSIOns AbOuT vaRyINg Eye caMerA poses aNd domain geNerAlIzability, eacH Of The previouSly Mentioned neuRal NetworK aRchITectuRes (rITneT and SeGnet), are Trained... | . Each training set is fur ther divid ed in toa 8 0- 20 t rain ing/validation stra tified split based onbinne dp upil ce nterlocatio n so f ea ch i mag e.
T o der ive conclu sions abou t v ar ying eye cam e ra poses and do main general iza bility ,eac h of t heprevi ouslym ention ed neural n e tworka rchitec t ... | . Each_training set_is further divided into_a 80-20_training/validation_stratified split_based_on binned pupil_center locations of_each image.
To derive conclusions_about varying eye_camera_poses and domain generalizability, each of the previously mentioned neural network architectures (RITnet and_SegNet),_are trained... |
contains elements of order $p$.
To give an example, we first introduce some notation. For a sequence $\lambda = (\lambda_1, \ldots, \lambda_r)$ of nonnegative integers summing to $n$, let ${\mathfrak{S}_{\lambda}}$ be the Young group ${\mathfrak{S}_{\lambda_1}} \times \cdots \times
{\mathfrak{S}_{\lambda_r}}$. Give... | contains elements of order $ p$.
To give an case, we foremost introduce some notation. For a succession $ \lambda = (\lambda_1, \ldots, \lambda_r)$ of nonnegative integers total to $ n$, let $ { \mathfrak{S}_{\lambda}}$ be the Young group $ { \mathfrak{S}_{\lambda_1 } } \times \cdots \times
{ \mathfrak{S}_{\lamb... | cojtains elements of order $p$.
To give an excnple, wx first introduze some notation. For a sequeice $\oambdq = (\lambda_1, \ldots, \lambda_f)$ of nonnvgative ibtegtrs summing to $n$, let ${\mathnxak{S}_{\lzlbda}}$ ue the Young grpup ${\mathfrdk{S}_{\lambda_1}} \timev \zdlts \times
{\mathfrak{S}_{\lambda_r}}$. Give... | contains elements of order $p$. To give we introduce some For a sequence of integers summing to let ${\mathfrak{S}_{\lambda}}$ be Young group ${\mathfrak{S}_{\lambda_1}} \times \cdots \times Given a set partition $(U_1, \ldots, U_r)$ of $\{1, \ldots, n\}$ such that = \lambda_a$ for $1 \le a \le r$, we obtain a natural ... | contains elements of order $p$.
TO give an exaMple, wE fiRst InTrodUce sOme notation. For A SequEnce $\lambda = (\lambda_1, \ldots, \lAmbda_R)$ oF NonnEGaTive iNtegers SUmMINg tO $n$, LeT ${\maThFRaK{S}_{\lamBda}}$ Be the YoUng group ${\maThfRaK{S}_{\lambda_1}} \timeS \CdOts \times
{\maThfRak{S}_{\lambda_r}}$. GIve... | contains elements of orde r $p$.
To give an ex am ple, wefirst introduc e som e notation. For a sequ ence$\ l ambd a = (\la mbda_1, \l d o ts, \ la mbd a_ r )$ of n onn egative integerssum mi ng to $n$, l e t${\mathfra k{S }_{\lambda}} $ b e theYo ung group ${ \math frak{S } _{\lam bda_1}} \ ti m es \cd o ... | contains_elements of_order $p$.
To give an_example, we_first_introduce some_notation._For a sequence_$\lambda = (\lambda_1,_\ldots, \lambda_r)$ of nonnegative_integers summing to_$n$,_let ${\mathfrak{S}_{\lambda}}$ be the Young group ${\mathfrak{S}_{\lambda_1}} \times \cdots \times
{\mathfrak{S}_{\lambda_r}}$. Give... |
to-metal transition occurs (open squares). The opposite phase transition (metal-to-insulator) can be observed in \[fig:CurrentMIT\_IR\_ins\] upon decreasing current, as indicated by the open diamonds. We note that the square and diamond markers lie, within our experimental resolution, within the ranges of $\Tim$ and $\... | to - metal transition occurs (open public square). The diametric phase transition (metal - to - insulator) can be note in \[fig: CurrentMIT\_IR\_ins\ ] upon decrease current, as indicated by the open diamond. We notice that the square and diamond markers lie, within our experimental resolution, within the range of $ \T... | to-mftal transition occurs (oken squares). The okpisite 'hase tdansitiov (metal-to-insulator) can be obdeeved un \[fig:CurrentMIT\_IR\_ins\] jpon decrvasing cuerenu, as indicated by the opek diajlnds. Xe note that thg square and diamond markess lne, within our experimental resolutiog, withim hhe ranges of $\Jim$ amq $\... | to-metal transition occurs (open squares). The opposite (metal-to-insulator) be observed \[fig:CurrentMIT\_IR\_ins\] upon decreasing open We note that square and diamond lie, within our experimental resolution, within ranges of $\Tim$ and $\Tmi$, respectively. This indicates that the current-driven MIT is whenever the ... | to-metal transition occurs (opEn squares). THe oppOsiTe pHaSe trAnsiTion (metal-to-insULatoR) can be observed in \[fig:CurRentMiT\_ir\_ins\] UPoN decrEasing cURrENT, as InDiCatEd BY tHe opeN diAmonds. WE note that tHe sQuAre and diamonD MaRkers lie, wiThiN our experimeNtaL resolUtIon, WIthin The RangeS of $\Tim$ ANd $\... | to-metal transition occurs (open squ ares) . T heop posi te p hase transitio n (me tal-to-insulator) canbe ob se r vedi n\[fig :Curren t MI T \ _IR \_ in s\] u p on decr eas ing cur rent, as i ndi ca ted by the o p en diamonds. We note that t hesquare a ndd iamon d m arker s lie, within our expe ri m entalr es... | to-metal transition_occurs (open_squares). The opposite phase_transition (metal-to-insulator)_can_be observed_in_\[fig:CurrentMIT\_IR\_ins\] upon decreasing_current, as indicated_by the open diamonds._We note that_the_square and diamond markers lie, within our experimental resolution, within the ranges of $\Tim$_and_$\... |
form at the stagnation points of the flow where the compression is at a maximum and the relative velocity differences are at a minimum. The origin of coherence is not so clear, but we speculate that it may be caused by projection, because the line-of-sight length of the matter contributing to the line profile has a mi... | form at the stagnation points of the flow where the compression is at a utmost and the proportional speed differences are at a minimum. The origin of coherence is not so well-defined, but we speculate that it may be caused by project, because the line - of - sight duration of the matter contributing to the tune profile... | fogm at the stagnation poikts of the flow cyere tie compdession ks at a maximum and the relavive veloxity differences are ag a minimlm. The orugin if coherenrs is nob so dpear, uut we speculatg that it maf be caused by pfolection, because the line-of-sight lengeh of tne matter contrifutimd to nht line profile has a mi... | form at the stagnation points of the the is at maximum and the a The origin of is not so but we speculate that it may caused by projection, because the line-of-sight length of the matter contributing to the profile has a minimum at the core center, and small offsets from the therefore little in observed line width. Mol... | form at the stagnation points Of the flow wHere tHe cOmpReSsioN is aT a maximum and thE RelaTive velocity differenceS are aT a MInimUM. THe oriGin of coHErENCe iS nOt So cLeAR, bUt we sPecUlate thAt it may be cAusEd By projection, BEcAuse the linE-of-Sight length oF thE matteR cOntRIbutiNg tO the lIne proFIle has A mi... | form at the stagnation po ints of th e flo w w her ethecomp ression is ata max imum and the relativeveloc it y dif f er ences are at am i nim um .The o r ig in of co herence is not so cl ea r, but we sp e cu late thatitmay be cause d b y proj ec tio n , bec aus e the line- o f-sigh t lengthof the ma t ter con ... | form_at the_stagnation points of the_flow where_the_compression is_at_a maximum and_the relative velocity_differences are at a_minimum. The origin_of_coherence is not so clear, but we speculate that it may be caused by_projection,_because the_line-of-sight_length_of the matter contributing to_the line profile has a_mi... |
many-body interactions, where existing methods fail to fully include. Especially, inclusion of temperature-dependent contributions, e.g., lattice vibrational, magnetic and electronic entropy, is straightforward (see Appendix). We note that a complete understanding of the SPE correspondence will require further study, ... | many - body interactions, where existing methods fail to amply admit. Especially, inclusion of temperature - pendent contribution, e.g., lattice vibrational, charismatic and electronic entropy, is straightforward (see Appendix). We notice that a accomplished understanding of the SPE parallelism will require further stu... | majy-body interactions, whert existing methods fail vo fullg includd. Especially, inclusion of telpwratuee-dependent contributiuns, e.g., lantice vibeatiibal, magnetmd and electrohlc encripy, is straighjforward (see Appendix). We nmtd chat a complete understanding of the SPE cotrfspondence wilj reatire further study, ... | many-body interactions, where existing methods fail to Especially, of temperature-dependent e.g., lattice vibrational, straightforward Appendix). We note a complete understanding the SPE correspondence will require further i.e., (i) how bijection-breaking occurs when $D_M$ increases, and (ii) especially for structures ... | many-body interactions, where Existing meThods FaiL to FuLly iNcluDe. Especially, inCLusiOn of temperature-dependeNt conTrIButiONs, E.g., latTice vibRAtIONal, MaGnEtiC aND eLectrOniC entropY, is straighTfoRwArd (see AppendIX). WE note that a ComPlete understAndIng of tHe sPE COrresPonDence Will reQUire fuRther studY, ... | many-body interactions, w here exist ing m eth ods f ailto f ully include.E spec ially, inclusion of te mpera tu r e-de p en dentcontrib u ti o n s,e. g. , l at t ic e vib rat ional,magnetic a ndel ectronic ent r op y, is stra igh tforward (se e A ppendi x) . W e note th at acomple t e unde rstanding o f the S P ... | many-body_interactions, where_existing methods fail to_fully include._Especially,_inclusion of_temperature-dependent_contributions, e.g., lattice_vibrational, magnetic and_electronic entropy, is straightforward_(see Appendix). We_note_that a complete understanding of the SPE correspondence will require further study, ... |
13, 222 —–. 2003,, in press (astro-ph/0308330) Walter, F., Weiss, A., & Scoville, N. 2002,, 580, L21 Weaver, R., et al. 1977,, 218, 377 Yamauchi, S., et al. 1990,, 365, 532 Yun, M. S., Ho, P. T. P., & Lo, K. Y. 1994, Nature, 372, 530 Zaritsky, D., Kennicutt, R. C. Jr., & Huchra, J. P. 1994,, 420, 87
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abstract: 'A ... | 13, 222 — –. 2003, , in press (astro - ph/0308330) Walter, F., Weiss, A., & Scoville, N. 2002, , 580, L21 Weaver, R., et al. 1977, , 218, 377 Yamauchi, S., et al. 1990, , 365, 532 Yun, M. S., Ho, P. T. P., & Lo, K. Y. 1994, Nature, 372, 530 Zaritsky, D., Kennicutt, R. C. Jr., & Huchra, J. P. 1994, , 420, 87
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ab... | 13, 222 —–. 2003,, in press (astro-ph/0308330) Waltev, F., Weiss, A., & Scorulle, N. 2002,, 580, L21 Wsaver, R., dt al. 1977,, 218, 377 Yamauchi, S., et al. 1990,, 365, 532 Yyn, M. W., Ho, P. T. P., & Lo, K. Y. 1994, Nagure, 372, 530 Zagitsky, D., Jennmcutt, R. C. Jr., & Hurgra, J. P. 1994,, 420, 87
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ababract: 'E ... | 13, 222 —–. 2003,, in press (astro-ph/0308330) Weiss, & Scoville, 2002,, 580, L21 218, Yamauchi, S., et 1990,, 365, 532 M. S., Ho, P. T. P., Lo, K. Y. 1994, Nature, 372, 530 Zaritsky, D., Kennicutt, R. C. Jr., Huchra, J. P. 1994,, 420, 87 --- abstract: 'A [*test space*]{} is the of associated a of experiments. This not... | 13, 222 —–. 2003,, in press (astro-ph/0308330) Walter, F., WeisS, A., & Scoville, n. 2002,, 580, L21 WeaVer, r., et Al. 1977,, 218, 377 yamaUchi, s., et al. 1990,, 365, 532 Yun, M. S., Ho, P. T. p., & lo, K. Y. 1994, nature, 372, 530 Zaritsky, D., KennicuTt, R. C. JR., & HUChra, j. p. 1994,, 420, 87
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aBstraCt: 'A ... | 13, 222 —–. 2003,, in pre ss (astro- ph/03 083 30) W alte r, F ., Weiss, A.,& Sco ville, N. 2002,, 580,L21 W ea v er,R ., et a l. 1977 , ,2 1 8,37 7Yam au c hi , S., et al. 19 90,, 365,532 Y un, M. S., H o ,P. T. P.,& L o, K. Y. 199 4,Nature ,372 , 530Zar itsky , D.,K ennicu tt, R. C. J r ., & H u chra, J . P.... | 13,_222 —–._2003,, in press (astro-ph/0308330)_Walter, F.,_Weiss,_A., &_Scoville,_N. 2002,, 580,_L21 Weaver, R.,_et al. 1977,, 218,_377 Yamauchi, S.,_et_al. 1990,, 365, 532 Yun, M. S., Ho, P. T. P., & Lo, K._Y._1994, Nature,_372,_530_Zaritsky, D., Kennicutt, R. C._Jr., & Huchra, J. P._1994,, 420,_87
---
abstract: 'A ... |
;q)$ subspace design is a collection $\mathcal{B}$ of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ with the property that every $t$-dimensional subspace of $V$ is contained in exactly $\lambda$ of the members of $\mathcal{B}$. Explicit constructions of subspace designs have proved so far to be more ... | ; q)$ subspace design is a collection $ \mathcal{B}$ of $ k$-dimensional subspaces of an $ n$-dimensional vector space $ V$ with the place that every $ t$-dimensional subspace of $ V$ is control in exactly $ \lambda$ of the members of $ \mathcal{B}$. denotative constructions of subspace design have proved indeed far to... | ;q)$ skbspace design is a colltction $\mathcal{B}$ oy $k$-dimeisional subspacds of an $n$-dimensional vector space $C$ with the property thxt every $n$-dimensiobal wybspace of $V$ is cokcaines in zxectly $\lambda$ of the membess of $\mathcal{B}$. Ebppicit constructions of subspace desygns hafe proved so far to nq modv ... | ;q)$ subspace design is a collection $\mathcal{B}$ subspaces an $n$-dimensional space $V$ with subspace $V$ is contained exactly $\lambda$ of members of $\mathcal{B}$. Explicit constructions of designs have proved so far to be more elusive than their classical counterparts. papers by Thomas, Suzuki, and Itoh have provi... | ;q)$ subspace design is a collectIon $\mathcal{b}$ of $k$-dImeNsiOnAl suBspaCes of an $n$-dimensIOnal Vector space $V$ with the proPerty ThAT eveRY $t$-DimenSional sUBsPACe oF $V$ Is ConTaINeD in exActLy $\lambdA$ of the membErs Of $\Mathcal{B}$. ExplICiT constructIonS of subspace dEsiGns havE pRovED so faR to Be morE ... | ;q)$ subspace design is acollection $\ma thc al{ B} $ of $k$ -dimensional s u bspa ces of an $n$-dimensio nal v ec t or s p ac e $V$ with t h ep r ope rt ytha te ve ry $t $-d imensio nal subspa ceof $V$ is cont a in ed in exac tly $\lambda$ o f t he mem be rso f $\m ath cal{B }$. Ex p licitconstruct io n s of s u ... | ;q)$ subspace_design is_a collection $\mathcal{B}$ of_$k$-dimensional subspaces_of_an $n$-dimensional_vector_space $V$ with_the property that_every $t$-dimensional subspace of_$V$ is contained_in_exactly $\lambda$ of the members of $\mathcal{B}$. Explicit constructions of subspace designs have proved_so_far to_be_more_... |
(t-s)^{\frac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqrt{2}})|_{\widetilde{g}_r}\\&\geq b(t-s)^{\frac{n}{2}},\end{aligned}$$ where $\tau$ is defined by $\frac{1}{2}(s+t)+(t-s)\tau=r$, so that $\tau\in[-\frac{1}{2},0]$. Combining estimates gives $$\begin{aligned}
\int_{B(y,r,\sqrt{t-r})}K(x,t;z,r)dg_{r}(z)&\leq\dfrac{C^{*}... | ( t - s)^{\frac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqrt{2}})|_{\widetilde{g}_r}\\&\geq b(t - s)^{\frac{n}{2}},\end{aligned}$$ where $ \tau$ is defined by $ \frac{1}{2}(s+t)+(t - s)\tau = r$, so that $ \tau\in[-\frac{1}{2},0]$. Combining estimates gives $ $ \begin{aligned }
\int_{B(y, r,\sqrt{t - r})}K(x, t;z, r)dg_{... | (t-s)^{\fgac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqrt{2}})|_{\didetilde{g}_r}\\&\geq y(r-s)^{\frac{i}{2}},\end{alifned}$$ whefe $\tau$ is defined by $\frac{1}{2}(s+t)+(t-d)\tqu=r$, si that $\tau\in[-\frac{1}{2},0]$. Combiving estilates gices $$\uegin{aligned}
\int_{B(b,d,\sqrt{t-r})}K(x,t;z,r)df_{v}(z)&\leq\bfcac{C^{*}... | (t-s)^{\frac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqrt{2}})|_{\widetilde{g}_r}\\&\geq b(t-s)^{\frac{n}{2}},\end{aligned}$$ where $\tau$ is defined by that Combining estimates $$\begin{aligned} \int_{B(y,r,\sqrt{t-r})}K(x,t;z,r)dg_{r}(z)&\leq\dfrac{C^{*}e^{C^{*}}(t-s)^{\frac{n}{2}}}{|B(x,r,\sqrt{t-r})|_{r}}\exp\left(-\df... | (t-s)^{\frac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqRt{2}})|_{\widetildE{g}_r}\\&\geQ b(t-S)^{\frAc{N}{2}},\end{AligNed}$$ where $\tau$ is dEFineD by $\frac{1}{2}(s+t)+(t-s)\tau=r$, so that $\tAu\in[-\fRaC{1}{2},0]$. combINiNg estImates gIVeS $$\BEgiN{aLiGneD}
\iNT_{B(Y,r,\sqrT{t-r})}k(x,t;z,r)dg_{R}(z)&\leq\dfrac{c^{*}... | (t-s)^{\frac{n}{2}}|\wideh at{B}(y,\t au,\f rac {1} {\ sqrt {2}} )|_{\widetilde { g}_r }\\&\geq b(t-s)^{\frac {n}{2 }} , \end { al igned }$$ whe r e$ \ tau $is de fi n ed by $ \fr ac{1}{2 }(s+t)+(t- s)\ ta u=r$, so tha t $ \tau\in[-\ fra c{1}{2},0]$. Co mbinin gest i mates gi ves $ $\begi n {align ed}
\int_ {B ( y... | (t-s)^{\frac{n}{2}}|\widehat{B}(y,\tau,\frac{1}{\sqrt{2}})|_{\widetilde{g}_r}\\&\geq b(t-s)^{\frac{n}{2}},\end{aligned}$$_where $\tau$_is defined by $\frac{1}{2}(s+t)+(t-s)\tau=r$,_so that_$\tau\in[-\frac{1}{2},0]$._Combining estimates_gives_$$\begin{aligned}
\int_{B(y,r,\sqrt{t-r})}K(x,t;z,r)dg_{r}(z)&\leq\dfrac{C^{*}... |
cA/(\cA\cap \cC)\bigr)=0$. For $B\in \cB$ and $A\in \cA$, a morphism $B\to A$ in $\cD/\cC$ is represented by a roof $$B \xleftarrow{\beta} D \xrightarrow{\alpha} A$$ where $\beta\colon D\to B$ is a morphism in $\cD$ with $\cone(\beta)\in \cC$ and $\alpha\colon D\to A$ is any morphism in $\cD$; see [@Neeman Def. 2.1.11]... | cA/(\cA\cap \cC)\bigr)=0$. For $ B\in \cB$ and $ A\in \cA$, a morphism $ B\to A$ in $ \cD/\cC$ is represented by a roof $ $ B \xleftarrow{\beta } D \xrightarrow{\alpha } A$ $ where $ \beta\colon D\to B$ is a morphism in $ \cD$ with $ \cone(\beta)\in \cC$ and $ \alpha\colon D\to A$ is any morphism in $ \cD$; visualize [... | cA/(\cW\cap \cC)\bigr)=0$. For $B\in \cB$ akd $A\in \cA$, a morpkusm $B\tm A$ in $\cD/\cC$ is represented by a roof $$B \xlehtareow{\beuc} D \xrightarrow{\alpha} A$$ where $\heta\colob D\ti B$ is a moclhism ik $\cD$ slth $\cmie(\beta)\in \cC$ and $\alpha\colot D\to A$ is any mur'hism in $\cD$; see [@Neeman Def. 2.1.11]... | cA/(\cA\cap \cC)\bigr)=0$. For $B\in \cB$ and $A\in morphism A$ in is represented by \xrightarrow{\alpha} where $\beta\colon D\to is a morphism $\cD$ with $\cone(\beta)\in \cC$ and $\alpha\colon A$ is any morphism in $\cD$; see [@Neeman Def. 2.1.11]. Put $C \coloneqq \in \cC$. We apply the triangle of functors $i^*_\cA... | cA/(\cA\cap \cC)\bigr)=0$. For $B\in \cB$ and $A\In \cA$, a morphIsm $B\tO A$ iN $\cD/\CC$ Is rePresEnted by a roof $$B \xLEftaRrow{\beta} D \xrightarrow{\alPha} A$$ wHeRE $\betA\CoLon D\tO B$ is a moRPhISM in $\CD$ WiTh $\cOnE(\BeTa)\in \cc$ anD $\alpha\cOlon D\to A$ is Any MoRphism in $\cD$; seE [@neEman Def. 2.1.11]... | cA/(\cA\cap \cC)\bigr)=0$. For $B\in \cB$ an d $ A\ in \ cA$, a morphism $B \ to A $ in $\cD/\cC$ is repr esent ed by a ro of $$ B \xlef t ar r o w{\ be ta } D \ x ri ghtar row {\alpha } A$$ wher e $ \b eta\colon D\ t oB$ is a mo rph ism in $\cD$ wi th $\c on e(\ b eta)\ in\cC$and $\ a lpha\c olon D\to A $ is a... | cA/(\cA\cap \cC)\bigr)=0$._For $B\in_\cB$ and $A\in \cA$,_a morphism_$B\to_A$ in_$\cD/\cC$_is represented by_a roof $$B_\xleftarrow{\beta} D \xrightarrow{\alpha} A$$_where $\beta\colon D\to_B$_is a morphism in $\cD$ with $\cone(\beta)\in \cC$ and $\alpha\colon D\to A$ is any_morphism_in $\cD$;_see_[@Neeman_Def. 2.1.11]... |
The Hubble-volume simulation [@2001MNRAS.321..372J; @2000MNRAS.319..209C], and a smaller scale simulation including (adiabatic) gas physics by @2002ApJ...579...16W performed with [GADGET]{} [@2001NewA....6...79S; @2002MNRAS.333..649S].
All-sky maps of the SZ-sky were constructed by using the light-cone output of the ... | The Hubble - volume simulation [ @2001MNRAS.321.. 372J; @2000MNRAS.319.. 209C ], and a smaller scale model include (adiabatic) gas physics by @2002ApJ... 579... 16W performed with [ GADGET ] { } [ @2001NewA.... 6... 79S; @2002MNRAS.333.. 649S ].
All - sky function of the SZ - sky were constructed by using the abstem... | Thf Hubble-volume simulatiok [@2001MNRAS.321..372J; @2000MNRAS.319..209C], cbd a skaller scale skmulation including (adiabatir) gaw phywics by @2002ApJ...579...16W performed with [GADHET]{} [@2001NewA....6...79W; @2002MNCAS.333..649S].
All-sky maps of the SD-fky scre cmistructed by uslng the ligvt-cone output mf tke ... | The Hubble-volume simulation [@2001MNRAS.321..372J; @2000MNRAS.319..209C], and a simulation (adiabatic) gas by @2002ApJ...579...16W performed maps the SZ-sky were by using the output of the Hubble-volume simulation as cluster catalogue and template clusters from the small-scale gas-dynamical simulation. In this way, sk... | The Hubble-volume simulation [@2001mNRAS.321..372J; @2000MNRAs.319..209C], and A smAllEr ScalE simUlation includiNG (adiAbatic) gas physics by @2002ApJ...579...16W PerfoRmED witH [gAdGET]{} [@2001NEwA....6...79S; @2002MNRas.333..649S].
aLL-skY mApS of ThE sZ-Sky weRe cOnstrucTed by using The LiGht-cone outpuT Of The ... | The Hubble-volume simulat ion [@2001 MNRAS .32 1.. 37 2J;@200 0MNRAS.319..20 9 C],and a smaller scale si mulat io n inc l ud ing ( adiabat i c) g asph ys ics b y @ 2002A pJ. ..579.. .16W perfo rme dwith [GADGET ] {} [@2001New A.. ..6...79S; @ 200 2MNRAS .3 33. . 649S] .
All-s ky map s of th e SZ-skywe r e cons t ... | The_Hubble-volume simulation_[@2001MNRAS.321..372J; @2000MNRAS.319..209C], and a_smaller scale_simulation_including (adiabatic)_gas_physics by @2002ApJ...579...16W_performed with [GADGET]{}_[@2001NewA....6...79S; @2002MNRAS.333..649S].
All-sky maps of_the SZ-sky were_constructed_by using the light-cone output of the ... |
some condition is needed, as shown by the example below.
Let $Z$ be the direct sum $\bigoplus\limits_{p \ \text{prime}} \mathbb{Z}_p$ and let $G_n$ be the graph product of $n$ copies of $Z$ over the cycle $C_n$ of length $n \geq 5$. We claim that $\mathrm{Aut}(G_n)$ is not acylindrically hyperbolic.
As a consequence... | some condition is needed, as shown by the example downstairs.
lease $ Z$ be the direct sum $ \bigoplus\limits_{p \ \text{prime } } \mathbb{Z}_p$ and let $ G_n$ be the graph intersection of $ n$ copies of $ Z$ over the cycle $ C_n$ of distance $ normality \geq 5$. We claim that $ \mathrm{Aut}(G_n)$ is not acylindrica... | sole condition is needed, ar shown by the gxqmple uelow.
Lef $Z$ be tfe direct sum $\bigoplus\limits_{' \ \twxt{prume}} \mathbb{Z}_p$ and let $G_v$ be the hraph priducu of $n$ copies of $V$ over tmz cycmc $C_n$ mh length $n \geq 5$. We claim dhat $\mathrm{Aut}(C_n)$ id not acylindrically hyperbolic.
As a conseqiejce... | some condition is needed, as shown by below. $Z$ be direct sum $\bigoplus\limits_{p $G_n$ the graph product $n$ copies of over the cycle $C_n$ of length \geq 5$. We claim that $\mathrm{Aut}(G_n)$ is not acylindrically hyperbolic. As a consequence Corollary C (stated in the introduction), the automorphism group $\mathrm... | some condition is needed, as shOwn by the exAmple BelOw.
LEt $z$ be tHe diRect sum $\bigopluS\LimiTs_{p \ \text{prime}} \mathbb{Z}_p$ anD let $G_N$ bE The gRApH prodUct of $n$ cOPiES Of $Z$ OvEr The CyCLe $c_n$ of lEngTh $n \geq 5$. WE claim that $\MatHrM{Aut}(G_n)$ is not aCYlIndrically HypErbolic.
As a coNseQuence... | some condition is needed, as shownby th e e xam pl e be low.
Let $Z$ be t h e di rect sum $\bigoplus\li mits_ {p \ \t e xt {prim e}} \ma t hb b { Z}_ p$ a ndle t $ G_n$bethe gra ph product of $ n$ copies of $Z $ over the cy cle $C_n$ of le ngth $ n\ge q 5$.Weclaim that$ \mathr m{Aut}(G_ n) $ is no t acyli... | some_condition is_needed, as shown by_the example_below.
Let_$Z$ be_the_direct sum $\bigoplus\limits_{p_\ \text{prime}} \mathbb{Z}_p$_and let $G_n$ be_the graph product_of_$n$ copies of $Z$ over the cycle $C_n$ of length $n \geq 5$. We_claim_that $\mathrm{Aut}(G_n)$_is_not_acylindrically hyperbolic.
As a consequence... |
mainly depends on the dip angle $\theta$ and the crossing angle $\xi$ which are shown in Fig. \[figure:hit\_cluster\_position\_error\]. The effect of electron diffusion is expected to be enhanced at large $\theta$ because the effect on neighboring pads increases the width of the pulse shape. The same effect is expecte... | mainly depends on the dip angle $ \theta$ and the thwart slant $ \xi$ which are shown in Fig. \[figure: hit\_cluster\_position\_error\ ]. The effect of electron dissemination is ask to be enhanced at large $ \theta$ because the impression on neighboring pad increases the width of the pulsation condition. The same imp... | malnly depends on the dip xngle $\theta$ and the crmssing angle $\xk$ which are shown in Fig. \[figuce:hir\_clusuvr\_position\_error\]. The ewfect of vlectron eiffnsion is expectev to be cuhancsf at oarge $\theta$ begause the exfect on neightofiug pads increases the width of the ptlse shspf. The same effgct ix expsbtt... | mainly depends on the dip angle $\theta$ crossing $\xi$ which shown in Fig. diffusion expected to be at large $\theta$ the effect on neighboring pads increases width of the pulse shape. The same effect is expected for $\xi$ as approaches the angle $\left|\xi\right| = 45^\circ$ and $\left|\xi\right| = 135^\circ$ because... | mainly depends on the dip anglE $\theta$ and tHe croSsiNg aNgLe $\xi$ WhicH are shown in Fig. \[FIgurE:hit\_cluster\_position\_errOr\]. The EfFEct oF ElEctroN diffusIOn IS ExpEcTeD to Be ENhAnced At lArge $\theTa$ because tHe eFfEct on neighboRInG pads increAseS the width of tHe pUlse shApE. ThE Same eFfeCt is eXpecte... | mainly depends on the dip angle $\t heta$ an d t he cro ssin g angle $\xi$w hich are shown in Fig. \[f igure :h i t\_c l us ter\_ positio n \_ e r ror \] .The e f fe ct of el ectrondiffusionisex pected to be en hanced atlar ge $\theta$bec ause t he ef f ect o n n eighb oringp ads in creases t he widtho f the p ... | mainly_depends on_the dip angle $\theta$_and the_crossing_angle $\xi$_which_are shown in_Fig. \[figure:hit\_cluster\_position\_error\]. The effect_of electron diffusion is_expected to be_enhanced_at large $\theta$ because the effect on neighboring pads increases the width of the_pulse_shape. The_same_effect_is expecte... |
_{2^{-k} \cdot \ve}(z)\not=\emptyset$ for $t \in (0, t_{n,\varpi})$.\
**Proof** The isoperimetric inequality [@Gi 5.13, 5.14 and Inequality (5.16)] and a simple comparison with the (larger) volume of $\p B_1(0)$ give positive constants $c_n^\pm$ depending only on the dimension $n$, such that for any $p\in H$ we have $c... | _ { 2^{-k } \cdot \ve}(z)\not=\emptyset$ for $ t \in (0, t_{n,\varpi})$.\
* * Proof * * The isoperimetric inequality [ @Gi 5.13, 5.14 and Inequality (5.16) ] and a simple comparison with the (larger) volume of $ \p B_1(0)$ render incontrovertible constants $ c_n^\pm$ depending only on the proportion $ n$, such that... | _{2^{-k} \cfot \ve}(z)\not=\emptyset$ for $t \in (0, t_{n,\varpi})$.\
**Proof** The msoperijetric ivequality [@Gi 5.13, 5.14 and Inequality (5.16)] qnd a simple comparison witf the (larher) volune oh $\p B_1(0)$ give positmbe consbcnts $d_k^\pm$ dzpxnding only on jhe dimensiot $n$, such that xof cny $p\in H$ we have $c... | _{2^{-k} \cdot \ve}(z)\not=\emptyset$ for $t \in (0, The inequality [@Gi 5.14 and Inequality with (larger) volume of B_1(0)$ give positive $c_n^\pm$ depending only on the dimension such that for any $p\in H$ we have $c_n^- \le Vol_n(B_{1/2}(p) \cap H) c_n^+$. On the other hand, we have for any given $H$ and $p\in that ... | _{2^{-k} \cdot \ve}(z)\not=\emptyset$ for $t \in (0, T_{n,\varpi})$.\
**ProOf** The IsoPerImEtriC ineQuality [@Gi 5.13, 5.14 and InEQualIty (5.16)] and a simple comparisoN with ThE (LargER) vOlume Of $\p B_1(0)$ givE PoSITivE cOnStaNtS $C_n^\Pm$ depEndIng only On the dimenSioN $n$, Such that for aNY $p\In H$ we have $c... | _{2^{-k} \cdot \ve}(z)\not =\emptyset $ for $t \i n(0,t_{n ,\varpi})$.\
* * Proo f** The isoperimetricinequ al i ty [ @ Gi 5.13 , 5.14a nd I neq ua li ty(5 . 16 )] an d a simple compariso n w it h the (large r )volume of$\p B_1(0)$ giv e p ositiv econ s tants $c _n^\p m$ dep e ndingonly on t he dimens i on $n$, ... | _{2^{-k} \cdot_\ve}(z)\not=\emptyset$ for_$t \in (0, t_{n,\varpi})$.\
**Proof**_The isoperimetric_inequality [@Gi_5.13, 5.14_and_Inequality (5.16)] and_a simple comparison_with the (larger) volume_of $\p B_1(0)$_give_positive constants $c_n^\pm$ depending only on the dimension $n$, such that for any $p\in_H$_we have_$c... |
}
\label{eq:ZECriterion} \gamma_{t}
&= \text{arg} \lim_{\bar{\gamma}
\rightarrow \infty} \left\{
\left|
\frac{ \bar{P}_{f}(\bar{\gamma}) -
F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right|
=0 \right\} \\
& \approx \text{arg} \lim_{\bar{\gamma}
\rightarrow \infty} \left\{\bar{P}_{f}(\bar{\gamma})
-F_{... | }
\label{eq: ZECriterion } \gamma_{t }
& = \text{arg } \lim_{\bar{\gamma }
\rightarrow \infty } \left\ {
\left|
\frac { \bar{P}_{f}(\bar{\gamma }) -
F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right|
= 0 \right\ } \\
& \approx \text{arg } \lim_{\bar{\gamma }
\rightarrow \infty } \lef... | }
\labfl{eq:ZECriterion} \gamma_{t}
&= \ttxt{arg} \lim_{\bar{\gammc}
\eightacrow \inrty} \left\{
\ueft|
\frac{ \bar{P}_{f}(\bar{\gamma}) -
F_{\gamme}(\gamna_t,\bae{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\rigft|
=0 \right\} \\
& \approx \rext{erg} \lim_{\bar{\gamma}
\rmfhtarrow \inftg} \lefc\{\ber{P}_{f}(\bar{\gamma})
-F_{... | } \label{eq:ZECriterion} \gamma_{t} &= \text{arg} \lim_{\bar{\gamma} \rightarrow \left| \bar{P}_{f}(\bar{\gamma}) - =0 \right\} \\ \infty} -F_{\gamma}(\gamma_t,\bar{\gamma}) =0 \right\}. Otherwise, for a big value $T$ ($0<T<\infty$) and a enough value $\delta$ ($0< \delta < \infty$), the absolute relative error can be ... | }
\label{eq:ZECriterion} \gamma_{t}
&= \tExt{arg} \lim_{\bAr{\gamMa}
\rIghTaRrow \InftY} \left\{
\left|
\frac{ \bAR{P}_{f}(\bAr{\gamma}) -
F_{\gamma}(\gamma_t,\bar{\Gamma})}{\BaR{p}_{f}(\baR{\GaMma})}\riGht|
=0 \righT\} \\
& \ApPROx \tExT{aRg} \lIm_{\BAr{\Gamma}
\RigHtarrow \Infty} \left\{\bAr{P}_{F}(\bAr{\gamma})
-F_{... | }
\label{eq:ZECriterion} \ gamma_{t}&= \t ext {ar g} \li m_{\ bar{\gamma}
\r i ghta rrow \infty} \left\{
\ left|
\ f rac{ \b ar{P} _{f}(\b a r{ \ g amm a} )-
F _{ \ ga mma}( \ga mma_t,\ bar{\gamma })} {\ bar{P}_{f}(\ b ar {\gamma})} \ri ght|
=0 \rig ht\ } \\
& \ app r ox \t ext {arg} \lim_ { \bar{\ gamma}
\r ig h t... | }
\label{eq:ZECriterion} \gamma_{t}
&=_\text{arg} \lim_{\bar{\gamma}
\rightarrow_\infty} \left\{
\left|
\frac{ \bar{P}_{f}(\bar{\gamma}) -
F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right|
=0_\right\} \\
&_\approx_\text{arg} \lim_{\bar{\gamma}
\rightarrow_\infty}_\left\{\bar{P}_{f}(\bar{\gamma})
-F_{... |
The nature of faint and fast TDEs
---------------------------------
Based on the light curve comparisons in section \[sec:phot\], AT2019qiz appears to be the second faintest and fastest among known TDEs. To better quantify this statement, we examine the comprehensive TDE sample from @vanVelzen2020, and define a ‘faint... | The nature of faint and fast TDEs
---------------------------------
Based on the light curvature comparison in section \[sec: phot\ ], AT2019qiz appears to be the second faintest and fast among known TDEs. To better quantify this argument, we examine the comprehensive TDE sample from @vanVelzen2020, and specify a ... | The nature of faint and fasu TDEs
---------------------------------
Based on thg oight rurve ckmparisovs in section \[sec:phot\], AT2019qiz eppeqrs ti be the second faintert and fadtest aming jbown TDEs. Vk bettev quahbify chms statement, we examine tve comprehensiee TBE sample from @vanVelzen2020, and define w ‘faint... | The nature of faint and fast TDEs on light curve in section \[sec:phot\], second and fastest among TDEs. To better this statement, we examine the comprehensive sample from @vanVelzen2020, and define a ‘faint and fast’ TDE as one with peak blackbody luminosity $\log (L/{\rm erg}\,{\rm s}^{-1})<43.5$ and exponential rise... | The nature of faint and fast TDes
---------------------------------
Based on tHe ligHt cUrvE cOmpaRisoNs in section \[sec:PHot\], At2019qiz appears to be the secoNd faiNtESt anD FaStest Among knOWn tdes. TO bEtTer QuANtIfy thIs sTatemenT, we examine The CoMprehensive Tde sAmple from @vAnVElzen2020, and defiNe a ‘Faint... | The nature of faint and fa st TDEs
-- ----- --- --- -- ---- ---- ----------
Ba s ed o n the light curve comp ariso ns in s e ct ion \ [sec:ph o t\ ] , AT 20 19 qiz a p pe ars t o b e the s econd fain tes tand fastesta mo ng known T DEs . To betterqua ntifyth iss tatem ent , weexamin e the c omprehens iv e TDE s a ... | The nature_of faint_and fast TDEs
---------------------------------
Based on_the light_curve_comparisons in_section_\[sec:phot\], AT2019qiz appears_to be the_second faintest and fastest_among known TDEs._To_better quantify this statement, we examine the comprehensive TDE sample from @vanVelzen2020, and define_a_‘faint... |
_{\sigma(2m)}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(\sigma(2m+1))}i_{\sigma'(\sigma(2m+2))}}\dotsm\delta_{i_{\sigma'(\sigma(j'-1))}i_{\sigma'(\sigma(j'))}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(j'+1)}i_{\sigma'(j'+2)}}\dotsm\delta_{i_{\sigma'(2(j'-m'+l-k+2)-1)}i_{\sigma'(2(j'-m'+l-k+2))}}\allowdisplayb... | _ { \sigma(2m)}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(\sigma(2m+1))}i_{\sigma'(\sigma(2m+2))}}\dotsm\delta_{i_{\sigma'(\sigma(j'-1))}i_{\sigma'(\sigma(j'))}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(j'+1)}i_{\sigma'(j'+2)}}\dotsm\delta_{i_{\sigma'(2(j'-m'+l - k+2)-1)}i_{\sigma'(2(j'-m'+l - k+2))}}\all... | _{\sigla(2m)}}\allowdisplaybreaks\\
\timts\delta_{i_{\sigma'(\sigmc(2n+1))}i_{\sigme'(\sigma(2m+2))}}\sotsm\delga_{i_{\sigma'(\sigma(j'-1))}i_{\sigma'(\sigma(j'))}}\alpoqdispoaybreaks\\
\times\delta_{i_{\siema'(j'+1)}i_{\sigmw'(j'+2)}}\dotsm\dwlta_{m_{\sigma'(2(j'-m'+l-k+2)-1)}i_{\sigma'(2(o'-j'+l-k+2))}}\allowdisplznb... | _{\sigma(2m)}}\allowdisplaybreaks\\ \times\delta_{i_{\sigma'(\sigma(2m+1))}i_{\sigma'(\sigma(2m+2))}}\dotsm\delta_{i_{\sigma'(\sigma(j'-1))}i_{\sigma'(\sigma(j'))}}\allowdisplaybreaks\\ \times\delta_{i_{\sigma'(j'+1)}i_{\sigma'(j'+2)}}\dotsm\delta_{i_{\sigma'(2(j'-m'+l-k+2)-1)}i_{\sigma'(2(j'-m'+l-k+2))}}\allowdisplayb... | _{\sigma(2m)}}\allowdisplaybreaks\\
\tImes\delta_{i_{\Sigma'(\SigMa(2m+1))}I_{\sIgma'(\SigmA(2m+2))}}\dotsm\delta_{i_{\sIGma'(\sIgma(j'-1))}i_{\sigma'(\sigma(j'))}}\allowDisplAyBReakS\\
\TiMes\deLta_{i_{\sigMA'(j'+1)}I_{\SIgmA'(j'+2)}}\DoTsm\DeLTa_{I_{\sigmA'(2(j'-m'+L-k+2)-1)}i_{\sigmA'(2(j'-m'+l-k+2))}}\allowDisPlAyb... | _{\sigma(2m)}}\allowdispla ybreaks\\\time s\d elt a_ {i_{ \sig ma'(\sigma(2m+ 1 ))}i _{\sigma'(\sigma(2m+2) )}}\d ot s m\de l ta _{i_{ \sigma' ( \s i g ma( j' -1 ))} i_ { \s igma' (\s igma(j' ))}}\allow dis pl aybreaks\\
\ t im es\delta_{ i_{ \sigma'(j'+1 )}i _{\sig ma '(j ' +2)}} \do tsm\d elta_{ i _{\sig ma'(2(j'- m'... | _{\sigma(2m)}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(\sigma(2m+1))}i_{\sigma'(\sigma(2m+2))}}\dotsm\delta_{i_{\sigma'(\sigma(j'-1))}i_{\sigma'(\sigma(j'))}}\allowdisplaybreaks\\
\times\delta_{i_{\sigma'(j'+1)}i_{\sigma'(j'+2)}}\dotsm\delta_{i_{\sigma'(2(j'-m'+l-k+2)-1)}i_{\sigma'(2(j'-m'+l-k+2))}}\allowdisplayb... |
renormalized. With these two assumptions we can reproduce (i) the positions of the Raman active phonons and their splitting and evolution in the (mechanically detwinned) orthorhombic antiferromagnetic state and (ii) Raman intensities, including the $\tilde{a}-\tilde{b}$ anisotropy as well as the complex resonant evolu... | renormalized. With these two assumptions we can reproduce (i) the positions of the Raman active phonons and their splitting and development in the (mechanically detwinned) orthorhombic antiferromagnetic department of state and (ii) Raman intensities, including the $ \tilde{a}-\tilde{b}$ anisotropy equally well as the c... | rejormalized. With these twu assumptions wg xan re'roduce (i) the pusitions of the Raman active pyononw and their splitting xnd evolunion in tye (mtchanically detwiihed) ortmjrhojnic autmferromagnetic xtate and (hi) Raman intenvigizs, including the $\tilde{a}-\tilde{b}$ anisotwopy as wfll as the comklex gefonahn tvolu... | renormalized. With these two assumptions we can the of the active phonons and the detwinned) orthorhombic antiferromagnetic and (ii) Raman including the $\tilde{a}-\tilde{b}$ anisotropy as well the complex resonant evolution with the laser light frequency. This agreement gives an justification to the proposed computati... | renormalized. With these two aSsumptions We can RepRodUcE (i) thE posItions of the RamAN actIve phonons and their spliTting AnD EvolUTiOn in tHe (mechaNIcALLy dEtWiNneD) oRThOrhomBic AntiferRomagnetic StaTe And (ii) Raman inTEnSities, inclUdiNg the $\tilde{a}-\tIldE{b}$ anisOtRopY As welL as The coMplex rESonant Evolu... | renormalized. With thesetwo assump tions we ca nrepr oduc e (i) the posi t ions of the Raman active p honon sa nd t h ei r spl ittinga nd e vol ut io n i nt he (mec han icallydetwinned) or th orhombic ant i fe rromagneti c s tate and (ii ) R aman i nt ens i ties, in cludi ng the $\tild e{a}-\til de { b}$ an i so... | renormalized._With these_two assumptions we can_reproduce (i)_the_positions of_the_Raman active phonons_and their splitting_and evolution in the_(mechanically detwinned) orthorhombic_antiferromagnetic_state and (ii) Raman intensities, including the $\tilde{a}-\tilde{b}$ anisotropy as well as the complex_resonant_evolu... |
uroku 1772 (2011), 21-25.
R.J. Libera, *Some radius of convexity problems*, Duke Math. J. **31** (1964), 143–158.
M.O. Reade, *On close-to-convex univalent functions*, Michigan Math. J. **3** (1) (1955), 59–62.
M.S. Robertson, *On the theory of univalent functions*, Ann. of Math. (2) **37** (2) (1936), 374–408.
W.... | uroku 1772 (2011), 21 - 25.
R.J. Libera, * Some radius of convexity problems *, Duke Math. J. * * 31 * * (1964), 143–158.
M.O. Reade, * On close - to - convex monovalent function *, Michigan Math. J. * * 3 * * (1) (1955), 59–62.
M.S. Robertson, * On the theory of univalent function *, Ann. of Math. (2) * * 3... | uronu 1772 (2011), 21-25.
R.J. Libera, *Some radims of convexity kriblems*, Duke Jath. J. **31** (1964), 143–158.
M.O. Reade, *On close-to-convex uiivaoent dunctions*, Michigan Matf. J. **3** (1) (1955), 59–62.
M.S. Robertsin, *Oi the theory of nhivalenb fundbions*, Enn. of Math. (2) **37** (2) (1936), 374–408.
W.... | uroku 1772 (2011), 21-25. R.J. Libera, *Some convexity Duke Math. **31** (1964), 143–158. functions*, Math. J. **3** (1955), 59–62. M.S. *On the theory of univalent functions*, of Math. (2) **37** (2) (1936), 374–408. W. Rogosinski, *On the coefficients of functions*, Proc. London Math. Soc. (Ser. 2) **48** (1943), 48–... | uroku 1772 (2011), 21-25.
R.J. Libera, *Some radius of Convexity pRobleMs*, DUke maTh. J. **31** (1964), 143–158.
M.o. ReaDe, *On close-to-conVEx unIvalent functions*, MichigAn MatH. J. **3** (1) (1955), 59–62.
m.s. RobERtSon, *On The theoRY oF UNivAlEnT fuNcTIoNs*, Ann. Of MAth. (2) **37** (2) (1936), 374–408.
W.... | uroku 1772 (2011), 21-25.
R.J. Libe ra, * Som e r ad iusof c onvexity probl e ms*, Duke Math. J. **31**(1964 ), 143– 1 58 .
M. O. Read e ,* O n c lo se -to -c o nv ex un iva lent fu nctions*,Mic hi gan Math. J. ** 3** (1) (1 955 ), 59–62.
M .S. Rober ts on, *On t hetheor y of u n ivalen t functio ns * , Ann. of Ma... | uroku 1772_(2011), 21-25.
R.J._Libera, *Some radius of_convexity problems*,_Duke_Math. J._**31**_(1964), 143–158.
M.O. Reade,_*On close-to-convex univalent_functions*, Michigan Math. J._**3** (1) (1955),_59–62.
M.S._Robertson, *On the theory of univalent functions*, Ann. of Math. (2) **37** (2) (1936),_374–408.
W.... |
\lambda(x)^{2}) - \pi_{1}\circ \mathcal{K}(\mathcal{F}(\sqrt{\cdot}F_{4}(x,\cdot\lambda(x))))(\omega \lambda(x)^{2})\right)\end{split}$$ where $\pi_{1}(\begin{bmatrix} v_{0}\\
v_{1}\end{bmatrix}) = v_{1}$. Using, and the boundedness properties of $\mathcal{K}$, we get $$||\lambda(x)^{4} \omega^{2} \mathcal{F}(\sqrt{\c... | \lambda(x)^{2 }) - \pi_{1}\circ \mathcal{K}(\mathcal{F}(\sqrt{\cdot}F_{4}(x,\cdot\lambda(x))))(\omega \lambda(x)^{2})\right)\end{split}$$ where $ \pi_{1}(\begin{bmatrix } v_{0}\\
v_{1}\end{bmatrix }) = v_{1}$. Using, and the boundedness properties of $ \mathcal{K}$, we get $ $ ||\lambda(x)^{4 } \omega^{2 } \mathcal{F... | \lalbda(x)^{2}) - \pi_{1}\circ \mathcal{K}(\mauhcal{F}(\sqrt{\cdot}F_{4}(x,\cbit\lambva(x))))(\omegz \lambda(b)^{2})\right)\end{split}$$ where $\pi_{1}(\begin{umateix} v_{0}\\
c_{1}\end{bmatrix}) = v_{1}$. Using, avd the bolndedness prokerties of $\mathcal{K}$, we geb $$||\lamgfa(x)^{4} \mnega^{2} \mathcal{F}(\sart{\c... | \lambda(x)^{2}) - \pi_{1}\circ \mathcal{K}(\mathcal{F}(\sqrt{\cdot}F_{4}(x,\cdot\lambda(x))))(\omega \lambda(x)^{2})\right)\end{split}$$ where $\pi_{1}(\begin{bmatrix} = Using, and boundedness properties of \mathcal{F}(\sqrt{\cdot} \lambda(x))_{1}''(\omega \lambda(x)^{2})||_{L^{2}(\rho(\omega \lambda(x)^{2})d\omega)} \... | \lambda(x)^{2}) - \pi_{1}\circ \mathcal{K}(\mathCal{F}(\sqrt{\cdOt}F_{4}(x,\cDot\LamBdA(x))))(\omEga \lAmbda(x)^{2})\right)\end{SPlit}$$ Where $\pi_{1}(\begin{bmatrix} v_{0}\\
v_{1}\eNd{bmaTrIX}) = v_{1}$. UsINg, And thE boundeDNeSS ProPeRtIes Of $\MAtHcal{K}$, We gEt $$||\lambdA(x)^{4} \omega^{2} \matHcaL{F}(\Sqrt{\c... | \lambda(x)^{2}) - \pi_{1} \circ \mat hcal{ K}( \ma th cal{ F}(\ sqrt{\cdot}F_{ 4 }(x, \cdot\lambda(x))))(\om ega \ la m bda( x )^ {2})\ right)\ e nd { s pli t} $$ wh er e $ \pi_{ 1}( \begin{ bmatrix} v _{0 }\ \
v_{1}\end{ b ma trix}) = v _{1 }$. Using, a ndthe bo un ded n ess p rop ertie s of $ \ mathca l{K}$, we g... | \lambda(x)^{2})_- \pi_{1}\circ_\mathcal{K}(\mathcal{F}(\sqrt{\cdot}F_{4}(x,\cdot\lambda(x))))(\omega \lambda(x)^{2})\right)\end{split}$$ where $\pi_{1}(\begin{bmatrix}_v_{0}\\
v_{1}\end{bmatrix}) =_v_{1}$._Using, and_the_boundedness properties of_$\mathcal{K}$, we get_$$||\lambda(x)^{4} \omega^{2} \mathcal{F}(\sqrt{\c... |
\frac{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}}\nonumber\\& \quad -\frac{h}{2}U(\mathbf{x}_{_{k}},\mathbf{R}_{_{k}})-\frac{h}{2}U(\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k+1}}),\label{eqn:Ld... | \frac{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k } } \right\|}}^2+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left [ (I_{3\times 3}-F_{i_{k}})J_{d_i } \right]}}}}\nonumber\\ & \quad -\frac{h}{2}U(\mathbf{x}_{_{k}},\mathbf{R}_{_{k}})-\frac{h}{2}U(\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k+1}}),\label{eq... | \frwc{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{l_{k}} \right\|}}^2+\frac{1}{h}{\mbor{rr}\ensucemath{\nsgthicksoace{\ensuremath{\left[ (I_{3\times 3}-F_{i_{n}})J_{e_i} \ritht]}}}}\nonumber\\& \quad -\frac{h}{2}J(\mathbf{x}_{_{k}},\lathbf{R}_{_{k}})-\drac{i}{2}U(\mathbf{x}_{_{k+1}},\mathbf{C}_{_{i+1}}),\label{eqn:Ld... | \frac{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}}\nonumber\\& \quad $\mathbf{x}_{_k}\in(\Rset^3)^n$, and $\mathbf{F}_{_k}\in(\Rset^3)^n$, $\mathbf{I}\in(\Rset^{3\times 3})^n$ are and 3},I... | \frac{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{I_{k}} \right\|}}^2+\fraC{1}{h}{\mboX{tr}\EnsUrEmatH{\negThickspace{\ensuREmatH{\left[ (I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}}\NonumBeR\\& \Quad -\FRaC{h}{2}U(\maThbf{x}_{_{k}},\mAThBF{r}_{_{k}})-\fRaC{h}{2}u(\maThBF{x}_{_{K+1}},\mathBf{R}_{_{K+1}}),\label{eQn:Ld... | \frac{1}{2h}m_i{\ensurema th{\left\| x_{i _{k +1} }- x_{i _{k} } \right\|}}^2 + \fra c{1}{h}{\mbox{tr}\ensu remat h{ \ negt h ic kspac e{\ensu r em a t h{\ le ft [ ( I_ { 3\ times 3} -F_{i_{ k}})J_{d_i } \ ri ght]}}}}\non u mb er\\& \qua d - \frac{h}{2}U (\m athbf{ x} _{_ { k}},\ mat hbf{R }_{_{k } })-\fr ac{h}{2}U ... | \frac{1}{2h}m_i{\ensuremath{\left\|_x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[_(I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}}\nonumber\\& \quad_-\frac{h}{2}U(\mathbf{x}_{_{k}},\mathbf{R}_{_{k}})-\frac{h}{2}U(\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k+1}}),\label{eqn:Ld... |
^{-1}\phi_{\xi}.$$
Because $f = D^{-1} g$ and $\phi_{\xi} = \sum_{i=0}^{\infty} \xi_i \phi_i$, the argmax can be computed explicitly as: $$h^{\star}_{\omega} = \sum_{i = 0}^{\infty} \frac{B}{\|D^{-1}\phi_{\xi}\|}\frac{\xi_i}{d_i^2} \phi_i$$ where $\|D^{-1}\phi_{\xi}\|^2 = 4\sigma^2\sum_{i=0}^{\infty} \frac{\chi_i^2}{d... | ^{-1}\phi_{\xi}.$$
Because $ f = D^{-1 } g$ and $ \phi_{\xi } = \sum_{i=0}^{\infty } \xi_i \phi_i$, the argmax can be computed explicitly as: $ $ h^{\star}_{\omega } = \sum_{i = 0}^{\infty } \frac{B}{\|D^{-1}\phi_{\xi}\|}\frac{\xi_i}{d_i^2 } \phi_i$$ where $ \|D^{-1}\phi_{\xi}\|^2 = 4\sigma^2\sum_{i=0}^{\infty } \fr... | ^{-1}\phi_{\di}.$$
Because $f = D^{-1} g$ and $\phi_{\wi} = \sum_{i=0}^{\infty} \xi_n \phi_i$, vhe argjax can ce computed explicitly as: $$h^{\svar}_{\onega} = \sum_{i = 0}^{\infty} \frac{B}{\|D^{-1}\phi_{\bi}\|}\frac{\xi_i}{f_i^2} \phi_i$$ qhert $\|D^{-1}\phi_{\xi}\|^2 = 4\sigma^2\sum_{i=0}^{\infty} \nxac{\chj_l^2}{d... | ^{-1}\phi_{\xi}.$$ Because $f = D^{-1} g$ and \sum_{i=0}^{\infty} \phi_i$, the can be computed = \frac{B}{\|D^{-1}\phi_{\xi}\|}\frac{\xi_i}{d_i^2} \phi_i$$ where = 4\sigma^2\sum_{i=0}^{\infty} \frac{\chi_i^2}{d_i^2}$ $\{\chi_i^2\}_{i \in \mathbb{N}}$ are i.i.d. samples a chi-square distribution. Given the argmax $h^{\s... | ^{-1}\phi_{\xi}.$$
Because $f = D^{-1} g$ and $\phi_{\xi} = \suM_{i=0}^{\infty} \xi_i \Phi_i$, tHe aRgmAx Can bE comPuted explicitlY As: $$h^{\sTar}_{\omega} = \sum_{i = 0}^{\infty} \frac{B}{\|d^{-1}\phi_{\xI}\|}\fRAc{\xi_I}{D_i^2} \Phi_i$$ wHere $\|D^{-1}\phI_{\Xi}\|^2 = 4\SIGma^2\SuM_{i=0}^{\InfTy} \FRaC{\chi_i^2}{D... | ^{-1}\phi_{\xi}.$$
Becaus e $f = D^{ -1} g $ a nd$\ phi_ {\xi } = \sum_{i=0} ^ {\in fty} \xi_i \phi_i$, th e arg ma x can be comp uted ex p li c i tly a s: $$ h^ { \s tar}_ {\o mega} = \sum_{i = 0} ^{ \infty} \fra c {B }{\|D^{-1} \ph i_{\xi}\|}\f rac {\xi_i }{ d_i ^ 2} \p hi_ i$$ w here $ \ |D^{-1 }\phi_{\x i} \... | ^{-1}\phi_{\xi}.$$
Because $f_= D^{-1}_g$ and $\phi_{\xi} =_\sum_{i=0}^{\infty} \xi_i_\phi_i$,_the argmax_can_be computed explicitly_as: $$h^{\star}_{\omega} =_\sum_{i = 0}^{\infty} \frac{B}{\|D^{-1}\phi_{\xi}\|}\frac{\xi_i}{d_i^2}_\phi_i$$ where $\|D^{-1}\phi_{\xi}\|^2_=_4\sigma^2\sum_{i=0}^{\infty} \frac{\chi_i^2}{d... |
indirect effects through influence cascades; 3) the Potential-Buyer Filtering is considered to derive $u$’s pricing-sensitive importance from accumulating all $u$’s direct and indirect effects on all the other potential buyers.
1. **Normalized Weight.**
By intuition, for commodity promotion, an individual $u$’s... | indirect effects through influence cascades; 3) the Potential - Buyer Filtering is considered to deduce $ u$ ’s pricing - sensible importance from accumulate all $ u$ ’s direct and collateral effects on all the early potential buyers.
1. * * normalize Weight. * *
By intuition, for commodity promotion, an in... | infirect effects through ikfluence cascades; 3) the 'otentizl-Buyer Wiltering is considered to dxrivw $u$’s kgicing-sensitive imporgance frol accumuoatiig all $u$’s direct and indlxect snfectv on all the otmer potentidl buyers.
1. **Norkauived Weight.**
By intuition, for commoqity prpmltion, an indivyduak $u$’s... | indirect effects through influence cascades; 3) the is to derive pricing-sensitive importance from indirect on all the potential buyers. 1. Weight.** By intuition, for commodity promotion, individual $u$’s importance towards another individual $v$ can be evaluated from how much valuation approaches the pricing due to $... | indirect effects through infLuence cascAdes; 3) tHe POteNtIal-BUyer filtering is conSIderEd to derive $u$’s pricing-senSitivE iMPortANcE from AccumulATiNG All $U$’s DiRecT aND iNdireCt eFfects oN all the othEr pOtEntial buyers.
1. **nOrMalized WeiGht.**
by intuition, fOr cOmmodiTy ProMOtion, An iNdiviDual $u$’s... | indirect effects throughinfluencecasca des ; 3 )thePote ntial-Buyer Fi l teri ng is considered to de rive$u $ ’s p r ic ing-s ensitiv e i m p ort an ce fr om ac cumul ati ng all$u$’s dire ctan d indirect e f fe cts on all th e other pote nti al buy er s.1. * *No rmali zed We i ght.**
Byin t uition , for co ... | indirect_effects through_influence cascades; 3) the_Potential-Buyer Filtering_is_considered to_derive_$u$’s pricing-sensitive importance_from accumulating all_$u$’s direct and indirect_effects on all_the_other potential buyers.
1. **Normalized Weight.**
By intuition, for commodity promotion,_an_individual $u$’s... |
$x\ge0$ and $p(x)\le0$, then $x\le r_2$. Taking $w=r_2$, we conclude that $\mu(V)\le w$, and we have the formula . It is easy to see that $w$ is a continuous function of $m_1$, $m_2$ and $\delta$.
\[lem:stepII\] Let $c<d$ be real numbers. For matrices $a_1$ and $a_2$, consider the maps ${{\mathrm{Tr}}}\,\circ\,{{\mat... | $ x\ge0 $ and $ p(x)\le0 $, then $ x\le r_2$. Taking $ w = r_2 $, we conclude that $ \mu(V)\le w$, and we have the rule . It is comfortable to see that $ w$ is a continuous function of $ m_1 $, $ m_2 $ and $ \delta$.
\[lem: stepII\ ] lease $ speed of light < d$ be real numbers. For matrix $ a_1 $ and $ a_2 $, regar... | $x\gf0$ and $p(x)\le0$, then $x\le r_2$. Taying $w=r_2$, we conclude thet $\mu(V)\ls w$, and de have the formula . It is eady to stv that $w$ is a continuuus functpon of $m_1$, $n_2$ anv $\delta$.
\[lem:stepII\] Let $c<d$ nz ream numyecs. For matrices $a_1$ and $a_2$, cmnsider the ma[s ${{\mcthrm{Tr}}}\,\circ\,{{\mat... | $x\ge0$ and $p(x)\le0$, then $x\le r_2$. Taking conclude $\mu(V)\le w$, we have the to that $w$ is continuous function of $m_2$ and $\delta$. \[lem:stepII\] Let $c<d$ real numbers. For matrices $a_1$ and $a_2$, consider the maps ${{\mathrm{Tr}}}\,\circ\,{{\mathrm{ev}}}_{a_1,a_2}:\mathbb{C}\langle x_1,x_2\rangle\to\math... | $x\ge0$ and $p(x)\le0$, then $x\le r_2$. Taking $w=R_2$, we concludE that $\Mu(V)\Le w$, AnD we hAve tHe formula . It is eASy to See that $w$ is a continuous fUnctiOn OF $m_1$, $m_2$ aND $\dElta$.
\[lEm:stepIi\] leT $C<D$ be ReAl NumBeRS. FOr matRicEs $a_1$ and $a_2$, Consider thE maPs ${{\Mathrm{Tr}}}\,\circ\,{{\MAt... | $x\ge0$ and $p(x)\le0$, t hen $x\ler_2$. Ta kin g$w=r _2$, we conclude t h at $ \mu(V)\le w$, and we h ave t he form u la . It is eas y t o see t ha t $ w$ is a co nti nuous f unction of $m _1 $, $m_2$ and $\ delta$.
\ [le m:stepII\] L et$c<d$be re a l num ber s. Fo r matr i ces $a _1$ and $ a_ 2 $, con s id... | $x\ge0$_and $p(x)\le0$,_then $x\le r_2$. Taking_$w=r_2$, we_conclude_that $\mu(V)\le_w$,_and we have_the formula . It_is easy to see_that $w$ is_a_continuous function of $m_1$, $m_2$ and $\delta$.
\[lem:stepII\] Let $c<d$ be real numbers. For matrices_$a_1$_and $a_2$,_consider_the_maps ${{\mathrm{Tr}}}\,\circ\,{{\mat... |
band. The peak of the X-ray emission is coincident with the position of the cD galaxy. Although the Chandra detector does not cover the entire cluster X-ray extension, a connection between the mini-halo and the X-ray emission is evident, which is better investigated in the next section.
Comparison between mini-halos ... | band. The peak of the X - ray discharge is coincident with the placement of the cD galaxy. Although the Chandra detector does not cover the entire bunch X - ray extension, a association between the mini - halo and the ten - ray emission is discernible, which is better investigated in the adjacent incision.
Compariso... | bajd. The peak of the X-ray tmission is coincneent wmth the positiov of the cD galaxy. Although vhe Xhandea detector does not cuver the vntire clystec X-ray extension, a connegcion gctweeu vhe mini-halo anc the X-ray emission is eeiaeut, which is better investigated in tre next sfction.
Comparisjn bttwqen jpnl-halos ... | band. The peak of the X-ray emission with position of cD galaxy. Although cover entire cluster X-ray a connection between mini-halo and the X-ray emission is which is better investigated in the next section. Comparison between mini-halos and the emitting gas ======================================================== By s... | band. The peak of the X-ray emissIon is coincIdent WitH thE pOsitIon oF the cD galaxy. AlTHougH the Chandra detector doeS not cOvER the ENtIre clUster X-rAY eXTEnsIoN, a ConNeCTiOn betWeeN the minI-halo and thE X-rAy Emission is evIDeNt, which is bEttEr investigatEd iN the neXt SecTIon.
CoMpaRison BetweeN Mini-haLos ... | band. The peak of the X-r ay emissio n iscoi nci de nt w iththe position o f the cD galaxy. Although t he Ch an d ra d e te ctordoes no t c o v erth eent ir e c luste r X -ray ex tension, a co nn ection betwe e nthe mini-h alo and the X-r ayemissi on is evide nt, whic h is b e tter i nvestigat ed in the next s... | band._The peak_of the X-ray emission_is coincident_with_the position_of_the cD galaxy._Although the Chandra_detector does not cover_the entire cluster_X-ray_extension, a connection between the mini-halo and the X-ray emission is evident, which is_better_investigated in_the_next_section.
Comparison between mini-halos ... |
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