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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)&\to M(k) A(x,k^{-1}[y])\\
C(x,y)&\to N(k)C(x,k^{-1}[y])\endaligned\right.,$$ are physical. Here $M(k)$ and $N(k)$ are matrix representations of $K$ and $k[y]$ is the image of the point $y\in\mathbb{S}^7$ under the operation of $k\in K$. But unlike the standard orbifolding conditions, we suppose that $M(K)\ne N(K)$.
... | ) & \to M(k) A(x, k^{-1}[y])\\
C(x, y)&\to N(k)C(x, k^{-1}[y])\endaligned\right. ,$$ are physical. Here $ M(k)$ and $ N(k)$ are matrix representations of $ K$ and $ k[y]$ is the image of the point $ y\in\mathbb{S}^7 $ under the process of $ k\in K$. But unlike the standard orbifolding condition, we suppose that $ M(K... | )&\to L(k) A(x,k^{-1}[y])\\
C(x,y)&\to N(k)C(x,k^{-1}[y])\endallgned\right.,$$ are pktsical. Here $J(k)$ and $N(y)$ are matrix representations od $K$ abd $k[y]$ is the image of ghe point $y\in\mathvb{S}^7$ nnder the operatmkn of $k\lu K$. Bhb unlnkx the standard prbifoldinc conditions, wa ru'pose that $M(K)\ne N(K)$.
... | )&\to M(k) A(x,k^{-1}[y])\\ C(x,y)&\to N(k)C(x,k^{-1}[y])\endaligned\right.,$$ are physical. and are matrix of $K$ and the $y\in\mathbb{S}^7$ under the of $k\in K$. unlike the standard orbifolding conditions, we that $M(K)\ne N(K)$. In order to determine automorphisms that are responsible for the breaking (\[2-04\]), w... | )&\to M(k) A(x,k^{-1}[y])\\
C(x,y)&\to N(k)C(x,k^{-1}[y])\endaliGned\right.,$$ aRe phySicAl. HErE $M(k)$ aNd $N(k)$ Are matrix repreSEntaTions of $K$ and $k[y]$ is the imagE of thE pOInt $y\IN\mAthbb{s}^7$ under tHE oPERatIoN oF $k\iN K$. bUt UnlikE thE standaRd orbifoldIng CoNditions, we suPPoSe that $M(K)\ne n(K)$.
... | )&\to M(k) A(x,k^{-1}[y])\ \
C(x,y)&\ to N( k)C (x, k^ {-1} [y]) \endaligned\ri g ht., $$ are physical. Here$M(k) $a nd $ N (k )$ ar e matri x r e p res en ta tio ns of $K$and $k[y]$ is the im age o f the point$ y\ in\mathbb{ S}^ 7$ under the op eratio nof$ k\inK$. Butunlike the st andard or bi f olding conditi ... | )&\to M(k)_A(x,k^{-1}[y])\\
C(x,y)&\to N(k)C(x,k^{-1}[y])\endaligned\right.,$$_are physical. Here $M(k)$_and $N(k)$_are_matrix representations_of_$K$ and $k[y]$_is the image_of the point $y\in\mathbb{S}^7$_under the operation_of_$k\in K$. But unlike the standard orbifolding conditions, we suppose that $M(K)\ne N(K)$.
... |
forcing $r$ to be odd. This is significant as the fields $\mathbb{F}_{2^m}$ are the most commonly used in modern engineering.
In Section 2 we considered irreducible composed products of the form $f \odot
\Phi_m.$ In particular, we derived the construction of a new class of irreducible polynomials in Theorem \[thm 3\]... | forcing $ r$ to be odd. This is significant as the fields $ \mathbb{F}_{2^m}$ are the about normally practice in modern engineering.
In department 2 we considered irreducible composed product of the shape $ f \odot
\Phi_m.$ In particular, we deduce the construction of a new course of irreducible polynomials in The... | fogcing $r$ to be odd. This ir significant as the fmelds $\mzthbb{F}_{2^m}$ xre the most commonly used ii moeern tugineering.
In Section 2 we conspdered ireedurible composed pckducts of the norm $y \idot
\Phi_m.$ In patticular, we gerived the cotsgrbction of a new class of irreducible polynokiwls in Theorem \[thm 3\]... | forcing $r$ to be odd. This is the $\mathbb{F}_{2^m}$ are most commonly used 2 considered irreducible composed of the form \odot \Phi_m.$ In particular, we derived construction of a new class of irreducible polynomials in Theorem \[thm 3\]. It natural to consider other classes of polynomials and substitute them for $\P... | forcing $r$ to be odd. This is signIficant as tHe fieLds $\MatHbB{F}_{2^m}$ aRe thE most commonly uSEd in Modern engineering.
In SecTion 2 wE cONsidEReD irreDucible COmPOSed PrOdUctS oF ThE form $F \odOt
\Phi_m.$ IN particulaR, we DeRived the consTRuCtion of a neW clAss of irreducIblE polynOmIalS In TheOreM \[thm 3\]... | forcing $r$ to be odd. Th is is sign ifica ntasth e fi elds $\mathbb{F}_{ 2 ^m}$ are the most commonly used i n mod e rn engi neering .
I n Se ct io n 2 w e c onsid ere d irred ucible com pos ed products of th e form $f\od ot
\Phi_m.$Inpartic ul ar, we de riv ed th e cons t ructio n of a ne wc lass o f irr... | forcing_$r$ to_be odd. This is_significant as_the_fields $\mathbb{F}_{2^m}$_are_the most commonly_used in modern_engineering.
In Section 2 we_considered irreducible composed_products_of the form $f \odot
\Phi_m.$ In particular, we derived the construction of a new_class_of irreducible_polynomials_in_Theorem \[thm 3\]... |
\|_2 \lesssim
\|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2.
\label{prod_rule}$$ A classical Leibnitz type inequality for fractional derivatives is the following (see [@KPV1]).
\[stand\_prod\] Let $0 < \alpha, \alpha_1, \alpha_2 < 1$, $\alpha=\alpha_1+\alpha_2$, $1 <p,p_1,p_2 < \infty$, and $\frac{1}{p}=\frac{1}{p... | \|_2 \lesssim
\|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2.
\label{prod_rule}$$ A classical Leibnitz type inequality for fractional derivatives is the following (visualize [ @KPV1 ]).
\[stand\_prod\ ] lease $ 0 < \alpha, \alpha_1, \alpha_2 < 1 $, $ \alpha=\alpha_1+\alpha_2 $, $ 1 < p, p_1,p_2 < \infty$, and... | \|_2 \ledssim
\|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|n\|_2.
\label{prod_rule}$$ A classiral Leignitz tyoe inequality for fractional dwrivaupves is the following (see [@KPV1]).
\[snand\_prod\] Oet $0 < \alpha, \alpiz_1, \alpha_2 < 1$, $\alpgw=\alpka_1+\elpha_2$, $1 <p,p_1,p_2 < \infjy$, and $\frac{1}{p}=\xrac{1}{p... | \|_2 \lesssim \|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2. \label{prod_rule}$$ A classical Leibnitz for derivatives is following (see [@KPV1]). \alpha_1, < 1$, $\alpha=\alpha_1+\alpha_2$, <p,p_1,p_2 < \infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. In addition, the $\alpha_1=\alpha$, $p=p_2$, $p_1=\infty$ is ... | \|_2 \lesssim
\|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_n^1}\|f\|_2.
\label{proD_rule}$$ a clAssIcAl LeIbniTz type inequaliTY for Fractional derivatives iS the fOlLOwinG (SeE [@KPV1]).
\[sTand\_proD\] leT $0 < \ALphA, \aLpHa_1, \aLpHA_2 < 1$, $\aLpha=\aLphA_1+\alpha_2$, $1 <p,P_1,p_2 < \infty$, and $\FraC{1}{p}=\Frac{1}{p... | \|_2 \lesssim
\|Q_ND_{x}^{ \alpha}g\| _{L_x ^{\ inf ty }l_N ^1}\ |f\|_2.
\label { prod _rule}$$ A classical L eibni tz type in equal ity for fr a c tio na lder iv a ti ves i s t he foll owing (see [@ KP V1]).
\[sta n d\ _prod\] Le t $ 0 < \alpha,\al pha_1, \ alp h a_2 < 1$ , $\a lpha=\ a lpha_1 +\alpha_2 $, $1 <p, ... | \|_2 \lesssim
\|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2.
\label{prod_rule}$$_A classical_Leibnitz type inequality for_fractional derivatives_is_the following_(see_[@KPV1]).
\[stand\_prod\] Let $0_< \alpha, \alpha_1,_\alpha_2 < 1$, $\alpha=\alpha_1+\alpha_2$,_$1 <p,p_1,p_2 <_\infty$,_and $\frac{1}{p}=\frac{1}{p... |
}}&:=\left(u_i^{h,n}+u_i^{h,n-1}\right)/2,\\
F_{iJ}^{h,n-\frac{1}{2}}&:=\delta_{iJ}+u_{i,J}^{h,n-\frac{1}{2}},\\
F_{iJ,K}^{h,n-\frac{1}{2}}&:=u_{i,JK}^{h,n-\frac{1}{2}},\\
v_i^{h,n-\frac{1}{2}}&:=(u_i^{h,n}-u_i^{h,n-1})/\Delta t,\\
\Delta^n u_i^h&:=\left(u_i^{h,n+1}-u_i^{h,n-1}\right)/2,\\
\Delta^n\!F^h_{iJ}&:=\Delta^n... | } } &: = \left(u_i^{h, n}+u_i^{h, n-1}\right)/2,\\
F_{iJ}^{h, n-\frac{1}{2}}&:=\delta_{iJ}+u_{i, J}^{h, n-\frac{1}{2}},\\
F_{iJ, K}^{h, n-\frac{1}{2}}&:=u_{i, JK}^{h, n-\frac{1}{2}},\\
v_i^{h, n-\frac{1}{2}}&:=(u_i^{h, n}-u_i^{h, n-1})/\Delta t,\\
\Delta^n u_i^h&:=\left(u_i^{h, n+1}-u_i^{h, n-1}\right)/2,\\
\... | }}&:=\lefh(u_i^{h,n}+u_i^{h,n-1}\right)/2,\\
F_{iJ}^{h,n-\frac{1}{2}}&:=\dtlta_{iJ}+u_{i,J}^{h,n-\frac{1}{2}},\\
F_{iL,J}^{h,n-\frar{1}{2}}&:=u_{i,JK}^{h,n-\rrac{1}{2}},\\
v_i^{h,n-\wrac{1}{2}}&:=(u_i^{h,n}-u_i^{h,n-1})/\Delta t,\\
\Delta^n u_i^i&:=\lefr(u_i^{h,n+1}-y_i^{h,n-1}\right)/2,\\
\Delta^n\!F^h_{iJ}&:=\Delga^n... | }}&:=\left(u_i^{h,n}+u_i^{h,n-1}\right)/2,\\ F_{iJ}^{h,n-\frac{1}{2}}&:=\delta_{iJ}+u_{i,J}^{h,n-\frac{1}{2}},\\ F_{iJ,K}^{h,n-\frac{1}{2}}&:=u_{i,JK}^{h,n-\frac{1}{2}},\\ v_i^{h,n-\frac{1}{2}}&:=(u_i^{h,n}-u_i^{h,n-1})/\Delta t,\\ \Delta^n u_i^h&:=\left(u_i^{h,n+1}-u_i^{h,n-1}\right)/2,\\ \Delta^n\!F^h_{iJ,K}&:=\Delta... | }}&:=\left(u_i^{h,n}+u_i^{h,n-1}\right)/2,\\
F_{iJ}^{h,n-\fraC{1}{2}}&:=\delta_{iJ}+u_{i,j}^{h,n-\frAc{1}{2}},\\
F_{IJ,K}^{H,n-\Frac{1}{2}}&:=U_{i,JK}^{H,n-\frac{1}{2}},\\
v_i^{h,n-\frac{1}{2}}&:=(U_I^{h,n}-u_I^{h,n-1})/\Delta t,\\
\Delta^n u_i^h&:=\left(U_i^{h,n+1}-u_I^{h,N-1}\RighT)/2,\\
\deLta^n\!F^H_{iJ}&:=\DeltA^N... | }}&:=\left(u_i^{h,n}+u_i^{ h,n-1}\rig ht)/2 ,\\
F_ {i J}^{ h,n- \frac{1}{2}}&: = \del ta_{iJ}+u_{i,J}^{h,n-\ frac{ 1} { 2}}, \ \F_{iJ ,K}^{h, n -\ f r ac{ 1} {2 }}& := u _{ i,JK} ^{h ,n-\fra c{1}{2}},\ \
v _i ^{h,n-\frac{ 1 }{ 2}}&:=(u_i ^{h ,n}-u_i^{h,n -1} )/\Del ta t, \ \
\De lta ^n u_ i^h&:= \ left(u _i^{h,n+1 }-... | }}&:=\left(u_i^{h,n}+u_i^{h,n-1}\right)/2,\\
F_{iJ}^{h,n-\frac{1}{2}}&:=\delta_{iJ}+u_{i,J}^{h,n-\frac{1}{2}},\\
F_{iJ,K}^{h,n-\frac{1}{2}}&:=u_{i,JK}^{h,n-\frac{1}{2}},\\
v_i^{h,n-\frac{1}{2}}&:=(u_i^{h,n}-u_i^{h,n-1})/\Delta t,\\
\Delta^n_u_i^h&:=\left(u_i^{h,n+1}-u_i^{h,n-1}\right)/2,\\
\Delta^n\!F^h_{iJ}&:=\Delta^n... |
x;{\boldsymbol}{\lambda})
=\phi_{\text{v}}(x;{\boldsymbol}{\lambda}),{\nonumber}\\
&\Delta_{\text{v}}^+=-h+\text{v}>0,\quad
\Delta_{\text{v}}^-=-\Delta_{\text{v}}^+<0,
\quad
\tilde{\mathcal{E}}^{\text{os}}_{\text{v}}({\boldsymbol}{\lambda})
<\mathcal{E}_0({\boldsymbol}{\lambda}) \quad(\text{v}>2h).
\label{... | x;{\boldsymbol}{\lambda })
= \phi_{\text{v}}(x;{\boldsymbol}{\lambda}),{\nonumber}\\
& \Delta_{\text{v}}^+=-h+\text{v}>0,\quad
\Delta_{\text{v}}^-=-\Delta_{\text{v}}^+<0,
\quad
\tilde{\mathcal{E}}^{\text{os}}_{\text{v}}({\boldsymbol}{\lambda })
< \mathcal{E}_0({\boldsymbol}{\lambda }) \quad(\text{v}>2... | x;{\bopdsymbol}{\lambda})
=\phi_{\text{v}}(w;{\boldsymbol}{\lambdc}),{\bonumbxr}\\
&\Delta_{\fext{v}}^+=-h+\tebt{v}>0,\quad
\Delta_{\text{v}}^-=-\Delta_{\text{v}}^+<0,
\euqd
\tulde{\mathcal{E}}^{\text{os}}_{\text{x}}({\boldsymbll}{\lambda})
<\mauhcal{E}_0({\boldsymbol}{\lejbda}) \quad(\text{b}>2m).
\label{... | x;{\boldsymbol}{\lambda}) =\phi_{\text{v}}(x;{\boldsymbol}{\lambda}),{\nonumber}\\ &\Delta_{\text{v}}^+=-h+\text{v}>0,\quad \Delta_{\text{v}}^-=-\Delta_{\text{v}}^+<0, \quad \tilde{\mathcal{E}}^{\text{os}}_{\text{v}}({\boldsymbol}{\lambda}) <\mathcal{E}_0({\boldsymbol}{\lambda}) #### scatterings A (‘pseudo’) virtual st... | x;{\boldsymbol}{\lambda})
=\phi_{\text{v}}(X;{\boldsymboL}{\lambDa}),{\nOnuMbEr}\\
&\DeLta_{\tExt{v}}^+=-h+\text{v}>0,\quad
\dElta_{\Text{v}}^-=-\Delta_{\text{v}}^+<0,
\quad
\tilDe{\matHcAL{E}}^{\teXT{oS}}_{\text{V}}({\boldsyMBoL}{\LAmbDa})
<\MaThcAl{e}_0({\BoLdsymBol}{\Lambda}) \qUad(\text{v}>2h).
\lAbeL{... | x;{\boldsymbol}{\lambda}) =\phi_{\ text{ v}} (x; {\ bold symb ol}{\lambda}), { \non umber}\\
&\Delta_{\tex t{v}} ^+ = -h+\ t ex t{v}> 0,\quad
\ D e lta _{ \t ext {v } }^ -=-\D elt a_{\tex t{v}}^+<0,
\q ua d
\tilde{\ m at hcal{E}}^{ \te xt{os}}_{\te xt{ v}}({\ bo lds y mbol} {\l ambda })
< \ mathca l{E}_0({\ bo l... | x;{\boldsymbol}{\lambda})
_=\phi_{\text{v}}(x;{\boldsymbol}{\lambda}),{\nonumber}\\
&\Delta_{\text{v}}^+=-h+\text{v}>0,\quad
\Delta_{\text{v}}^-=-\Delta_{\text{v}}^+<0,
\quad
_ \tilde{\mathcal{E}}^{\text{os}}_{\text{v}}({\boldsymbol}{\lambda})
<\mathcal{E}_0({\boldsymbol}{\lambda})_\quad(\text{v}>2h).
__ __\label{... |
matrices acting in spin space, $$\begin{gathered}
{\bf \sigma_i}=
\begin{pmatrix}
&\underline{\sigma_i}&0\\
&0&\underline{\sigma_i}
\label{sigmai}
\end{pmatrix}.\end{gathered}$$ Using (\[conj\]), it follows that $$\begin{aligned}
\label{l}
[C G({{\bf{r}}}',{{\bf{r}}};\tau )C]_{\alpha \beta } &=& - \langle T_\tau C\Psi... | matrices acting in spin space, $ $ \begin{gathered }
{ \bf \sigma_i}=
\begin{pmatrix }
& \underline{\sigma_i}&0\\
& 0&\underline{\sigma_i }
\label{sigmai }
\end{pmatrix}.\end{gathered}$$ use (\[conj\ ]), it take after that $ $ \begin{aligned }
\label{l }
[ C G({{\bf{r}}}',{{\bf{r}}};\tau) C]_{\alpha \be... | mahrices acting in spin spxce, $$\begin{gathergd}
{\vf \sigka_i}=
\begjn{pmatrib}
&\underline{\sigma_i}&0\\
&0&\underline{\sigla_u}
\labeo{sigmai}
\end{pmatrix}.\end{gaghered}$$ Uspng (\[conj\]), ut fiolows that $$\begin{allyned}
\lznel{l}
[C J({{\bf{r}}}',{{\bf{r}}};\tau )C]_{\alpma \beta } &=& - \nangle T_\tau C\Pvi... | matrices acting in spin space, $$\begin{gathered} {\bf &\underline{\sigma_i}&0\\ \label{sigmai} \end{pmatrix}.\end{gathered}$$ (\[conj\]), it follows )C]_{\alpha } &=& - T_\tau C\Psi ({{\bf{r}}}',\tau)\Psi{^{\dagger \beta } \cr &=& - \langle }^{*} ({{\bf{r}}}',\tau)\Psi_{\beta }^{T} ({{\bf{r}}},0)\rangle\cr &=& \langle... | matrices acting in spin space, $$\Begin{gatheRed}
{\bf \SigMa_i}=
\BeGin{pMatrIx}
&\underline{\sigMA_i}&0\\
&0&\unDerline{\sigma_i}
\label{sigmAi}
\end{PmATrix}.\ENd{GatheRed}$$ UsinG (\[CoNJ\]), It fOlLoWs tHaT $$\BeGin{alIgnEd}
\label{L}
[C G({{\bf{r}}}',{{\bf{r}}};\tAu )C]_{\AlPha \beta } &=& - \langlE t_\tAu C\Psi... | matrices acting in spin s pace, $$\b egin{ gat her ed }
{\ bf \ sigma_i}=
\beg i n{pm atrix}
&\underline{\si gma_i }& 0 \\
& 0 &\ under line{\s i gm a _ i}\l ab el{ si g ma i}
\e nd{ pmatrix }.\end{gat her ed }$$ Using (\ [ co nj\]), itfol lows that $$ \be gin{al ig ned }
\lab el{ l}
[C G({{\ b f{r}}} ',{{\bf{r }} }... | matrices_acting in_spin space, $$\begin{gathered}
{\bf \sigma_i}=
\begin{pmatrix}
&\underline{\sigma_i}&0\\
&0&\underline{\sigma_i}
\label{sigmai}
\end{pmatrix}.\end{gathered}$$_Using (\[conj\]),_it_follows that_$$\begin{aligned}
\label{l}
[C_G({{\bf{r}}}',{{\bf{r}}};\tau )C]_{\alpha \beta_} &=& -_\langle T_\tau C\Psi... |
to the functional $$J = A + \lambda \, {\rm Vol}$$ for unconstrained compactly supported deformations, and moreover, an oriented immersion is stationary with respect to $J$ if and only if, with respect to some choice of unit normal, it has scalar mean curvature equal to $\lambda.$ Here $A,$ ${\rm Vol}$ denote the area... | to the functional $ $ J = A + \lambda \, { \rm Vol}$$ for unconstrained compactly supported contortion, and furthermore, an oriented immersion is stationary with obedience to $ J$ if and entirely if, with respect to some choice of whole normal, it have scalar mean curvature adequate to $ \lambda.$ Here $ A,$ $ { \rm Vo... | to the functional $$J = A + \laobda \, {\rm Vol}$$ fot ynconsvrained compactuy supported deformations, anv moeeovee, an oriented immersiov is statponary wirh rtspect to $J$ if anv only in, witg res'ert to some choige of unit tormal, it has vcxlcr mean curvature equal to $\lambda.$ Hewe $A,$ ${\rm Vll}$ denote the wrea... | to the functional $$J = A + {\rm for unconstrained supported deformations, and stationary respect to $J$ and only if, respect to some choice of unit it has scalar mean curvature equal to $\lambda.$ Here $A,$ ${\rm Vol}$ denote area functional and the enclosed volume functional respectively. The main regularity result o... | to the functional $$J = A + \lambda \, {\rm vol}$$ for uncoNstraIneD coMpActlY supPorted deformatIOns, aNd moreover, an oriented imMersiOn IS staTIoNary wIth respECt TO $j$ if AnD oNly If, WItH respEct To some cHoice of uniT noRmAl, it has scalaR MeAn curvaturE eqUal to $\lambda.$ HEre $a,$ ${\rm Vol}$ DeNotE The arEa... | to the functional $$J = A + \lambda \, { \rm Vo l} $$ f or u nconstrained c o mpac tly supported deformat ions, a n d mo r eo ver,an orie n te d imm er si onis st ation ary with r espect to$J$ i f and only i f ,with respe ctto some choi ceof uni tnor m al, i t h as sc alar m e an cur vature eq ua l to $\ l ambd... | to_the functional_$$J = A +_\lambda \,_{\rm_Vol}$$ for_unconstrained_compactly supported deformations,_and moreover, an_oriented immersion is stationary_with respect to_$J$_if and only if, with respect to some choice of unit normal, it has_scalar_mean curvature_equal_to_$\lambda.$ Here $A,$ ${\rm Vol}$_denote the area... |
a well-trained annotator three minutes to complete a bounding box annotation on one image). We strike a middle ground by grouping items of the same types, that often also appear in similar location (e.g. *jacket, coat, t-shirt*) into a high-level class (e.g. *top*). Our complete list of top-level detection class for a... | a well - trained annotator three minutes to complete a bounding box note on one prototype). We strike a middle earth by group items of the same type, that much also look in exchangeable location (for example * jacket, coat, t - shirt *) into a high - horizontal surface class (for example * top *). Our accomplished tilt... | a aell-trained annotator thvee minutes to complete a bouhding bob annotation on one image). We srrike a middle ground by gruuping itvms of thw sanw types, thef often also zipear mn similar locajion (e.g. *jackat, coat, t-shirt*) ivtl a high-level class (e.g. *top*). Our compjete lixt of top-level dgtectpog clzss for a... | a well-trained annotator three minutes to complete box on one We strike a of same types, that also appear in location (e.g. *jacket, coat, t-shirt*) into high-level class (e.g. *top*). Our complete list of top-level detection class for apparel is as follows: *headwear, eyewear, earring, belt, bottom, dress, top, suit, ... | a well-trained annotator threE minutes to ComplEte A boUnDing Box aNnotation on one IMage). we strike a middle ground bY grouPiNG iteMS oF the sAme typeS, ThAT OftEn AlSo aPpEAr In simIlaR locatiOn (e.g. *jacket, CoaT, t-Shirt*) into a hiGH-lEvel class (e.G. *toP*). Our complete LisT of top-LeVel DEtectIon Class For a... | a well-trained annotatorthree minu tes t o c omp le te a bou nding box anno t atio n on one image). We st rikeam iddl e g round by gro u pi n g it em softh e s ame t ype s, that often als o a pp ear in simil a rlocation ( e.g . *jacket, c oat , t-sh ir t*) intoa h igh-l evel c l ass (e .g. *top* ). Our co m plet... | a_well-trained annotator_three minutes to complete_a bounding_box_annotation on_one_image). We strike_a middle ground_by grouping items of_the same types,_that_often also appear in similar location (e.g. *jacket, coat, t-shirt*) into a high-level class_(e.g._*top*). Our_complete_list_of top-level detection class for_a... |
R}^p \rightarrow \mathbf{R}$ such that for all $i \in N$, $$\begin{aligned}
\mathbf{E}[Y_{i,1}|\overline Y_{i,0},Y_{i,0},X,G] = g_1(\overline Y_{i,0},Y_{i,0},X_i),
\end{aligned}$$ where $$\begin{aligned}
\label{Y bar}
\overline Y_{i,0} = \frac{1}{|N(i)|}\sum_{j \in N(i)} Y_{j,0}.
\end{aligned}$$... | R}^p \rightarrow \mathbf{R}$ such that for all $ i \in N$, $ $ \begin{aligned }
\mathbf{E}[Y_{i,1}|\overline Y_{i,0},Y_{i,0},X, G ] = g_1(\overline Y_{i,0},Y_{i,0},X_i),
\end{aligned}$$ where $ $ \begin{aligned }
\label{Y bar }
\overline Y_{i,0 } = \frac{1}{|N(i)|}\sum_{j \in N(i) } Y_{j,0 }... | R}^p \gightarrow \mathbf{R}$ such uhat for all $i \in N$, $$\begii{alignes}
\mathbf{E}[Y_{i,1}|\overline Y_{i,0},Y_{i,0},X,G] = g_1(\lvwrlint Y_{i,0},Y_{i,0},X_i),
\end{aligved}$$ where $$\begin{alugnev}
\label{Y bar}
\overllue Y_{i,0} = \frae{1}{|N(m)|}\sum_{j \in N(i)} Y_{j,0}.
\end{aligted}$$... | R}^p \rightarrow \mathbf{R}$ such that for all N$, \mathbf{E}[Y_{i,1}|\overline Y_{i,0},Y_{i,0},X,G] g_1(\overline Y_{i,0},Y_{i,0},X_i), \end{aligned}$$ Y_{i,0} \frac{1}{|N(i)|}\sum_{j \in N(i)} \end{aligned}$$ The assumption that the conditional expectation of $Y_{i,1}$ $\overline Y_{i,0}, Y_{i,0},X,G$ depend on $G$ o... | R}^p \rightarrow \mathbf{R}$ such thAt for all $i \iN N$, $$\begIn{aLigNeD}
\matHbf{E}[y_{i,1}|\overline Y_{i,0},Y_{i,0},x,g] = g_1(\ovErline Y_{i,0},Y_{i,0},X_i),
\end{aligned}$$ Where $$\BeGIn{alIGnEd}
\labEl{Y bar}
\oVErLINe Y_{I,0} = \fRaC{1}{|N(i)|}\SuM_{J \iN N(i)} Y_{j,0}.
\End{Aligned}$$... | R}^p \rightarrow \mathbf{R }$ such th at fo r a ll$i \in N$, $$\begin{alig n ed} \mathbf{E}[Y_{ i,1}| \o v erli n eY_{i, 0},Y_{i , 0} , X ,G] = g _1( \o v er lineY_{ i,0},Y_ {i,0},X_i) ,
\end{aligne d }$ $ where $$ \be gin{aligned}
\lab el {Yb ar}
\over line Y _ {i,0}= \frac{1 }{ | N(i)|} \ sum_{j\ ... | R}^p \rightarrow_\mathbf{R}$ such_that for all $i_\in N$,_$$\begin{aligned}
_ __ _ \mathbf{E}[Y_{i,1}|\overline Y_{i,0},Y_{i,0},X,G]_= g_1(\overline Y_{i,0},Y_{i,0},X_i),
_ \end{aligned}$$_where_$$\begin{aligned}
\label{Y bar}
\overline Y_{i,0} = \frac{1}{|N(i)|}\sum_{j \in_N(i)}_Y_{j,0}.
___\end{aligned}$$... |
given in Section 2.
Weighted Hardy inequalities
===========================
Let $\mu$ a weight function in ${{\mathbb}{R}}^N$. We define the weighted Sobolev space $H^1_\mu=H^1({{\mathbb}{R}}^N, \mu(x)dx))$ as the space of functions in $L^2_\mu:=L^2({{\mathbb}{R}}^N, \mu(x)dx)$ whose weak derivatives belong to $(L_\... | given in Section 2.
Weighted Hardy inequalities
= = = = = = = = = = = = = = = = = = = = = = = = = = =
lease $ \mu$ a system of weights function in $ { { \mathbb}{R}}^N$. We define the weighted Sobolev distance $ H^1_\mu = H^1({{\mathbb}{R}}^N, \mu(x)dx))$ as the space of functions in $ L^2_\mu:=L^2({{\mathbb}{R... | gigen in Section 2.
Weighted Mardy inequalitigs
===========================
Oet $\mu$ a weifht funcgion in ${{\mathbb}{R}}^N$. We define tie wwighttb Sobolev space $H^1_\mu=H^1({{\oathbb}{R}}^N, \lu(x)dx))$ as the wpace of fnhctions in $L^2_\mh:=P^2({{\matkbu}{R}}^N, \mu(x)dx)$ whose weak derieatives belong tu $(P_\... | given in Section 2. Weighted Hardy inequalities $\mu$ weight function ${{\mathbb}{R}}^N$. We define \mu(x)dx))$ the space of in $L^2_\mu:=L^2({{\mathbb}{R}}^N, \mu(x)dx)$ weak derivatives belong to $(L_\mu^2)^N$. As step we consider the following conditions on $\mu$ which we need to state preliminary weighted Hardy ine... | given in Section 2.
Weighted HarDy inequaliTies
===========================
LEt $\mU$ a wEiGht fUnctIon in ${{\mathbb}{R}}^N$. WE DefiNe the weighted Sobolev spAce $H^1_\mU=H^1({{\MAthbB}{r}}^N, \Mu(x)dx))$ As the spACe OF FunCtIoNs iN $L^2_\MU:=L^2({{\MathbB}{R}}^N, \Mu(x)dx)$ whOse weak derIvaTiVes belong to $(L_\... | given in Section 2.
Weig hted Hardy ineq ual iti es
=== ==== ============== = ==== =
Let $\mu$ a weightfunct io n in$ {{ \math bb}{R}} ^ N$ . Wede fi neth e w eight edSobolev space $H^ 1_\ mu =H^1({{\math b b} {R}}^N, \m u(x )dx))$ as th e s pace o ffun c tions in $L^2 _\mu:= L ^2({{\ mathbb}{R }} ^ N, \mu ( ... | given_in Section_2.
Weighted Hardy inequalities
===========================
Let $\mu$_a weight_function_in ${{\mathbb}{R}}^N$._We_define the weighted_Sobolev space $H^1_\mu=H^1({{\mathbb}{R}}^N,_\mu(x)dx))$ as the space_of functions in_$L^2_\mu:=L^2({{\mathbb}{R}}^N,_\mu(x)dx)$ whose weak derivatives belong to $(L_\... |
Chen, J. Han, Y. Su, A class of simple weight modules over the twisted Heisenberg-Virasoro algebra, [*J. Math. Phys.*]{} [**57**]{} (2016), 101705.
H. Chen, X. Guo, A new family of modules over the Virasoro algebra, [*J. Algebra*]{} [**457**]{} (2016), 73-105.
H. Chen, X. Guo, K. Zhao, Tensor product weight modules ... | Chen, J. Han, Y. Su, A class of simple weight modules over the distorted Heisenberg - Virasoro algebra, [ * J. Math. Phys. * ] { } [ * * 57 * * ] { } (2016), 101705.
H. Chen, X. Guo, A newfangled family of modules over the Virasoro algebra, [ * J. Algebra * ] { } [ * * 457 * * ] { } (2016), 73 - 105.
H. Chen, X. ... | Chfn, J. Han, Y. Su, A class of simple weight modules over fhe twisged Heisenberg-Virasoro algebca, [*J. Math. Phys.*]{} [**57**]{} (2016), 101705.
H. Chen, X. Guo, A new famipy of moeulew over the Tjrasoro algebdw, [*J. Cljebra*]{} [**457**]{} (2016), 73-105.
H. Chen, W. Guo, K. Zham, Tensor produwt wzight modules ... | Chen, J. Han, Y. Su, A class weight over the Heisenberg-Virasoro algebra, [*J. H. X. Guo, A family of modules the Virasoro algebra, [*J. Algebra*]{} [**457**]{} 73-105. H. Chen, X. Guo, K. Zhao, Tensor product weight modules over the algebra, [*J. Lond. Math. Soc.*]{} [**88**]{} (2013), 829-844. C. H. Conley, C. Martin... | Chen, J. Han, Y. Su, A class of simple Weight moduLes ovEr tHe tWiSted heisEnberg-Virasoro ALgebRa, [*J. Math. Phys.*]{} [**57**]{} (2016), 101705.
H. Chen, X. Guo, A nEw famIlY Of moDUlEs oveR the VirASoRO AlgEbRa, [*j. AlGeBRa*]{} [**457**]{} (2016), 73-105.
h. Chen, x. GuO, K. Zhao, TEnsor produCt wEiGht modules ... | Chen, J. Han, Y. Su, A cl ass of sim ple w eig htmo dule s ov er the twisted Heis enberg-Virasoro algebr a, [* J. Math . P hys.* ]{} [** 5 7* * ] {}(2 01 6), 1 0 17 05.
H.Chen, X . Guo, A n ewfa mily of modu l es over theVir asoro algebr a,[*J. A lg ebr a *]{}[** 457** ]{} (2 0 16), 7 3-105.
H .C hen, X . Guo, K ... | Chen,_J. Han,_Y. Su, A class_of simple_weight_modules over_the_twisted Heisenberg-Virasoro algebra,_[*J. Math. Phys.*]{}_[**57**]{} (2016), 101705.
H. Chen,_X. Guo, A_new_family of modules over the Virasoro algebra, [*J. Algebra*]{} [**457**]{} (2016), 73-105.
H. Chen, X._Guo,_K. Zhao,_Tensor_product_weight modules ... |
93(01)00918-2) [****, ()](\doibase 10.1016/S0375-9474(02)01402-1) [****, ()](\doibase 10.1103/PhysRevC.88.044001) [****, ()](\doibase 10.1088/0954-3899/43/5/055106) [****, ()](\doibase 10.1016/S0375-9474(02)01270-8) [****, ()](\doibase 10.1016/j.physletb.2003.07.049) [****, ()](\doibase 10.1016/S0375-9474(98)00612-... | 93(01)00918 - 2) [ * * * *, () ] (\doibase 10.1016 / S0375 - 9474(02)01402 - 1) [ * * * *, () ] (\doibase 10.1103 / PhysRevC.88.044001) [ * * * *, () ] (\doibase 10.1088/0954 - 3899/43/5/055106) [ * * * *, () ] (\doibase 10.1016 / S0375 - 9474(02)01270 - 8) [ * * * *, () ] (\doibase 10.1016 / j.physletb.2003.0... | 93(01)00918-2) [****, ()](\dlibase 10.1016/S0375-9474(02)01402-1) [****, ()](\doibase 10.1103/PhysReyC.88.044001) [****, ()](\doibase 10.1088/0954-3899/43/5/055106) [****, ()](\donvase 10.1016/S0375-9474(02)01270-8) [****, ()](\doibzse 10.1016/j.phyrletb.2003.07.049) [****, ()](\doibase 10.1016/S0375-9474(98)00612-... | 93(01)00918-2) [****, ()](\doibase 10.1016/S0375-9474(02)01402-1) [****, ()](\doibase 10.1103/PhysRevC.88.044001) 10.1088/0954-3899/43/5/055106) ()](\doibase 10.1016/S0375-9474(02)01270-8) ()](\doibase 10.1016/j.physletb.2003.07.049) [****, [****, 10.1016/j.nuclphysa.2004.08.012) [****, ()](\doibase [****, ()](\doibase... | 93(01)00918-2) [****, ()](\doibase 10.1016/S0375-9474(02)01402-1) [****, ()](\doibase 10.1103/PhysRevC.88.044001) [****, ()](\doIbase 10.1088/0954-3899/43/5/055106) [****, ()](\doibaSe 10.1016/S0375-9474(02)01270-8) [****, ()](\doIbaSe 10.1016/j.PhYsleTb.2003.07.049) [****, ()](\doIbase 10.1016/S0375-9474(98)00612-... | 93(01)00918-2) [****, ()]( \doibase 1 0.101 6/S 037 5- 9474 (02) 01402-1) [**** , () ](\doibase 10.1103/Phy sRevC .8 8 .044 0 01 ) [** **, () ] (\ d o iba se 1 0.1 08 8 /0 954-3 899 /43/5/0 55106) [** **, ()](\doibase 10 .1016/S037 5-9 474(02)01270 -8) [**** ,()] ( \doib ase 10.1 016/j. p hyslet b.2003.07 .0 4 9... | 93(01)00918-2) [****,_()](\doibase 10.1016/S0375-9474(02)01402-1)_[****, ()](\doibase 10.1103/PhysRevC.88.044001) [****, _()](\doibase 10.1088/0954-3899/43/5/055106)_[****, _()](\doibase 10.1016/S0375-9474(02)01270-8)_[****,_()](\doibase 10.1016/j.physletb.2003.07.049) [****, _()](\doibase 10.1016/S0375-9474(98)00612-... |
2$-AGL rings for the further studies, we investigate three topics on $2$-AGL rings, which are closely studied already for the case of AGL rings. The first topic concerns minimal presentations of canonical ideals. In Section 2, we will give a necessary and sufficient condition for a given one-dimensional Cohen-Macaulay ... | 2$-AGL rings for the further studies, we investigate three topics on $ 2$-AGL rings, which are close study already for the case of AGL rings. The inaugural subject concerns minimal presentations of canonical ideal. In Section 2, we will give a necessary and sufficient circumstance for a give one - dimensional Cohen - M... | 2$-AGL rings for the further suudies, we investiyqte thcee topjcs on $2$-AEL rings, which are closely svudiwd aleeady for the case of XGL rings. The firwt tipic conceria minimal preacntatnois of canonical ideals. In Section 2, we whlu yive a necessary and sufficient condytion fpr a given one-diienspogal Dohen-Macaulay ... | 2$-AGL rings for the further studies, we topics $2$-AGL rings, are closely studied AGL The first topic minimal presentations of ideals. In Section 2, we will a necessary and sufficient condition for a given one-dimensional Cohen-Macaulay local ring $R$ be a $2$-AGL ring, in terms of minimal presentations of canonical f... | 2$-AGL rings for the further studIes, we invesTigatE thRee ToPics On $2$-AGl rings, which are CLoseLy studied already for the Case oF Agl rinGS. THe firSt topic COnCERns MiNiMal PrESeNtatiOns Of canonIcal ideals. in SEcTion 2, we will giVE a Necessary aNd sUfficient conDitIon for A gIveN One-diMenSionaL Cohen-mAcaulaY ... | 2$-AGL rings for the furth er studies , weinv est ig atethre e topics on $2 $ -AGL rings, which are clos ely s tu d ieda lr eadyfor the ca s e of A GL ri ng s .The f irs t topic concernsmin im al presentat i on s of canon ica l ideals. In Se ction2, we willgiv e a n ecessa r y andsufficien tc onditi o n for a g ... | 2$-AGL rings_for the_further studies, we investigate_three topics_on_$2$-AGL rings,_which_are closely studied_already for the_case of AGL rings._The first topic_concerns_minimal presentations of canonical ideals. In Section 2, we will give a necessary and_sufficient_condition for_a_given_one-dimensional Cohen-Macaulay ... |
the abundance data (cf. Copi, Schramm, & Turner 1995b). The nature of such errors is unclear, and this remains controversial. Other authors have reacted to the impending crisis in self-consistency by simply omitting one or more of the four nuclides in making the fit (Dar 1995; Olive & Thomas 1997; Hata et al. 1996, 19... | the abundance data (cf. Copi, Schramm, & Turner 1995b). The nature of such errors is unclear, and this remain controversial. early authors have reacted to the at hand crisis in self - consistency by simply neglect one or more of the four nuclides in making the fit (Dar 1995; Olive & Thomas 1997; Hata et al. 1996, 1997;... | thf abundance data (cf. Copi, Schramm, & Turnet 1995v). The iature kf such drrors is unclear, and this rxmaibs cobtroversial. Other authurs have geacted ti tht impending crisis in seln-eonsiabency uy simply omittlng one or kore of the fogr nbclides in making the fit (Dar 1995; Olive & Thomax 1997; Hata et al. 1996, 19... | the abundance data (cf. Copi, Schramm, & The of such is unclear, and have to the impending in self-consistency by omitting one or more of the nuclides in making the fit (Dar 1995; Olive & Thomas 1997; Hata et 1996, 1997; Fields et al. 1996). This controversy has been sharpened by new giving deuterium on lines of sight ... | the abundance data (cf. Copi, SchRamm, & Turner 1995B). The nAtuRe oF sUch eRrorS is unclear, and tHIs reMains controversial. OtheR authOrS Have REaCted tO the impENdING crIsIs In sElF-CoNsistEncY by simpLy omitting One Or More of the fouR NuClides in maKinG the fit (Dar 1995; OlIve & thomas 1997; haTa eT Al. 1996, 19... | the abundance data (cf. C opi, Schra mm, & Tu rne r1995 b).The nature ofs ucherrors is unclear, and this r e main s c ontro versial . O t h erau th ors h a ve reac ted to the impending cr is is in self-c o ns istency by si mply omittin g o ne ormo reo f the fo ur nu clides in mak ing the f it (Dar 1 9 95; ... | the_abundance data_(cf. Copi, Schramm, &_Turner 1995b)._The_nature of_such_errors is unclear,_and this remains_controversial. Other authors have_reacted to the_impending_crisis in self-consistency by simply omitting one or more of the four nuclides in_making_the fit_(Dar_1995;_Olive & Thomas 1997; Hata_et al. 1996, 19... |
pm$ 0.030 0.815$\pm$ 0.024 0.881$\pm$ 0.020
(b)\[$50$/$\sigma$\] $1$ 0.875$\pm$ 0.020 0.558$\pm$ 0.031 0.814$\pm$ 0.024 0.907 $\pm$ 0.018
(c)\[$20$/$\sigma$\] $15$ 0.875$\pm$ 0.020 0.608$\pm$ 0.030 0.769$\pm$ 0.026 0.893$\pm$ 0.019
(d)\[$20$/$\sigma$\] ... | pm$ 0.030 0.815$\pm$ 0.024 0.881$\pm$ 0.020
(b)\[$50$/$\sigma$\ ] $ 1 $ 0.875$\pm$ 0.020 0.558$\pm$ 0.031 0.814$\pm$ 0.024 0.907 $ \pm$ 0.018
(c)\[$20$/$\sigma$\ ] $ 15 $ 0.875$\pm$ 0.020 0.608$\pm$ 0.030 0.769$\pm$ 0.026 0.893$\pm$ 0.019
... | pm$ 0.030 0.815$\pm$ 0.024 0.881$\pm$ 0.020
(b)\[$50$/$\sinma$\] $1$ 0.875$\km$ 0.020 0.558$\pm$ 0.031 0.814$\pj$ 0.024 0.907 $\po$ 0.018
(c)\[$20$/$\sigma$\] $15$ 0.875$\pm$ 0.020 0.608$\pm$ 0.030 0.769$\pm$ 0.026 0.893$\pm$ 0.019
(d)\[$20$/$\rigma$\] ... | pm$ 0.030 0.815$\pm$ 0.024 0.881$\pm$ 0.020 (b)\[$50$/$\sigma$\] 0.020 0.031 0.814$\pm$ 0.907 $\pm$ 0.018 0.030 0.026 0.893$\pm$ 0.019 $1$ 0.900$\pm$ 0.019 0.030 0.793$\pm$ 0.025 0.892$\pm$ 0.019 : frequencies \[with precision $\pm 95\%$\] of the CoRP, CoLP, CoRLaP, CENeP, the Early-Stopped and the $2$-PN CoLP based on... | pm$ 0.030 0.815$\pm$ 0.024 0.881$\pm$ 0.020
(b)\[$50$/$\sigma$\] $1$ 0.875$\pm$ 0.020 0.558$\pm$ 0.031 0.814$\pm$ 0.024 0.907 $\pm$ 0.018
(c)\[$20$/$\sigmA$\] $15$ 0.875$\pm$ 0.020 0.608$\pm$ 0.030 0.769$\pm$ 0.026 0.893$\pm$ 0.019
(d)\[$20$/$\Sigma$\] ... | pm$ 0.030 0.815$\pm$ 0. 024 0. 881$\ pm$ 0. 02 0
(b)\[$50$/$\si g ma$\ ] $1$ 0.875 $\pm$ 0 . 020 0 .558$ \pm$ 0. 0 31 0 .8 14 $\p m$ 0. 024 0 .907 $\ pm$ 0.018 (c)\[$20$/$ \ si gma$\] $15$ 0.8 75$ \pm$ 0 .0 20 0.60 8$\ pm$ 0 .030 0.769 $\pm$ 0.0 26 0. 8 93$\pm$ 0 .0 19
(... | pm$ 0.030_ _ 0.815$\pm$ 0.024 _ __0.881$\pm$ 0.020
__ _ (b)\[$50$/$\sigma$\] _ _ $1$ __ 0.875$\pm$ 0.020 0.558$\pm$ 0.031 0.814$\pm$ 0.024__ _0.907_$\pm$_0.018
_ (c)\[$20$/$\sigma$\] _ _ $15$ 0.875$\pm$ 0.020__ 0.608$\pm$ 0.030_ 0.769$\pm$ 0.026 _0.893$\pm$ 0.019
_ (d)\[$20$/$\sigma$\] ___ ... |
much above the electroweak scale. However, if the family symmetry is global, and broken only spontaneously, then a massless Goldstone boson, the familon, will appear [@w82].
As an application of the mechanism of Fig. 3, consider the non-Abelian discrete symmetry $A_4$, the group of the even permutation of 4 objects w... | much above the electroweak scale. However, if the family symmetry is ball-shaped, and break in only spontaneously, then a massless Goldstone boson, the familon, will look [ @w82 ].
As an application of the mechanism of Fig. 3, regard the non - Abelian discrete symmetry $ A_4 $, the group of the tied permutation of... | mufh above the electroweak scale. However, nd the hamily aymmetry is global, and broken only s'ontqneouwly, then a massless Goudstone blson, the fammlon, will appear [@w82].
As an appliczbion mh the mechanism of Fig. 3, cotsider the non-Dbdlnan discrete symmetry $A_4$, the group of the evrn permutation os 4 ontecta w... | much above the electroweak scale. However, if symmetry global, and only spontaneously, then familon, appear [@w82]. As application of the of Fig. 3, consider the non-Abelian symmetry $A_4$, the group of the even permutation of 4 objects which is the symmetry group of the tetrahedron. It has been discussed [@mr01; @bmv0... | much above the electroweak scAle. However, If the FamIly SyMmetRy is Global, and brokeN Only Spontaneously, then a massLess GOlDStonE BoSon, thE familoN, WiLL AppEaR [@w82].
as aN aPPlIcatiOn oF the mecHanism of FiG. 3, coNsIder the non-AbELiAn discrete SymMetry $A_4$, the groUp oF the evEn PerMUtatiOn oF 4 objeCts w... | much above the electrowea k scale. H oweve r,ifth e fa mily symmetry is g l obal , and broken only spon taneo us l y, t h en a ma sslessG ol d s ton ebo son ,t he fami lon , willappear [@w 82] .
As an appli c at ion of the me chanism of F ig. 3, co ns ide r thenon -Abel ian di s cretesymmetry$A _ 4$, th e group... | much_above the_electroweak scale. However, if_the family_symmetry_is global,_and_broken only spontaneously,_then a massless_Goldstone boson, the familon,_will appear [@w82].
As_an_application of the mechanism of Fig. 3, consider the non-Abelian discrete symmetry $A_4$, the group_of_the even_permutation_of_4 objects w... |
_B(R/B)+1\leq n$. Then, by the same argument of proof of Theorem \[thB\], it follows that $\m_B$ is a self-dual ideal of $R$ satisfying $\ell_R(R/\m_B)\leq n$. In this case, Question \[Q1\] is affirmative for $R$.
0
Let $R=k[[t^3,t^{10},t^{11}]]$ be a numerical semigroup ring, where $k$ is a field. We give some obser... | _ B(R / B)+1\leq n$. Then, by the same argument of proof of Theorem \[thB\ ], it follows that $ \m_B$ is a self - double ideal of $ R$ meet $ \ell_R(R/\m_B)\leq n$. In this case, Question \[Q1\ ] is affirmative for $ R$.
0
Let $ gas constant = k[[t^3,t^{10},t^{11}]]$ be a numerical semigroup ring, where $ k$ is a... | _B(R/B)+1\peq n$. Then, by the same avgument of proof of Themrem \[tgB\], it foulows that $\m_B$ is a self-dual mdeao of $E$ satisfying $\ell_R(R/\m_B)\led n$. In thps case, Qyestmon \[Q1\] is affirmavjve for $R$.
0
Let $D=n[[t^3,t^{10},t^{11}]]$ ue a numerical xemigroup sing, where $k$ iv x yield. We give some obser... | _B(R/B)+1\leq n$. Then, by the same argument of \[thB\], it that $\m_B$ is satisfying n$. In this Question \[Q1\] is for $R$. 0 Let $R=k[[t^3,t^{10},t^{11}]]$ be numerical semigroup ring, where $k$ is a field. We give some observation on \[Q1\] fo $R$. $R$ has minimal multiplicity and is not almost Gorenstein. By $bg(R... | _B(R/B)+1\leq n$. Then, by the same argumEnt of proof Of TheOreM \[thb\], iT folLows That $\m_B$ is a self-dUAl idEal of $R$ satisfying $\ell_R(R/\m_b)\leq n$. in THis cASe, questIon \[Q1\] is aFFiRMAtiVe FoR $R$.
0
LEt $r=K[[t^3,T^{10},t^{11}]]$ be a NumErical sEmigroup riNg, wHeRe $k$ is a field. WE GiVe some obseR... | _B(R/B)+1\leq n$. Then, by the sameargum ent of p roof ofTheorem \[thB\ ] , it follows that $\m_B$ i s a s el f -dua l i dealof $R$s at i s fyi ng $ \el l_ R (R /\m_B )\l eq n$.In this ca se, Q uestion \[Q1 \ ]is affirma tiv e for $R$.
0
Let $R =k [[t ^ 3,t^{ 10} ,t^{1 1}]]$b e a nu merical s em i groupr ing, w... | _B(R/B)+1\leq n$._Then, by_the same argument of_proof of_Theorem_\[thB\], it_follows_that $\m_B$ is_a self-dual ideal_of $R$ satisfying $\ell_R(R/\m_B)\leq_n$. In this_case,_Question \[Q1\] is affirmative for $R$.
0
Let $R=k[[t^3,t^{10},t^{11}]]$ be a numerical semigroup ring, where $k$_is_a field._We_give_some obser... |
O_h$) space group symmetry. In this [*whole-crystal*]{} symmetry, we first focus on the Co lattice by ignoring the first-neighboring Mn and Si atoms. The lattice is assumed to be a simple cubic composed by the second-neighboring Co at different sublattices in the primitive cell, which leads to the Co sitting at $O_h$ [... | O_h$) space group symmetry. In this [ * whole - crystal * ] { } isotropy, we foremost concenter on the Co lattice by ignore the beginning - neighboring Mn and Si atoms. The wicket is assumed to be a simple cubic composed by the second - neighbor Co at different sublattices in the archaic cell, which leads to the cobalt... | O_h$) dpace group symmetry. In uhis [*whole-crystal*]{} symmetcy, we fjrst focjs on the Co lattice by ignocing the dirst-neighboring Mn ana Si atomd. The lartict is assumed to bx a simple cubjg com'owed by the secpnd-neighbosing Co at difxefeut sublattices in the primitive cell, which kewds to the Co fittpnd at $O_h$ [... | O_h$) space group symmetry. In this [*whole-crystal*]{} first on the lattice by ignoring atoms. lattice is assumed be a simple composed by the second-neighboring Co at sublattices in the primitive cell, which leads to the Co sitting at $O_h$ symmetry. Second, our focus is turned on the tetrahedral ($T_d$) [*site*]{} sy... | O_h$) space group symmetry. In thiS [*whole-crysTal*]{} syMmeTry, We FirsT focUs on the Co lattiCE by iGnoring the first-neighboRing MN aND Si aTOmS. The lAttice iS AsSUMed To Be A siMpLE cUbic cOmpOsed by tHe second-neIghBoRing Co at diffEReNt sublattiCes In the primitiVe cEll, whiCh LeaDS to thE Co SittiNg at $O_h$ [... | O_h$) space group symmetry . In this[*who le- cry st al*] {} s ymmetry, we fi r st f ocus on the Co lattice by i gn o ring th e fir st-neig h bo r i ngMn a ndSi at oms.The lattic e is assum edto be a simple cu bic compos edby the secon d-n eighbo ri ngC o atdif feren t subl a ttices in the p ri m itivec ell, wh i ... | O_h$) space_group symmetry._In this [*whole-crystal*]{} symmetry,_we first_focus_on the_Co_lattice by ignoring_the first-neighboring Mn_and Si atoms. The_lattice is assumed_to_be a simple cubic composed by the second-neighboring Co at different sublattices in the_primitive_cell, which_leads_to_the Co sitting at $O_h$_[... |
aligned}$$
The above properties (\[eab0\]), (\[eab4\]), (\[eab\]), and $\mu^\emptyset$ an identity, are all we need from $\otimes$ to get the results of this work.
Note that commutation and associativity imply that $\bigotimes_{L\in {{\cal D}}}\mu_L\in {\cal P}_I$ is well-defined for a partition ${{\cal D}}$ of $I$.
... | aligned}$$
The above properties (\[eab0\ ]), (\[eab4\ ]), (\[eab\ ]), and $ \mu^\emptyset$ an identity, are all we need from $ \otimes$ to get the results of this workplace.
notice that commutation and associativity imply that $ \bigotimes_{L\in { { \cal D}}}\mu_L\in { \cal P}_I$ is well - define for a division $... | alihned}$$
The above properties (\[eab0\]), (\[eab4\]), (\[eab\]), anb $\mu^\empvyset$ ah identigy, are all we need from $\otimxs$ ti get the results of this wurk.
Note tjat commytatmon and associatmbity imijy tgwt $\bngitimes_{L\in {{\cal C}}}\mu_L\in {\cal P}_I$ is well-defhndd for a partition ${{\cal D}}$ of $I$.
... | aligned}$$ The above properties (\[eab0\]), (\[eab4\]), (\[eab\]), an are all need from $\otimes$ this Note that commutation associativity imply that {{\cal D}}}\mu_L\in {\cal P}_I$ is well-defined a partition ${{\cal D}}$ of $I$. For $\otimes$ the product measure we have x\in \prod_{i\in I}A_i: \quad \bigotimes_{L\in ... | aligned}$$
The above properties (\[Eab0\]), (\[eab4\]), (\[eab\]), aNd $\mu^\eMptYseT$ aN ideNtitY, are all we need fROm $\otImes$ to get the results of tHis woRk.
nOte tHAt CommuTation aND aSSOciAtIvIty ImPLy That $\bIgoTimes_{L\iN {{\cal D}}}\mu_L\in {\Cal p}_I$ Is well-defineD FoR a partitioN ${{\caL D}}$ of $I$.
... | aligned}$$
The above prop erties (\[ eab0\ ]), (\ [e ab4\ ]),(\[eab\]), and $\mu ^\emptyset$ an identit y, ar ea ll w e n eed f rom $\o t im e s $ t oge t t he re sults of this w ork.
Note th at commutation an d associat ivi ty imply tha t $ \bigot im es_ { L\in{{\ cal D }}}\mu _ L\in { \cal P}_I $i s well - de... | aligned}$$
The above_properties (\[eab0\]),_(\[eab4\]), (\[eab\]), and $\mu^\emptyset$_an identity,_are_all we_need_from $\otimes$ to_get the results_of this work.
Note that_commutation and associativity_imply_that $\bigotimes_{L\in {{\cal D}}}\mu_L\in {\cal P}_I$ is well-defined for a partition ${{\cal D}}$ of_$I$.
... |
& &&&1&&\Red{\rightarrow n+1}\\
& & & &+ \Green{n}\ *\quad&{{\textstyle\frac{1}{2}}} & & {{\textstyle\frac{1}{2}}} &\\
& & & & & \Green{0} & & \Green{0}&
\end{array}$$ The supergravity sector contains the graviton, 2 gravitini and ... | & & & & 1&&\Red{\rightarrow n+1}\\
& & & & + \Green{n}\ * \quad&{{\textstyle\frac{1}{2 } } } & & { { \textstyle\frac{1}{2 } } } & \\
& & & & & \Green{0 } & & \Green{0 } &
\end{array}$$ The supergravity sector contains the graviton, 2 ... | & &&&1&&\Red{\vightarrow n+1}\\
& & & &+ \Green{n}\ *\quad&{{\textstyle\frar{1}{2}}} & & {{\rextsujle\frac{1}{2}}} &\\
& & & & & \Grxsn{0} & & \Nxeen{0}&
\ehf{arrcy}$$ The supergravlty sector wontains the gsaxicon, 2 gravitini and ... | & &&&1&&\Red{\rightarrow n+1}\\ & & & &+ & {{\textstyle\frac{1}{2}}} &\\ & & & \end{array}$$ supergravity sector contains graviton, 2 gravitini a so-called graviphoton. That spin-1 field by coupling to $n$ vector multiplets, part of a set of vectors, which be uniformly described by the special [Kähler]{} geometry. The ... | & &&&1&&\Red{\rightarrow n+1}\\
& & & &+ \Green{n}\ *\quad&{{\tExtstyle\frAc{1}{2}}} & & {{\texTstYle\FrAc{1}{2}}} &\\
& & & & & \GrEen{0} & & \GReen{0}&
\end{array}$$ ThE SupeRgravity sector contains The grAvITon, 2 gRAvItini And ... | & & &&1&&\Red{ \righ tar row n +1}\ \
& & & &+ \Green{n }\ *\ qu a d&{{ \ te xtsty le\frac { 1} { 2 }}} & & {{ \t e xt style \fr ac{1}{2 }}} &\\
& & & & & \G reen{0 } && \Gr een {0}&\end{a r ray}$$ The supe rg r avitys ector c o n ta i... | _ _ _ __ __ _ & _ _&&&1&&\Red{\rightarrow n+1}\\
__ & & __ ___ & _ &+ \Green{n}\ *\quad&{{\textstyle\frac{1}{2}}}_& &_{{\textstyle\frac{1}{2}}} &\\
& & __ _ _ & _ _&__ _ &_ \Green{0} & &_ \Green{0}&
\end{array}$$ The supergravity sector contains the_graviton, 2 gravitini and ... |
+0.010\pm0.007$ $0.002\pm0.001$
2 0.250 – 0.500 $0.146\pm 0.009$ $0.321\pm0.008$ $-0.022\pm0.010$ $0.019\pm0.002$
3 0.500 – 0.625 $0.108\pm 0.008$ $0.224\pm0.011$ $+0.031\pm0.011$ $0.033\pm0.004$
4 0.625 – 0.750 $0.107\pm 0.008$ $0.157\pm0.010$ $+0.002\pm0.011$ $0.051\pm0.005... | +0.010\pm0.007 $ $ 0.002\pm0.001 $
2 0.250 – 0.500 $ 0.146\pm 0.009 $ $ 0.321\pm0.008 $ $ -0.022\pm0.010 $ $ 0.019\pm0.002 $
3 0.500 – 0.625 $ 0.108\pm 0.008 $ $ 0.224\pm0.011 $ $ +0.031\pm0.011 $ $ 0.033\pm0.004 $
4 0.625 – 0.750 $ 0.107\pm 0.008 $ $ 0.157\pm... | +0.010\pm0.007$ $0.002\pm0.001$
2 0.250 – 0.500 $0.146\pm 0.009$ $0.321\pm0.008$ $-0.022\pm0.010$ $0.019\pm0.002$
3 0.500 – 0.625 $0.108\pm 0.008$ $0.224\lm0.011$ $+0.031\pm0.011$ $0.033\pm0.004$
4 0.625 – 0.750 $0.107\pm 0.008$ $0.157\pm0.010$ $+0.002\pl0.011$ $0.051\pm0.005... | +0.010\pm0.007$ $0.002\pm0.001$ 2 0.250 – 0.500 $0.146\pm $-0.022\pm0.010$ 3 0.500 0.625 $0.108\pm 0.008$ – $0.107\pm 0.008$ $0.157\pm0.010$ $0.051\pm0.005$ 5 0.750 0.875 $0.098\pm 0.007$ $0.109\pm0.009$ $-0.028\pm0.011$ $0.060\pm0.005$ 0.875 – 1.000 $0.144\pm 0.009$ $0.016\pm0.005$ $+0.007\pm0.007$ $0.135\pm0.009$ \[t... | +0.010\pm0.007$ $0.002\pm0.001$
2 0.250 – 0.500 $0.146\pm 0.009$ $0.321\pm0.008$ $-0.022\pm0.010$ $0.019\pm0.002$
3 0.500 – 0.625 $0.108\pm 0.008$ $0.224\pm0.011$ $+0.031\pm0.011$ $0.033\pm0.004$
4 0.625 – 0.750 $0.107\pm 0.008$ $0.157\pm0.010$ $+0.002\pM0.011$ $0.051\pm0.005... | +0.010\pm0.007$ $0.002\p m0.001$
2 0 .25 0– 0. 500 $0.146\pm 0. 0 09$ $0.321\pm0.008$ $-0 .022\ pm 0 .010 $ $0.0 19\pm0. 0 02 $ 3 0 .5 0 0– 0.6 25 $0.10 8\pm 0.008 $ $0 .224\pm0.011 $ $+0.031\p m0. 011$ $0.03 3\p m0.004 $ 4 0.6 25 –0.750 $0.10 7\pm 0.00 8$ $0.15 7 \pm0.01 0 $ $+0 .002\pm0.0... | +0.010\pm0.007$ _ $0.002\pm0.001$
_ 2_ __0.250 –_0.500_ $0.146\pm_0.009$ $0.321\pm0.008$_ $-0.022\pm0.010$ _ $0.019\pm0.002$
__ 3 0.500 – 0.625 $0.108\pm 0.008$ $0.224\pm0.011$__ $+0.031\pm0.011$___$0.033\pm0.004$
4_ 0.625 –_0.750 _ $0.107\pm 0.008$ $0.157\pm0.010$ $+0.002\pm0.011$__ $0.051\pm0.005... |
$\$_{A}$ increases from one to two. With an completely corrupt source, $r=1$, the payoffs become $(\$_{A},\$_{B})=(2,1)$. The reason for this is the same as explained for PD. When the quantum strategies with and without corrupt source are compared to the classical mixed strategy without noise, it is seen that the form... | $ \$_{A}$ increases from one to two. With an completely corrupt source, $ r=1 $, the return become $ (\$_{A},\$_{B})=(2,1)$. The rationality for this is the same as explained for PD. When the quantum strategies with and without crooked source are compared to the classical assorted strategy without noise, it is see that... |
$\$_{A}$ ijcreases from one to two. With an complejeoy corcupt sohrce, $r=1$, tfe payoffs become $(\$_{A},\$_{B})=(2,1)$. The readob for this is the same as ebplained vor PD. Wyen uhe quantum stratxfies wibk and aithmnt corrupt sourge are compdred to the cldsrieal mixed strategy without noise, it ys seen tjat the form... | $\$_{A}$ increases from one to two. With corrupt $r=1$, the become $(\$_{A},\$_{B})=(2,1)$. The same explained for PD. the quantum strategies and without corrupt source are compared the classical mixed strategy without noise, it is seen that the former ones give better payoffs to the players. However, when the source b... |
$\$_{A}$ increases from one to two. WitH an completEly coRruPt sOuRce, $r=1$, The pAyoffs become $(\$_{A},\$_{B})=(2,1)$. tHe reAson for this is the same as ExplaInED for pd. WHen thE quantuM StRATegIeS wIth AnD WiThout CorRupt souRce are compAreD tO the classicaL MiXed strategY wiThout noise, it Is sEen thaT tHe fORm... |
$\$_{A}$ increases from o ne to two. With an co mp lete ly c orrupt source, $r=1 $, the payoffs become$(\$_ {A } ,\$_ { B} )=(2, 1)$. Th e r e a son f or th is is thesam e as ex plained fo r P D. When the qu a nt um strateg ies with and wi tho ut cor ru pts ource ar e com paredt o theclassical m i xed st r ... |
$\$_{A}$ increases_from one_to two. With an_completely corrupt_source,_$r=1$, the_payoffs_become $(\$_{A},\$_{B})=(2,1)$. The_reason for this_is the same as_explained for PD._When_the quantum strategies with and without corrupt source are compared to the classical mixed_strategy_without noise,_it_is_seen that the form... |
1,2,-4,1)$, we find that $$\begin{gathered}
F(a):=
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{}F_{1}\biggl(\genfrac..{0pt}{}{a;2a,1-4a}{a+1/2};\frac{80}{81},
\frac{16}{15}\biggr) \endgroup
}\end{gathered}$$ has a closed form, and it s... | 1,2,-4,1)$, we find that $ $ \begin{gathered }
F(a):=
{ \begingroup
\begingroup\lccode`~= `,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip } \mathcode`,=\string"8000
{ } _ { } F_{1}\biggl(\genfrac.. {0pt}{}{a;2a,1 - 4a}{a+1/2};\frac{80}{81 },
\frac{16}{15}\biggr) \endgroup
} \end{gathere... | 1,2,-4,1)$, we find that $$\begin{gathered}
N(a):=
{ \begingroup
\ywgingrmup\lcckde`~=`,
\luwercase{\endgroup\def~}{\pFcomma\mkxrn\pDqskik} \mathcode`,=\string"8000
{}_{}W_{1}\biggl(\genvrac..{0pt}{}{a;2a,1-4q}{a+1/2};\frec{80}{81},
\frac{16}{15}\biggr) \endjdoup
}\end{ncthersf}$$ hav a closed form, and it s... | 1,2,-4,1)$, we find that $$\begin{gathered} F(a):= { \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} {}_{}F_{1}\biggl(\genfrac..{0pt}{}{a;2a,1-4a}{a+1/2};\frac{80}{81}, \frac{16}{15}\biggr) }\end{gathered}$$ has a the closed-form relation: $$\begin{gathered} \label{ex2_closed1}\end{gathered}$$ This implies $$\begin{... | 1,2,-4,1)$, we find that $$\begin{gathered}
F(a):=
{ \Begingroup
\BeginGroUp\lCcOde`~=`,
\lOwerCase{\endgroup\deF~}{\PFcoMma\mkern\pFqskip} \mathcodE`,=\striNg"8000
{}_{}f_{1}\BiggL(\GeNfrac..{0Pt}{}{a;2a,1-4a}{a+1/2};\fRAc{80}{81},
\FRAc{16}{15}\bIgGr) \EndGrOUp
}\End{gaTheRed}$$ has a Closed form, And It S... | 1,2,-4,1)$, we find that $ $\begin{ga there d}F(a ): =
{ \be gingroup
\be g ingr oup\lccode`~=`,
\l owerc as e {\en d gr oup\d ef~}{\p F co m m a\m ke rn \pF qs k ip } \m ath code`,= \string"80 00 {}_{}F_{1}\b i gg l(\genfrac ..{ 0pt}{}{a;2a, 1-4 a}{a+1 /2 };\ f rac{8 0}{ 81},\frac{ 1 6}{15} \biggr) \e n dgroup ... | 1,2,-4,1)$, we_find that_$$\begin{gathered}
F(a):=
{ \begingroup
_\begingroup\lccode`~=`,
__ \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip}__\mathcode`,=\string"8000
{}_{}F_{1}\biggl(\genfrac..{0pt}{}{a;2a,1-4a}{a+1/2};\frac{80}{81},
\frac{16}{15}\biggr)_ \endgroup
}\end{gathered}$$ has_a closed form, and_it s... |
R_g(x^\prime)\leq R_g(x)) \rho(x^\prime |\lambda)$, where $R_g(x)$ is radius of gyration for the polymer configuration $x$ and where $\Theta(a \leq b)$ is unity when the inequality in the argument is satisfied and zero otherwise. With this definition, $f(x)$ is the probability that another configuration drawn at random... | R_g(x^\prime)\leq R_g(x) ) \rho(x^\prime |\lambda)$, where $ R_g(x)$ is radius of gyration for the polymer configuration $ x$ and where $ \Theta(a \leq b)$ is unity when the inequality in the controversy is quenched and zero otherwise. With this definition, $ f(x)$ is the probability that another configuration trace at... | R_g(x^\orime)\leq R_g(x)) \rho(x^\prime |\lxmbda)$, where $R_g(x)$ is radmus of fyration for the polymer configuratiln $x$ ane where $\Theta(a \leq b)$ ir unity wjen the uneqnality in the arjhment is satianied cnv zero otherwisg. With this gefinition, $f(x)$ hs tke probability that another configurwtion dtaan at random... | R_g(x^\prime)\leq R_g(x)) \rho(x^\prime |\lambda)$, where $R_g(x)$ is gyration the polymer $x$ and where when inequality in the is satisfied and otherwise. With this definition, $f(x)$ is probability that another configuration drawn at random from $\rho$ has a larger radius gyration than the configuration $x$. With all... | R_g(x^\prime)\leq R_g(x)) \rho(x^\prime |\laMbda)$, where $R_G(x)$ is rAdiUs oF gYratIon fOr the polymer coNFiguRation $x$ and where $\Theta(a \lEq b)$ is UnITy whEN tHe ineQuality IN tHE ArgUmEnT is SaTIsFied aNd zEro otheRwise. With tHis DeFinition, $f(x)$ is THe ProbabilitY thAt another conFigUratioN dRawN At ranDom... | R_g(x^\prime)\leq R_g(x))\rho(x^\pr ime | \la mbd a) $, w here $R_g(x)$ is r a dius of gyration for the p olyme rc onfi g ur ation $x$ an d w h e re$\ Th eta (a \l eq b) $ i s unity when theine qu ality in the ar gument issat isfied and z ero other wi se. Withthi s def initio n , $f(x )$ is the p r obabil i ty t... | R_g(x^\prime)\leq R_g(x))_\rho(x^\prime |\lambda)$,_where $R_g(x)$ is radius_of gyration_for_the polymer_configuration_$x$ and where_$\Theta(a \leq b)$_is unity when the_inequality in the_argument_is satisfied and zero otherwise. With this definition, $f(x)$ is the probability that another_configuration_drawn at_random... |
$$|\psi _m^{(\beta )}(x) - \sigma_{1 - 1/\beta }^{(\beta )}(m)x| = O \bigg( x^{1/2} \log^2 T + \frac{x^{1+\varepsilon}\log x}{T} + \frac{{x{{\log }^2}T}}{T} \bigg)$$ provided that $x$ is an integer. Taking $T = x^{1/2}$ leads to $$\begin{aligned}
\psi _m^{(\beta )}(x) & = \sigma_{1 - 1/\beta }^{(\beta )}(m)x + O({x^{1... | $ $ |\psi _ m^{(\beta) } (x) - \sigma_{1 - 1/\beta } ^{(\beta) } (m)x| = O \bigg (x^{1/2 } \log^2 T + \frac{x^{1+\varepsilon}\log x}{T } + \frac{{x{{\log } ^2}T}}{T } \bigg)$$ provided that $ x$ is an integer. Taking $ T = x^{1/2}$ lead to $ $ \begin{aligned }
\psi _ m^{(\beta) } (x) & = \sigma_{1 - 1/\beta } ^{(\bet... | $$|\psl _m^{(\beta )}(x) - \sigma_{1 - 1/\beta }^{(\btta )}(m)x| = O \bigg( x^{1/2} \log^2 T + \hrac{x^{1+\vadepsilon}\uog x}{T} + \frac{{x{{\log }^2}T}}{T} \bigg)$$ protidee thau $x$ is an integer. Txking $T = q^{1/2}$ leads ti $$\bejin{aligned}
\psi _m^{(\bxfa )}(x) & = \sigma_{1 - 1/\neta }^{(\yeva )}(m)x + O({x^{1... | $$|\psi _m^{(\beta )}(x) - \sigma_{1 - 1/\beta = \bigg( x^{1/2} T + \frac{x^{1+\varepsilon}\log provided $x$ is an Taking $T = leads to $$\begin{aligned} \psi _m^{(\beta )}(x) = \sigma_{1 - 1/\beta }^{(\beta )}(m)x + O({x^{1/2}}{\log ^2}{x} + x^{1/2+\varepsilon}\log x ) \sigma_{1 - 1/\beta }^{(\beta )}(m)x + O({x^{1/2+... | $$|\psi _m^{(\beta )}(x) - \sigma_{1 - 1/\beta }^{(\beta )}(m)x| = O \Bigg( x^{1/2} \log^2 T + \fRac{x^{1+\vArePsiLoN}\log X}{T} + \frAc{{x{{\log }^2}T}}{T} \bigg)$$ prOVideD that $x$ is an integer. TakinG $T = x^{1/2}$ leAdS To $$\beGIn{AlignEd}
\psi _m^{(\bETa )}(X) & = \SIgmA_{1 - 1/\bEtA }^{(\beTa )}(M)X + O({X^{1... | $$|\psi _m^{(\beta )}(x)- \sigma_{ 1 - 1 /\b eta } ^{(\ beta )}(m)x| = O \ b igg( x^{1/2} \log^2 T + \f rac{x ^{ 1 +\va r ep silon }\log x } {T } + \ fr ac {{x {{ \ lo g }^2 }T} }{T} \b igg)$$ pro vid ed that $x$ is an integer.Tak ing $T = x^{ 1/2 }$ lea ds to $$\be gin {alig ned}
\ p si _m^ {(\beta ) }( x ) & =... | $$|\psi__m^{(\beta )}(x)_- \sigma_{1 - 1/\beta_}^{(\beta )}(m)x|_=_O \bigg(_x^{1/2}_\log^2 T +_\frac{x^{1+\varepsilon}\log x}{T} +_\frac{{x{{\log }^2}T}}{T} \bigg)$$ provided_that $x$ is_an_integer. Taking $T = x^{1/2}$ leads to $$\begin{aligned}
\psi _m^{(\beta )}(x) & = \sigma_{1 -_1/\beta_}^{(\beta )}(m)x_+_O({x^{1... |
$ hadronisation corrections might not have made it past early childhood). Rather, various theoretical objections ([@NasonSeymour]) and the gradual appearance of new data, especially for the broadenings, forced people to refine their ideas.
[0.42]{}
Among the developments was the realisation that to control the normal... | $ hadronisation corrections might not have made it past early childhood). Rather, various theoretical protest ([ @NasonSeymour ]) and the gradual appearance of newfangled data, especially for the broadenings, impel people to refine their estimate.
[ 0.42 ] { }
Among the development was the realization that to con... | $ hafronisation corrections oight not have made it past sarly chkldhood). Rather, various theorxticql obhections ([@NasonSeymour]) xnd the ggadual appearence of new data, especially fod the uroadenings, forged people do refine theis kdzas.
[0.42]{}
Among the developments was the rewlisatipn that to contrjl tnq nodmal... | $ hadronisation corrections might not have made early Rather, various objections ([@NasonSeymour]) and data, for the broadenings, people to refine ideas. [0.42]{} Among the developments was realisation that to control the normalisation of the $c_{\mathcal{V}}$ it is necessary to into account the decay of the massive, v... | $ hadronisation corrections mIght not havE made It pAst EaRly cHildHood). Rather, variOUs thEoretical objections ([@NasOnSeyMoUR]) and THe GraduAl appeaRAnCE Of nEw DaTa, eSpECiAlly fOr tHe broadEnings, forcEd pEoPle to refine tHEiR ideas.
[0.42]{}
AmonG thE developmentS waS the reAlIsaTIon thAt tO contRol the NOrmal... | $ hadronisation correction s might no t hav e m ade i t pa st e arly childhood ) . Ra ther, various theoreti cal o bj e ctio n s([@Na sonSeym o ur ] ) an dth e g ra d ua l app ear ance of new data, es pe cially for t h ebroadening s,forced peopl e t o refi ne th e ir id eas .
[0 .42]{}
Among the deve lo p ments... | $ hadronisation_corrections might_not have made it_past early_childhood)._Rather, various_theoretical_objections ([@NasonSeymour]) and_the gradual appearance_of new data, especially_for the broadenings,_forced_people to refine their ideas.
[0.42]{}
Among the developments was the realisation that to control the_normal... |
_i + b_0$$ where $b_0$ is the average rating, $b_u$ and $b_i$ are user and item biases, respectively. The estimation of parameters is to minimize the rating prediction error in the training dataset. The optimization objective function is $$\label{eq:ojf}
\begin{split}
\vspace{-2pt}
\underset{p*,q*}{... | _ i + b_0$$ where $ b_0 $ is the average rating, $ b_u$ and $ b_i$ are user and item bias, respectively. The estimate of parameters is to minimize the rating prediction erroneousness in the training dataset. The optimization objective function is $ $ \label{eq: ojf }
\begin{split }
\vspace{-2pt }
... | _i + h_0$$ where $b_0$ is the average rating, $b_u$ and $y_u$ are nser ans item bkases, respectively. The estimetiob of kcrameters is to minioize the gating prwdicuion error in the traininn datzdet. Chx optimization pbjective xunction is $$\lateu{ee:ojf}
\begin{split}
\vspace{-2pt}
\inferset{p*,q*}{... | _i + b_0$$ where $b_0$ is the $b_u$ $b_i$ are and item biases, is minimize the rating error in the dataset. The optimization objective function is \begin{split} \vspace{-2pt} \underset{p*,q*}{\text{min}} \frac{1}{2}\sum_{u,i} (r_{u,i} & -\hat{r}_{u,i})^2 + \frac{\mu_u}{2} ||\bm{p_u}||_2^2 + \frac{\mu_i}{2} ||\bm{q_i}||... | _i + b_0$$ where $b_0$ is the average ratinG, $b_u$ and $b_i$ arE user And IteM bIaseS, resPectively. The esTImatIon of parameters is to minImize ThE RatiNG pRedicTion errOR iN THe tRaInIng DaTAsEt. The OptImizatiOn objectivE fuNcTion is $$\label{eQ:OjF}
\begin{spliT}
\vsPace{-2pt}
\undersEt{p*,Q*}{... | _i + b_0$$ where $b_0$ isthe averag e rat ing , $ b_ u$ a nd $ b_i$ are usera nd i tem biases, respective ly. T he esti m at ion o f param e te r s is t omin im i ze therat ing pre diction er ror i n the traini n gdataset. T heoptimization ob jectiv efun c tionis$$\la bel{eq : ojf}
\begin {s p lit}
\ v s ... | _i +_b_0$$ where_$b_0$ is the average_rating, $b_u$_and_$b_i$ are_user_and item biases,_respectively. The estimation_of parameters is to_minimize the rating_prediction_error in the training dataset. The optimization objective function is $$\label{eq:ojf}
_\begin{split}
_ ___ \vspace{-2pt}
_ _ \underset{p*,q*}{... |
alpha]. Both dots show excited states at $\Delta \epsilon^* \approx 120~\mu$eV. In Fig. \[properties\](a) the charging diagram of the coupled dot is plotted in a linear grayscale representation in the weak coupling regime ($G_{c} \approx 0.08~e^2/h$). In linear transport, only two ground states participate, e.g., the g... | alpha ]. Both dots show excited states at $ \Delta \epsilon^ * \approx 120~\mu$eV. In Fig. \[properties\](a) the charge diagram of the conjugate dot is plotted in a analogue grayscale representation in the decrepit coupling regimen ($ G_{c } \approx 0.08 ~ e^2 / h$). In analogue transport, only two ground states part... | alpja]. Both dots show excitea states at $\Delja \epsilmn^* \appdox 120~\mu$eV. In Fig. \[properties\](a) the chargmng eiagrqm of the coupled dot ks plottef in a luneac grayscale reprxaentation in fme weck coupling regike ($G_{c} \apprmx 0.08~e^2/h$). In lineas grcnsport, only two ground states partisipate, r.g., the g... | alpha]. Both dots show excited states at \approx In Fig. the charging diagram plotted a linear grayscale in the weak regime ($G_{c} \approx 0.08~e^2/h$). In linear only two ground states participate, e.g., the ground state $E_l$ in the left and $E_r$ in the right dot, as schematically shown in Fig. \[properties\](b) [@... | alpha]. Both dots show excited sTates at $\DelTa \epsIloN^* \apPrOx 120~\mu$EV. In fig. \[properties\](a) THe chArging diagram of the coupLed doT iS PlotTEd In a liNear graYScALE rePrEsEntAtIOn In the WeaK coupliNg regime ($G_{c} \AppRoX 0.08~e^2/h$). In linear tRAnSport, only tWo gRound states pArtIcipatE, e.G., thE G... | alpha]. Both dots show exc ited state s at$\D elt a\eps ilon ^* \approx 120 ~ \mu$ eV. In Fig. \[properti es\]( a) thec ha rging diagra m o f the c ou ple dd ot is p lot ted ina linear g ray sc ale represen t at ion in the we ak couplingreg ime ($ G_ {c} \appr ox0.08~ e^2/h$ ) . In l inear tra ns p ort, o n ly two... | alpha]. Both_dots show_excited states at $\Delta_\epsilon^* \approx_120~\mu$eV._In Fig. \[properties\](a)_the_charging diagram of_the coupled dot_is plotted in a_linear grayscale representation_in_the weak coupling regime ($G_{c} \approx 0.08~e^2/h$). In linear transport, only two ground states_participate,_e.g., the_g... |
Geom.* **25** (1987), no.1, 55–73.
A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization, *J. Reine Angew. Math.*, **417** (1991), 159–189.
A. Weinstein, Lagrangian mechanics and groupoids, *Fields Inst. Commun.*, **7**, (1996), 207–231.
A. Weinstein, Linearization of regular proper groupoids, *J.... | Geom. * * * 25 * * (1987), no.1, 55–73.
A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization, * J. Reine Angew. Math. *, * * 417 * * (1991), 159–189.
A. Weinstein, Lagrangian mechanics and groupoids, * Fields Inst. Commun. *, * * 7 * *, (1996), 207–231.
A. Weinstein, Linearization o... | Geol.* **25** (1987), no.1, 55–73.
A. Weinstein, P. Xu, Exuensions of symplgcric grmupoida and quxntization, *J. Reine Angew. Mati.*, **417** (1991), 159–189.
Q. Weinwtein, Lagrangian mechavics and hroupoidw, *Fitlds Inst. Commun.*, **7**, (1996), 207–231.
A. Weinsbzin, Ljkearivavion of regular proper grmupoids, *J.... | Geom.* **25** (1987), no.1, 55–73. A. Weinstein, Extensions symplectic groupoids quantization, *J. Reine A. Lagrangian mechanics and *Fields Inst. Commun.*, (1996), 207–231. A. Weinstein, Linearization of proper groupoids, *J. Inst. Math. Jussieu* **1** (2002), no.3, 493–511. M. Zambon, Submanifolds Poisson geometry: a... | Geom.* **25** (1987), no.1, 55–73.
A. Weinstein, P. Xu, ExtensIons of sympLectiC grOupOiDs anD quaNtization, *J. ReinE angeW. Math.*, **417** (1991), 159–189.
A. Weinstein, LagrangIan meChANics ANd GroupOids, *FieLDs iNSt. COmMuN.*, **7**, (1996), 207–231.
A. WEiNStEin, LiNeaRizatioN of regular ProPeR groupoids, *J.... | Geom.* **25** (1987), no.1 , 55–73.
A. We ins tei n, P.Xu,Extensions ofs ympl ectic groupoids and qu antiz at i on,* J. Rein e Angew . M a t h.* ,** 417 ** (1 991), 15 9–189.
A. Weinst ein ,Lagrangian m e ch anics andgro upoids, *Fie lds Inst. C omm u n.*,**7 **, ( 1996), 207–23 1.
A. We in s tein,L ineariz a t ... | Geom.* **25**_(1987), no.1,_55–73.
A. Weinstein, P. Xu, Extensions of_symplectic groupoids_and_quantization, *J._Reine_Angew. Math.*, **417**_(1991), 159–189.
A. Weinstein, Lagrangian_mechanics and groupoids, *Fields_Inst. Commun.*, **7**,_(1996),_207–231.
A. Weinstein, Linearization of regular proper groupoids, *J.... |
) = \min_{\pi'} \max_{1 \leq t \leq k'} \pi'(t)$, where $\pi'$ ranges over all paths starting at $\pi'(1) = \pi(t_1)$ and ending at $\pi'(k') = \pi(t_2)$.
$P$ and $Q$ are curves with $m$ and $n$ edges A locally correct discrete [Fréchet matching]{} for $P$ and $Q$
Construct grid $G$ for $P$ and $Q$ Let $T$ be a tree ... | ) = \min_{\pi' } \max_{1 \leq t \leq k' } \pi'(t)$, where $ \pi'$ ranges over all paths depart at $ \pi'(1) = \pi(t_1)$ and end at $ \pi'(k') = \pi(t_2)$.
$ P$ and $ Q$ are curves with $ m$ and $ n$ edges A locally correct discrete [ Fréchet matching ] { } for $ P$ and $ Q$
Construct grid $ G$ for $ P$ and $ Q$... | ) = \mln_{\pi'} \max_{1 \leq t \leq k'} \pi'(t)$, where $\pi'$ ranges over ell patgs startkng at $\pi'(1) = \pi(t_1)$ and ending at $\pu'(k') = \pu(t_2)$.
$P$ and $Q$ are curves wkth $m$ and $n$ edges A lixally corrxdt discvzte [Fdéghet ketching]{} for $P$ anc $Q$
Construwt grid $G$ for $[$ xnb $Q$ Let $T$ be a tree ... | ) = \min_{\pi'} \max_{1 \leq t \leq where ranges over paths starting at at = \pi(t_2)$. $P$ $Q$ are curves $m$ and $n$ edges A locally discrete [Fréchet matching]{} for $P$ and $Q$ Construct grid $G$ for $P$ and Let $T$ be a tree consisting only of the root $G[0,0]$ Add $G[i,0]$ $T$ $G[0,j]$ $T$ G, i, j)$ path in $T$ b... | ) = \min_{\pi'} \max_{1 \leq t \leq k'} \pi'(t)$, where $\pI'$ ranges oveR all pAthS stArTing At $\pi'(1) = \Pi(t_1)$ and ending at $\PI'(k') = \pi(T_2)$.
$P$ and $Q$ are curves with $m$ anD $n$ edgEs a LocaLLy CorreCt discrETe [fRÉchEt MaTchInG]{} FoR $P$ and $q$
CoNstruct Grid $G$ for $P$ aNd $Q$ leT $T$ be a tree ... | ) = \min_{\pi'} \max_{1 \l eq t \leqk'} \ pi' (t) $, whe re $ \pi'$ ranges o v er a ll paths starting at $ \pi'( 1) = \p i (t _1)$and end i ng a t $ \p i' (k' )= \ pi(t_ 2)$ .
$P$and $Q$ ar e c ur ves with $m$ an d $n$ edge s A locally cor rec t disc re te[ Fréch etmatch ing]{} for $P $ and $Q$
C onstru c t grid$ ... | ) =_\min_{\pi'} \max_{1_\leq t \leq k'}_\pi'(t)$, where_$\pi'$_ranges over_all_paths starting at_$\pi'(1) = \pi(t_1)$_and ending at $\pi'(k')_= \pi(t_2)$.
$P$ and_$Q$_are curves with $m$ and $n$ edges A locally correct discrete [Fréchet matching]{} for $P$_and_$Q$
Construct grid_$G$_for_$P$ and $Q$ Let $T$_be a tree ... |
&\leq \norm{T_{{\bm{x}}^{-1}}} \cdot \norm{{\Delta{\bm{x}}}- {{\overline{{\Delta{\bm{x}}}}}}}_F \leq \xi, \text{ and } \\
\norm{{{{\Delta{\bm{s}}}'}}- {{{{\overline{{\Delta{\bm{s}}}}}'}}}}_F &= \mu^{-1}\norm{T_{{\bm{x}}} {\Delta{\bm{s}}}- T_{{\bm{x}}} {{\overline{{\Delta{\bm{s}}}}}}}_F \\
&\leq \mu^{-1}\nor... | & \leq \norm{T_{{\bm{x}}^{-1 } } } \cdot \norm{{\Delta{\bm{x}}}- { { \overline{{\Delta{\bm{x}}}}}}}_F \leq \xi, \text { and } \\
\norm{{{{\Delta{\bm{s}}}'}}- { { { { \overline{{\Delta{\bm{s}}}}}'}}}}_F & = \mu^{-1}\norm{T_{{\bm{x } } } { \Delta{\bm{s}}}- T_{{\bm{x } } } { { \overline{{\Delta{\bm{s}}}}}}}_F \\
... | &\leq \norm{T_{{\bm{x}}^{-1}}} \cdot \norm{{\Dtlta{\bm{x}}}- {{\overline{{\Dglra{\bm{x}}}}}}}_F \leq \xj, \text{ avd } \\
\norm{{{{\Delta{\bm{s}}}'}}- {{{{\overlinx{{\Delra{\bm{s}}}}}'}}}}_D &= \mu^{-1}\norm{T_{{\bm{x}}} {\Delta{\bm{s}}}- T_{{\bm{x}}} {{\ovegline{{\Deltq{\bm{s}}}}}}}_H \\
&\leq \mu^{-1}\nor... | &\leq \norm{T_{{\bm{x}}^{-1}}} \cdot \norm{{\Delta{\bm{x}}}- {{\overline{{\Delta{\bm{x}}}}}}}_F \leq \xi, } \norm{{{{\Delta{\bm{s}}}'}}- {{{{\overline{{\Delta{\bm{s}}}}}'}}}}_F \mu^{-1}\norm{T_{{\bm{x}}} {\Delta{\bm{s}}}- T_{{\bm{x}}} {{\overline{{\Delta{\bm{s}}}}}}}_F &= \mu^{-1} \norm{{\bm{x}}}_2 {{\overline{{\Delta{... | &\leq \norm{T_{{\bm{x}}^{-1}}} \cdot \norm{{\Delta{\bM{x}}}- {{\overline{{\delta{\Bm{x}}}}}}}_f \leQ \xI, \texT{ and } \\
\Norm{{{{\Delta{\bm{s}}}'}}- {{{{\ovERlinE{{\Delta{\bm{s}}}}}'}}}}_F &= \mu^{-1}\norm{T_{{\bm{x}}} {\DeLta{\bm{S}}}- T_{{\BM{x}}} {{\ovERlIne{{\DeLta{\bm{s}}}}}}}_F \\
&\LEq \MU^{-1}\Nor... | &\leq \norm{T_{{\bm{x} }^{-1}}} \ cdot\no rm{ {\ Delt a{\b m{x}}}- {{\ove r line {{\Delta{\bm{x}}}}}}}_ F \le q\ xi,\ te xt{ a nd } \\ \no rm {{ {{\ De l ta {\bm{ s}} }'}}- { {{{\overli ne{ {\ Delta{\bm{s} } }} }'}}}}_F & = \ mu^{-1}\norm {T_ {{\bm{ x} }}{ \Delt a{\ bm{s} }}- T_ { {\bm{x }}} {{\ov er l ine{{\ D el... | _ _&\leq \norm{T_{{\bm{x}}^{-1}}} \cdot \norm{{\Delta{\bm{x}}}-_{{\overline{{\Delta{\bm{x}}}}}}}_F \leq_\xi,_\text{ and_}_\\
_ \norm{{{{\Delta{\bm{s}}}'}}- {{{{\overline{{\Delta{\bm{s}}}}}'}}}}_F_&= \mu^{-1}\norm{T_{{\bm{x}}} {\Delta{\bm{s}}}- T_{{\bm{x}}}_{{\overline{{\Delta{\bm{s}}}}}}}_F \\
__ &\leq \mu^{-1}\nor... |
for different time steps and extrapolating to zero time step or by simply choosing a small enough time step. An example of time-step extrapolation is shown in figure \[fig:time\_step\_errors\].
- Population control bias. The $f$ distribution is represented by a finite population of configurations which fluctuates d... | for different time steps and extrapolating to zero meter gradation or by simply choosing a small adequate time step. An example of clock time - step extrapolation is shown in human body \[fig: time\_step\_errors\ ].
- Population control bias. The $ f$ distribution is represent by a finite population of configurat... | fog different time steps akd extrapolating to zerm time step or by simply choosing a small xnouth tine step. An example of gime-step vxtrapolarion us shown ii figure \[fig:tijc\_step\_zrcors\].
- Populatipn control bias. The $f$ divtfiyution is represented by a finite po[ulatiom lf configuratijns erich fluctuates d... | for different time steps and extrapolating to step by simply a small enough time-step is shown in \[fig:time\_step\_errors\]. - Population bias. The $f$ distribution is represented a finite population of configurations which fluctuates due to branching. The population may controlled in various ways, but this introduces... | for different time steps and eXtrapolatiNg to zEro TimE sTep oR by sImply choosing a SMall Enough time step. An examplE of tiMe-STep eXTrApolaTion is sHOwN IN fiGuRe \[Fig:TiME\_sTep\_erRorS\].
- PopulaTion controL biAs. the $f$ distribuTIoN is represeNteD by a finite poPulAtion oF cOnfIGuratIonS whicH fluctUAtes d... | for different time stepsand extrap olati ngtoze ro t imestep or by sim p ly c hoosing a small enough time s t ep.A nexamp le of t i me - s tep e xt rap ol a ti on is sh own infigure \[f ig: ti me\_step\_er r or s\].
- Pop ulation cont rol bias. T he$ f$ di str ibuti on isr eprese nted by a f i nite p o pulati... | for_different time_steps and extrapolating to_zero time_step_or by_simply_choosing a small_enough time step._An example of time-step_extrapolation is shown_in_figure \[fig:time\_step\_errors\].
- Population control bias. The $f$ distribution is represented by a_finite_population of_configurations_which_fluctuates d... |
The probability of measuring some set of burst arrival times $t_\mathrm{1}, t_\mathrm{2}, \dots t_\mathrm{N}$ in a single observation of duration $T$ can be split into three parts:
1. The probability of the interval between the start of the observation and the first burst: $P(t_1)$
2. The probabilities of the inter... | The probability of measuring some set of burst arrival times $ t_\mathrm{1 }, t_\mathrm{2 }, \dots t_\mathrm{N}$ in a single notice of duration $ T$ can be separate into three parts:
1. The probability of the interval between the beginning of the notice and the first burst: $ P(t_1)$
2. The probabilities of t... | The probability of measurinn some set of butsr arrital timss $t_\mathfm{1}, t_\mathrm{2}, \dots t_\mathrm{N}$ in e sibgle ibservation of duratiov $T$ can bv split ibto uhree parts:
1. The 'dobabillcy of bhe iutxrval between tme start of the observatimn aud the first burst: $P(t_1)$
2. The probabilieies of tje inter... | The probability of measuring some set of times t_\mathrm{2}, \dots in a single be into three parts: The probability of interval between the start of the and the first burst: $P(t_1)$ 2. The probabilities of the intervals between subsequent in a single observation: $P(t_2 \dots t_\mathrm{N}) = \prod_\mathrm{i=1}^\mathrm... | The probability of measuring Some set of bUrst aRriVal TiMes $t_\MathRm{1}, t_\mathrm{2}, \dots t_\MAthrM{N}$ in a single observation Of durAtIOn $T$ cAN bE spliT into thREe PARts:
1. thE pRobAbILiTy of tHe iNterval Between the StaRt Of the observaTIoN and the firSt bUrst: $P(t_1)$
2. The proBabIlitieS oF thE Inter... | The probability of measuri ng some se t ofbur star riva l ti mes $t_\mathrm { 1},t_\mathrm{2}, \dots t_ \math rm { N}$i na sin gle obs e rv a t ion o fdur at i on $T$can be spl it into th ree p arts:
1. T h eprobabilit y o f the interv albetwee nthe start of theobserv a tion a nd the fi rs t burst : $P(t_1 ) ... | The probability_of measuring_some set of burst_arrival times_$t_\mathrm{1},_t_\mathrm{2}, \dots_t_\mathrm{N}$_in a single_observation of duration_$T$ can be split_into three parts:
1.__The probability of the interval between the start of the observation and the first_burst:_$P(t_1)$
2. _The_probabilities_of the inter... |
D}}_0$ which makes the left diagram in the following display commutative: $$\xymatrix{
{\tilde{C}} \ar[r]^-{{\tilde{\eta}}_{{\tilde{C}}}} \ar[d]_-{{\tilde{\varphi}}} & {\tilde{G}}({\tilde{D}}) \ar[dl]^-{{\tilde{G}}({\tilde{\delta}})}
\\
{\tilde{G}}({\tilde{D}}_0) & {}
}
\qquad \qquad
\xymatrix{
C ... | D}}_0 $ which makes the left diagram in the following display commutative: $ $ \xymatrix {
{ \tilde{C } } \ar[r]^-{{\tilde{\eta}}_{{\tilde{C } } } } \ar[d]_-{{\tilde{\varphi } } } & { \tilde{G}}({\tilde{D } }) \ar[dl]^-{{\tilde{G}}({\tilde{\delta } }) }
\\
{ \tilde{G}}({\tilde{D}}_0) & { }
}
\... | D}}_0$ wjich makes the left diagvam in the follocung divplay dommutatkve: $$\xymatrix{
{\tilde{C}} \ar[r]^-{{\tipdw{\eta}}_{{\tulde{C}}}} \ar[d]_-{{\tilde{\varphi}}} & {\gilde{G}}({\tilfe{D}}) \ar[dl]^-{{\rildt{G}}({\tilde{\delta}})}
\\
{\tilde{N}}({\cilde{S}}_0) & {}
}
\qquad \qquad
\wymatrix{
C ... | D}}_0$ which makes the left diagram in display $$\xymatrix{ {\tilde{C}} \ar[d]_-{{\tilde{\varphi}}} & {\tilde{G}}({\tilde{D}}) } \qquad \xymatrix{ C \ar[d]_-{\eta_C} & G(D) \\ G(D) & {} }$$ The diagram above is obtained from the left one by applying the functor ${U_{{\mathcal{C}}}}$. the unit $\eta_C \colon C \to G(D)$... | D}}_0$ which makes the left diagram In the folloWing dIspLay CoMmutAtivE: $$\xymatrix{
{\tilde{c}} \Ar[r]^-{{\tIlde{\eta}}_{{\tilde{C}}}} \ar[d]_-{{\tilde{\vArphi}}} & {\TiLDe{G}}({\tILdE{D}}) \ar[dL]^-{{\tilde{G}}({\TIlDE{\DelTa}})}
\\
{\TiLde{g}}({\tILdE{D}}_0) & {}
}
\qquAd \qQuad
\xymAtrix{
C ... | D}}_0$ which makes the lef t diagramin th e f oll ow ingdisp lay commutativ e : $$ \xymatrix{
{\tilde {C}}\a r [r]^ - {{ \tild e{\eta} } _{ { \ til de {C }}} }\ ar [d]_- {{\ tilde{\ varphi}}}& { \t ilde{G}}({\t i ld e{D}}) \ar [dl ]^-{{\tilde{ G}} ({\til de {\d e lta}} )} \ \
{ \tilde {G}}({\ti ld e {D}}_0 ) ... | D}}_0$ which_makes the_left diagram in the_following display_commutative:_$$\xymatrix{
__ {\tilde{C}} \ar[r]^-{{\tilde{\eta}}_{{\tilde{C}}}}_\ar[d]_-{{\tilde{\varphi}}} & {\tilde{G}}({\tilde{D}})_\ar[dl]^-{{\tilde{G}}({\tilde{\delta}})}
_\\
__{\tilde{G}}({\tilde{D}}_0) & {}
}
\qquad \qquad
\xymatrix{
C_... |
exists an $a\in A{\backslash{\{0\}}}$ such that $ae_3, ae_2\in A{\backslash{\{0\}}}$. So we get $f(a_3e_3)=a_3e_3\neq 0$ and yet $$f(a_3e_3)=f(a_3(1-e_1)e_2) = f(a_3e_2)f(1-e_1) = a_3e_2(1-e_2)=0$$ a contradiction. If the former case holds, then we get a contradiction in a similar manner.
This has been discussed thro... | exists an $ a\in A{\backslash{\{0\}}}$ such that $ ae_3, ae_2\in A{\backslash{\{0\}}}$. So we get $ f(a_3e_3)=a_3e_3\neq 0 $ and yet $ $ f(a_3e_3)=f(a_3(1 - e_1)e_2) = f(a_3e_2)f(1 - e_1) = a_3e_2(1 - e_2)=0$$ a contradiction. If the erstwhile subject holds, then we get a contradiction in a similar manner.
This has ... | exlsts an $a\in A{\backslash{\{0\}}}$ smch that $ae_3, ae_2\in A{\backsnash{\{0\}}}$. Sk we get $f(a_3e_3)=a_3e_3\neq 0$ and yet $$f(a_3e_3)=f(a_3(1-e_1)e_2) = h(a_3e_2)f(1-w_1) = a_3e_2(1-t_2)=0$$ a contradiction. Iw the forler case holvs, then we get a contradletion ln a vmmilar manner.
Thls has been discussed thrm... | exists an $a\in A{\backslash{\{0\}}}$ such that $ae_3, So get $f(a_3e_3)=a_3e_3\neq and yet $$f(a_3e_3)=f(a_3(1-e_1)e_2) contradiction. the former case then we get contradiction in a similar manner. This been discussed throughout our study of regular rings, but we shall give a prove for all reduced rings here. The keyw... | exists an $a\in A{\backslash{\{0\}}}$ such That $ae_3, ae_2\in a{\backSlaSh{\{0\}}}$. SO wE get $F(a_3e_3)=a_3E_3\neq 0$ and yet $$f(a_3e_3)=f(A_3(1-E_1)e_2) = f(a_3E_2)f(1-e_1) = a_3e_2(1-e_2)=0$$ a contradiction. If The foRmER casE HoLds, thEn we get A CoNTRadIcTiOn iN a SImIlar mAnnEr.
This hAs been discUssEd Thro... | exists an $a\in A{\backsl ash{\{0\}} }$ su chtha t$ae_ 3, a e_2\in A{\back s lash {\{0\}}}$. So we get $ f(a_3 e_ 3 )=a_ 3 e_ 3\neq 0$ and ye t $$f (a _3 e_3 )= f (a _3(1- e_1 )e_2) = f(a_3e_2) f(1 -e _1) = a_3e_2 ( 1- e_2)=0$$ a co ntradiction. If the f or mer casehol ds, t hen we get acontradic ti o n in a simi... | exists_an $a\in_A{\backslash{\{0\}}}$ such that $ae_3,_ae_2\in A{\backslash{\{0\}}}$._So_we get_$f(a_3e_3)=a_3e_3\neq_0$ and yet_$$f(a_3e_3)=f(a_3(1-e_1)e_2) = f(a_3e_2)f(1-e_1)_= a_3e_2(1-e_2)=0$$ a contradiction._If the former_case_holds, then we get a contradiction in a similar manner.
This has been discussed thro... |
For instance, we perform entanglement swapping protocol for $n$ times to connect the entanglement between the remote locations A and K, we will get $$\begin{aligned}
|\Psi^{n+1}\rangle_{ak}=(\alpha^{n+1}a^{\dagger}\pm\beta^{n+1}k^{\dagger}e^{i\theta_{AK}})|0\rangle\label{state8}.\end{aligned}$$ For $\alpha\neq\beta$, ... | For instance, we perform entanglement swapping protocol for $ n$ time to plug in the entanglement between the remote locations A and K, we will receive $ $ \begin{aligned }
|\Psi^{n+1}\rangle_{ak}=(\alpha^{n+1}a^{\dagger}\pm\beta^{n+1}k^{\dagger}e^{i\theta_{AK}})|0\rangle\label{state8}.\end{aligned}$$ For $ \alpha\ne... | Fog instance, we perform enuanglement swappiut protmcol fkr $n$ timds to connect the entanglemeit bwtweeb the remote locations A and K, ae will tet $$\uegin{aligned}
|\Psi^{n+1}\czngle_{ak}=(\alpha^{n+1}z^{\faggzr}\'m\beta^{n+1}k^{\dagger}e^{l\theta_{AK}})|0\rancle\label{state8}.\etd{xlngned}$$ For $\alpha\neq\beta$, ... | For instance, we perform entanglement swapping protocol times connect the between the remote will $$\begin{aligned} |\Psi^{n+1}\rangle_{ak}=(\alpha^{n+1}a^{\dagger}\pm\beta^{n+1}k^{\dagger}e^{i\theta_{AK}})|0\rangle\label{state8}.\end{aligned}$$ For the entanglement decreases and more, and we will fail establish a perf... | For instance, we perform entanGlement swaPping ProTocOl For $n$ TimeS to connect the eNTangLement between the remote LocatIoNS A anD k, wE will Get $$\begiN{AlIGNed}
|\psI^{n+1}\RanGlE_{Ak}=(\Alpha^{N+1}a^{\dAgger}\pm\Beta^{n+1}k^{\daggEr}e^{I\tHeta_{AK}})|0\rangle\LAbEl{state8}.\end{AliGned}$$ For $\alpha\Neq\Beta$, ... | For instance, we performentangleme nt sw app ing p roto colfor $n$ timest o co nnect the entanglement betw ee n the re motelocatio n sA and K ,wewi l lget $ $\b egin{al igned}
|\P si^ {n +1}\rangle_{ a k} =(\alpha^{ n+1 }a^{\dagger} \pm \beta^ {n +1} k ^{\da gge r}e^{ i\thet a _{AK}} )|0\rangl e\ l abel{s t ate8}.... | For_instance, we_perform entanglement swapping protocol_for $n$_times_to connect_the_entanglement between the_remote locations A_and K, we will_get $$\begin{aligned}
|\Psi^{n+1}\rangle_{ak}=(\alpha^{n+1}a^{\dagger}\pm\beta^{n+1}k^{\dagger}e^{i\theta_{AK}})|0\rangle\label{state8}.\end{aligned}$$ For_$\alpha\neq\beta$,_... |
s \;
[\partial_s^2 f(s) ]^2.\end{aligned}$$ Which verifies that the energy is constant in a model-independent manner. Furthermore if $f(s)$ is smooth and satisfies suitable falloff conditions at $s\to\pm\infty$ then the wavelet will be nonsingular and of finite energy.
To obtain a specific example it only remains to... | s \;
[ \partial_s^2 f(s) ] ^2.\end{aligned}$$ Which verifies that the energy is constant in a model - autonomous manner. Furthermore if $ f(s)$ is legato and satisfies suitable falloff condition at $ s\to\pm\infty$ then the ripple will be nonsingular and of finite energy.
To obtain a specific example it merely rem... | s \;
[\partial_s^2 f(s) ]^2.\end{aligned}$$ Dhich verifies jhqt the energg is conrtant in a model-independent labner. Durthermore if $f(s)$ is soooth and satisfiws snitable falloff rknditiokf at $d\to\pk\mnfty$ then the eavelet winl be nonsingunaf cnd of finite energy.
To obtain a specyfic exsmole it only reiainx to... | s \; [\partial_s^2 f(s) ]^2.\end{aligned}$$ Which verifies energy constant in model-independent manner. Furthermore satisfies falloff conditions at then the wavelet be nonsingular and of finite energy. obtain a specific example it only remains to finish the complete specification of potential $\psi(r,t)$. One particula... | s \;
[\partial_s^2 f(s) ]^2.\end{aligned}$$ WhicH verifies tHat thE enErgY iS conStanT in a model-indepENdenT manner. Furthermore if $f(s)$ Is smoOtH And sATiSfies SuitablE FaLLOff CoNdItiOnS At $S\to\pm\InfTy$ then tHe wavelet wIll Be Nonsingular aND oF finite eneRgy.
to obtain a speCifIc examPlE it ONly reMaiNs to... | s \;
[\partial_s^2 f(s)]^2.\end{a ligne d}$ $ W hi ch v erif ies that the e n ergy is constant in a mode l-ind ep e nden t m anner . Furth e rm o r e i f$f (s) $i ssmoot h a nd sati sfies suit abl efalloff cond i ti ons at $s\ to\ pm\infty$ th enthe wa ve let willbenonsi ngular and of finite e ne r gy.
T o obtai... | s_\;
[\partial_s^2_f(s) ]^2.\end{aligned}$$ Which verifies_that the_energy_is constant_in_a model-independent manner._Furthermore if $f(s)$_is smooth and satisfies_suitable falloff conditions_at_$s\to\pm\infty$ then the wavelet will be nonsingular and of finite energy.
To obtain a specific_example_it only_remains_to... |
in L^2([0,1])$ with $f''\in
L^2([0,1])$ and with periodic boundary conditions. Then $B$ has (complex-valued) eigenfunctions $u_k(x)=\exp(2\pi kix)$ with eigenvalues $\nu_k=(2\pi k)^2$ such that $$H^r=\{f\in L^2([0,1])\,:\,
\sum_{k\in\operatorname{{\mathbb Z}}}\nu_k^{2r}{\lvert {\langle f,u_k \rangle} \rvert}^2<\infty\}... | in L^2([0,1])$ with $ f''\in
L^2([0,1])$ and with periodic boundary conditions. Then $ B$ has (complex - valued) eigenfunctions $ u_k(x)=\exp(2\pi kix)$ with eigenvalues $ \nu_k=(2\pi k)^2 $ such that $ $ H^r=\{f\in L^2([0,1])\,:\,
\sum_{k\in\operatorname{{\mathbb Z}}}\nu_k^{2r}{\lvert { \langle f, u_k \rangle } \r... | in P^2([0,1])$ with $f''\in
L^2([0,1])$ and with perlodic boundary conditiois. Then $B$ has (cumplex-valued) eigenfunctions $n_k(x)=\ezp(2\pi jix)$ with eigenvalues $\nj_k=(2\pi k)^2$ subh that $$H^e=\{f\in O^2([0,1])\,:\,
\sum_{k\in\opecztornamc{{\iathgn Z}}}\nu_n^{2c}{\lvert {\langle f,o_k \rangle} \rvart}^2<\infty\}... | in L^2([0,1])$ with $f''\in L^2([0,1])$ and with conditions. $B$ has eigenfunctions $u_k(x)=\exp(2\pi kix)$ that L^2([0,1])\,:\, \sum_{k\in\operatorname{{\mathbb Z}}}\nu_k^{2r}{\lvert f,u_k \rangle} \rvert}^2<\infty\}$$ the classical $L^2$-Sobolev space $H^{2r}_{per}$ of (smoothness) $2r$ with periodic boundary conditi... | in L^2([0,1])$ with $f''\in
L^2([0,1])$ and with periodiC boundary cOnditIonS. ThEn $b$ has (CompLex-valued) eigenFUnctIons $u_k(x)=\exp(2\pi kix)$ with eigEnvalUeS $\Nu_k=(2\pI K)^2$ sUch thAt $$H^r=\{f\in l^2([0,1])\,:\,
\SuM_{K\In\oPeRaTorNaME{{\mAthbb z}}}\nu_K^{2r}{\lvert {\Langle f,u_k \rAngLe} \Rvert}^2<\infty\}... | in L^2([0,1])$ with $f''\i n
L^2([0,1 ])$ a ndwit hperi odic boundary cond i tion s. Then $B$ has (compl ex-va lu e d) e i ge nfunc tions $ u _k ( x )=\ ex p( 2\p ik ix )$ wi theigenva lues $\nu_ k=( 2\ pi k)^2$ suc h t hat $$H^r= \{f \in L^2([0,1 ])\ ,:\,
\ su m_{ k \in\o per atorn ame{{\ m athbbZ}}}\nu_k ^{ 2 r}{\l... | in L^2([0,1])$_with $f''\in
L^2([0,1])$_and with periodic boundary_conditions. Then_$B$_has (complex-valued)_eigenfunctions_$u_k(x)=\exp(2\pi kix)$ with_eigenvalues $\nu_k=(2\pi k)^2$_such that $$H^r=\{f\in L^2([0,1])\,:\,
\sum_{k\in\operatorname{{\mathbb_Z}}}\nu_k^{2r}{\lvert {\langle f,u_k_\rangle}_\rvert}^2<\infty\}... |
\right) \,.
\end{split}$$ This expression can be written as $I=\sum_r^d f(a,b,c,d,r)$, where $f$ is a well-behaved function. Furthermore, we can write this result as $$I_{abcd}=\sum_{r=0}^d \exp{\log{[f(a,b,c,d,r)]}}\,,$$ where $f$ must be positive.\
\
The logarithm $\log{(f)}$ is given by $$\begin{split}
\log{(f)}... | \right) \, .
\end{split}$$ This expression can be written as $ I=\sum_r^d f(a, b, c, d, r)$, where $ f$ is a well - behaved function. Furthermore, we can spell this resultant role as $ $ I_{abcd}=\sum_{r=0}^d \exp{\log{[f(a, b, c, d, r)]}}\,,$$ where $ f$ must be positive.\
\
The logarithm $ \log{(f)}$ is given b... | \rigjt) \,.
\end{split}$$ This expresslon can be writtgn as $I=\snm_r^d f(a,g,c,d,r)$, whefe $f$ is a well-behaved functiln. Furtyermore, we can write tfis resuln as $$I_{abce}=\sum_{c=0}^d \exp{\log{[f(a,b,c,d,r)]}}\,,$$ xgere $f$ must bs posntmve.\
\
The logarithk $\log{(f)}$ is civen by $$\begin{vpuic}
\log{(f)}... | \right) \,. \end{split}$$ This expression can be $I=\sum_r^d where $f$ a well-behaved function. result $$I_{abcd}=\sum_{r=0}^d \exp{\log{[f(a,b,c,d,r)]}}\,,$$ where must be positive.\ The logarithm $\log{(f)}$ is given by \log{(f)}=&-\frac{1}{2}\log{(2)}-2\log{(\pi)}\\ &+\frac{1}{2}\left(\log{(c!)}+\log{(d!)} -\log{(a!... | \right) \,.
\end{split}$$ This expressiOn can be wriTten aS $I=\sUm_r^D f(A,b,c,d,R)$, wheRe $f$ is a well-behaVEd fuNction. Furthermore, we can Write ThIS resULt As $$I_{abCd}=\sum_{r=0}^d \EXp{\LOG{[f(a,B,c,D,r)]}}\,,$$ WheRe $F$ MuSt be pOsiTive.\
\
The Logarithm $\lOg{(f)}$ Is Given by $$\begin{SPlIt}
\log{(f)}... | \right) \,.
\end{split}$$This expre ssion ca n b ewrit tenas $I=\sum_r^d f(a, b,c,d,r)$, where $f$ i s a w el l -beh a ve d fun ction.F ur t h erm or e, we c a nwrite th is resu lt as $$I_ {ab cd }=\sum_{r=0} ^ d\exp{\log{ [f( a,b,c,d,r)]} }\, ,$$ wh er e $ f $ mus t b e pos itive. \
\
The logarith m$ \log{( f )}... | \right) \,.
\end{split}$$_This expression_can be written as_$I=\sum_r^d f(a,b,c,d,r)$,_where_$f$ is_a_well-behaved function. Furthermore,_we can write_this result as $$I_{abcd}=\sum_{r=0}^d_\exp{\log{[f(a,b,c,d,r)]}}\,,$$ where $f$_must_be positive.\
\
The logarithm $\log{(f)}$ is given by $$\begin{split}
\log{(f)}... |
. Here we are interested in neutral stoichiometric clusters of typical materials with ionic bonding, that is $(AX)_n$ clusters, where $A$ is an alkali and X a halide atom. As [*ab initio*]{} studies on these clusters are computationally expensive, the first theoretical calculations were based on pairwise interaction mo... | . Here we are interested in neutral stoichiometric clusters of typical fabric with ionic bonding, that is $ (AX)_n$ bunch, where $ A$ is an alkali and X a halide atom. As [ * ab initio * ] { } report on these bunch are computationally expensive, the first theoretical calculations were based on pairwise interaction exem... | . Hege we are interested in keutral stoichiometric rlustera of typkcal materials with ionic boidint, thau is $(AX)_n$ clusters, wfere $A$ is an alkaoi aid X a halide atom. As [*ab initik*]{} stubixs on these cluxters are womputationallf dx'ensive, the first theoretical calculwtions eege based on payrwixq infvrcction mo... | . Here we are interested in neutral of materials with bonding, that is an and X a atom. As [*ab studies on these clusters are computationally the first theoretical calculations were based on pairwise interaction models [@Mar83; @Die85; @Phi91]. experimentalists moved forward using several techniques to produce and inve... | . Here we are interested in neutRal stoichiOmetrIc cLusTeRs of TypiCal materials wiTH ionIc bonding, that is $(AX)_n$ clusTers, wHeRE $A$ is AN aLkali And X a haLIdE ATom. as [*Ab IniTiO*]{} StUdies On tHese cluSters are coMpuTaTionally expeNSiVe, the first TheOretical calcUlaTions wErE baSEd on pAirWise iNteracTIon mo... | . Here we are interested i n neutralstoic hio met ri c cl uste rs of typicalm ater ials with ionic bondin g, th at is $ ( AX )_n$cluster s ,w h ere $ A$ is a n a lkali an d X a h alide atom . A s[*ab initio* ] {} studies o n t hese cluster s a re com pu tat i onall y e xpens ive, t h e firs t theoret ic a l calc ... | . Here_we are_interested in neutral stoichiometric_clusters of_typical_materials with_ionic_bonding, that is_$(AX)_n$ clusters, where_$A$ is an alkali_and X a_halide_atom. As [*ab initio*]{} studies on these clusters are computationally expensive, the first theoretical_calculations_were based_on_pairwise_interaction mo... |
($\Delta
E_{Arr}^*\!=\!\Delta E_{Arr} / T_{MCT}$) for different model systems. The data of MLJ, MSS, BMLJ and BMLJ$_2$ are from this work (except the $\Delta E_{Arr}^*$ for BMLJ, that is from Ref. [@sastry_pisa]).
Having in mind that $\Delta E_{Arr}^* \! \simeq \! 2 \Delta E^*$ and that $\Delta E^*\! \simeq\!10$ (i... | ($ \Delta
E_{Arr}^*\!=\!\Delta E_{Arr } / T_{MCT}$) for different model systems. The data of MLJ, MSS, BMLJ and BMLJ$_2 $ are from this workplace (except the $ \Delta E_{Arr}^*$ for BMLJ, that is from Ref. [ @sastry_pisa ]).
have in mind that $ \Delta E_{Arr}^ * \! \simeq \! 2 \Delta E^*$ and that $ \Delta E^*\!... | ($\Depta
E_{Arr}^*\!=\!\Delta E_{Arr} / T_{MCU}$) for different model syvtems. Fhe data of MLJ, MSS, BMLJ and BMLJ$_2$ arx frim thus work (except the $\Delga E_{Arr}^*$ flr BMLJ, rhat us from Reh. [@sastry_insa]).
Hablng iu nind that $\Delts E_{Arr}^* \! \sikeq \! 2 \Delta E^*$ dna chat $\Delta E^*\! \simeq\!10$ (i... | ($\Delta E_{Arr}^*\!=\!\Delta E_{Arr} / T_{MCT}$) for different The of MLJ, BMLJ and BMLJ$_2$ the E_{Arr}^*$ for BMLJ, is from Ref. Having in mind that $\Delta E_{Arr}^* \simeq \! 2 \Delta E^*$ and that $\Delta E^*\! \simeq\!10$ (i.e. $\Delta E_{Arr}^* 20$), we can try to analyze what is observed for other model potent... | ($\Delta
E_{Arr}^*\!=\!\Delta E_{Arr} / T_{MCT}$) for Different mOdel sYstEms. thE datA of MlJ, MSS, BMLJ and BMlj$_2$ are From this work (except the $\DElta E_{arR}^*$ For BmlJ, That iS from ReF. [@SaSTRy_pIsA]).
HAviNg IN mInd thAt $\DElta E_{ArR}^* \! \simeq \! 2 \DeltA E^*$ aNd That $\Delta E^*\! \siMEq\!10$ (I... | ($\Delta
E_{Arr}^*\!=\! \Delta E_{ Arr}/ T _{M CT }$)fordifferent mode l sys tems. The data of MLJ, MSS, B M LJ a n dBMLJ$ _2$ are fr o m th is w ork ( e xc ept t he$\Delta E_{Arr}^* $ f or BMLJ, thati sfrom Ref.[@s astry_pisa]) .
Having i n m i nd th at$\Del ta E_{ A rr}^*\! \simeq \ ! 2 \De l ta E^*$ a ... | ($\Delta
_ E_{Arr}^*\!=\!\Delta_E_{Arr} / T_{MCT}$) for_different model_systems._The data_of_MLJ, MSS, BMLJ_and BMLJ$_2$ are_from this work (except_the $\Delta E_{Arr}^*$_for_BMLJ, that is from Ref. [@sastry_pisa]).
Having in mind that $\Delta E_{Arr}^* \! \simeq \!_2_\Delta E^*$_and_that_$\Delta E^*\! \simeq\!10$ (i... |
Rudolf T, Kant C, Mayr F and Loidl A 2008 [*Phys. Rev. B*]{} [**77**]{}(2) 024421 <http://link.aps.org/doi/10.1103/PhysRevB.77.024421>
Yamashita Y and Ueda K 2000 [*Physical Review Letters*]{} [**85**]{} 4960
Rudolf T, Kant C, Mayr F, Hemberger J, Tsurkan V and Loidl A 2007 [*New Journal of Physics*]{} [**9**]{} 76
... | Rudolf T, Kant C, Mayr F and Loidl A 2008 [ * Phys. Rev. B * ] { } [ * * 77**]{}(2) 024421 < http://link.aps.org/doi/10.1103/PhysRevB.77.024421 >
Yamashita Y and Ueda K 2000 [ * Physical Review Letters * ] { } [ * * 85 * * ] { } 4960
Rudolf T, Kant C, Mayr F, Hemberger J, Tsurkan V and Loidl A 2007 [ * New Journa... | Rudllf T, Kant C, Mayr F and Uoidl A 2008 [*Phys. Rgv. B*]{} [**77**]{}(2) 024421 <hvtp://link.zps.org/dok/10.1103/PhysRevB.77.024421>
Yamashita Y and Uede K 2000 [*Physucal Review Letters*]{} [**85**]{} 4960
Rjdolf T, Kwnt C, Matr F, Yemberger O, Tsurkak V ahf Londo A 2007 [*New Journsl of Physhcs*]{} [**9**]{} 76
... | Rudolf T, Kant C, Mayr F and 2008 Rev. B*]{} 024421 <http://link.aps.org/doi/10.1103/PhysRevB.77.024421> Yamashita [*Physical Letters*]{} [**85**]{} 4960 T, Kant C, F, Hemberger J, Tsurkan V and A 2007 [*New Journal of Physics*]{} [**9**]{} 76 Schaack G 1977 [*Zeitschrift f[ü]{}r B Condensed Matter*]{} [**26**]{} 49–58... | Rudolf T, Kant C, Mayr F and Loidl a 2008 [*Phys. Rev. B*]{} [**77**]{}(2) 024421 <hTtp://liNk.aPs.oRg/Doi/10.1103/PHysREvB.77.024421>
Yamashita Y aND UedA K 2000 [*Physical Review LetterS*]{} [**85**]{} 4960
RudoLf t, kant c, maYr F, HeMberger j, tsURKan v aNd loiDl a 2007 [*neW JourNal Of PhysiCs*]{} [**9**]{} 76
... | Rudolf T, Kant C, Mayr F a nd Loidl A 2008 [* Phy s. Rev . B* ]{} [**77**]{} ( 2) 0 24421 <http://link.aps .org/ do i /10. 1 10 3/Phy sRevB.7 7 .0 2 4 421 >
Y ama sh i ta Y an d U eda K 2 000 [*Phys ica lReview Lette r s* ]{} [**85* *]{ } 4960
Rudo lfT, Kan tC,M ayr F , H ember ger J, Tsurka n V and L oi d l A 20 ... | Rudolf T,_Kant C,_Mayr F and Loidl_A 2008_[*Phys._Rev. B*]{}_[**77**]{}(2)_024421 <http://link.aps.org/doi/10.1103/PhysRevB.77.024421>
Yamashita Y_and Ueda K_2000 [*Physical Review Letters*]{}_[**85**]{} 4960
Rudolf T,_Kant_C, Mayr F, Hemberger J, Tsurkan V and Loidl A 2007 [*New Journal of_Physics*]{}_[**9**]{} 76
... |
, V. Mannings, A. Boss, & S. Russell (Tucson:Univ. Arizona Press), 59
Ballesteros-Paredes, J. 2003. In “From Observations to Self-Consistent Modeling of the Interstellar Medium”. Ed. M. Avillez & D. Breitschwerdt (Kluwer), in press.
Ballesteros-Paredes, J., Hartmann, L., & V[' a]{}zquez-Semadeni, E.1999b,, 527, 285
... | , V. Mannings, A. Boss, & S. Russell (Tucson: Univ. Arizona Press), 59
Ballesteros - Paredes, J. 2003. In “ From Observations to Self - Consistent Modeling of the Interstellar Medium ”. Ed. M. Avillez & D. Breitschwerdt (Kluwer), in press.
Ballesteros - Paredes, J., Hartmann, L., & V [' a]{}zquez - Semade... | , V. Lannings, A. Boss, & S. Russeul (Tucson:Univ. Atizona Pcess), 59
Bamlesteror-Paredes, J. 2003. In “From Observatmons to Stjf-Consistent Modelkng of thv Interstwllac Medium”. Ed. M. Avillxa & D. Brelcschwsvdt (Knnwer), in press.
Baklesteros-Pdredes, J., Hartmdnv, P., & V[' a]{}zquez-Semadeni, E.1999b,, 527, 285
... | , V. Mannings, A. Boss, & S. Arizona 59 Ballesteros-Paredes, 2003. In “From the Medium”. Ed. M. & D. Breitschwerdt in press. Ballesteros-Paredes, J., Hartmann, L., V[' a]{}zquez-Semadeni, E.1999b,, 527, 285 Ballesteros-Paredes, J. & Mac Low, M. 2002,, 570, Ballesteros-Paredes, J., V[' a]{}zquez-Semadeni, E., & Scalo, J... | , V. Mannings, A. Boss, & S. Russell (TucSon:Univ. AriZona PResS), 59
BaLlEsteRos-PAredes, J. 2003. In “From OBServAtions to Self-Consistent modelInG Of thE inTerstEllar MeDIuM”. eD. M. AViLlEz & D. brEItSchweRdt (kluwer), iN press.
BallEstErOs-Paredes, J., HaRTmAnn, L., & V[' a]{}zqueZ-SeMadeni, E.1999b,, 527, 285
... | , V. Mannings, A. Boss, &S. Russell (Tuc son :Un iv . Ar izon a Press), 59
B alle steros-Paredes, J. 200 3. In “ F romO bs ervat ions to Se l f -Co ns is ten tM od eling of the In terstellar Me di um”. Ed. M.A vi llez & D.Bre itschwerdt ( Klu wer),in pr e ss.
Bal leste ros-Pa r edes,J., Hartm an n , L.,& V[' a] ... | , V._Mannings, A._Boss, & S. Russell_(Tucson:Univ. Arizona_Press),_59
Ballesteros-Paredes, J._2003._In “From Observations_to Self-Consistent Modeling_of the Interstellar Medium”. Ed. M. Avillez_& D. Breitschwerdt (Kluwer),_in_press.
Ballesteros-Paredes, J., Hartmann, L., & V[' a]{}zquez-Semadeni, E.1999b,, 527, 285
... |
for massless HISQ quarks with NRQCD formulated in a moving frame (mNRQCD) was undertaken for the vector and tensor heavy-light currents in [@mueller11].
For massive quarks similar matching calculations using the same lattice action for both quarks have been carried out for Wilson quarks in [@kuramashi98] and for vari... | for massless HISQ quarks with NRQCD formulated in a moving frame (mNRQCD) was undertake for the vector and tensor dense - light currents in [ @mueller11 ].
For massive quarks like matching calculations using the like lattice action for both quarks have been carry out for Wilson quarks in [ @kuramashi98 ] and for var... | fog massless HISQ quarks wlth NRQCD formulcred in a movjng framd (mNRQCD) was undertaken for vhe cectoe and tensor heavy-lighg currentd in [@mueoler11].
Hor massive quarks similar matdming eaoculations usikg the same lattice actiot wox both quarks have been carried out sor Wilxoj quarks in [@kutamasny98] ans for vari... | for massless HISQ quarks with NRQCD formulated moving (mNRQCD) was for the vector [@mueller11]. massive quarks similar calculations using the lattice action for both quarks have carried out for Wilson quarks in [@kuramashi98] and for various implementations of NRQCD [@braaten95; @jones99; @boyle00; @hart07]. To our kno... | for massless HISQ quarks with nRQCD formuLated In a MovInG fraMe (mNrQCD) was undertaKEn foR the vector and tensor heaVy-ligHt CUrreNTs In [@mueLler11].
For MAsSIVe qUaRkS siMiLAr MatchIng CalculaTions using The SaMe lattice actIOn For both quaRks Have been carrIed Out for wiLsoN QuarkS in [@KuramAshi98] anD For varI... | for massless HISQ quarkswith NRQCD form ula ted i n amovi ng frame (mNRQ C D) w as undertaken for thevecto ra nd t e ns or he avy-lig h tc u rre nt sin[@ m ue ller1 1].
For m assive qua rks s imilar match i ng calculati ons using the s ame latti ce ac t ion f orbothquarks have b een carri ed out fo r Wilson ... | for_massless HISQ_quarks with NRQCD formulated_in a_moving_frame (mNRQCD)_was_undertaken for the_vector and tensor_heavy-light currents in [@mueller11].
For_massive quarks similar_matching_calculations using the same lattice action for both quarks have been carried out for_Wilson_quarks in_[@kuramashi98]_and_for vari... |
_q}\over{\beta(1-\beta)Q^2}}),$$ and $$\begin{aligned}
\frac{1}{z}=1+\frac{\kappa_{t}^{2}+m_q^2}{(1-\beta)Q^2}+\frac{k_t^2+\kappa_t^2-2{\bf{\kappa_t}}.{\bf{k_t}}+m_q^2}{\beta
Q^2}
\label{eq:a}.\end{aligned}$$ As in the reference [@4kimber], the scale $\mu$ which controls the $unintegrated$ gluon and the $QCD$ coupling... | _ q}\over{\beta(1-\beta)Q^2}}),$$ and $ $ \begin{aligned }
\frac{1}{z}=1+\frac{\kappa_{t}^{2}+m_q^2}{(1-\beta)Q^2}+\frac{k_t^2+\kappa_t^2 - 2{\bf{\kappa_t}}.{\bf{k_t}}+m_q^2}{\beta
Q^2 }
\label{eq: a}.\end{aligned}$$ As in the reference [ @4kimber ], the scale $ \mu$ which controls the $ unintegrated$ gluon and ... | _q}\ovfr{\beta(1-\beta)Q^2}}),$$ and $$\begin{alinned}
\frac{1}{z}=1+\frac{\kapka_{r}^{2}+m_q^2}{(1-\bete)Q^2}+\frac{k_f^2+\kappa_t^2-2{\bw{\kappa_t}}.{\bf{k_t}}+m_q^2}{\beta
Q^2}
\label{eq:a}.\eid{alugned}$$ As in the reference [@4kkmber], the scale $\my$ whmch controls the $unintegvcted$ fpuon end the $QCD$ coukling... | _q}\over{\beta(1-\beta)Q^2}}),$$ and $$\begin{aligned} \frac{1}{z}=1+\frac{\kappa_{t}^{2}+m_q^2}{(1-\beta)Q^2}+\frac{k_t^2+\kappa_t^2-2{\bf{\kappa_t}}.{\bf{k_t}}+m_q^2}{\beta Q^2} \label{eq:a}.\end{aligned}$$ As reference the scale which controls the coupling $\alpha_s$, is chosen follows: $$\begin{aligned} \mu^2=k_t^2... | _q}\over{\beta(1-\beta)Q^2}}),$$ and $$\begin{aliGned}
\frac{1}{z}=1+\fRac{\kaPpa_{T}^{2}+m_q^2}{(1-\BeTa)Q^2}+\fRac{k_T^2+\kappa_t^2-2{\bf{\kappa_T}}.{\Bf{k_t}}+M_q^2}{\beta
Q^2}
\label{eq:a}.\end{aligNed}$$ As In THe reFErEnce [@4kImber], thE ScALE $\mu$ WhIcH coNtROlS the $uNinTegrateD$ gluon and tHe $QcD$ Coupling... | _q}\over{\beta(1-\beta)Q^2 }}),$$ and $$\b egi n{a li gned }
\f rac{1}{z}=1+\f r ac{\ kappa_{t}^{2}+m_q^2}{( 1-\be ta ) Q^2} + \f rac{k _t^2+\k a pp a _ t^2 -2 {\ bf{ \k a pp a_t}} .{\ bf{k_t} }+m_q^2}{\ bet aQ^2}
\label { eq :a}.\end{a lig ned}$$ As in th e refe re nce [@4ki mbe r], t he sca l e $\mu $ which c on t... | _q}\over{\beta(1-\beta)Q^2}}),$$ and_$$\begin{aligned}
\frac{1}{z}=1+\frac{\kappa_{t}^{2}+m_q^2}{(1-\beta)Q^2}+\frac{k_t^2+\kappa_t^2-2{\bf{\kappa_t}}.{\bf{k_t}}+m_q^2}{\beta
Q^2}
\label{eq:a}.\end{aligned}$$_As in the reference_[@4kimber], the_scale_$\mu$ which_controls_the $unintegrated$ gluon_and the $QCD$_coupling... |
mathbf{n}\times \mathbf{I},\ \xi\in\mathbb{R}$ [@HarringtonMautz; @ColtonKress], and thus those operators $\mathcal{R}$ are not regularizing operators for $\mathcal{T}_k$. If $\mathcal{R}$ is chosen as a right regularizing operator for $\mathcal{T}_k$, the integral operator on the left hand side of (\[eq:CFIE\_R\]) is ... | mathbf{n}\times \mathbf{I},\ \xi\in\mathbb{R}$ [ @HarringtonMautz; @ColtonKress ], and thus those operators $ \mathcal{R}$ are not regularizing operators for $ \mathcal{T}_k$. If $ \mathcal{R}$ is choose as a proper regularizing operator for $ \mathcal{T}_k$, the integral operator on the leftover hand side of (\[eq: ... | matjbf{n}\times \mathbf{I},\ \xi\in\mauhbb{R}$ [@HarringtonMaotz; @ColtmnKresa], and thjs those operators $\mathcal{R}$ ere bot rtyularizing operators for $\mathbal{T}_k$. If $\nathral{R}$ is chosen as a righb reghpariviig operator for $\mathcal{T}_k$, the integral mpdrctor on the left hand side of (\[eq:CFIE\_W\]) is ... | mathbf{n}\times \mathbf{I},\ \xi\in\mathbb{R}$ [@HarringtonMautz; @ColtonKress], and thus $\mathcal{R}$ not regularizing for $\mathcal{T}_k$. If right operator for $\mathcal{T}_k$, integral operator on left hand side of (\[eq:CFIE\_R\]) is second kind Fredholm operator, and, thus, the unique solvability of equation (\[... | mathbf{n}\times \mathbf{I},\ \xi\in\maThbb{R}$ [@HarriNgtonmauTz; @COlTonKRess], And thus those opERatoRs $\mathcal{R}$ are not regulaRizinG oPEratORs For $\maThcal{T}_k$. iF $\mATHcaL{R}$ Is ChoSeN As A righT reGularizIng operatoR foR $\mAthcal{T}_k$, the iNTeGral operatOr oN the left hand SidE of (\[eq:CfIe\_R\]) iS ... | mathbf{n}\times \mathbf{I} ,\ \xi\in\ mathb b{R }$[@ Harr ingt onMautz; @Colt o nKre ss], and thus those op erato rs $\ma t hc al{R} $ are n o tr e gul ar iz ing o p er ators fo r $\mat hcal{T}_k$ . I f$\mathcal{R} $ i s chosen a s a right regul ari zing o pe rat o r for $\ mathc al{T}_ k $, the integral o p era... | mathbf{n}\times \mathbf{I},\_\xi\in\mathbb{R}$ [@HarringtonMautz; @ColtonKress],_and thus those operators_$\mathcal{R}$ are_not_regularizing operators_for_$\mathcal{T}_k$. If $\mathcal{R}$_is chosen as_a right regularizing operator_for $\mathcal{T}_k$, the_integral_operator on the left hand side of (\[eq:CFIE\_R\]) is ... |
{{\cal V}},{{\cal B}},{{\cal O}})$-1dta. By $\mathrm{1QFA}$, we indicate the family of all languages recognized by bounded-error 1qfa’s. We then obtain $\mathrm{1QFA} = {\mathrm{REG}}$.
[**(vii) Deterministic Pushdown Automata.**]{} A *1-way deterministic pushdown automaton* (or a 1dpda) $M$ can be seen as a $({{\cal ... | { { \cal V}},{{\cal B}},{{\cal O}})$-1dta. By $ \mathrm{1QFA}$, we indicate the family of all languages recognized by restrict - mistake 1qfa ’s. We then obtain $ \mathrm{1QFA } = { \mathrm{REG}}$.
[ * * (vii) Deterministic Pushdown Automata. * * ] { } A * 1 - way deterministic pushdown automaton * (or a 1dpda) $ M$... | {{\cal V}},{{\cal B}},{{\cal O}})$-1dta. By $\mathro{1QFA}$, we indicatg rhe fakily or all lavguages recognized by boundev-errir 1qfq’s. We then obtain $\mathfm{1QFA} = {\manhrm{REG}}$.
[**(viu) Deuerministic Pushdown Automata.**]{} A *1-aay beverministic pusmdown automdton* (or a 1dpda) $M$ ccn be seen as a $({{\cal ... | {{\cal V}},{{\cal B}},{{\cal O}})$-1dta. By $\mathrm{1QFA}$, we family all languages by bounded-error 1qfa’s. {\mathrm{REG}}$. Deterministic Pushdown Automata.**]{} *1-way deterministic pushdown (or a 1dpda) $M$ can be as a $({{\cal V}},{{\cal B}},{{\cal O}})$-1dta when $({{\cal V}},{{\cal B}},{{\cal O}})$ satisfies th... | {{\cal V}},{{\cal B}},{{\cal O}})$-1dta. By $\mathrm{1QFa}$, we indicatE the fAmiLy oF aLl laNguaGes recognized bY BounDed-error 1qfa’s. We then obtaIn $\matHrM{1qFA} = {\mAThRm{REG}}$.
[**(Vii) DeteRMiNIStiC PUsHdoWn aUtOmata.**]{} a *1-waY determInistic pusHdoWn Automaton* (or a 1DPdA) $M$ can be seeN as A $({{\cal ... | {{\cal V}},{{\cal B}},{{\c al O}})$-1 dta.By$\m at hrm{ 1QFA }$, we indicat e the family of all languag es re co g nize d b y bou nded-er r or 1 qfa ’s .Weth e nobtai n $ \mathrm {1QFA} = { \ma th rm{REG}}$.
[ ** (vii) Dete rmi nistic Pushd own Autom at a.* * ]{} A *1 -waydeterm i nistic pushdown a u tomato n * ... | {{\cal V}},{{\cal_B}},{{\cal O}})$-1dta._By $\mathrm{1QFA}$, we indicate_the family_of_all languages_recognized_by bounded-error 1qfa’s._We then obtain_$\mathrm{1QFA} = {\mathrm{REG}}$.
[**(vii) Deterministic_Pushdown Automata.**]{} A_*1-way_deterministic pushdown automaton* (or a 1dpda) $M$ can be seen as a $({{\cal ... |
5.
\nonumber\end{aligned}$$
The spin 1 polarization vector $\epsilon^{(\lambda)}_\mu(p)$ satisfies the constraints:
$$\def\arraystretch{2.0}
\begin{array}{ll}
\displaystyle \epsilon^{(\lambda)}_\mu(p) \, p^\mu = 0 &
\hspace*{1cm} {\rm transversality},
\\
\displaystyle \sum\limits_{\lambda=0,\pm}\epsilon^{(\lambda)}_\... | 5.
\nonumber\end{aligned}$$
The spin 1 polarization vector $ \epsilon^{(\lambda)}_\mu(p)$ satisfies the constraints:
$ $ \def\arraystretch{2.0 }
\begin{array}{ll }
\displaystyle \epsilon^{(\lambda)}_\mu(p) \, p^\mu = 0 &
\hspace*{1 cm } { \rm transversality },
\\
\displaystyle \sum\limits_{\lambda=0,\... | 5.
\nonkmber\end{aligned}$$
The spin 1 polarization vgcror $\epvilon^{(\lzmbda)}_\mu(p)$ satisfies the constraints:
$$\deh\arrqystrtnch{2.0}
\begin{array}{ll}
\displahstyle \epdilon^{(\lamvda)}_\mn(p) \, p^\mu = 0 &
\hspace*{1rj} {\rm transverawlitv},
\\
\dmsplaystyle \sum\kimits_{\lambga=0,\pm}\epsilon^{(\lamtdx)}_\... | 5. \nonumber\end{aligned}$$ The spin 1 polarization vector the $$\def\arraystretch{2.0} \begin{array}{ll} \epsilon^{(\lambda)}_\mu(p) \, p^\mu transversality}, \displaystyle \sum\limits_{\lambda=0,\pm}\epsilon^{(\lambda)}_\mu(p) \epsilon^{\dagger\,(\lambda)}_\nu =-g_{\mu\nu}+\frac{p_\mu\,p_\nu}{m^2} & \hspace*{1cm} com... | 5.
\nonumber\end{aligned}$$
The spin 1 PolarizatiOn vecTor $\EpsIlOn^{(\laMbda)}_\Mu(p)$ satisfies thE ConsTraints:
$$\def\arraystretch{2.0}
\Begin{ArRAy}{ll}
\DIsPlaysTyle \epsILoN^{(\LAmbDa)}_\Mu(P) \, p^\mU = 0 &
\hSPaCe*{1cm} {\rM trAnsversAlity},
\\
\displAysTyLe \sum\limits_{\lAMbDa=0,\pm}\epsiloN^{(\laMbda)}_\... | 5.
\nonumber\end{aligned}$ $
The spi n 1 p ola riz at ionvect or $\epsilon^{ ( \lam bda)}_\mu(p)$ satisfie s the c o nstr a in ts:
$$\def\ a rr a y str et ch {2. 0} \b egin{ arr ay}{ll}
\displays tyl e\epsilon^{(\ l am bda)}_\mu( p)\, p^\mu = 0 &\hspac e* {1c m } {\r m t ransv ersali t y},
\\
\display st y le \su ... | 5.
\nonumber\end{aligned}$$
The spin_1 polarization_vector $\epsilon^{(\lambda)}_\mu(p)$ satisfies the_constraints:
$$\def\arraystretch{2.0}
\begin{array}{ll}
\displaystyle \epsilon^{(\lambda)}_\mu(p)_\,_p^\mu =_0_&
\hspace*{1cm} {\rm transversality},
\\
\displaystyle_\sum\limits_{\lambda=0,\pm}\epsilon^{(\lambda)}_\... |
(y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Big|\\
&&\quad\lesssim \Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\nonumber\\
&&\quad\lesssim\Big(\int^2_1\Big(\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|\Bi... | ( y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Big|\\
& & \quad\lesssim \Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j - l}*f_2(y)-K^j_t*\phi_{j - l}*f_2(x_0)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\nonumber\\
& & \quad\lesssim\Big(\int^2_1\Big(\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j - l}*f_2(y)-K^j_t*\phi_{j - l}... | (y)-\wifetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Blg|\\
&&\quad\lesssim \Biy(\unt^2_1\sum_{o\in\mathgb{Z}}\big|K^j_g*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|^2{\rm d}t\Bij)^{\frax{1}{2}}\nonunber\\
&&\quad\lesssim\Big(\int^2_1\Bkg(\sum_{j\in\mwthbb{Z}}\bit|K^j_t*\khi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\ujg|\Bi... | (y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Big|\\ &&\quad\lesssim \Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\nonumber\\ &&\quad\lesssim\Big(\int^2_1\Big(\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|\Bi... | (y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(Y_0)\Big|\\
&&\quad\leSssim \big(\Int^2_1\SuM_{j\in\MathBb{Z}}\big|K^j_t*\phi_{j-l}*F_2(Y)-K^j_t*\Phi_{j-l}*f_2(x_0)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\nOnumbEr\\
&&\QUad\lESsSim\BiG(\int^2_1\Big(\SUm_{J\IN\maThBb{z}}\biG|K^J_T*\pHi_{j-l}*f_2(Y)-K^j_T*\phi_{j-l}*f_2(X_0)\big|\Bi... | (y)-\widetilde{\mathcal{M} }_{\Omega} ^lf_2 (y_ 0)\ Bi g|\\
&&\ quad\lesssim \ B ig(\ int^2_1\sum_{j\in\math bb{Z} }\ b ig|K ^ j_ t*\ph i_{j-l} * f_ 2 ( y)- K^ j_ t*\ ph i _{ j-l}* f_2 (x_0)\b ig|^2{\rmd}t \B ig)^{\frac{1 } {2 }}\nonumbe r\\
&&\quad\les ssi m\Big( \i nt^ 2 _1\Bi g(\ sum_{ j\in\m a thbb{Z }}\big|K^ j_... | (y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Big|\\
&&\quad\lesssim \Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|^2{\rm_d}t\Big)^{\frac{1}{2}}\nonumber\\
&&\quad\lesssim\Big(\int^2_1\Big(\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|\Bi... |
P. Erd[ő]{}s, A. R[é]{}nyi, On random graphs i., Publ. Math. Debrecen 6 (1959) 290–297.
G. Palla, I. Der[é]{}nyi, I. Farkas, T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society, Nature 435 (7043) (2005) 814–818.
E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-... | P. Erd[ő]{}s, A. R[é]{}nyi, On random graphs i., Publ. Math. Debrecen 6 (1959) 290–297.
G. Palla, I. Der[é]{}nyi, I. Farkas, T. Vicsek, Uncovering the overlapping community social organization of complex net in nature and society, Nature 435 (7043) (2005) 814–818.
E. Ravasz, A. L. Somera, D. A... |
P. Erf[ő]{}s, A. R[é]{}nyi, On random graphr i., Publ. Math. Dgbeecen 6 (1959) 290–297.
G. Palma, I. Der[é]{}nhi, I. Farkas, T. Vicsek, Uncoverinj thw oveelapping community strjcture of complex netxorks in nature ehd sociccy, Nafmre 435 (7043) (2005) 814–818.
X. Ravasz, A. L. Someta, D. A. Mongru, Z. N. Oltvai, A.-... | P. Erd[ő]{}s, A. R[é]{}nyi, On random graphs Math. 6 (1959) G. Palla, I. Uncovering overlapping community structure complex networks in and society, Nature 435 (7043) (2005) E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-L. Barab[á]{}si, organization of modularity in metabolic networks, Science 297 (5586) (200... |
P. Erd[ő]{}s, A. R[é]{}nyi, On random graphS i., Publ. Math. debreCen 6 (1959) 290–297.
g. PaLlA, I. DeR[é]{}nyI, I. Farkas, T. VicseK, uncoVering the overlapping coMmuniTy STrucTUrE of coMplex neTWoRKS in NaTuRe aNd SOcIety, NAtuRe 435 (7043) (2005) 814–818.
E. RavaSz, A. L. Somera, d. A. MOnGru, Z. N. Oltvai, A.-... |
P. Erd[ő]{}s, A. R[é]{}n yi, On ran dom g rap hsi. , Pu bl.Math. Debrecen 6 (1 959) 290–297.
G. Pall a, I. D e r[é] { }n yi, I . Farka s ,T . Vi cs ek , U nc o ve ringthe overla pping comm uni ty structure o f c omplex net wor ks in nature an d soci et y,N ature 43 5 (70 43) (2 0 05) 81 4–818.
E .R avasz, A.... |
P. Erd[ő]{}s, A. R[é]{}nyi,_On random_graphs i., Publ. Math._Debrecen 6_(1959)_290–297.
G. Palla, I. Der[é]{}nyi,_I. Farkas,_T. Vicsek, Uncovering the_overlapping community structure_of complex networks in_nature and society,_Nature_435 (7043) (2005) 814–818.
E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-... |
vec y_i\rangle = 0 \textrm{ for all } i \}$.
For a Boolean function $f: \F_2^n \rightarrow \F_2^n$ we denote its [*universal (quantum) embedding*]{} by $$U_f: \F_2^{2n} \rightarrow \F_2^{2n} \textrm{ with } (\vec x, \vec y) \mapsto (\vec x, f(\vec x)+\vec y).$$ Notice that $U_f(U_f(\vec x,\vec y)) = (\vec x, \vec y)$.... | vec y_i\rangle = 0 \textrm { for all } i \}$.
For a Boolean function $ f: \F_2^n \rightarrow \F_2^n$ we denote its [ * universal (quantum) embedding * ] { } by $ $ U_f: \F_2^{2n } \rightarrow \F_2^{2n } \textrm { with } (\vec x, \vec y) \mapsto (\vec x, f(\vec x)+\vec y).$$ detect that $ U_f(U_f(\vec x,\vec y) ) = (... | vec y_i\rangle = 0 \textrm{ for aul } i \}$.
For a Boolean fuiction $r: \F_2^n \rigftarrow \F_2^n$ we denote its [*unitersql (quqntum) embedding*]{} by $$U_f: \W_2^{2n} \rightagrow \F_2^{2n} \twxtrn{ with } (\vec x, \vec y) \mapstk (\vec e, f(\vec x)+\vec y).$$ Nptice that $U_f(U_f(\vec x,\vec f)) = (\vzc x, \vec y)$.... | vec y_i\rangle = 0 \textrm{ for all \}$. a Boolean $f: \F_2^n \rightarrow (quantum) by $$U_f: \F_2^{2n} \F_2^{2n} \textrm{ with (\vec x, \vec y) \mapsto (\vec f(\vec x)+\vec y).$$ Notice that $U_f(U_f(\vec x,\vec y)) = (\vec x, \vec y)$. $\ket{x} \in \mathbb{C}^2$ with $x \in \F_2$ be a qubit. We denote by the function... | vec y_i\rangle = 0 \textrm{ for all } i \}$.
FOr a Boolean FunctIon $F: \F_2^n \RiGhtaRrow \f_2^n$ we denote its [*uNIverSal (quantum) embedding*]{} by $$U_F: \F_2^{2n} \riGhTArroW \f_2^{2n} \TextrM{ with } (\veC X, \vEC Y) \maPsTo (\Vec X, f(\VEc X)+\vec y).$$ notIce that $u_f(U_f(\vec x,\veC y)) = (\vEc X, \vec y)$.... | vec y_i\rangle = 0 \textrm { for all} i \ }$.
F or a B oole an function $f : \F_ 2^n \rightarrow \F_2^n $ wede n otei ts [*un iversal (q u a ntu m) e mbe dd i ng *]{}by$$U_f:\F_2^{2n}\ri gh tarrow \F_2^ { 2n } \textrm{ wi th } (\vec x , \ vec y) \ map s to (\ vec x, f (\vecx )+\vec y).$$ No ti c e that $U_f(U... | vec y_i\rangle_= 0_\textrm{ for all }_i \}$.
For_a_Boolean function_$f:_\F_2^n \rightarrow \F_2^n$_we denote its_[*universal (quantum) embedding*]{} by_$$U_f: \F_2^{2n} \rightarrow_\F_2^{2n}_\textrm{ with } (\vec x, \vec y) \mapsto (\vec x, f(\vec x)+\vec y).$$ Notice_that_$U_f(U_f(\vec x,\vec_y))_=_(\vec x, \vec y)$.... |
ient Shared Control {#sec:NGSC}
===============================
Expressing Shared Control as an Optimization Problem
----------------------------------------------------
Let $s \in \mathcal{S}$ be the state of the system. Let $a^H \in \mathcal{A^H}$ as the user action, $a^R \in \mathcal{A^R}$ be the autonomous robot ... | ient Shared Control { # sec: NGSC }
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Expressing Shared Control as an Optimization Problem
----------------------------------------------------
Let $ s \in \mathcal{S}$ be the state of the arrangement. lease $ a^H \in \mathcal{A^H}$ as the user actio... | ienh Shared Control {#sec:NGSC}
===============================
Txpressing Shared Contron as ah Optimixation Problem
----------------------------------------------------
Let $s \in \mathcel{S}$ ve tht state of the systdm. Let $a^H \in \mathxal{A^I}$ as the user acvjon, $a^R \lu \matggal{A^R}$ ue the autonomoos robot ... | ient Shared Control {#sec:NGSC} =============================== Expressing Shared an Problem ---------------------------------------------------- $s \in \mathcal{S}$ system. $a^H \in \mathcal{A^H}$ the user action, \in \mathcal{A^R}$ be the autonomous robot and $u \in \mathcal{U}$ be the shared control action. The huma... | ient Shared Control {#sec:NGSC}
===============================
EXpressing SHared conTroL aS an OPtimIzation Problem
----------------------------------------------------
lEt $s \iN \mathcal{S}$ be the state of tHe sysTeM. let $a^h \In \MathcAl{A^H}$ as tHE uSER acTiOn, $A^R \iN \mAThCal{A^R}$ Be tHe autonOmous robot ... | ient Shared Control {#sec: NGSC}
==== ===== === === == ==== ==== ======
Expres s ingShared Control as an O ptimi za t ionP ro blem------- - -- - - --- -- -- --- -- - -- ----- --- ------- ----------
L et $s \in \mat h ca l{S}$ be t hestate of the sy stem.Le t $ a ^H \i n \ mathc al{A^H } $ as t he user a ct i on, $... | ient Shared_Control {#sec:NGSC}
===============================
Expressing_Shared Control as an_Optimization Problem
----------------------------------------------------
Let_$s_\in \mathcal{S}$_be_the state of_the system. Let_$a^H \in \mathcal{A^H}$ as_the user action,_$a^R_\in \mathcal{A^R}$ be the autonomous robot ... |
TP-AGB model may be changed in future calculations.
Anyhow, various tests indicate that the present version of the `COLIBRI` models already yields a fairly good description of the TP-AGB phase. Compared to our previously calibrated sets [@MarigoGirardi_07; @Marigo_etal08; @Girardi_etal10] the new TP-AGB models yield ... | TP - AGB model may be changed in future calculations.
Anyhow, various examination bespeak that the present version of the 'COLIBRI 'models already yields a reasonably good description of the TP - AGB phase. Compared to our previously calibrated set [ @MarigoGirardi_07; @Marigo_etal08; @Girardi_etal10 ] the new TP - ... | TP-WGB model may be changed in future calcolqtions.
Enyhow, barious gests indicate that the presxnt cersiin of the `COLIBRI` modeus alreadj yields q famrly good descri'fion of the TL-WGB 'hese. Compared to our previmusly calibratad szts [@MarigoGirardi_07; @Marigo_etal08; @Girardi_qtal10] thr jew TP-AGB modejs ypejd ... | TP-AGB model may be changed in future various indicate that present version of a good description of TP-AGB phase. Compared our previously calibrated sets [@MarigoGirardi_07; @Marigo_etal08; the new TP-AGB models yield somewhat shorter, but still comparable, TP-AGB lifetimes, and successfully recover various observatio... | TP-AGB model may be changed in fUture calcuLatioNs.
ANyhOw, VariOus tEsts indicate thAT the Present version of the `COLiBRI` mOdELs alREaDy yieLds a faiRLy GOOd dEsCrIptIoN Of The TP-aGB Phase. CoMpared to ouR prEvIously calibrATeD sets [@MarigOGiRardi_07; @Marigo_eTal08; @girardI_eTal10] THe new tP-AgB modEls yieLD ... | TP-AGB model may be chang ed in futu re ca lcu lat io ns.
Any how, various t e stsindicate that the pres ent v er s iono fthe ` COLIBRI ` m o d els a lr ead yy ie lds a fa irly go od descrip tio nof the TP-AG B p hase. Comp are d to our pre vio usly c al ibr a ted s ets [@Ma rigoGi r ardi_0 7; @Marig o_ e tal08; ... | TP-AGB_model may_be changed in future_calculations.
Anyhow, various_tests_indicate that_the_present version of_the `COLIBRI` models_already yields a fairly_good description of_the_TP-AGB phase. Compared to our previously calibrated sets [@MarigoGirardi_07; @Marigo_etal08; @Girardi_etal10] the new TP-AGB_models_yield ... |
at least of order $\simg
10^{15}$ G. Indeed, it is not obvious why the fields cannot become even higher. Note that the virial limit is $B_v \sim 10^{17}$ G.
After magnetic fields have built up to some fraction of the equipartition value with the shear motion, a magnetic viscosity develops. Assuming that $B_rB_\phi\si... | at least of order $ \simg
10^{15}$ G. Indeed, it is not obvious why the fields cannot become even higher. Note that the virial terminus ad quem is $ B_v \sim 10^{17}$ G.
After charismatic discipline have build up to some fraction of the equipartition value with the shear movement, a magnetic viscosity develops. as... | at least of order $\simg
10^{15}$ G. Ikdeed, it is not obvious why tge fieldr cannot become even higher. Iote that the virial limit is $B_x \sim 10^{17}$ G.
Avter magbetir fields have bummt up to some nractnoi of the equipattition valua with the shedr mltion, a magnetic viscosity develops. Assumimg that $B_rB_\phi\si... | at least of order $\simg 10^{15}$ G. is obvious why fields cannot become virial is $B_v \sim G. After magnetic have built up to some fraction the equipartition value with the shear motion, a magnetic viscosity develops. Assuming that B^2$, it can be characterized in the usual way by the parameter $\alpha\sim \rho ) 10^... | at least of order $\simg
10^{15}$ G. Indeed, It is not obvIous wHy tHe fIeLds cAnnoT become even higHEr. NoTe that the virial limit is $b_v \sim 10^{17}$ g.
AFTer mAGnEtic fIelds haVE bUILt uP tO sOme FrACtIon of The EquiparTition valuE wiTh The shear motiON, a Magnetic viScoSity develops. assUming tHaT $B_rb_\Phi\si... | at least of order $\simg10^{15}$ G . Ind eed , i tis n ot o bvious why the fiel ds cannot become evenhighe r. Note th at th e viria l l i m itis $ B_v \ s im 10^{ 17} $ G.
A fter magne tic f ields have b u il t up to so mefraction ofthe equip ar tit i on va lue with the s h ear mo tion, a m ag n etic v i scos... | at_least of_order $\simg
10^{15}$ G. Indeed,_it is_not_obvious why_the_fields cannot become_even higher. Note_that the virial limit_is $B_v \sim_10^{17}$_G.
After magnetic fields have built up to some fraction of the equipartition value with_the_shear motion,_a_magnetic_viscosity develops. Assuming that $B_rB_\phi\si... |
pa}^{\alpha}\; ds.$$ We first transform $$\int_{-D}^D {\lpa u + n - s \rpa}^{H'}\psi (u)\, du$$ through $\ell$ successive integrations by parts. Recalling (\[border\]), we see that each time the border terms $I^m\psi(\pm D)$, $0\leq m\leq \ell$, vanish. In the end, we obtain $$\begin{aligned}
\label{ijn1}
I^1_{n}& = & ... | pa}^{\alpha}\; ds.$$ We first transform $ $ \int_{-D}^D { \lpa u + n - s \rpa}^{H'}\psi (u)\, du$$ through $ \ell$ successive integration by part. Recalling (\[border\ ]), we see that each time the margin terms $ I^m\psi(\pm D)$, $ 0\leq m\leq \ell$, vanish. In the conclusion, we prevail $ $ \begin{aligned }
\label{i... | pa}^{\appha}\; ds.$$ We first transfovm $$\int_{-D}^D {\lpa u + u - s \rpe}^{H'}\psi (u)\, du$$ throjgh $\ell$ successive integratilnw by kcrts. Recalling (\[bordef\]), we see nhat each timt the border terms $I^m\psi(\pm D)$, $0\les m\lex \ell$, vanish. In the end, wa obtain $$\begin{dlkgued}
\label{ijn1}
I^1_{n}& = & ... | pa}^{\alpha}\; ds.$$ We first transform $$\int_{-D}^D {\lpa n s \rpa}^{H'}\psi du$$ through $\ell$ (\[border\]), see that each the border terms D)$, $0\leq m\leq \ell$, vanish. In end, we obtain $$\begin{aligned} \label{ijn1} I^1_{n}& = & C\,\int_0^{n - 2D} {\lpa \int_{-D}^D u + n - s \rpa}^{H' - \ell}I^{\ell}\psi(u) \... | pa}^{\alpha}\; ds.$$ We first transform $$\Int_{-D}^D {\lpa u + n - S \rpa}^{H'}\Psi (U)\, du$$ ThRougH $\ell$ Successive inteGRatiOns by parts. Recalling (\[borDer\]), we SeE That EAcH time The bordER tERMs $I^M\pSi(\Pm D)$, $0\LeQ M\lEq \ell$, VanIsh. In thE end, we obtaIn $$\bEgIn{aligned}
\labEL{iJn1}
I^1_{n}& = & ... | pa}^{\alpha}\; ds.$$ We fi rst transf orm $ $\i nt_ {- D}^D {\l pa u + n - s \ r pa}^ {H'}\psi (u)\, du$$ th rough $ \ ell$ su ccess ive int e gr a t ion sby pa rt s .Recal lin g (\[bo rder\]), w e s ee that each t i me the borde r t erms $I^m\ps i(\ pm D)$ ,$0\ l eq m\ leq \ell $, van i sh. In the end, w e ob... | pa}^{\alpha}\; ds.$$_We first_transform $$\int_{-D}^D {\lpa u_+ n_-_s \rpa}^{H'}\psi_(u)\,_du$$ through $\ell$_successive integrations by_parts. Recalling (\[border\]), we_see that each_time_the border terms $I^m\psi(\pm D)$, $0\leq m\leq \ell$, vanish. In the end, we obtain_$$\begin{aligned}
\label{ijn1}
I^1_{n}&_= &_... |
Scientific, 2004.
Y. Aharonov and D. Bohm, Phys. Rev. [**115**]{}, 485 (1959).
---
abstract: 'The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genu... | Scientific, 2004.
Y. Aharonov and D. Bohm, Phys. Rev. [ * * 115 * * ] { }, 485 (1959).
---
abstract:' The CM class number one problem for elliptic curve ask to find all elliptic curves define over the rationals with non - trivial endomorphism band. For genus-2 crook it is the problem of determine all C... | Sclentific, 2004.
Y. Aharonov and D. Nohm, Phys. Rev. [**115**]{}, 485 (1959).
---
cvstracv: 'The CJ class vumber one problem for ellipvic xurvew asked to find all eluiptic cugves defibed icer the ravjonals with nkk-trivnao endomorphism ring. For cenus-2 curves id ks the problem of determining all CM surves pf genu... | Scientific, 2004. Y. Aharonov and D. Bohm, [**115**]{}, (1959). --- 'The CM class curves to find all curves defined over rationals with non-trivial endomorphism ring. For curves it is the problem of determining all CM curves of genus $2$ over the *reflex field*. We solve the problem by showing that the list in [@bouyer... | Scientific, 2004.
Y. Aharonov and D. BoHm, Phys. Rev. [**115**]{}, 485 (1959).
---
aBstraCt: 'THe Cm cLass NumbEr one problem foR ElliPtic curves asked to find aLl ellIpTIc cuRVeS defiNed over THe RATioNaLs WitH nON-tRiviaL enDomorphIsm ring. For GenUs-2 Curves it is thE PrOblem of detErmIning all CM cuRveS of genU... | Scientific, 2004.
Y. Aha ronov andD. Bo hm, Ph ys . Re v. [**115**]{}, 4 8 5 (1 959).
---
abstract: ' The C Mc lass nu mberone pro b le m for e ll ipt ic cu rvesask ed to f ind all el lip ti c curves def i ne d over the ra tionals with no n-triv ia l e n domor phi sm ri ng. Fo r genus -2 curves i t is th e p... | Scientific,_2004.
Y. Aharonov and_D. Bohm, Phys. Rev. [**115**]{}, 485_(1959).
---
abstract:_'The_CM class_number_one problem for_elliptic curves asked_to find all elliptic_curves defined over_the_rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining_all_CM curves_of_genu... |
this channel will provide a strong indication of the nuclear sticking effect. Thus, an increase in the average charge state of the nuclei, together with sharper positron peaks, could be used to demonstrate the existence of long sticking times and the existence of charged vacuum decay.
The calculations presented in th... | this channel will provide a strong indication of the nuclear sticking consequence. therefore, an increase in the average cathexis country of the nuclei, together with sharp positron peak, could be used to show the existence of long sticking time and the existence of charged vacuum decay.
The calculations presented i... | thls channel will provide x strong indicajiin of vhe nucmear stizking effect. Thus, an increasx in the qverage charge state ow the nucpei, togerher qith sharpxd positvjn pswks, eonld be used to cemonstrata the existenca uf long sticking times and the existegce of vhwrged vacuum dgcay.
Tnq cambuoations presented in th... | this channel will provide a strong indication nuclear effect. Thus, increase in the nuclei, with sharper positron could be used demonstrate the existence of long sticking and the existence of charged vacuum decay. The calculations presented in this paper single-electron Dirac calculations with spin degeneracy. Thus, fo... | this channel will provide a stRong indicaTion oF thE nuClEar sTickIng effect. Thus, aN IncrEase in the average charge State Of THe nuCLeI, togeTher witH ShARPer PoSiTroN pEAkS, coulD be Used to dEmonstrate The ExIstence of lonG StIcking timeS anD the existencE of ChargeD vAcuUM decaY.
ThE calcUlatioNS preseNted in th... | this channel will provide a strongindic ati onof the nuc lear stickinge ffec t. Thus, an increase i n the a v erag e c harge stateo ft h e n uc le i,to g et her w ith sharpe r positron pe ak s, could beu se d to demon str ate the exis ten ce oflo ngs ticki ngtimes and t h e exis tence ofch a rged v a cuum de ... | this_channel will_provide a strong indication_of the_nuclear_sticking effect._Thus,_an increase in_the average charge_state of the nuclei,_together with sharper_positron_peaks, could be used to demonstrate the existence of long sticking times and the_existence_of charged_vacuum_decay.
The_calculations presented in th... |
which for power-law inflation can be derived using a trick of integrating by parts $$\tau \equiv \int \frac{dt}{a(t)} = \int \frac{da}{a^2 H} =
-\frac{1}{aH} + \int \frac{\epsilon \, da}{a^2 H} \,,$$ which for constant $\epsilon$ implies $$\tau = -\frac{1}{aH} \, \frac{1}{1-\epsilon} \,.$$
With these results, Eq... | which for power - law inflation can be derived use a whoremaster of integrating by parts $ $ \tau \equiv \int \frac{dt}{a(t) } = \int \frac{da}{a^2 H } =
-\frac{1}{aH } + \int \frac{\epsilon \, da}{a^2 henry } \,,$$ which for constant $ \epsilon$ implies $ $ \tau = -\frac{1}{aH } \, \frac{1}{1-\epsilon } \,.$$
... | whlch for power-law inflatiun can be derivgd using a tridk of ingegrating by parts $$\tau \equiv \ibt \frqc{dt}{a(t)} = \int \frac{da}{a^2 H} =
-\frac{1}{wH} + \int \drac{\tpsilon \, da}{a^2 H} \,,$$ wijch for constzkt $\epvmlon$ implies $$\tao = -\frac{1}{aH} \, \fsac{1}{1-\epsilon} \,.$$
Witv ghzse results, Eq... | which for power-law inflation can be derived trick integrating by $$\tau \equiv \int = + \int \frac{\epsilon da}{a^2 H} \,,$$ for constant $\epsilon$ implies $$\tau = \, \frac{1}{1-\epsilon} \,.$$ With these results, Eq. (\[ufield\]) for the perturbations reduces to Bessel equation $$\label{bessel} \left[ \frac{d^2}{d\... | which for power-law inflation Can be derivEd usiNg a TriCk Of inTegrAting by parts $$\taU \EquiV \int \frac{dt}{a(t)} = \int \frac{da}{a^2 h} =
-\frac{1}{AH} + \INt \frAC{\ePsiloN \, da}{a^2 H} \,,$$ whICh FOR coNsTaNt $\ePsILoN$ implIes $$\Tau = -\frac{1}{AH} \, \frac{1}{1-\epsiLon} \,.$$
wiTh these resulTS, EQ... | which for power-law infla tion can b e der ive d u si ng a tri ck of integrat i ng b y parts $$\tau \equiv\int\f r ac{d t }{ a(t)} = \int \f r a c{d a} {a ^2H} =
-\f rac{1}{ aH} + \int \f ra c{\epsilon \ , d a}{a^2 H}\,, $$ which for co nstant $ \ep s ilon$ im plies $$\ta u = -\f rac{1}{aH }\ , \fra c {1}{... | which_for power-law_inflation can be derived_using a_trick_of integrating_by_parts $$\tau \equiv_\int \frac{dt}{a(t)} =_\int \frac{da}{a^2 H} =_
__-\frac{1}{aH} + \int \frac{\epsilon \, da}{a^2 H} \,,$$ which for constant $\epsilon$ implies $$\tau_=_-\frac{1}{aH} \,_\frac{1}{1-\epsilon}_\,.$$
With_these results, Eq... |
& 1.474 & 0.0027 & 28.300 & 17.038 & 12.474\
\
.01 & 1.575 & 0.0033 & 28.096 & 16.281 & 12.491 &.02 & 1.480 & 0.0029 & 28.099 & 16.353 & 12.354\
.03 & 1.527 & 0.0033 & 28.111 & 16.315 & 12.734 &.04 & 1.519 & 0.0030 & 28.106 & 16.342 & 12.453\
.05 & 1.541 & 0.0030 & 28.109 & 16.400 & 12.495 &.06 & 1.526 & 0.0029 & 28.1... | & 1.474 & 0.0027 & 28.300 & 17.038 & 12.474\
\
.01 & 1.575 & 0.0033 & 28.096 & 16.281 & 12.491 & .02 & 1.480 & 0.0029 & 28.099 & 16.353 & 12.354\
.03 & 1.527 & 0.0033 & 28.111 & 16.315 & 12.734 & .04 & 1.519 & 0.0030 & 28.106 & 16.342 & 12.453\
.05 & 1.541 & 0.0030 & 28.109 & 16.400 & 12.495 & .06 & 1.526 & 0.0... | & 1.474 & 0.0027 & 28.300 & 17.038 & 12.474\
\
.01 & 1.575 & 0.0033 & 28.096 & 16.281 & 12.491 &.02 & 1.480 & 0.0029 & 28.099 & 16.353 & 12.354\
.03 & 1.527 & 0.0033 & 28.111 & 16.315 & 12.734 &.04 & 1.519 & 0.0030 & 28.106 & 16.342 & 12.453\
.05 & 1.541 & 0.0030 & 28.109 & 16.400 & 12.495 &.06 & 1.526 & 0.0029 & 28.1... | & 1.474 & 0.0027 & 28.300 & 12.474\ .01 & & 0.0033 & &.02 1.480 & 0.0029 28.099 & 16.353 12.354\ .03 & 1.527 & 0.0033 28.111 & 16.315 & 12.734 &.04 & 1.519 & 0.0030 & 28.106 & & 12.453\ .05 & 1.541 & 0.0030 & 28.109 & 16.400 & 12.495 & & & & 16.374 & 12.524\ .07 & 1.532 & 0.0032 & 28.104 & 16.348 & 12.641 &.08 1.564 & ... | & 1.474 & 0.0027 & 28.300 & 17.038 & 12.474\
\
.01 & 1.575 & 0.0033 & 28.096 & 16.281 & 12.491 &.02 & 1.480 & 0.0029 & 28.099 & 16.353 & 12.354\
.03 & 1.527 & 0.0033 & 28.111 & 16.315 & 12.734 &.04 & 1.519 & 0.0030 & 28.106 & 16.342 & 12.453\
.05 & 1.541 & 0.0030 & 28.109 & 16.400 & 12.495 &.06 & 1.526 & 0.0029 & 28.1... | & 1.474 & 0.0027 & 28.300 & 17.038& 12. 474 \
\
. 01 & 1.5 75 & 0.0033 &2 8.09 6 & 16.281 & 12.491 &. 02 &1. 4 80 & 0. 0029& 28.09 9 & 1 6.3 53 & 12 .3 5 4\
.03& 1 .527 &0.0033 & 2 8.1 11 & 16.315 &1 2. 734 &.04 & 1. 519 & 0.0030 &28.106 & 16 . 342 & 12 .453\
.05 & 1.541& 0.0030&2 8.109& 16.400 & 1 2.4... | &_1.474 &_0.0027 & 28.300 &_17.038 &_12.474\
\
.01_& 1.575_&_0.0033 & 28.096_& 16.281 &_12.491 &.02 & 1.480_& 0.0029 &_28.099_& 16.353 & 12.354\
.03 & 1.527 & 0.0033 & 28.111 & 16.315 & 12.734_&.04_& 1.519_&_0.0030_& 28.106 & 16.342 &_12.453\
.05 & 1.541 & 0.0030_& 28.109_& 16.400 & 12.495 &.06 & 1.526 &_0.0029_& 28.1... |
\theta };q)_{k}}{(q,abq^{\alpha },acq^{\alpha
},adq^{\alpha },abcdq^{\alpha -1};q)_{k}}\ q^{k} \notag \\
& \qquad \times ~_{10}W_{9}(abcdq^{2\alpha +k-1};q^{\alpha },bcq^{\alpha
-1},bdq^{\alpha -1},cdq^{\alpha -1},q^{k+1},abcdq^{2\alpha
+n+k-1},q^{k-n};q,a^{2}). \label{aaw4}\end{aligned}$$There is another useful repr... | \theta }; q)_{k}}{(q, abq^{\alpha }, acq^{\alpha
}, adq^{\alpha }, abcdq^{\alpha -1};q)_{k}}\ q^{k } \notag \\
& \qquad \times ~_{10}W_{9}(abcdq^{2\alpha + k-1};q^{\alpha }, bcq^{\alpha
-1},bdq^{\alpha -1},cdq^{\alpha -1},q^{k+1},abcdq^{2\alpha
+ n+k-1},q^{k - n};q, a^{2 }). \label{aaw4}\end{aligned}$$There... | \theha };q)_{k}}{(q,abq^{\alpha },acq^{\alpha
},aaq^{\alpha },abcdq^{\alkhq -1};q)_{k}}\ q^{n} \notzg \\
& \qquaa \times ~_{10}W_{9}(abcdq^{2\alpha +k-1};q^{\alpha },ucq^{\aopha
-1},beq^{\alpha -1},cdq^{\alpha -1},q^{k+1},abcaq^{2\alpha
+n+k-1},e^{k-n};q,a^{2}). \lqbel{eaw4}\end{aligned}$$Thecs is another hdefun repr... | \theta };q)_{k}}{(q,abq^{\alpha },acq^{\alpha },adq^{\alpha },abcdq^{\alpha -1};q)_{k}}\ q^{k} & \times ~_{10}W_{9}(abcdq^{2\alpha },bcq^{\alpha -1},bdq^{\alpha -1},cdq^{\alpha useful of the associated polynomials in terms a double series due to Rahman, p_{n}^{\alpha }(x)& =p_{n}^{\alpha }(x;a,b,c,d|q)\smallskip \notag... | \theta };q)_{k}}{(q,abq^{\alpha },acq^{\alpha
},aDq^{\alpha },abcDq^{\alpHa -1};q)_{K}}\ q^{k} \NoTag \\
& \qQuad \Times ~_{10}W_{9}(abcdq^{2\alpHA +k-1};q^{\aLpha },bcq^{\alpha
-1},bdq^{\alpha -1},cdQ^{\alphA -1},q^{K+1},AbcdQ^{2\AlPha
+n+k-1},Q^{k-n};q,a^{2}). \laBEl{AAW4}\enD{aLiGneD}$$THErE is anOthEr usefuL repr... | \theta };q)_{k}}{(q,abq^{\ alpha },ac q^{\a lph a
} ,a dq^{ \alp ha },abcdq^{\a l pha-1};q)_{k}}\ q^{k} \n otag\\ & \q q ua d \ti mes ~_{ 1 0} W _ {9} (a bc dq^ {2 \ al pha + k-1 };q^{\a lpha },bcq ^{\ al pha
-1},bdq^ { \a lpha -1},c dq^ {\alpha -1}, q^{ k+1},a bc dq^ { 2\alp ha+n+k- 1},q^{ k -n};q, a^{2}). \l a bel{a... | \theta };q)_{k}}{(q,abq^{\alpha_},acq^{\alpha
},adq^{\alpha },abcdq^{\alpha_-1};q)_{k}}\ q^{k} \notag_\\
& \qquad_\times_~_{10}W_{9}(abcdq^{2\alpha +k-1};q^{\alpha_},bcq^{\alpha
-1},bdq^{\alpha_-1},cdq^{\alpha -1},q^{k+1},abcdq^{2\alpha
+n+k-1},q^{k-n};q,a^{2}). _\label{aaw4}\end{aligned}$$There is another_useful repr... |
label{eq:py}\end{aligned}$$ $Z = \sum_{{\mathbf{y}}} \sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right)$ is the partition function.
In the case of our Embedded Latent CRF model, as in the LDCRF model, latent states are deterministically partitioned to correspond to output val... | label{eq: py}\end{aligned}$$ $ Z = \sum_{{\mathbf{y } } } \sum_{{\mathbf{z } } } \exp \left (\mathcal{E}({\mathbf{y } }, { \mathbf{z}}| { \mathbf{x } }) \right)$ is the partition function.
In the case of our Embedded Latent CRF model, as in the LDCRF exemplar, latent state are deterministically partitioned to corres... | labfl{eq:py}\end{aligned}$$ $Z = \sum_{{\mxthbf{y}}} \sum_{{\mathby{z}}} \exp \neft( \mzthcal{E}({\mxthbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right)$ iw the partition function.
In ghe case lf our Enbedved Latent CRF model, as lu the PDCRY nodel, latent sjates are dederministicallf oaxtitioned to correspond to output vaj... | label{eq:py}\end{aligned}$$ $Z = \sum_{{\mathbf{y}}} \sum_{{\mathbf{z}}} \exp \left( {\mathbf{x}}) is the function. In the CRF as in the model, latent states deterministically partitioned to correspond to output That is, the number of latent states is a multiple of the number output values, and $\psi_{zy}=0$ for pairs ... | label{eq:py}\end{aligned}$$ $Z = \sum_{{\maThbf{y}}} \sum_{{\maThbf{z}}} \Exp \LefT( \mAthcAl{E}({\mAthbf{y}}, {\mathbf{z}}| {\mAThbf{X}}) \right)$ is the partition fuNctioN.
IN The cASe Of our embeddeD laTENt CrF MoDel, As IN tHe LDCrF mOdel, latEnt states aRe dEtErministicalLY pArtitioned To cOrrespond to oUtpUt val... | label{eq:py}\end{aligned}$ $ $Z = \su m_{{\ mat hbf {y }}}\sum _{{\mathbf{z}} } \ex p \left( \mathcal{E}({ \math bf { y}}, {\ mathb f{z}}|{ \m a t hbf {x }} ) \ ri g ht )$ is th e parti tion funct ion .
In the case of our Embed ded Latent CRFmod el, as i n t h e LDC RFmodel , late n t stat es are de te r minist i ... | label{eq:py}\end{aligned}$$ $Z_= \sum_{{\mathbf{y}}}_\sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}},_{\mathbf{z}}| {\mathbf{x}})_\right)$_is the_partition_function.
In the case_of our Embedded_Latent CRF model, as_in the LDCRF_model,_latent states are deterministically partitioned to correspond to output val... |
V/\partial z\right)=
\left( \partial V^* /\partial z^* \right)$, we have from Eq. (\[A1\]) for the quadratic variables $(\delta z)^2$ and $|\delta z|^2$ the following equations of motion $$\begin{aligned}
\label{A2}
i \frac{d}{d\tau}\left(\delta z\right)^2 &=&
-i\Gamma \left(\delta z\right)^2 +
2\frac{\partial V}{\par... | V/\partial z\right)=
\left (\partial V^ * /\partial z^ * \right)$, we have from Eq. (\[A1\ ]) for the quadratic variables $ (\delta z)^2 $ and $ |\delta z|^2 $ the following equations of apparent motion $ $ \begin{aligned }
\label{A2 }
i \frac{d}{d\tau}\left(\delta z\right)^2 & = &
-i\Gamma \left(\delta z\right... | V/\pwrtial z\right)=
\left( \partiau V^* /\partial z^* \rntht)$, we have rrom Eq. (\[X1\]) for the quadratic variabled $(\eelta z)^2$ and $|\delta z|^2$ the foluowing eqlations od mouion $$\begin{aligned}
\label{A2}
i \nxac{d}{d\fwu}\leyt(\velta z\right)^2 &=&
-i\Gsmma \left(\dalta z\right)^2 +
2\frdc{\oaxtial V}{\par... | V/\partial z\right)= \left( \partial V^* /\partial z^* have Eq. (\[A1\]) the quadratic variables the equations of motion \label{A2} i \frac{d}{d\tau}\left(\delta &=& -i\Gamma \left(\delta z\right)^2 + 2\frac{\partial z} \left(\delta z\right)^2 + 2\frac{\partial V}{\partial z^*} |\delta z|^2,\nonumber \\ % i \frac{d}{d\... | V/\partial z\right)=
\left( \partial v^* /\partial z^* \rIght)$, wE haVe fRoM Eq. (\[A1\]) For tHe quadratic varIAbleS $(\delta z)^2$ and $|\delta z|^2$ the folLowinG eQUatiONs Of motIon $$\begiN{AlIGNed}
\LaBeL{A2}
i \FrAC{d}{D\tau}\lEft(\Delta z\rIght)^2 &=&
-i\Gamma \LefT(\dElta z\right)^2 +
2\frAC{\pArtial V}{\par... | V/\partial z\right)=
\lef t( \partia l V^* /\ par ti al z ^* \ right)$, we ha v e fr om Eq. (\[A1\]) for th e qua dr a ticv ar iable s $(\de l ta z )^2 $an d $ |\ d el ta z| ^2$ the fo llowing eq uat io ns of motion $$ \begin{ali gne d}
\label{A2 }
i \frac {d }{d \ tau}\ lef t(\de lta z\ r ight)^ 2 &=&
-i\ Ga m ma ... | V/\partial_z\right)=
\left( \partial_V^* /\partial z^* \right)$,_we have_from_Eq. (\[A1\])_for_the quadratic variables_$(\delta z)^2$ and_$|\delta z|^2$ the following_equations of motion_$$\begin{aligned}
\label{A2}
i_\frac{d}{d\tau}\left(\delta z\right)^2 &=&
-i\Gamma \left(\delta z\right)^2 +
2\frac{\partial V}{\par... |
eless such that ${\bf u}_1(x)={\bf m}(A,e,f)$ and ${\bf u}_\kappa(x)={\bf m}(A,E,F)$.
We shall exhibit a matrix ${\bf m}(A,0,0)={\bf n}(A,0,0)$ whose projective order is a multiple of $4$ and it is bigger than $8$. This will prove the statement for both $t=1,\kappa$.
Let ${{\mathbb F}}^\times_{q^2}=\langle \xi \rangl... | eless such that $ { \bf u}_1(x)={\bf m}(A, e, f)$ and $ { \bf u}_\kappa(x)={\bf m}(A, E, F)$.
We shall exhibit a matrix $ { \bf m}(A,0,0)={\bf n}(A,0,0)$ whose projective order is a multiple of $ 4 $ and it is bigger than $ 8$. This will rise the instruction for both $ t=1,\kappa$.
Let $ { { \mathbb F}}^\times_{q... | eleds such that ${\bf u}_1(x)={\bf m}(A,e,n)$ and ${\bf u}_\kappa(x)={\yd m}(A,E,F)$.
Xe shalm exhibig a matrix ${\bf m}(A,0,0)={\bf n}(A,0,0)$ whose 'rojwctivt order is a multipue of $4$ anf it is viggtr than $8$. This will prove bke stzbemenc hor both $t=1,\kappa$.
Ket ${{\mathbb F}}^\times_{q^2}=\langle \xk \xangl... | eless such that ${\bf u}_1(x)={\bf m}(A,e,f)$ and m}(A,E,F)$. shall exhibit matrix ${\bf m}(A,0,0)={\bf a of $4$ and is bigger than This will prove the statement for $t=1,\kappa$. Let ${{\mathbb F}}^\times_{q^2}=\langle \xi \rangle$ and consider the matrix $z={\operatorname{diag}}(\xi^{\frac{q-1}{2}},-\xi^{\frac{1-q}{2... | eless such that ${\bf u}_1(x)={\bf m}(A,e,f)$ anD ${\bf u}_\kappa(x)={\Bf m}(A,E,f)$.
We ShaLl ExhiBit a Matrix ${\bf m}(A,0,0)={\bf n}(A,0,0)$ WHose Projective order is a multIple oF $4$ aND it iS BiGger tHan $8$. This WIlL PRovE tHe StaTeMEnT for bOth $T=1,\kappa$.
LEt ${{\mathbb F}}^\tImeS_{q^2}=\Langle \xi \rangL... | eless such that ${\bf u}_1 (x)={\bf m }(A,e ,f) $ a nd ${\ bf u }_\kappa(x)={\ b f m} (A,E,F)$.
We shall ex hibit a matr i x${\bf m}(A,0 , 0) = { \bf n }( A,0 ,0 ) $whose pr ojectiv e order is amu ltiple of $4 $ a nd it is b igg er than $8$. Th is wil lpro v e the st ateme nt for both $ t=1,\kapp a$ .
Let$ ... | eless such_that ${\bf_u}_1(x)={\bf m}(A,e,f)$ and ${\bf_u}_\kappa(x)={\bf m}(A,E,F)$.
We_shall_exhibit a_matrix_${\bf m}(A,0,0)={\bf n}(A,0,0)$_whose projective order_is a multiple of_$4$ and it_is_bigger than $8$. This will prove the statement for both $t=1,\kappa$.
Let ${{\mathbb F}}^\times_{q^2}=\langle \xi_\rangl... |
contribution. If such an enhancement were observed in the experiment, it would provide an important and unambiguous determination of the weak $\pi NN$ coupling constant $h_{\pi}^{1}$. However, a recent schematic calculation of $A_{\gamma}$ by Khriplovich and Korkin [@Khriplovich:2000mb], partly suggested by one of the... | contribution. If such an enhancement were observed in the experiment, it would provide an important and unambiguous decision of the fallible $ \pi NN$ coupling constant $ h_{\pi}^{1}$. However, a recent conventional calculation of $ A_{\gamma}$ by Khriplovich and Korkin [ @Khriplovich:2000 mb ], partly suggested by one... | cojtribution. If such an enmancement were oywerved in ths experioent, it would provide an implrrant qnd unambiguous determknation ov the weqk $\pm NN$ coupling coiatant $h_{\in}^{1}$. Howsyer, a cecent schematig calculatimn of $A_{\gamma}$ bf Yhxiplovich and Korkin [@Khriplovich:2000mb], pwrtly sighested by one jf tnq... | contribution. If such an enhancement were observed experiment, would provide important and unambiguous NN$ constant $h_{\pi}^{1}$. However, recent schematic calculation $A_{\gamma}$ by Khriplovich and Korkin [@Khriplovich:2000mb], suggested by one of the present author, showed critical contradiction to Oka’s result, a ... | contribution. If such an enhanCement were ObserVed In tHe ExpeRimeNt, it would proviDE an iMportant and unambiguous DeterMiNAtioN Of The weAk $\pi NN$ cOUpLINg cOnStAnt $H_{\pI}^{1}$. hoWever, A reCent schEmatic calcUlaTiOn of $A_{\gamma}$ by kHrIplovich anD KoRkin [@KhriplovIch:2000Mb], partLy SugGEsted By oNe of tHe... | contribution. If such anenhancemen t wer e o bse rv ed i n th e experiment,i t wo uld provide an importa nt an du namb i gu ous d etermin a ti o n of t he we ak $\ pi NN $ c oupling constant$h_ {\ pi}^{1}$. Ho w ev er, a rece ntschematic ca lcu lation o f $ A _{\ga mma }$ by Khrip l ovichand Korki n[ @Khrip l ov... | contribution._If such_an enhancement were observed_in the_experiment,_it would_provide_an important and_unambiguous determination of_the weak $\pi NN$_coupling constant $h_{\pi}^{1}$._However,_a recent schematic calculation of $A_{\gamma}$ by Khriplovich and Korkin [@Khriplovich:2000mb], partly suggested by_one_of the... |
has allowed us to confirm the peculiar red colors of five sources in the sample. We can impose modest upper limits of 0.9% and 1.8% on the linear polarization degree for seven targets with a confidence of 99%. Only one source, 2MASSJ02411151$-$0326587 (L0), appears to be strongly polarized ($P \sim 3\,\%$) in the $J$-... | has allowed us to confirm the peculiar red colors of five source in the sample distribution. We can impose modest upper limit of 0.9% and 1.8% on the analogue polarization degree for seven targets with a confidence of 99% . merely one source, 2MASSJ02411151$-$0326587 (L0), appears to be powerfully polarize ($ P \si... | had allowed us to confirm uhe peculiar red eilors mf fivs sourcer in the sample. We can imposx moeest ypper limits of 0.9% and 1.8% on the linewr polaruzatmon degree for sxben tarnzts wjbh a eoifidence of 99%. Onky one sousce, 2MASSJ02411151$-$0326587 (L0), ap[exrd to be strongly polarized ($P \sim 3\,\%$) ig the $J$-... | has allowed us to confirm the peculiar of sources in sample. We can 0.9% 1.8% on the polarization degree for targets with a confidence of 99%. one source, 2MASSJ02411151$-$0326587 (L0), appears to be strongly polarized ($P \sim 3\,\%$) in $J$-band with a significance level of $P/\sigma_P \sim 10$. The likely origin of ... | has allowed us to confirm the pEculiar red ColorS of FivE sOurcEs in The sample. We can IMposE modest upper limits of 0.9% anD 1.8% on thE lINear POlArizaTion degREe FOR seVeN tArgEtS WiTh a coNfiDence of 99%. only one souRce, 2mAsSJ02411151$-$0326587 (L0), appears tO Be Strongly poLarIzed ($P \sim 3\,\%$) in thE $J$-... | has allowed us to confirm the pecul iar r edcol or s of fiv e sources in t h e sa mple. We can impose mo destup p er l i mi ts of 0.9% a n d1 . 8%on t heli n ea r pol ari zationdegree for se ve n targets wi t ha confiden ceof 99%. Only on e sour ce , 2 M ASSJ0 241 1151$ -$0326 5 87 (L0 ), appear st o be s t rong... | has_allowed us_to confirm the peculiar_red colors_of_five sources_in_the sample. We_can impose modest_upper limits of 0.9% and_1.8% on the linear_polarization_degree for seven targets with a confidence of 99%. Only one source, 2MASSJ02411151$-$0326587 (L0),_appears_to be_strongly_polarized_($P \sim 3\,\%$) in the_$J$-... |
,j})-\phi(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^k|^{3/2}).\end{aligned}$$ In the last line, $\big(\phi(c_{1,i})-\phi(c_{2,i})\big)\big(\phi(c_{1,j})-\phi(c_{2,j})\big)a_ia_j$ is bounded by some constant except on a countable collection of measure zero sets. Let $C_i$ be defined as the set $\{z_{t/2}+\eta_i+\mu_i=0\... | , j})-\phi(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^k|^{3/2}).\end{aligned}$$ In the last line, $ \big(\phi(c_{1,i})-\phi(c_{2,i})\big)\big(\phi(c_{1,j})-\phi(c_{2,j})\big)a_ia_j$ is bounded by some constant except on a countable solicitation of standard zero sets. Let $ C_i$ be define as the fixed $ \{z_{t/2}+\eta_i+... | ,j})-\phl(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^k|^{3/2}).\ekd{aligned}$$ In the last lmne, $\big(\lhi(c_{1,i})-\phi(z_{2,i})\big)\big(\phi(c_{1,j})-\phi(c_{2,j})\big)a_ia_j$ is biundee by some constant excdpt on a bountable colowction of measure dzro ssbs. Lec $R_i$ be defined ax the set $\{s_{t/2}+\eta_i+\mu_i=0\... | ,j})-\phi(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^k|^{3/2}).\end{aligned}$$ In the last line, $\big(\phi(c_{1,i})-\phi(c_{2,i})\big)\big(\phi(c_{1,j})-\phi(c_{2,j})\big)a_ia_j$ is some except on countable collection of be as the set On the set $\big(\phi(c_{1,i})-\phi(c_{2,i})\big)a_i$ converges to zero as $a_i\right... | ,j})-\phi(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^K|^{3/2}).\end{aligneD}$$ In thE laSt lInE, $\big(\Phi(c_{1,I})-\phi(c_{2,i})\big)\big(\phI(C_{1,j})-\phI(c_{2,j})\big)a_ia_j$ is bounded by sOme coNsTAnt eXCePt on a CountabLE cOLLecTiOn Of mEaSUrE zero SetS. Let $C_i$ bE defined as The SeT $\{z_{t/2}+\eta_i+\mu_i=0\... | ,j})-\phi(c_{2,j})\Big)a_i a_jcov_{ij }^k+O (|c ov_ {i j}^k |^{3 /2}).\end{alig n ed}$ $ In the last line, $\ big(\ ph i (c_{ 1 ,i })-\p hi(c_{2 , i} ) \ big )\ bi g(\ ph i (c _{1,j })- \phi(c_ {2,j})\big )a_ ia _j$ is bound e dby some co nst ant except o n a count ab lec ollec tio n ofmeasur e zerosets. Let $ C _i$... | ,j})-\phi(c_{2,j})\Big)a_ia_jcov_{ij}^k+O(|cov_{ij}^k|^{3/2}).\end{aligned}$$ In_the last_line, $\big(\phi(c_{1,i})-\phi(c_{2,i})\big)\big(\phi(c_{1,j})-\phi(c_{2,j})\big)a_ia_j$ is bounded_by some_constant_except on_a_countable collection of_measure zero sets._Let $C_i$ be defined_as the set_$\{z_{t/2}+\eta_i+\mu_i=0\... |
.[]{data-label="fig:VoterResults"}](VoterResults.png){width="\columnwidth"}
Agent Based Models on a spatial domain
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In the previous subsection, we have detailed how to create an ABM on a graph, how to create an associated problem, how to generate its code and how to run it. Simfl... | .[]{data - label="fig: VoterResults"}](VoterResults.png){width="\columnwidth " }
Agent Based Models on a spatial domain
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In the previous subsection, we have detailed how to create an ABM on a graph, how to make an associated trouble, how to generate its code and how to run ... | .[]{datw-label="fig:VoterResults"}](VottrResults.png){width="\eilumnwmdth"}
Ageht Based Models on a spatial domain
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Ii thw precious subsection, we haxe detailvd how to creete an ABM on a jdaph, how to cdcate cn associated prpblem, how do generate itv zobe and how to run it. Simfl... | .[]{data-label="fig:VoterResults"}](VoterResults.png){width="\columnwidth"} Agent Based Models on a spatial In previous subsection, have detailed how a how to create associated problem, how generate its code and how to it. Simflowny 2 also includes the new family of ABM on a spatial which we validate in this section. A... | .[]{data-label="fig:VoterResults"}](VOterResultS.png){wIdtH="\coLuMnwiDth"}
AGent Based ModelS On a sPatial domain
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In the previOus suBsECtioN, We Have dEtailed HOw TO CreAtE aN ABm oN A gRaph, hOw tO create An associatEd pRoBlem, how to genERaTe its code aNd hOw to run it. SimFl... | .[]{data-label="fig:VoterR esults"}]( Voter Res ult s. png) {wid th="\columnwid t h"}
Agent Based Models on a sp at i al d o ma in
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In t he previou s s ub section, weh av e detailed ho w to createanABM on a gr a ph, h owto cr eate a n assoc iated pro bl e m, ho... | .[]{data-label="fig:VoterResults"}](VoterResults.png){width="\columnwidth"}
Agent Based_Models on_a spatial domain
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In the_previous subsection,_we_have detailed_how_to create an_ABM on a_graph, how to create_an associated problem,_how_to generate its code and how to run it. Simfl... |
quez-Ramírez, D. Veberič, C. Ventura, I.D. Vergara Quispe, V. Verzi, J. Vicha, L. Villaseñor, J. Vink, S. Vorobiov, H. Wahlberg, A.A. Watson, M. Weber, A. Weindl, L. Wiencke, H. Wilczyński, T. Winchen, M. Wirtz, D. Wittkowski, B. Wundheiler, A. Yushkov, E. Zas, D. Zavrtanik, M. Zavrtanik, L. Zehrer, A. Zepeda, M. Ziolk... | quez - Ramírez, D. Veberič, C. Ventura, I.D. Vergara Quispe, V. Verzi, J. Vicha, L. Villaseñor, J. Vink, S. Vorobiov, H. Wahlberg, A.A. Watson, M. Weber, A. Weindl, L. Wiencke, H. Wilczyński, T. Winchen, M. Wirtz, D. Wittkowski, B. Wundheiler, A. Yushkov, E. Zas, D. Zavrtanik, ... | quex-Ramírez, D. Veberič, C. Ventura, I.D. Vergara Quiske, V. Verzm, J. Vichz, L. Villareñor, J. Vink, S. Vorobiov, H. Wahlbecg, A.Q. Watsin, M. Weber, A. Weindl, L. Wievcke, H. Wilbzyński, T. Wunchtn, M. Wirtz, D. Wittkoxaki, B. Wukbheilsv, A. Yuvikov, E. Zas, D. Zavrjanik, M. Zavrtdnik, L. Zehrer, A. Seoeba, M. Ziolk... | quez-Ramírez, D. Veberič, C. Ventura, I.D. Vergara Verzi, Vicha, L. J. Vink, S. M. A. Weindl, L. H. Wilczyński, T. M. Wirtz, D. Wittkowski, B. Wundheiler, Yushkov, E. Zas, D. Zavrtanik, M. Zavrtanik, L. Zehrer, A. Zepeda, M. Ziolkowski, Zuccarello' title: | Cosmic-ray anisotropies in right ascension\ measured by the Pi... | quez-Ramírez, D. Veberič, C. VentuRa, I.D. VergarA QuisPe, V. verZi, j. VicHa, L. VIllaseñor, J. Vink, s. voroBiov, H. Wahlberg, A.A. Watson, M. weber, a. WEIndl, l. wiEncke, h. WilczyŃSkI, t. winChEn, m. WiRtZ, d. WIttkoWskI, B. WundhEiler, A. YushKov, e. ZAs, D. Zavrtanik, m. zaVrtanik, L. ZeHreR, A. Zepeda, M. ZioLk... | quez-Ramírez, D. Veberič,C. Ventura , I.D . V erg ar a Qu ispe , V. Verzi, J. Vich a, L. Villaseñor, J. V ink,S. Voro b io v, H. Wahlbe r g, A .A. W at son ,M .Weber , A . Weind l, L. Wien cke ,H. Wilczyńsk i ,T. Winchen , M . Wirtz, D.Wit tkowsk i, B. Wundh eil er, A . Yush k ov, E. Zas, D.Za v rtanik , M. Zav r ... | quez-Ramírez, D. Veberič,_C. Ventura, I.D. Vergara_Quispe, V. Verzi, J. Vicha, L. Villaseñor,_J. Vink, S. Vorobiov,_H. Wahlberg,_A.A. Watson, M. Weber,_A. Weindl,_L. Wiencke, H. Wilczyński, T. Winchen,_M. Wirtz, D. Wittkowski, B. Wundheiler,_A. Yushkov, E. Zas, D. Zavrtanik, M. Zavrtanik,_L. Zehrer, A. Zepeda, M. Ziolk... |
]. On the one hand such an effect of doping is not to be expected in the case of a high-$T$ peak originating from optical phonons, since the lattice impurities induced by the Sr-ions are similar to the Eu-impurities discussed above. On the other hand the strong frustration of antiferromagnetism upon doping of mobile ho... | ]. On the one hand such an effect of doping is not to be expected in the case of a high-$T$ acme originate from optical phonons, since the lattice impurities induce by the Sr - ions are similar to the Eu - impurity hash out above. On the other hired hand the strong frustration of antiferromagnetism upon doping of mobil... | ]. On the one hand such an efnect of doping is not tm be espected kn the case of a high-$T$ peak lruginaupng from optical phonuns, since the latrice umpurities induced by ths Sr-imis are similar jo the Eu-impgrities discusvea cbove. On the other hand the strong fwustratooj of antiferroiagnttifm ulon doping of mobile ho... | ]. On the one hand such an doping not to expected in the originating optical phonons, since lattice impurities induced the Sr-ions are similar to the discussed above. On the other hand the strong frustration of antiferromagnetism upon doping mobile holes (cf. Ref. [@Hucker02] and references therein) would provide for a... | ]. On the one hand such an effect oF doping is nOt to bE exPecTeD in tHe caSe of a high-$T$ peak ORigiNating from optical phonoNs, sinCe THe laTTiCe impUrities INdUCEd bY tHe sr-iOnS ArE simiLar To the Eu-Impurities DisCuSsed above. On tHE oTher hand thE stRong frustratIon Of antiFeRroMAgnetIsm Upon dOping oF Mobile Ho... | ]. On the one hand such an effect of dopi ngisno t to beexpected in th e cas e of a high-$T$ peak o rigin at i ng f r om opti cal pho n on s , si nc ethe l a tt ice i mpu ritiesinduced by th eSr-ions ares im ilar to th e E u-impurities di scusse dabo v e. On th e oth er han d the s trong fru st r ationo f an... | ]. On_the one_hand such an effect_of doping_is_not to_be_expected in the_case of a_high-$T$ peak originating from_optical phonons, since_the_lattice impurities induced by the Sr-ions are similar to the Eu-impurities discussed above. On_the_other hand_the_strong_frustration of antiferromagnetism upon doping_of mobile ho... |
"fig:"){width="240pt"}![The dynamical measure synchronization of (a)(b) $S_{c}(t)$ and (c)(d) the mutual information $I_{AB}(t)$ in the whole Hilbert space of $\bigoplus_{N}%
\mathcal{H}_{N}$ for $N=0,1,2,\cdots,n$. The truncated number $n=15$ for (a) (c) and $n=10$ for (b)(d). The other parameters are the same as tha... | " fig:"){width="240pt"}![The dynamical measure synchronization of (a)(b) $ S_{c}(t)$ and (c)(d) the mutual data $ I_{AB}(t)$ in the solid Hilbert space of $ \bigoplus_{N}%
\mathcal{H}_{N}$ for $ N=0,1,2,\cdots, n$. The truncated numeral $ n=15 $ for (a) (speed of light) and $ n=10 $ for (b)(d). The other parameters a... | "fih:"){width="240pt"}![The dynamical mexsure synchroniearion oh (a)(b) $S_{c}(f)$ and (c)(d) the mutual information $I_{AB}(t)$ ib the whole Hilbert space ow $\bigoplud_{N}%
\mathcao{H}_{N}$ hor $N=0,1,2,\cdots,n$. The vduncated numbsv $n=15$ fmc (a) (c) and $n=10$ for (b)(d). The otver parameters afe the same as tha... | "fig:"){width="240pt"}![The dynamical measure synchronization of (a)(b) $S_{c}(t)$ the information $I_{AB}(t)$ the whole Hilbert $N=0,1,2,\cdots,n$. truncated number $n=15$ (a) (c) and for (b)(d). The other parameters are same as that in Fig.\[figure8\] and Fig.\[figure9\]. Insets: the zoomed-in images from the rectang... | "fig:"){width="240pt"}![The dynamical meaSure synchrOnizaTioN of (A)(b) $s_{c}(t)$ aNd (c)(d) The mutual inforMAtioN $I_{AB}(t)$ in the whole Hilbert Space Of $\BIgopLUs_{n}%
\mathCal{H}_{N}$ foR $n=0,1,2,\cDOTs,n$. thE tRunCaTEd NumbeR $n=15$ fOr (a) (c) and $N=10$ for (b)(d). The oTheR pArameters are THe Same as tha... | "fig:"){width="240pt"}![T he dynamic al me asu resy nchr oniz ation of (a)(b ) $S_ {c}(t)$ and (c)(d) the mutu al info r ma tion$I_{AB} ( t) $ inth ewho le Hi lbert sp ace of$\bigoplus _{N }%
\mathcal{H} _ {N }$ for $N= 0,1 ,2,\cdots,n$ . T he tru nc ate d numb er$n=15 $ for( a) (c) and $n=1 0$ for (b ) (d). Th ... | "fig:"){width="240pt"}![The_dynamical measure_synchronization of (a)(b) $S_{c}(t)$_and (c)(d)_the_mutual information_$I_{AB}(t)$_in the whole_Hilbert space of_$\bigoplus_{N}%
\mathcal{H}_{N}$ for $N=0,1,2,\cdots,n$. The_truncated number $n=15$_for_(a) (c) and $n=10$ for (b)(d). The other parameters are the same as tha... |
(E_1)=m(E_0)\otimes m(\ker A)^*$).
Let us consider an example when $M_0=CP^n$. For every $k\in{\bf Z}$ we construct a line bundle $\alpha _k$ over $CP^n$ in the following way. We define the total space $E_k$ of the bundle $\alpha _k$ taking quotient of ${\bf C}^{n+2}\setminus \{ 0\} $ with respect to equivalence relat... | ( E_1)=m(E_0)\otimes m(\ker A)^*$).
Let us consider an example when $ M_0 = CP^n$. For every $ k\in{\bf Z}$ we construct a course package $ \alpha _ k$ over $ CP^n$ in the following way. We define the full space $ E_k$ of the bundle $ \alpha _ k$ take quotient of $ { \bf C}^{n+2}\setminus \ { 0\ } $ with respect to ... | (E_1)=m(E_0)\ltimes m(\ker A)^*$).
Let us conslder an example cyen $M_0=C'^n$. For svery $k\iv{\bf Z}$ we construct a line buidle $\alphq _k$ over $CP^n$ in the foulowing wwy. We dedine rhe total space $E_k$ of ths bunblx $\alpha _k$ takinn quotient mf ${\bf C}^{n+2}\setmings \{ 0\} $ with respect to equivalence relat... | (E_1)=m(E_0)\otimes m(\ker A)^*$). Let us consider an $M_0=CP^n$. every $k\in{\bf we construct a $CP^n$ the following way. define the total $E_k$ of the bundle $\alpha _k$ quotient of ${\bf C}^{n+2}\setminus \{ 0\} $ with respect to equivalence relation $$(z_1,...,z_{n+2})\sim z_1,...,\lambda z_{n+1}, \lambda ^kz_{n+2}... | (E_1)=m(E_0)\otimes m(\ker A)^*$).
Let us considEr an examplE when $m_0=CP^N$. FoR eVery $K\in{\bF Z}$ we construct a LIne bUndle $\alpha _k$ over $CP^n$ in thE follOwINg waY. we DefinE the totAL sPACe $E_K$ oF tHe bUnDLe $\Alpha _K$ taKing quoTient of ${\bf C}^{N+2}\seTmInus \{ 0\} $ with respECt To equivaleNce Relat... | (E_1)=m(E_0)\otimes m(\ker A)^*$).
Let u s c ons id er a n ex ample when $M_ 0 =CP^ n$. For every $k\in{\b f Z}$ w e con s tr uct a line b u nd l e $\ al ph a _ k$ ov er $C P^n $ in th e followin g w ay . We definet he total spa ce$E_k$ of the bu ndle $ \a lph a _k$tak ing q uotien t of ${ \bf C}^{n +2 } \setm... | (E_1)=m(E_0)\otimes m(\ker_A)^*$).
Let us_consider an example when_$M_0=CP^n$. For_every_$k\in{\bf Z}$_we_construct a line_bundle $\alpha _k$_over $CP^n$ in the_following way. We_define_the total space $E_k$ of the bundle $\alpha _k$ taking quotient of ${\bf C}^{n+2}\setminus_\{_0\} $_with_respect_to equivalence relat... |
inctions derived from a Miller & Scalo IMF and a Salpeter IMF, respectively. Therefore, the small GRB host sample does not allow us to determine whether or not GRB hosts have smaller intrinsic extinctions than young field starburst galaxies. Likewise, comparing GRB host extinctions with those derived for all types of f... | inctions derived from a Miller & Scalo IMF and a Salpeter IMF, respectively. Therefore, the small GRB host sample distribution does not leave us to determine whether or not GRB hosts get minor intrinsic extinctions than young playing field starburst galax. Likewise, comparing GRB host extinctions with those derive for ... | inchions derived from a Miluer & Scalo IMF cbd a Selpeter IMF, resoectively. Therefore, the smalp TRB hist sample does not aluow us to determibe wiether or not GRU hosts mcve sjwllex mntrinsic extingtions than young field sdafbbrst galaxies. Likewise, comparing GRB host ectlnctions with jhose qeribvd for all types of f... | inctions derived from a Miller & Scalo a IMF, respectively. the small GRB us determine whether or GRB hosts have intrinsic extinctions than young field starburst Likewise, comparing GRB host extinctions with those derived for all types of field does not allow us to determine whether they have different extinction distr... | inctions derived from a MilleR & Scalo IMF aNd a SaLpeTer iMf, resPectIvely. Therefore, THe smAll GRB host sample does noT alloW uS To deTErMine wHether oR NoT grB hOsTs HavE sMAlLer inTriNsic extInctions thAn yOuNg field starbURsT galaxies. LIkeWise, comparinG GRb host eXtIncTIons wIth Those DeriveD For all Types of f... | inctions derived from a Mi ller & Sca lo IM F a ndaSalp eter IMF, respecti v ely. Therefore, the smallGRB h os t sam p le does not al l ow u s t ode ter mi n ewheth eror notGRB hostshav esmaller intr i ns ic extinct ion s than young fi eld st ar bur s t gal axi es. L ikewis e , comp aring GRB h o st ext i nction... | inctions derived_from a_Miller & Scalo IMF_and a_Salpeter_IMF, respectively._Therefore,_the small GRB_host sample does_not allow us to_determine whether or_not_GRB hosts have smaller intrinsic extinctions than young field starburst galaxies. Likewise, comparing GRB_host_extinctions with_those_derived_for all types of f... |
2^{0\,0} \sim \Delta M^2_{\rm susy},$[^2] as explained in [@Alvarez-Gaume:2003]. The presence of such $\Pi_2$ effects will lead to unacceptable pathologies such as Lorentz-noninvariant dispersion relations giving mass to only one of the polarisations of the trace-U(1) gauge field, leaving the other polarisation massles... | 2^{0\,0 } \sim \Delta M^2_{\rm susy},$[^2 ] as explained in [ @Alvarez - Gaume:2003 ]. The presence of such $ \Pi_2 $ effects will lead to impossible pathology such as Lorentz - noninvariant dispersion relations give multitude to only one of the polarisations of the tracing - U(1) bore field, leaving the other polarisa... | 2^{0\,0} \sil \Delta M^2_{\rm susy},$[^2] as explxined in [@Alvaree-Gqume:2003]. Tie pressnce of ruch $\Pi_2$ effects will lead to ubacceknable pathologies sucf as Lorejtz-nonincarient dispersion rxmations givinf masv to only one on the polarhsations of tha grcce-U(1) gauge field, leaving the other pjlarisayiln massles... | 2^{0\,0} \sim \Delta M^2_{\rm susy},$[^2] as explained The of such effects will lead Lorentz-noninvariant relations giving mass only one of polarisations of the trace-U(1) gauge field, the other polarisation massless. The presence of the UV/IR effects in the trace-U(1) makes it pretty clear that a simple noncommutative... | 2^{0\,0} \sim \Delta M^2_{\rm susy},$[^2] as explaineD in [@Alvarez-gaume:2003]. the PreSeNce oF sucH $\Pi_2$ effects will LEad tO unacceptable pathologiEs sucH aS loreNTz-NoninVariant DIsPERsiOn ReLatIoNS gIving MasS to only One of the poLarIsAtions of the tRAcE-U(1) gauge fieLd, lEaving the othEr pOlarisAtIon MAssleS... | 2^{0\,0} \sim \Delta M^2_{ \rm susy}, $[^2] as ex pl aine d in [@Alvarez-Gau m e:20 03]. The presence of s uch $ \P i _2$e ff ectswill le a dt o un ac ce pta bl e p athol ogi es such as Lorent z-n on invariant di s pe rsion rela tio ns giving ma ssto onl yone of th e p olari sation s of th e trace-U (1 ) gauge ... | 2^{0\,0} \sim_\Delta M^2_{\rm_susy},$[^2] as explained in_[@Alvarez-Gaume:2003]. The_presence_of such_$\Pi_2$_effects will lead_to unacceptable pathologies_such as Lorentz-noninvariant dispersion_relations giving mass_to_only one of the polarisations of the trace-U(1) gauge field, leaving the other polarisation_massles... |
general graphs we show that fixed response dynamics converge to a steady state. On the complete graph we show that best response dynamics also converge to a steady state. Both of these results hold regardless of the initial conditions (i.e., the agents’ estimators at time $t=0$).
We show that the steady state of best... | general graphs we show that fixed reception moral force converge to a steady state. On the accomplished graph we testify that best response moral force also converge to a steady state. Both of these consequence hold regardless of the initial conditions (i.e., the agent ’ estimators at time $ t=0 $).
We read that the... | gejeral graphs we show thau fixed response btnamicv convsrge to x steady state. On the compleve geaph qe show that best respunse dynalics alsi coiverge to a steavg state. Both kn thevx results hold tegardless ox the initial wovdntions (i.e., the agents’ estimators at tyme $t=0$).
We sjow that the sjeady ftats of best... | general graphs we show that fixed response to steady state. the complete graph dynamics converge to a state. Both of results hold regardless of the initial (i.e., the agents’ estimators at time $t=0$). We show that the steady state best response dynamics is not necessarily optimal; there exist fixed response dynamics i... | general graphs we show that fiXed responsE dynaMicS coNvErge To a sTeady state. On thE CompLete graph we show that besT respOnSE dynAMiCs alsO converGE tO A SteAdY sTatE. BOTh Of theSe rEsults hOld regardlEss Of The initial coNDiTions (i.e., the AgeNts’ estimatorS at Time $t=0$).
WE sHow THat thE stEady sTate of BEst... | general graphs we show th at fixed r espon sedyn am icsconv erge to a stea d y st ate. On the complete g raphwe show th at be st resp o ns e dyn am ic s a ls o c onver geto a st eady state . B ot h of these r e su lts hold r ega rdless of th e i nitial c ond i tions (i .e.,the ag e nts’ e stimators a t time$ t=0$... | general_graphs we_show that fixed response_dynamics converge_to_a steady_state._On the complete_graph we show_that best response dynamics_also converge to_a_steady state. Both of these results hold regardless of the initial conditions (i.e., the_agents’_estimators at_time_$t=0$).
We_show that the steady state_of best... |
/\bigl(nzn^{-\gamma}
\bigr) \,\mathrm{d}z+\mathrm{o}_{{\mathbb{P}}}\bigl(n^{-\gamma/2}\bigr).\qquad\end{aligned}$$ Since $V_i$’s are i.i.d. standard Gaussian, a central limit theorem for $\lambda_n^*-\mathbf{B}_n^*$ can be easily derived. Now Theorem \[thmnewtest\] follows from [(\[eqnewbias\])]{} and [(\[eqnewvar\])]{... | /\bigl(nzn^{-\gamma }
\bigr) \,\mathrm{d}z+\mathrm{o}_{{\mathbb{P}}}\bigl(n^{-\gamma/2}\bigr).\qquad\end{aligned}$$ Since $ V_i$ ’s are i.i.d. standard Gaussian, a central limit theorem for $ \lambda_n^*-\mathbf{B}_n^*$ can be easily derived. nowadays Theorem \[thmnewtest\ ] take after from [ (\[eqnewbias\ ]) ] { } a... | /\bigp(nzn^{-\gamma}
\bigr) \,\mathrm{d}z+\mauhrm{o}_{{\mathbb{P}}}\bigl(n^{-\yqmma/2}\bijr).\qquad\snd{aligndd}$$ Since $V_i$’s are i.i.d. standarv Gayssiab, a central limit theofem for $\lwmbda_n^*-\marhbf{U}_n^*$ can be easily derived. Now Tgcorem \[vhmnewtest\] follpws from [(\[exnewbias\])]{} and [(\[exndwrar\])]{... | /\bigl(nzn^{-\gamma} \bigr) \,\mathrm{d}z+\mathrm{o}_{{\mathbb{P}}}\bigl(n^{-\gamma/2}\bigr).\qquad\end{aligned}$$ Since $V_i$’s are i.i.d. a limit theorem $\lambda_n^*-\mathbf{B}_n^*$ can be follows [(\[eqnewbias\])]{} and [(\[eqnewvar\])]{}. are omitted. [Proof Proposition \[proppower\]]{} By the Cauchy–Schwarz inequ... | /\bigl(nzn^{-\gamma}
\bigr) \,\mathrm{d}z+\mAthrm{o}_{{\mathBb{P}}}\biGl(n^{-\GamMa/2}\Bigr).\QquaD\end{aligned}$$ SinCE $V_i$’s Are i.i.d. standard Gaussian, A centRaL LimiT ThEorem For $\lambDA_n^*-\MAThbF{B}_N^*$ cAn bE eASiLy derIveD. Now TheOrem \[thmnewTesT\] fOllows from [(\[eqNEwBias\])]{} and [(\[eqnEwvAr\])]{... | /\bigl(nzn^{-\gamma}
\bigr ) \,\mathr m{d}z +\m ath rm {o}_ {{\m athbb{P}}}\big l (n^{ -\gamma/2}\bigr).\qqua d\end {a l igne d }$ $ Sin ce $V_i $ ’s a rei. i. d.st a nd ard G aus sian, a central l imi ttheorem for$ \l ambda_n^*- \ma thbf{B}_n^*$ ca n be e as ily deriv ed. NowTheore m \[thm newtest\] f o llowsf ro... | /\bigl(nzn^{-\gamma}
\bigr) \,\mathrm{d}z+\mathrm{o}_{{\mathbb{P}}}\bigl(n^{-\gamma/2}\bigr).\qquad\end{aligned}$$_Since $V_i$’s_are i.i.d. standard Gaussian,_a central_limit_theorem for_$\lambda_n^*-\mathbf{B}_n^*$_can be easily_derived. Now Theorem_\[thmnewtest\] follows from [(\[eqnewbias\])]{}_and [(\[eqnewvar\])]{... |
ell}\right)^\frac{4}{3}~,$$ being $g_H$ the number of degrees of freedom that become non relativistic between typical BBN temperatures and $T_D$. The authors of [@Blennow:2012de] have shown that the cosmological constraints on ${N_{\textrm{eff}}}$ can be translated into the required heavy degrees of freedom heating the... | ell}\right)^\frac{4}{3}~,$$ being $ g_H$ the number of degrees of freedom that become non relativistic between distinctive BBN temperature and $ T_D$. The authors of [ @Blennow:2012de ] have shown that the cosmologic restraint on $ { N_{\textrm{eff}}}$ can be translated into the required big degree of freedom heat the ... | ell}\gight)^\frac{4}{3}~,$$ being $g_H$ the nmmber of degrees of frexdom thzt becomd non relativistic between tbpicql BBB temperatures and $T_D$. Ghe authogs of [@Blebnow:2012ve] have shown thef the cosmoloflcal eoistraints on ${N_{\tgxtrm{eff}}}$ can be translated ivtl the required heavy degrees of freqdom hestlng the... | ell}\right)^\frac{4}{3}~,$$ being $g_H$ the number of degrees that non relativistic typical BBN temperatures [@Blennow:2012de] shown that the constraints on ${N_{\textrm{eff}}}$ be translated into the required heavy of freedom heating the light dark sector plasma $g_h$ as a function of dark sector decoupling temperatur... | ell}\right)^\frac{4}{3}~,$$ being $g_H$ the numBer of degreEs of fReeDom ThAt beCome Non relativistiC BetwEen typical BBN temperatuRes anD $T_d$. the aUThOrs of [@blennow:2012DE] hAVE shOwN tHat ThE CoSmoloGicAl constRaints on ${N_{\tExtRm{Eff}}}$ can be tranSLaTed into the ReqUired heavy deGreEs of frEeDom HEatinG thE... | ell}\right)^\frac{4}{3}~,$ $ being $g _H$ t henum be r of deg rees of freedo m tha t become non relativis tic b et w eent yp icalBBN tem p er a t ure san d $ T_ D $. Theaut hors of [@Blennow :20 12 de] have sho w nthat the c osm ological con str aintson ${ N _{\te xtr m{eff }}}$ c a n be t ranslated i n to the re... | ell}\right)^\frac{4}{3}~,$$ being_$g_H$ the_number of degrees of_freedom that_become_non relativistic_between_typical BBN temperatures_and $T_D$. The_authors of [@Blennow:2012de] have_shown that the_cosmological_constraints on ${N_{\textrm{eff}}}$ can be translated into the required heavy degrees of freedom heating_the... |
a{K(z,\bar z)}={{\rm i}}\left( X^A\frac{\partial}{\partial\bar X^A}\bar F(\bar X)- \bar X^A
\frac{\partial}{\partial X^A} F(X)\right)\,.$$ On overlap of charts these functions should be related by (inhomogeneous) symplectic transformations $ISp(2n,{{\mbox{\rm$\mbox{I}\!\mbox{R}$}}})$: $$\left( \begin{array}{c}
X \\ \... | a{K(z,\bar z)}={{\rm i}}\left (X^A\frac{\partial}{\partial\bar X^A}\bar F(\bar X)- \bar X^A
\frac{\partial}{\partial X^A } F(X)\right)\,.$$ On overlap of charts these functions should be related by (inhomogeneous) symplectic transformation $ ISp(2n,{{\mbox{\rm$\mbox{I}\!\mbox{R}$}}})$: $ $ \left (\begin{array}{c } ... | a{K(z,\har z)}={{\rm i}}\left( X^A\frac{\partlal}{\partial\bar X^A}\yqr F(\bac X)- \bad X^A
\frac{\oartial}{\partial X^A} F(X)\right)\,.$$ On ocerlak of charts these fjnctions dhould bw reoqted by (inikmogeneous) syjilectnc transformatioks $ISp(2n,{{\mbox{\sm$\mbox{I}\!\mbox{R}$}}})$: $$\lafg( \yegin{array}{c}
X \\ \... | a{K(z,\bar z)}={{\rm i}}\left( X^A\frac{\partial}{\partial\bar X^A}\bar F(\bar X)- \frac{\partial}{\partial F(X)\right)\,.$$ On of charts these (inhomogeneous) transformations $ISp(2n,{{\mbox{\rm$\mbox{I}\!\mbox{R}$}}})$: $$\left( X \\ \partial \end{array}\right)_{(i)} = e^{{{\rm i}}c_{ij}} M_{ij} \left( X \\ \partial ... | a{K(z,\bar z)}={{\rm i}}\left( X^A\frac{\partiAl}{\partial\bAr X^A}\bAr F(\Bar x)- \bAr X^A
\Frac{\Partial}{\partial x^a} F(X)\rIght)\,.$$ On overlap of charts tHese fUnCTionS ShOuld bE relateD By (INHomOgEnEouS) sYMpLectiC trAnsformAtions $ISp(2n,{{\MboX{\rM$\mbox{I}\!\mbox{R}$}}})$: $$\lEFt( \Begin{array}{C}
X \\ \... | a{K(z,\bar z)}={{\rm i}}\l eft( X^A\f rac{\ par tia l} {\pa rtia l\bar X^A}\bar F(\b ar X)- \bar X^A
\frac {\par ti a l}{\ p ar tialX^A} F( X )\ r i ght )\ ,. $$On ov erlap of charts these fun cti on s should ber el ated by (i nho mogeneous) s ymp lectic t ran s forma tio ns $I Sp(2n, { {\mbox {\rm$\mbo x{ I }\!\m... | a{K(z,\bar z)}={{\rm_i}}\left( X^A\frac{\partial}{\partial\bar_X^A}\bar F(\bar X)- _\bar X^A
\frac{\partial}{\partial_X^A}_F(X)\right)\,.$$ On_overlap_of charts these_functions should be_related by (inhomogeneous) symplectic_transformations $ISp(2n,{{\mbox{\rm$\mbox{I}\!\mbox{R}$}}})$: $$\left(_\begin{array}{c}
_X \\ \... |
$u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i b$), and for every $P_i'$, the following is true: $$\sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq
\sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x,F(P_i'P_{-i}))u_i(x).$$
For $|A|\geq 3$, $F$ has a weak dictator.
The notion of manipulation... | $ u_i$ consistent with $ P_i$ ($ u_i(a)>u_i(b)$ whenever $ a\succ_i b$), and for every $ P_i'$, the following is true: $ $ \sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x, F(P_iP_{-i}))u_i(x)\geq
\sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x, F(P_i'P_{-i}))u_i(x).$$
For $ |A|\geq 3 $, $ F$ has a fallible authoritarian.
... | $u_i$ consistent with $P_i$ ($u_i(a)>u_l(b)$ whenever $a\suce_u b$), anv for ebery $P_i'$, ghe following is true: $$\sum_{x\in F(P_iP_{-i})}p_u(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq
\sjm_{x\in F(P_i'I_{-i})}p_i(P_iP_{-i},x,D(P_i'P_{-m}))u_i(x).$$
For $|A|\geq 3$, $F$ izs a weak dicfwtor.
Chx notion of manlpulation... | $u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i for $P_i'$, the is true: $$\sum_{x\in 3$, has a weak The notion of used by @Duggan1992 is obviously more than that of @Taylor2002, and one is thus tempted to conclude that the theorem is weaker than Taylor’s reformulation.[^1] However, this would be errone... | $u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ wHenever $a\suCc_i b$), aNd fOr eVeRy $P_i'$, The fOllowing is true: $$\SUm_{x\iN F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\gEq
\sum_{X\iN f(P_i'P_{-I})}P_i(p_iP_{-i},x,f(P_i'P_{-i}))u_i(X).$$
foR $|a|\Geq 3$, $f$ hAs A weAk DIcTator.
the Notion oF manipulatIon... | $u_i$ consistent with $P_ i$ ($u_i(a )>u_i (b) $ w he neve r $a \succ_i b$), a n d fo r every $P_i'$, the fo llowi ng is t r ue : $$\ sum_{x\ i nF ( P_i P_ {- i}) }p _ i( P_iP_ {-i },x,F(P _iP_{-i})) u_i (x )\geq
\s u m_ {x\in F(P_ i'P _{-i})}p_i(P _iP _{-i}, x, F(P _ i'P_{ -i} ))u_i (x).$$
For $ |A|\geq 3 $, $F$... | $u_i$_consistent with_$P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i_b$), and_for_every $P_i'$,_the_following is true:_$$\sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq
_ \sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x,F(P_i'P_{-i}))u_i(x).$$
For_$|A|\geq 3$, $F$_has_a weak dictator.
The notion of manipulation... |
) e^{\beta N}$. The general solution of Eq. , in either of the aforementioned cases, can be written as: $$\left[
\begin{array}{c}
e^\alpha \cos(\tilde{\vartheta}/2) \\
e^\alpha \sin(\tilde{\vartheta}/2)
\end{array}
\right] \simeq
\left[ \begin{array}{cc}
\cos(\tilde{\omega} N) & \si... | ) e^{\beta N}$. The general solution of Eq. , in either of the aforementioned cases, can be written as: $ $ \left [
\begin{array}{c }
e^\alpha \cos(\tilde{\vartheta}/2) \\
e^\alpha \sin(\tilde{\vartheta}/2)
\end{array }
\right ] \simeq
\left [ \begin{array}{cc }
\cos(\t... | ) e^{\bfta N}$. The general solutiun of Eq. , in eitkwr of vhe afodementioved cases, can be written as: $$\pedt[
\begin{array}{c}
e^\aloha \cos(\tipde{\varthwta}/2) \\
e^\alphe \sin(\tilde{\vartgcta}/2)
\end{array}
\rigmt] \simeq
\neft[ \begin{arraf}{cz}
\cos(\tilde{\omega} N) & \si... | ) e^{\beta N}$. The general solution of in of the cases, can be \cos(\tilde{\vartheta}/2) e^\alpha \sin(\tilde{\vartheta}/2) \end{array} \simeq \left[ \begin{array}{cc} N) & \sin(\tilde{\omega} N) \\ -\sin(\tilde{\omega} & \cos(\tilde{\omega} N) \end{array} \right] \left[ \begin{array}{c} C_1 - \frac{\tilde{\omega}}{2(... | ) e^{\beta N}$. The general solution oF Eq. , in eitheR of thE afOreMeNtioNed cAses, can be writtEN as: $$\lEft[
\begin{array}{c}
e^\alpha \coS(\tildE{\vARtheTA}/2) \\
e^\Alpha \Sin(\tildE{\VaRTHetA}/2)
\eNd{ArrAy}
\RIgHt] \simEq
\lEft[ \begiN{array}{cc}
\coS(\tiLdE{\omega} N) & \si... | ) e^{\beta N}$. The genera l solution of E q., i neith er o f the aforemen t ione d cases, can be writte n as: $ $ \lef t [
\begin{ a rr a y }{c } e ^ \a lpha\co s(\tild e{\varthet a}/ 2) \\
e^ \ al pha \sin(\ til de{\vartheta }/2 )
\e nd{ a rray}
\righ t] \si m eq
\left[ \b eg i n{arra y }{cc}
... | ) e^{\beta_N}$. The_general solution of Eq. ,_in either_of_the aforementioned_cases,_can be written_as: $$\left[
_ \begin{array}{c}
_ __ e^\alpha \cos(\tilde{\vartheta}/2) \\
e^\alpha \sin(\tilde{\vartheta}/2)
_\end{array}
_ \right]_\simeq_
_ \left[ \begin{array}{cc}
_ _ _ \cos(\tilde{\omega} N) & \si... |
right)}}\bigg)\\
+{\left(}n-2k{\right)}\bigg(\frac{{\left(}k+1{\right)}{\left(}k-1{\right)}{\left(}n-2k-2l+4{\right)}}{18}\\
-\frac{{\left(}n-2{\right)}{\left(}2{\left(}n-2k+2{\right)}-n{\left(}n+2l-2k{\right)}{\right)}}{24}\bigg)\bigg)\\
\times\operatorname{B}{\left(}\frac{n}{2}-k-1,l+1{\right)}^{-1}\left\{\begin{alig... | right)}}\bigg)\\
+ { \left(}n-2k{\right)}\bigg(\frac{{\left(}k+1{\right)}{\left(}k-1{\right)}{\left(}n-2k-2l+4{\right)}}{18}\\
-\frac{{\left(}n-2{\right)}{\left(}2{\left(}n-2k+2{\right)}-n{\left(}n+2l-2k{\right)}{\right)}}{24}\bigg)\bigg)\\
\times\operatorname{B}{\left(}\frac{n}{2}-k-1,l+1{\right)}^{-1}\left\{\be... | rigjt)}}\bigg)\\
+{\left(}n-2k{\right)}\bigg(\frag{{\left(}k+1{\right)}{\left(}k-1{\titht)}{\lefv(}n-2k-2l+4{\riggt)}}{18}\\
-\frac{{\lewt(}n-2{\right)}{\left(}2{\left(}n-2k+2{\right)}-n{\left(}i+2l-2k{\rught)}{\rught)}}{24}\bigg)\bigg)\\
\times\operagorname{B}{\lvft(}\frac{n}{2}-k-1,o+1{\rigit)}^{-1}\left\{\begin{alig... | right)}}\bigg)\\ +{\left(}n-2k{\right)}\bigg(\frac{{\left(}k+1{\right)}{\left(}k-1{\right)}{\left(}n-2k-2l+4{\right)}}{18}\\ -\frac{{\left(}n-2{\right)}{\left(}2{\left(}n-2k+2{\right)}-n{\left(}n+2l-2k{\right)}{\right)}}{24}\bigg)\bigg)\\ \times\operatorname{B}{\left(}\frac{n}{2}-k-1,l+1{\right)}^{-1}\left\{\begin{alig... | right)}}\bigg)\\
+{\left(}n-2k{\right)}\bigg(\fRac{{\left(}k+1{\riGht)}{\leFt(}k-1{\RigHt)}{\Left(}N-2k-2l+4{\rIght)}}{18}\\
-\frac{{\left(}n-2{\rIGht)}{\lEft(}2{\left(}n-2k+2{\right)}-n{\left(}n+2l-2k{\Right)}{\RiGHt)}}{24}\biGG)\bIgg)\\
\tiMes\operAToRNAme{b}{\lEfT(}\frAc{N}{2}-K-1,l+1{\Right)}^{-1}\LefT\{\begin{aLig... | right)}}\bigg)\\
+{\left(} n-2k{\righ t)}\b igg (\f ra c{{\ left (}k+1{\right)} { \lef t(}k-1{\right)}{\left( }n-2k -2 l +4{\ r ig ht)}} {18}\\- \f r a c{{ \l ef t(} n- 2 {\ right )}{ \left(} 2{\left(}n -2k +2 {\right)}-n{ \ le ft(}n+2l-2 k{\ right)}{\rig ht) }}{24} \b igg ) \bigg )\\
\tim es\ope r atorna me{B}{\le ft... | right)}}\bigg)\\
+{\left(}n-2k{\right)}\bigg(\frac{{\left(}k+1{\right)}{\left(}k-1{\right)}{\left(}n-2k-2l+4{\right)}}{18}\\
-\frac{{\left(}n-2{\right)}{\left(}2{\left(}n-2k+2{\right)}-n{\left(}n+2l-2k{\right)}{\right)}}{24}\bigg)\bigg)\\
\times\operatorname{B}{\left(}\frac{n}{2}-k-1,l+1{\right)}^{-1}\left\{\begin{alig... |
k,\ldots,p_1],c) \mapsto [p_k,\ldots,p_1,p_1^{-1}\cdots p_k^{-1} c]$, which is easily seen to be well-defined and an isomorphism, then the diagram commutes.
We can now study the right-hand map $BS_{\vec\alpha} f$ by reversing the isomorphism on the top of the diagram. The fibers of the map from the northwest corner to... | k,\ldots, p_1],c) \mapsto [ p_k,\ldots, p_1,p_1^{-1}\cdots p_k^{-1 } c]$, which is easily seen to be well - defined and an isomorphism, then the diagram commutes.
We can nowadays analyze the right - hand map $ BS_{\vec\alpha } f$ by turn back the isomorphism on the top of the diagram. The fibers of the function from... | k,\ldlts,p_1],c) \mapsto [p_k,\ldots,p_1,p_1^{-1}\cduts p_k^{-1} c]$, which nw easiny seeh to be dell-defined and an isomorphidm, then the diagram commutes.
Wd can now study tye rmght-hand map $BS_{\vxd\alpha} n$ by dcversnnj the isomorphixm on the dop of the diacrxm. The fibers of the map from the norehwest vogner to... | k,\ldots,p_1],c) \mapsto [p_k,\ldots,p_1,p_1^{-1}\cdots p_k^{-1} c]$, which is to well-defined and isomorphism, then the study right-hand map $BS_{\vec\alpha} by reversing the on the top of the diagram. fibers of the map from the northwest corner to the southeast are just of the fibers of $f$ with Bott-Samelson manifol... | k,\ldots,p_1],c) \mapsto [p_k,\ldots,p_1,p_1^{-1}\cdOts p_k^{-1} c]$, whicH is eaSilY seEn To be Well-Defined and an isOMorpHism, then the diagram commUtes.
WE cAN now STuDy the Right-haND mAP $bS_{\vEc\AlPha} F$ bY ReVersiNg tHe isomoRphism on thE toP oF the diagram. THE fIbers of the Map From the northWesT corneR tO... | k,\ldots,p_1],c) \mapsto [ p_k,\ldots ,p_1, p_1 ^{- 1} \cdo ts p _k^{-1} c]$, w h ichis easily seen to be w ell-d ef i neda nd an i somorph i sm , the nth e d ia g ra m com mut es.
We can now s tud ythe right-ha n dmap $BS_{\ vec \alpha} f$ b y r eversi ng th e isom orp hismon the top of the diag ra m . Thef iber... | k,\ldots,p_1],c) \mapsto_[p_k,\ldots,p_1,p_1^{-1}\cdots p_k^{-1}_c]$, which is easily_seen to_be_well-defined and_an_isomorphism, then the_diagram commutes.
We can_now study the right-hand_map $BS_{\vec\alpha} f$_by_reversing the isomorphism on the top of the diagram. The fibers of the map_from_the northwest_corner_to... |
T. Kodama, Prog. Part. Nucl. Phys. [**86**]{}, 35 (2016) doi:10.1016/j.ppnp.2015.09.002 \[arXiv:1506.03863 \[nucl-th\]\].
T. Burch and G. Torrieri, Phys. Rev. D [**92**]{}, no. 1, 016009 (2015) doi:10.1103/PhysRevD.92.016009 \[arXiv:1502.05421 \[hep-lat\]\]. M. A. L. Capri, M. S. Guimaraes, I. F. Justo, L. F. Palhar... | T. Kodama, Prog. Part. Nucl. Phys. [ * * 86 * * ] { }, 35 (2016) doi:10.1016 / j.ppnp.2015.09.002 \[arXiv:1506.03863 \[nucl - th\]\ ].
T. Burch and G. Torrieri, Phys. Rev. D [ * * 92 * * ] { }, no. 1, 016009 (2015) doi:10.1103 / PhysRevD.92.016009 \[arXiv:1502.05421 \[hep - lat\]\ ]. M. A. L. ... | T. Kldama, Prog. Part. Nucl. Phys. [**86**]{}, 35 (2016) doi:10.1016/j.ppnp.2015.09.002 \[arXir:1506.03863 \[nucl-ti\]\].
T. Burch and G. Tofrieri, Phys. Rev. D [**92**]{}, no. 1, 016009 (2015) doi:10.1103/PhbsRecD.92.016009 \[arZiv:1502.05421 \[hep-lat\]\]. M. A. L. Capri, M. S. Euimaraes, I. F. Justo, L. F. Pelhar... | T. Kodama, Prog. Part. Nucl. Phys. [**86**]{}, doi:10.1016/j.ppnp.2015.09.002 \[nucl-th\]\]. T. and G. Torrieri, 1, (2015) doi:10.1103/PhysRevD.92.016009 \[arXiv:1502.05421 M. A. L. M. S. Guimaraes, I. F. Justo, F. Palhares and S. P. Sorella, Phys. Lett. B [**735**]{}, 277 (2014) doi:10.1016/j.physletb.2014.06.035 \[he... | T. Kodama, Prog. Part. Nucl. Phys. [**86**]{}, 35 (2016) doI:10.1016/j.ppnp.2015.09.002 \[arXiV:1506.03863 \[nucl-Th\]\].
T. burCh And G. torrIeri, Phys. Rev. D [**92**]{}, no. 1, 016009 (2015) DOi:10.1103/PhYsRevD.92.016009 \[arXiv:1502.05421 \[hep-lat\]\]. M. A. L. CaPri, M. S. guIMaraES, I. f. JustO, L. F. PalhAR... | T. Kodama, Prog. Part. Nu cl. Phys. [**8 6** ]{} ,35 ( 2016 ) doi:10.1016/ j .ppn p.2015.09.002 \[arXiv: 1506. 03 8 63 \ [ nu cl-th \]\].
T .B u rch a nd G. T o rr ieri, Ph ys. Rev . D [**92* *]{ }, no. 1, 0160 0 9(2015) doi :10 .1103/PhysRe vD. 92.016 00 9 \ [ arXiv :15 02.05 421 \[ h ep-lat \]\]. M.A. L. Cap r ... | T. Kodama,_Prog. Part. Nucl. Phys. [**86**]{},_35 (2016) doi:10.1016/j.ppnp.2015.09.002 \[arXiv:1506.03863_\[nucl-th\]\].
T. Burch and_G. Torrieri,_Phys. Rev. D [**92**]{},_no._1, 016009 (2015)_doi:10.1103/PhysRevD.92.016009 \[arXiv:1502.05421 \[hep-lat\]\]._M. A. L. Capri, M. S. Guimaraes, I. F. Justo, L. F. Palhar... |
, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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abstract: |
A new generalization of Pascal’s triangle, the so-called hyperbolic Pascal triangles were introduced in [@BNSz]. The mathematical background goes... | , which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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abstract: |
A new generalization of Pascal ’s triangulum, the therefore - called hyperbolic Pascal triangles were insert in [ @BNSz ]. The numerical ba... | , whlch is operated by the Arsociation of Uuuversivies fod Researzh in Astronomy, Inc., under colpwratice agreement with the Vational Dcience Dounvation.
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A new genedwlizctmon of Pascal’s jriangle, the so-called hypesbulnc Pascal triangles were introduced yn [@BNSz]. Tje mathematicaj babkdrouhd goes... | , which is operated by the Association for in Astronomy, under cooperative agreement --- | A new of Pascal’s triangle, so-called hyperbolic Pascal triangles were introduced [@BNSz]. The mathematical background goes back to the regular mosaics in the hyperbolic The alternating sum of elements in the rows was given in th... | , which is operated by the AssocIation of UnIversItiEs fOr reseArch In Astronomy, Inc., UNder Cooperative agreement wiTh the naTIonaL scIence foundatIOn.
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ABStrAcT: |
A New GeNErAlizaTioN of PascAl’s trianglE, thE sO-called hyperBOlIc Pascal trIanGles were intrOduCed in [@BnSZ]. ThE MatheMatIcal bAckgroUNd goes... | , which is operated by the Associati on of Un ive rs itie s fo r Research inA stro nomy, Inc., under coop erati ve agre e me nt wi th theN at i o nal S ci enc eF ou ndati on.
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a bstract: |
A new genera l iz ation of P asc al’s triangl e,the so -c all e d hyp erb olicPascal triang les werein t roduce d in [@... | , which_is operated_by the Association of_Universities for_Research_in Astronomy,_Inc.,_under cooperative agreement_with the National_Science Foundation.
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abstract: |
_ _A_new generalization of Pascal’s triangle, the so-called hyperbolic Pascal triangles were introduced in [@BNSz]._The_mathematical background_goes... |
& $-$203047 & 3180 & 1.4535 & 2.8893 & Severely warped\
B-24 & A 2333-1637 & 233305 & $-$163714 & & & &\
C-02 & A 0017+2212 & 001716.7 & $+$221200 & & & &\
C-03 & ESO 474-G26 & 004440 & $-$243836 & 16246 & 0.9131 & 1.7967 & Inner & outer rings?\
C-04 & A 0051-1323 & 005100.2 & $-$132304 & 10876 & & &\
C-05 & AM 0051-2... | & $ -$203047 & 3180 & 1.4535 & 2.8893 & Severely warped\
B-24 & A 2333 - 1637 & 233305 & $ -$163714 & & & & \
C-02 & A 0017 + 2212 & 001716.7 & $ + $ 221200 & & & & \
C-03 & ESO 474 - G26 & 004440 & $ -$243836 & 16246 & 0.9131 & 1.7967 & Inner & outer rings?\
C-04 & A 0051 - 1323 & 005100.2 & $ -$1323... | & $-$203047 & 3180 & 1.4535 & 2.8893 & Severely warped\
B-24 & A 2333-1637 & 233305 & $-$163714 & & & &\
C-02 & A 0017+2212 & 001716.7 & $+$221200 & & & &\
R-03 & ESO 474-G26 & 004440 & $-$243836 & 16246 & 0.9131 & 1.7967 & Inner & outer rings?\
C-04 & A 0051-1323 & 005100.2 & $-$132304 & 10876 & & &\
C-05 & AM 0051-2... | & $-$203047 & 3180 & 1.4535 & Severely B-24 & 2333-1637 & 233305 &\ & A 0017+2212 001716.7 & $+$221200 & & &\ C-03 & ESO & 004440 & $-$243836 & 16246 & 0.9131 & 1.7967 & Inner & rings?\ C-04 & A 0051-1323 & 005100.2 & $-$132304 & 10876 & & C-05 AM & & $-$234926 & 20156 & 0.4437 & 1.3828 &\ C-06 & NGC 304 & 005324 & & 4... | & $-$203047 & 3180 & 1.4535 & 2.8893 & Severely warped\
B-24 & A 2333-1637 & 233305 & $-$163714 & & & &\
C-02 & A 0017+2212 & 001716.7 & $+$221200 & & & &\
C-03 & ESO 474-G26 & 004440 & $-$243836 & 16246 & 0.9131 & 1.7967 & InNer & outer riNgs?\
C-04 & A 0051-1323 & 005100.2 & $-$132304 & 10876 & & &\
c-05 & AM 0051-2... | & $-$203047 & 3180 & 1.45 35 & 2.889 3 & S eve rel ywarp ed\B-24 & A 2333- 1 637& 233305 & $-$163714 & & &&\ C-02 &A 001 7+2212& 0 0 1 716 .7 & $+ $2 2 12 00 && & &\
C-0 3 & ESO 47 4-G 26 & 004440 &$ -$ 243836 & 1 624 6 & 0.9131 & 1. 7967 & I nne r & ou ter ring s?\
C- 0 4 & A0051-1323 & 005100 . 2 & $-$ 1 ... | &_$-$203047 &_3180 & 1.4535 &_2.8893 &_Severely_warped\
B-24 &_A 2333-1637_& 233305 &_$-$163714 & &_& &\
C-02 & A 0017+2212_& 001716.7 &_$+$221200_& & & &\
C-03 & ESO 474-G26 & 004440 & $-$243836 & 16246 & 0.9131_&_1.7967 &_Inner_& outer_rings?\
C-04 & A 0051-1323 & 005100.2_& $-$132304 & 10876 &_& &\
C-05_& AM 0051-2... |
open-string model in [@cudell].
As seen from the above discussion, an accurate determination of the ratio $T_h/T_c$ is of great phenomenological interest. However, such a study is not the main purpose of the present paper, where we aim at giving reliable predictions for the equation of state of YM theory with an arbi... | open - string model in [ @cudell ].
As seen from the above discussion, an accurate decision of the proportion $ T_h / T_c$ is of great phenomenological interest. However, such a discipline is not the independent purpose of the present paper, where we draw a bead on at impart reliable prediction for the equation of s... | opfn-string model in [@cudell].
Xs seen from thg qbove viscussjon, an azcurate determination of the rqtio $U_k/T_c$ is of great phenumenologibal interwst. Iowever, such a svhdy is kjt tgc maiu 'urpose of the kresent papes, where we aim ag yiving reliable predictions for the qquatiom lf state of YM thepwy wjnh an arbi... | open-string model in [@cudell]. As seen from discussion, accurate determination the ratio $T_h/T_c$ However, a study is the main purpose the present paper, where we aim giving reliable predictions for the equation of state of YM theory with an gauge algebra. As observed in [@meyer], typical values $T_h\approx T_c$ give... | open-string model in [@cudell].
As Seen from thE abovE diScuSsIon, aN accUrate determinaTIon oF the ratio $T_h/T_c$ is of great PhenoMeNOlogICaL inteRest. HowEVeR, SUch A sTuDy iS nOT tHe maiN puRpose of The present PapEr, Where we aim at GIvIng reliablE prEdictions for The EquatiOn Of sTAte of yM tHeory With an ARbi... | open-string model in [@cu dell].
As seen fr omth e ab ovediscussion, an accu rate determination ofthe r at i o $T _ h/ T_c$is of g r ea t phe no me nol og i ca l int ere st. How ever, such ast udy is not t h emain purpo seof the prese ntpaper, w her e we a imat gi ving r e liable predicti on s for t h e equat ... | open-string_model in_[@cudell].
As seen from the_above discussion,_an_accurate determination_of_the ratio $T_h/T_c$_is of great_phenomenological interest. However, such_a study is_not_the main purpose of the present paper, where we aim at giving reliable predictions_for_the equation_of_state_of YM theory with an_arbi... |
\]. Let $\Gamma '= \pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, by Proposition \[observation2\], the rank hypothesis of Corollary \[cor:n.5solvable\] is equivalent to the fact that $\mathcal{K}\Gamma '$-rank of $H_1(\pi_1(M);\mathcal{K}\Gamma ')$ is $\beta_1(M)-1$. Since $j_{n+1}: \Gamma '\to\G$ is a monomorphism, by [@CH1 Lemm... | \ ]. Let $ \Gamma' = \pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, by Proposition \[observation2\ ], the rank hypothesis of Corollary \[cor: n.5solvable\ ] is equivalent to the fact that $ \mathcal{K}\Gamma' $ -rank of $ H_1(\pi_1(M);\mathcal{K}\Gamma') $ is $ \beta_1(M)-1$. Since $ j_{n+1 }: \Gamma' \to\G$ is a monomorphism... | \]. Leh $\Gamma '= \pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, bn Proposition \[obsgrcation2\], the rznk hypoghesis of Corollary \[cor:n.5solvaule\] us eqyivalent to the fact tfat $\mathcwl{K}\Gamma '$-ranj of $H_1(\pi_1(M);\mavgcal{K}\Gamma ')$ ia $\betc_1(M)-1$. Since $j_{n+1}: \Gamms '\to\G$ is a monomorphism, ty [@CK1 Lemm... | \]. Let $\Gamma '= \pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, by the hypothesis of \[cor:n.5solvable\] is equivalent '$-rank $H_1(\pi_1(M);\mathcal{K}\Gamma ')$ is Since $j_{n+1}: \Gamma is a monomorphism, by [@CH1 Lemma this rank is the same as the $\mathcal{K}\Gamma$-rank of $H_1(\pi_1(M);\mathcal{K}\Gamma)$ associated to t... | \]. Let $\Gamma '= \pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, by ProPosition \[obServaTioN2\], thE rAnk hYpotHesis of CorollaRY \[cor:N.5solvable\] is equivalent tO the fAcT That $\MAtHcal{K}\gamma '$-raNK oF $h_1(\Pi_1(M);\MaThCal{k}\GAMmA ')$ is $\beTa_1(M)-1$. since $j_{n+1}: \gamma '\to\G$ is A moNoMorphism, by [@CH1 lEmM... | \]. Let $\Gamma '= \pi_1(M )/\pi_1(M) ^{(n+ 1)} _H$ .Then , by Proposition \ [ obse rvation2\], the rank h ypoth es i s of Co rolla ry \[co r :n . 5 sol va bl e\] i s e quiva len t to th e fact tha t $ \m athcal{K}\Ga m ma '$-rank o f $ H_1(\pi_1(M) ;\m athcal {K }\G a mma ' )$is $\ beta_1 ( M)-1$. Since $j _{ n +... | \]. Let_$\Gamma '=_\pi_1(M)/\pi_1(M)^{(n+1)}_H$. Then, by Proposition \[observation2\],_the rank_hypothesis_of Corollary \[cor:n.5solvable\]_is_equivalent to the_fact that $\mathcal{K}\Gamma_'$-rank of $H_1(\pi_1(M);\mathcal{K}\Gamma ')$_is $\beta_1(M)-1$. Since_$j_{n+1}:_\Gamma '\to\G$ is a monomorphism, by [@CH1 Lemm... |
close enough to permit the transition from two $j=0$ opposing simple loops into two opposing $j=\frac{1}{2}$ twisted loops. A possible geometric scenario for the transformation of two simple loops of current (yons) with opposite rotations into two $j=\frac{1}{2}$ twisted loops of current (preons) is shown in Fig. 3. T... | close enough to permit the transition from two $ j=0 $ opposing childlike loop into two opposing $ j=\frac{1}{2}$ twist loop. A possible geometric scenario for the transformation of two simple loop of current (yons) with opposite rotations into two $ j=\frac{1}{2}$ twist loops of current (preons) is shown in Fig. 3. To... | cllse enough to permit the transition from two $j=0$ opposjng simpue loops into two opposing $j=\hrac{1}{2}$ twisuvd loops. A possible gdometric dcenario for rhe transformation of twk sim'lx loops of currgnt (yons) witv opposite rotdtkous into two $j=\frac{1}{2}$ twisted loops of ctrrent (lrfons) is shown yn Fpg. 3. T... | close enough to permit the transition from opposing loops into opposing $j=\frac{1}{2}$ twisted for transformation of two loops of current with opposite rotations into two $j=\frac{1}{2}$ loops of current (preons) is shown in Fig. 3. To implement this scenario would expect to go beyond the earlier considerations of thi... | close enough to permit the traNsition froM two $j=0$ OppOsiNg SimpLe loOps into two oppoSIng $j=\Frac{1}{2}$ twisted loops. A possiBle geOmETric SCeNario For the tRAnSFOrmAtIoN of TwO SiMple lOopS of currEnt (yons) witH opPoSite rotationS InTo two $j=\frac{1}{2}$ TwiSted loops of cUrrEnt (preOnS) is SHown iN FiG. 3. T... | close enough to permit th e transiti on fr omtwo $ j=0$ opp osing simple l o opsinto two opposing $j=\ frac{ 1} { 2}$t wi stedloops.A p o s sib le g eom et r ic scen ari o for t he transfo rma ti on of two si m pl e loops of cu rrent (yons) wi th opp os ite rotat ion s int o two$ j=\fra c{1}{2}$tw i sted l o op... | close_enough to_permit the transition from_two $j=0$_opposing_simple loops_into_two opposing $j=\frac{1}{2}$_twisted loops. A_possible geometric scenario for_the transformation of_two_simple loops of current (yons) with opposite rotations into two $j=\frac{1}{2}$ twisted loops of_current_(preons) is_shown_in_Fig. 3. T... |
-F bonds in the eclipsed conformation. Figure 2 shows as function of rotation of the carbon-sulfur bond the atom centered partial charges obtained for the triflic acid molecule. The sulfur-oxygen(hydroxyl) rotational potential energy surface possesses two physically equivalent minima (Figure 3). The lowest barrier betw... | -F bonds in the eclipsed conformation. Figure 2 show as affair of rotation of the carbon - sulfur bond the atom centered fond charges obtained for the triflic acid molecule. The sulfur - oxygen(hydroxyl) rotational likely department of energy surface possesses two physically equivalent minima (Figure 3). The lowest bar... | -F blnds in the eclipsed connormation. Figure 2 shows as fuhction ow rotation of the carbon-sulfnr bind tye atom centered partixl charged obtainwd fie the triflic acid molechpe. Tke sulfur-oxygen(hidroxyl) rotadional potentidl euergy surface possesses two physicaljy equifapent minima (Fidure 3). The lowest barrier betw... | -F bonds in the eclipsed conformation. Figure as of rotation the carbon-sulfur bond obtained the triflic acid The sulfur-oxygen(hydroxyl) rotational energy surface possesses two physically equivalent (Figure 3). The lowest barrier between these minima corresponds to the proton directed out into the solvent and has an e... | -F bonds in the eclipsed conforMation. FiguRe 2 shoWs aS fuNcTion Of roTation of the carBOn-suLfur bond the atom centereD partIaL CharGEs ObtaiNed for tHE tRIFliC aCiD moLeCUlE. The sUlfUr-oxygeN(hydroxyl) rOtaTiOnal potentiaL EnErgy surfacE poSsesses two phYsiCally eQuIvaLEnt miNimA (FiguRe 3). The lOWest baRrier betw... | -F bonds in the eclipsed c onformatio n. Fi gur e 2 s hows asfunction of ro t atio n of the carbon-sulfur bond t h e at o mcente red par t ia l cha rg es ob ta i ne d for th e trifl ic acid mo lec ul e. The sulfu r -o xygen(hydr oxy l) rotationa l p otenti al en e rgy s urf ace p ossess e s twophysicall ye quiva... | -F bonds_in the_eclipsed conformation. Figure 2_shows as_function_of rotation_of_the carbon-sulfur bond_the atom centered_partial charges obtained for_the triflic acid_molecule._The sulfur-oxygen(hydroxyl) rotational potential energy surface possesses two physically equivalent minima (Figure 3). The_lowest_barrier betw... |
It implies $$f'_{i_0}\in \sum_{j\neq i_0}\mathcal{O}f'_j+\mathcal{O}f,$$ which is impossible in the case of an isolated singularity $f$.
If $E=1$, considering the leading term in $\bar{s}$ in the relation (\[eq3.3\]), we obtain that $1$ belongs to the maximal ideal of $W$, a contradiction.$\square$
*Remark :* After ... | It implies $ $ f'_{i_0}\in \sum_{j\neq i_0}\mathcal{O}f'_j+\mathcal{O}f,$$ which is impossible in the case of an isolated singularity $ f$.
If $ E=1 $, considering the lead condition in $ \bar{s}$ in the relation (\[eq3.3\ ]), we obtain that $ 1 $ belongs to the maximal ideal of $ W$, a contradiction.$\square$
* ... | It implies $$f'_{i_0}\in \sum_{j\neq i_0}\mxthcal{O}f'_j+\mathcal{O}f,$$ whirh is ijpossibld in the case of an isolated sungulqrity $f$.
If $E=1$, considerine the leafing tern in $\var{s}$ in thx relation (\[eq3.3\]), sc obtcii that $1$ belongs to the mafimal ideal of $W$, a contradiction.$\square$
*Remark :* After ... | It implies $$f'_{i_0}\in \sum_{j\neq i_0}\mathcal{O}f'_j+\mathcal{O}f,$$ which is the of an singularity $f$. If in in the relation we obtain that belongs to the maximal ideal of a contradiction.$\square$ *Remark :* After the preceding lemma, we have $\beta_1>n+1.$ Since in smooth case, $\beta_1=n+1$, we conclude that $... | It implies $$f'_{i_0}\in \sum_{j\neq i_0}\mathCal{O}f'_j+\mathCal{O}f,$$ WhiCh iS iMposSiblE in the case of an ISolaTed singularity $f$.
If $E=1$, consIderiNg THe leADiNg terM in $\bar{s}$ IN tHE RelAtIoN (\[eq3.3\]), We OBtAin thAt $1$ bElongs tO the maximaL idEaL of $W$, a contradICtIon.$\square$
*REmaRk :* After ... | It implies $$f'_{i_0}\in\sum_{j\ne q i_0 }\m ath ca l{O} f'_j +\mathcal{O}f, $ $ wh ich is impossible in t he ca se of a n i solat ed sing u la r i ty$f $.
I f$ E= 1$, c ons idering the leadi ngte rm in $\bar{ s }$ in the re lat ion (\[eq3.3 \]) , we o bt ain that$1$ belo ngs to the ma ximal ide al of $W$ , a c... | It_implies $$f'_{i_0}\in_\sum_{j\neq i_0}\mathcal{O}f'_j+\mathcal{O}f,$$ which is_impossible in_the_case of_an_isolated singularity $f$.
If_$E=1$, considering the_leading term in $\bar{s}$_in the relation_(\[eq3.3\]),_we obtain that $1$ belongs to the maximal ideal of $W$, a contradiction.$\square$
*Remark :*_After_... |
{p}}}$ belongs to $(\mathcal{V}_{\operatorname*{\mathfrak{p}}})_n \subseteq \mathcal{V}_n$, again using Lemma \[L:local\] for the last inclusion. The natural map $\iota: M \rightarrow \prod_{\operatorname*{\mathfrak{p}}\in \operatorname*{Spec}(R)}M_{\operatorname*{\mathfrak{p}}}$ is a monomorphism. Since $R \rightarrow... | { p}}}$ belongs to $ (\mathcal{V}_{\operatorname*{\mathfrak{p}}})_n \subseteq \mathcal{V}_n$, again using Lemma \[L: local\ ] for the last inclusion. The lifelike function $ \iota: M \rightarrow \prod_{\operatorname*{\mathfrak{p}}\in \operatorname*{Spec}(R)}M_{\operatorname*{\mathfrak{p}}}$ is a monomorphism. Since $... | {p}}}$ bflongs to $(\mathcal{V}_{\operaturname*{\mathfrak{p}}})_u \subseveq \matgcal{V}_n$, aeain using Lemma \[L:local\] for tie lqst ibclusion. The natural mxp $\iota: M \rightareow \krod_{\operatorname*{\mefhfrak{p}}\lu \opedwtoruane*{Spec}(R)}M_{\operatprname*{\mathxrak{p}}}$ is a monmmur'hism. Since $R \rightarrow... | {p}}}$ belongs to $(\mathcal{V}_{\operatorname*{\mathfrak{p}}})_n \subseteq \mathcal{V}_n$, again \[L:local\] the last The natural map is monomorphism. Since $R R_{\operatorname*{\mathfrak{p}}}$ is a ring epimorphism, the injective envelope $E(M_{\operatorname*{\mathfrak{p}}})$ $\operatorname*{Mod-R_{\operatorname*{\ma... | {p}}}$ belongs to $(\mathcal{V}_{\operatoRname*{\mathfRak{p}}})_n \SubSetEq \MathCal{V}_N$, again using LemMA \[L:loCal\] for the last inclusion. the naTuRAl maP $\IoTa: M \riGhtarroW \PrOD_{\OpeRaToRnaMe*{\MAtHfrak{P}}\in \OperatoRname*{Spec}(R)}m_{\opErAtorname*{\mathFRaK{p}}}$ is a monomOrpHism. Since $R \riGhtArrow... | {p}}}$ belongs to $(\mathc al{V}_{\op erato rna me* {\ math frak {p}}})_n \subs e teq\mathcal{V}_n$, againusing L e mma\ [L :loca l\] for th e las tin clu si o n. Thenat ural ma p $\iota:M \ ri ghtarrow \pr o d_ {\operator nam e*{\mathfrak {p} }\in \ op era t ornam e*{ Spec} (R)}M_ { \opera torname*{ \m a thfrak { p}... | {p}}}$ belongs_to $(\mathcal{V}_{\operatorname*{\mathfrak{p}}})_n_\subseteq \mathcal{V}_n$, again using_Lemma \[L:local\] for_the_last inclusion._The_natural map $\iota:_M \rightarrow \prod_{\operatorname*{\mathfrak{p}}\in_\operatorname*{Spec}(R)}M_{\operatorname*{\mathfrak{p}}}$ is a monomorphism._Since $R \rightarrow... |
After reconstructing the pre-clusters in the four projections, the final (full) cluster properties are determined by adding the pre-cluster energies and averaging the coordinates of the pre-cluster centres.
In case of more than one cluster per projection the situation is more difficult, as illustrated in figure \[fig:... | After reconstructing the pre - clusters in the four projections, the final (wide) bunch properties are determined by lend the pre - cluster energy and averaging the coordinates of the pre - cluster center.
In case of more than one cluster per project the situation is more difficult, as illustrate in figure \[fig: tw... | Aftfr reconstructing the prt-clusters in the yiur prmjectikns, the winal (full) cluster propertied qre dtnermined by adding thd pre-clusner energues end averaging thx coordikctes kn the 're-cluster centtes.
In case ox more than ona zlbster per projection the situation if more civficult, as illostraued in rpgmre \[fig:... | After reconstructing the pre-clusters in the four final cluster properties determined by adding the of the pre-cluster In case of than one cluster per projection the is more difficult, as illustrated in figure \[fig:twocl\]: ![Projected views of a two-cluster Two particles hitting the calorimeter lead to eight pre-clus... | After reconstructing the pre-Clusters in The foUr pRojEcTionS, the Final (full) clustER proPerties are determined by AddinG tHE pre-CLuSter eNergies ANd AVEraGiNg The CoORdInateS of The pre-cLuster centRes.
in Case of more thAN oNe cluster pEr pRojection the SitUation Is MorE DiffiCulT, as ilLustraTEd in fiGure \[fig:... | After reconstructing the p re-cluster s inthe fo ur pro ject ions, the fina l (fu ll) cluster properties arede t ermi n ed by a dding t h ep r e-c lu st eren e rg ies a ndaveragi ng the coo rdi na tes of the p r e- cluster ce ntr es.
In case of moreth ano ne cl ust er pe r proj e ctionthe situa ti o n is m o re... | After reconstructing_the pre-clusters_in the four projections,_the final_(full)_cluster properties_are_determined by adding_the pre-cluster energies_and averaging the coordinates_of the pre-cluster_centres.
In_case of more than one cluster per projection the situation is more difficult, as_illustrated_in figure_\[fig:... |
00:Sharing; @Whittaker94:Informal]. This idea was recently further confirmed by Pentland *et al.*, in a study of workplace communication patterns at a Prague bank [@Pentland12:New]. They discovered a key characteristic of successful teams: members periodically interact with others outside of their team, and bring back ... | 00: Sharing; @Whittaker94: Informal ]. This idea was recently further confirmed by Pentland * et al. *, in a discipline of workplace communication design at a Prague bank [ @Pentland12: New ]. They discovered a key feature of successful teams: members periodically interact with others outside of their team, and insti... | 00:Shaging; @Whittaker94:Informal]. Tmis idea was recgnrly fucther cknfirmed by Pentland *et al.*, in a studb of workkjace communication patterns at a Prqgue vank [@Pentlais12:New]. Thcv disdlverzd a key charactgristic of sgccessful teamv: oelbers periodically interact with otrers ouyslde of their tgam, amq brjng back ... | 00:Sharing; @Whittaker94:Informal]. This idea was recently further Pentland al.*, in study of workplace bank They discovered a characteristic of successful members periodically interact with others outside their team, and bring back new information. They dubbed this critical dimension of ‘exploration’, the tendency for... | 00:Sharing; @Whittaker94:Informal]. THis idea was RecenTly FurThEr coNfirMed by Pentland *eT Al.*, in A study of workplace commuNicatIoN PattERnS at a PRague baNK [@PENTlaNd12:neW]. ThEy DIsCoverEd a Key charActeristic Of sUcCessful teams: MEmBers periodIcaLly interact wIth Others OuTsiDE of thEir Team, aNd brinG Back ... | 00:Sharing; @Whittaker94:I nformal].Thiside a w as rec entl y further conf i rmed by Pentland *et al.*, in a s t udyo fworkp lace co m mu n i cat io npat te r ns at a Pr ague ba nk [@Pentl and 12 :New]. Theyd is covered akey characteris tic of su cc ess f ul te ams : mem bers p e riodic ally inte ra c t with ot... | 00:Sharing; @Whittaker94:Informal]._This idea_was recently further confirmed_by Pentland_*et_al.*, in_a_study of workplace_communication patterns at_a Prague bank [@Pentland12:New]. They_discovered a key_characteristic_of successful teams: members periodically interact with others outside of their team, and bring_back_... |
\over 2}(\cos(\theta_j) + \cos(\theta_{j+1}))\right],
\label{thetacenter}\end{aligned}$$ and the zone widths are $(\Delta r)_{i'} = r_{i+1} - r_i$ and $(\Delta \theta)_{j'} = \theta_{j+1} - \theta_j$. The definitions of $(\cos\vartheta)_{\beta'}$, $(\sin\vartheta)_{\beta'}$, and $(\cos\varphi)_{\gamma'}$ in (\[spaceDiv... | \over 2}(\cos(\theta_j) + \cos(\theta_{j+1}))\right ],
\label{thetacenter}\end{aligned}$$ and the zone widths are $ (\Delta r)_{i' } = r_{i+1 } - r_i$ and $ (\Delta \theta)_{j' } = \theta_{j+1 } - \theta_j$. The definitions of $ (\cos\vartheta)_{\beta'}$, $ (\sin\vartheta)_{\beta'}$, and $ (\cos\varphi)_{\gamma'}$ in... | \oveg 2}(\cos(\theta_j) + \cos(\theta_{j+1}))\rinht],
\label{thetacenjee}\end{almgned}$$ ahd the zune widths are $(\Delta r)_{i'} = r_{i+1} - r_u$ and $(\Delta \theta)_{j'} = \theta_{j+1} - \theta_j$. Tje definutiois of $(\cos\varthete)_{\geta'}$, $(\sik\rarthsba)_{\betc'}$, end $(\cos\varphi)_{\gakma'}$ in (\[spaweDiv... | \over 2}(\cos(\theta_j) + \cos(\theta_{j+1}))\right], \label{thetacenter}\end{aligned}$$ and the are r)_{i'} = - r_i$ and \theta_j$. definitions of $(\cos\vartheta)_{\beta'}$, and $(\cos\varphi)_{\gamma'}$ in will be given below. Finally, the of ${\cal N}$ on zone surfaces in (\[spaceDivergence\]) are given by a partic... | \over 2}(\cos(\theta_j) + \cos(\theta_{j+1}))\rigHt],
\label{theTacenTer}\End{AlIgneD}$$ and The zone widths aRE $(\DelTa r)_{i'} = r_{i+1} - r_i$ and $(\Delta \theta)_{j'} = \Theta_{J+1} - \tHEta_j$. tHe DefinItions oF $(\CoS\VArtHeTa)_{\BetA'}$, $(\sIN\vArtheTa)_{\bEta'}$, and $(\cOs\varphi)_{\gaMma'}$ In (\[SpaceDiv... | \over 2}(\cos(\theta_j) +\cos(\thet a_{j+ 1}) )\r ig ht],
\la bel{thetacente r }\en d{aligned}$$ and the z one w id t hs a r e$(\De lta r)_ { i' } = r _{ i+ 1}-r _i $ and $( \Delta\theta)_{j '}=\theta_{j+1} -\theta_j$. Th e definition s o f $(\c os \va r theta )_{ \beta '}$, $ ( \sin\v artheta)_ {\ b eta'}$ , and $... | \over 2}(\cos(\theta_j)_+ \cos(\theta_{j+1}))\right],
\label{thetacenter}\end{aligned}$$_and the zone widths_are $(\Delta_r)_{i'}_= r_{i+1}_-_r_i$ and $(\Delta_\theta)_{j'} = \theta_{j+1}_- \theta_j$. The definitions_of $(\cos\vartheta)_{\beta'}$, $(\sin\vartheta)_{\beta'}$,_and_$(\cos\varphi)_{\gamma'}$ in (\[spaceDiv... |
_P^k g_1^m \prod_{s=1}^{q-2} h_s^{n_s}$. Equations,, combined with Lemmas \[lem:g1\] and \[lem:hs\] imply that $f \in L((2g(\cX)-2)P_\infty)$ if $$i(q^3+1)+j(q^3-q^2+q)+k(q^3+1)+m(q^4+q)+\sum_{s=1}^{q-2} n_s ((s+1)q-s)(q^3+1) \leq q^5-2q^3+q^2-2,$$ which is equivalent to $$\label{eq:inquality1}
i(q+1)+jq+k(q+1)+mq(q+1... | _ P^k g_1^m \prod_{s=1}^{q-2 } h_s^{n_s}$. Equations, , combined with Lemmas \[lem: g1\ ] and \[lem: hs\ ] imply that $ f \in L((2g(\cX)-2)P_\infty)$ if $ $ i(q^3 + 1)+j(q^3 - q^2+q)+k(q^3 + 1)+m(q^4+q)+\sum_{s=1}^{q-2 } n_s (( s+1)q - s)(q^3 + 1) \leq q^5 - 2q^3+q^2 - 2,$$ which is equivalent to $ $ \label{eq: inqua... | _P^k g_1^m \prod_{s=1}^{q-2} h_s^{n_s}$. Equationr,, combined with Lemmas \[lem:g1\] znd \[lem:hr\] imply that $f \in L((2g(\cX)-2)P_\infty)$ id $$i(q^3+1)+j(w^3-q^2+q)+k(q^3+1)+m(q^4+q)+\sum_{s=1}^{q-2} n_s ((s+1)q-s)(q^3+1) \ueq q^5-2q^3+q^2-2,$$ wjich is wquitalent to $$\label{eq:inqualibv1}
i(q+1)+jq+i(e+1)+mq(q+1... | _P^k g_1^m \prod_{s=1}^{q-2} h_s^{n_s}$. Equations,, combined with and imply that \in L((2g(\cX)-2)P_\infty)$ if which equivalent to $$\label{eq:inquality1} n_s ((s+1)q-s)(q+1) \leq On the other hand we have n_s ((s+1)q^2).$$ Hence the claim follows from Lemma \[holom\]. \[obs:largestgapinG\] Inequality implies in that... | _P^k g_1^m \prod_{s=1}^{q-2} h_s^{n_s}$. Equations,, coMbined with lemmaS \[leM:g1\] aNd \[Lem:hS\] impLy that $f \in L((2g(\cX)-2)P_\INfty)$ If $$i(q^3+1)+j(q^3-q^2+q)+k(q^3+1)+m(q^4+q)+\sum_{s=1}^{q-2} n_s ((s+1)q-S)(q^3+1) \leq Q^5-2q^3+Q^2-2,$$ WhicH Is EquivAlent to $$\LAbEL{Eq:iNqUaLitY1}
i(Q+1)+Jq+K(q+1)+mq(q+1... | _P^k g_1^m \prod_{s=1}^{q -2} h_s^{n _s}$. Eq uat io ns,, com bined with Lem m as \ [lem:g1\] and \[lem:hs \] im pl y tha t $ f \in L((2g( \ cX ) - 2)P _\ in fty )$ if $$i( q^3 +1)+j(q ^3-q^2+q)+ k(q ^3 +1)+m(q^4+q) + \s um_{s=1}^{ q-2 } n_s ((s+1) q-s )(q^3+ 1) \l e q q^5 -2q ^3+q^ 2-2,$$ whichis equiva le n t... | _P^k _g_1^m \prod_{s=1}^{q-2}_h_s^{n_s}$. Equations,, combined with_Lemmas \[lem:g1\]_and_\[lem:hs\] imply_that_$f \in L((2g(\cX)-2)P_\infty)$_if $$i(q^3+1)+j(q^3-q^2+q)+k(q^3+1)+m(q^4+q)+\sum_{s=1}^{q-2} n_s_((s+1)q-s)(q^3+1) \leq q^5-2q^3+q^2-2,$$ which_is equivalent to_$$\label{eq:inquality1}
i(q+1)+jq+k(q+1)+mq(q+1... |
1+\frac{\epsilon}{2}.$$ With this choice, $\delta_0$ in depends only on $\epsilon$. Thus we have to verify that $$\left(\frac{1}{\log|\log{|t|}|}+|c|\right)^{\frac{1}{k}} \leq |\log|t||^{\frac{\epsilon}{2}},$$ but this obviously holds for small $|t|$ since the left-hand side is bounded. Hence $\tilde{\Omega} \subset \O... | 1+\frac{\epsilon}{2}.$$ With this choice, $ \delta_0 $ in depends only on $ \epsilon$. Thus we get to control that $ $ \left(\frac{1}{\log|\log{|t|}|}+|c|\right)^{\frac{1}{k } } \leq |\log|t||^{\frac{\epsilon}{2}},$$ but this obviously holds for small $ |t|$ since the left - handwriting side is bounded. therefore $ \ti... | 1+\fraf{\epsilon}{2}.$$ With this choict, $\delta_0$ in depends only mn $\epsjlon$. Thur we have to verify that $$\lefv(\frax{1}{\log|\lig{|t|}|}+|c|\right)^{\frac{1}{k}} \leq |\log|g||^{\frac{\epsipon}{2}},$$ but rhis ibviously iklds fov smamp $|t|$ vmnce the left-hakd side is tounded. Hence $\diudz{\Omega} \subset \O... | 1+\frac{\epsilon}{2}.$$ With this choice, $\delta_0$ in depends $\epsilon$. we have verify that $$\left(\frac{1}{\log|\log{|t|}|}+|c|\right)^{\frac{1}{k}} holds small $|t|$ since left-hand side is Hence $\tilde{\Omega} \subset \Omega$ and $(0,0)$ an irregular boundary point for $\Omega$ as well. This implies that the u... | 1+\frac{\epsilon}{2}.$$ With this choice, $\Delta_0$ in depEnds oNly On $\ePsIlon$. thus We have to verify THat $$\lEft(\frac{1}{\log|\log{|t|}|}+|c|\right)^{\frAc{1}{k}} \leQ |\lOG|t||^{\frAC{\ePsiloN}{2}},$$ but thiS ObVIOusLy HoLds FoR SmAll $|t|$ sIncE the lefT-hand side iS boUnDed. Hence $\tildE{\omEga} \subset \O... | 1+\frac{\epsilon}{2}.$$ Wi th this ch oice, $\ del ta _0$in d epends only on $\ep silon$. Thus we have t o ver if y tha t $ $\lef t(\frac { 1} { \ log |\ lo g{| t| } |} +|c|\ rig ht)^{\f rac{1}{k}} \l eq |\log|t||^{ \ fr ac{\epsilo n}{ 2}},$$ but t his obvio us lyh oldsfor smal l $|t| $ since the left -h a nd ... | 1+\frac{\epsilon}{2}.$$ With_this choice,_$\delta_0$ in depends only_on $\epsilon$._Thus_we have_to_verify that $$\left(\frac{1}{\log|\log{|t|}|}+|c|\right)^{\frac{1}{k}}_\leq |\log|t||^{\frac{\epsilon}{2}},$$ but_this obviously holds for_small $|t|$ since_the_left-hand side is bounded. Hence $\tilde{\Omega} \subset \O... |
)\to (E,V)\to (E_2,V_2) \to 0$$ in which
- $(E,V)$ is ${\alpha}^-_i$-stable and of type ${(n,d,k)}$,
- ${{\operatorname{rk}}}(E_1)=n_1$ and $\dim(V_1)={\lambda}n_1$,
- $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpha}_i}(E,V)$,
- $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}_i^-$-stable,
... | ) \to (E, V)\to (E_2,V_2) \to 0$$ in which
- $ (E, V)$ is $ { \alpha}^-_i$-stable and of type $ { (n, d, k)}$,
- $ { { \operatorname{rk}}}(E_1)=n_1 $ and $ \dim(V_1)={\lambda}n_1 $,
- $ \mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpha}_i}(E, V)$,
- $ (E_1,V_1)$ and $ (E_2,V_2)$ a... | )\to (F,V)\to (E_2,V_2) \to 0$$ in which
- $(E,Y)$ is ${\alpha}^-_i$-stablg qnd of type ${(h,d,k)}$,
- ${{\opdratorname{rk}}}(E_1)=n_1$ and $\dim(V_1)={\lambde}n_1$,
- $\mu_{{\alkka}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpfa}_i}(E,V)$,
- $(E_1,N_1)$ and $(E_2,V_2)$ qre uoth ${\alpha}_i^-$-stablx,
... | )\to (E,V)\to (E_2,V_2) \to 0$$ in which is and of ${(n,d,k)}$, - ${{\operatorname{rk}}}(E_1)=n_1$ $(E_1,V_1)$ $(E_2,V_2)$ are both - $\dim(V_1)$ and satisfy the min-min criteria given in and [(d)]{} of Lemma \[lem:vicente\](ii). Define $$W^+({\alpha}_i,n,d,k)=\bigsqcup_{{\lambda}<\frac{k}{n},\, n_1<n} W^+({\alpha}_i, ... | )\to (E,V)\to (E_2,V_2) \to 0$$ in which
- $(E,V)$ is ${\alphA}^-_i$-stable anD of tyPe ${(n,D,k)}$,
- ${{\oPeRatoRnamE{rk}}}(E_1)=n_1$ and $\dim(V_1)={\laMBda}n_1$,
- $\Mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\Alpha}_I}(E,v)$,
- $(e_1,V_1)$ anD $(e_2,V_2)$ Are boTh ${\alpha}_I^-$-StABLe,
... | )\to (E,V)\to (E_2,V_2) \t o 0$$ in w hich
- $ (E ,V)$ is${\alpha}^-_i$ - stab le and of type ${(n,d, k)}$,
- $ { {\ opera torname { rk } } }(E _1 )= n_1 $a nd $\di m(V _1)={\l ambda}n_1$ ,
- $\mu_{{\al p ha }_i}(E_1,V _1) =\mu_{{\alph a}_ i}(E_2 ,V _2) = \mu_{ {\a lpha} _i}(E, V )$,
- $(E_1, V_ 1 )$ and $(... | )\to (E,V)\to_(E_2,V_2) \to_0$$ in which
- _ $(E,V)$_is_${\alpha}^-_i$-stable and_of_type ${(n,d,k)}$,
- _ ${{\operatorname{rk}}}(E_1)=n_1$ and_$\dim(V_1)={\lambda}n_1$,
- $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpha}_i}(E,V)$,
-_ $(E_1,V_1)$_and_$(E_2,V_2)$ are both ${\alpha}_i^-$-stable,
... |
|\,,$$ we may write $$[\,f\,]_* =\,\int_0^R \,\sup_{ \,x \in\,{{\overline{\Omega}}}}\, {{\omega}}_f(x;\,r)\,
\,\frac{dr}{r}\,.\label{catriz}$$ So, together with $\,C_*({{\overline{\Omega}}})\,,$ we have considered a functional space $\,B_*({{\overline{\Omega}}})\,$ obtained by commuting *integral* and *sup* operators ... | |\,,$$ we may write $ $ [ \,f\, ] _ * = \,\int_0^R \,\sup _ { \,x \in\,{{\overline{\Omega}}}}\, { { \omega}}_f(x;\,r)\,
\,\frac{dr}{r}\,.\label{catriz}$$ So, together with $ \,C_*({{\overline{\Omega}}})\,,$ we have considered a running quad $ \,B_*({{\overline{\Omega}}})\,$ obtained by commuting * built-in * and * ... | |\,,$$ we may write $$[\,f\,]_* =\,\int_0^R \,\sup_{ \,x \in\,{{\overline{\Omegc}}}}\, {{\omega}}_h(x;\,r)\,
\,\frac{sr}{r}\,.\label{zatriz}$$ So, together with $\,C_*({{\oveclinw{\Omegq}}})\,,$ we have considered a functionwl space $\,B_*({{\ovtrline{\Omega}}})\,$ obtaiisd by commutihn *intzgcal* and *sup* opetators ... | |\,,$$ we may write $$[\,f\,]_* =\,\int_0^R \,\sup_{ {{\omega}}_f(x;\,r)\, So, together $\,C_*({{\overline{\Omega}}})\,,$ we have obtained commuting *integral* and operators in the hand side of definition : For $\,f \in\,C({{\overline{\Omega}}})\,,$ we defined the semi-norm $$\langle\,f\,\rangle_* = \,\sup_{ \,x \in\,{... | |\,,$$ we may write $$[\,f\,]_* =\,\int_0^R \,\sup_{ \,x \in\,{{\overLine{\Omega}}}}\, {{\oMega}}_f(X;\,r)\,
\,\fRac{Dr}{R}\,.\labEl{caTriz}$$ So, together WIth $\,C_*({{\Overline{\Omega}}})\,,$ we have conSiderEd A FuncTIoNal spAce $\,B_*({{\oveRLiNE{\omeGa}}})\,$ ObTaiNeD By CommuTinG *integrAl* and *sup* opEraToRs ... | |\,,$$ we may write $$[\,f \,]_* =\,\ int_0 ^R \, \s up_{ \,x \in\,{{\overl i ne{\ Omega}}}}\, {{\omega}} _f(x; \, r )\,\ ,\ frac{ dr}{r}\ , .\ l a bel {c at riz }$ $ S o, to get her wit h $\,C_*({ {\o ve rline{\Omega } }} )\,,$ we h ave considereda f unctio na l s p ace $ \,B _*({{ \overl i ne{\Om ega}}})\, $o bta... | |\,,$$ we_may write_$$[\,f\,]_* =\,\int_0^R \,\sup_{_\,x \in\,{{\overline{\Omega}}}}\,_{{\omega}}_f(x;\,r)\,
\,\frac{dr}{r}\,.\label{catriz}$$_So, together_with_$\,C_*({{\overline{\Omega}}})\,,$ we have_considered a functional_space $\,B_*({{\overline{\Omega}}})\,$ obtained by_commuting *integral* and_*sup*_operators ... |
_0$ is the normalization factor and $\Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots
{\bf S}_k)$ is the Brink wave function for the $C+k(^2n)$-cluster system consisting the core($C$) and $k$ dineutrons($^2n$) as, $$\Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots
{\bf S}_k)\equiv {\cal{A}}\left\{\phi^{\rm C}... | _ 0 $ is the normalization factor and $ \Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots
{ \bf S}_k)$ is the Brink wave function for the $ C+k(^2n)$-cluster system consist the core($C$) and $ k$ dineutrons($^2n$) as, $ $ \Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots
{ \bf S}_k)\equiv { \cal{A}}\left\{\... | _0$ is the normalization factov and $\Phi_{\rm Brink}({\bf S}_C,{\bh S}_1,{\bf S}_2,\ddots
{\bf R}_k)$ is the Brink wave functioi foe the $C+k(^2n)$-cluster system conristing tje core($C$) and $j$ dineutroia($^2n$) as, $$\Pmn_{\rm Bdlnk}({\bf W}_C,{\bf S}_1,{\bf S}_2,\cdotx
{\bf S}_k)\equie {\cal{A}}\left\{\phi^{\rk Z}... | _0$ is the normalization factor and $\Phi_{\rm S}_1,{\bf {\bf S}_k)$ the Brink wave consisting core($C$) and $k$ as, $$\Phi_{\rm Brink}({\bf S}_1,{\bf S}_2,\cdots {\bf S}_k)\equiv {\cal{A}}\left\{\phi^{\rm C}({\bf S}_1) \phi^{^2n}({\bf S}_2)\cdots \phi^{^2n}({\bf S}_k) \right \}.$$ Here, the wave function of the $^2n$,... | _0$ is the normalization factor aNd $\Phi_{\rm BriNk}({\bf S}_c,{\bf s}_1,{\bf s}_2,\cDots
{\Bf S}_k)$ Is the Brink wave FUnctIon for the $C+k(^2n)$-cluster sysTem coNsIStinG ThE core($c$) and $k$ diNEuTROns($^2N$) aS, $$\PHi_{\rM BRInK}({\bf S}_C,{\Bf S}_1,{\Bf S}_2,\cdotS
{\bf S}_k)\equiv {\Cal{a}}\lEft\{\phi^{\rm C}... | _0$ is the normalization f actor and$\Phi _{\ rmBr ink} ({\b f S}_C,{\bf S} _ 1,{\ bf S}_2,\cdots
{\bf S} _k)$is theB ri nk wa ve func t io n for t he $C +k ( ^2 n)$-c lus ter sys tem consis tin gthe core($C$ ) a nd $k$ din eut rons($^2n$)as, $$\Ph i_ {\r m Brin k}( {\bfS}_C,{ \ bf S}_ 1,{\bf S} _2 , \cdots {\bf S... | _0$ is_the normalization_factor and $\Phi_{\rm Brink}({\bf_S}_C,{\bf S}_1,{\bf_S}_2,\cdots
{\bf_S}_k)$ is_the_Brink wave function_for the $C+k(^2n)$-cluster_system consisting the core($C$)_and $k$ dineutrons($^2n$)_as,_$$\Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots
{\bf S}_k)\equiv {\cal{A}}\left\{\phi^{\rm C}... |
_{\varepsilon})$ is optimal in energy. We have $$\lim_{{\varepsilon}\to 0} \frac{1}{|\log {\varepsilon}|^2}\int_{\Omega}W(\beta_{\varepsilon}) \, dx= \lim_{{\varepsilon}\to 0} \frac{1}{|\log {\varepsilon}|^2}
\int_{\Omega}W(|\log {\varepsilon}| \beta + \hat \beta_{\varepsilon})\, dx.$$ Since $\hat \beta_{\varepsilon}/|... | _ { \varepsilon})$ is optimal in energy. We have $ $ \lim_{{\varepsilon}\to 0 } \frac{1}{|\log { \varepsilon}|^2}\int_{\Omega}W(\beta_{\varepsilon }) \, dx= \lim_{{\varepsilon}\to 0 } \frac{1}{|\log { \varepsilon}|^2 }
\int_{\Omega}W(|\log { \varepsilon}| \beta + \hat \beta_{\varepsilon})\, dx.$$ Since $ \hat \beta_{... | _{\varfpsilon})$ is optimal in entrgy. We have $$\lim_{{\vceepsilmn}\to 0} \rrac{1}{|\log {\xarepsilon}|^2}\int_{\Omega}W(\beta_{\varepdioon}) \, ex= \lim_{{\varepsilon}\to 0} \frxc{1}{|\log {\varvpsilon}|^2}
\inr_{\Omeja}W(|\log {\varepsiloi}| \beta + \mct \befw_{\varzpwilon})\, dx.$$ Since $\hat \beta_{\vdrepsilon}/|... | _{\varepsilon})$ is optimal in energy. We have \frac{1}{|\log \, dx= 0} \frac{1}{|\log {\varepsilon}|^2} \beta_{\varepsilon})\, Since $\hat \beta_{\varepsilon}/|\log {\rightharpoonup}0$ in $L^2({\Omega};{\mathbb{M}^{2\times taking into account also, we conclude 0} \frac{1}{|\log {\varepsilon}|^2}\int_{\Omega}W(\beta_{\... | _{\varepsilon})$ is optimal in enerGy. We have $$\liM_{{\varePsiLon}\To 0} \Frac{1}{|\Log {\vArepsilon}|^2}\int_{\OmEGa}W(\bEta_{\varepsilon}) \, dx= \lim_{{\varePsiloN}\tO 0} \Frac{1}{|\LOg {\VarepSilon}|^2}
\inT_{\omEGA}W(|\lOg {\VaRepSiLOn}| \Beta + \hAt \bEta_{\varePsilon})\, dx.$$ SiNce $\HaT \beta_{\varepsiLOn}/|... | _{\varepsilon})$ is optima l in energ y. We ha ve$$ \lim _{{\ varepsilon}\to 0} \ frac{1}{|\log {\vareps ilon} |^ 2 }\in t _{ \Omeg a}W(\be t a_ { \ var ep si lon }) \, dx=\li m_{{\va repsilon}\ to0} \frac{1}{|\ l og {\varepsi lon }|^2}
\int_{ \Om ega}W( |\ log {\var eps ilon} | \bet a + \ha t \beta_{ \v a repsil ... | _{\varepsilon})$ is_optimal in_energy. We have $$\lim_{{\varepsilon}\to_0} \frac{1}{|\log_{\varepsilon}|^2}\int_{\Omega}W(\beta_{\varepsilon})_\, dx=_\lim_{{\varepsilon}\to_0} \frac{1}{|\log {\varepsilon}|^2}
\int_{\Omega}W(|\log_{\varepsilon}| \beta +_\hat \beta_{\varepsilon})\, dx.$$ Since_$\hat \beta_{\varepsilon}/|... |
Table \[auc\_table\] are compared with other common link prediction methods in Table \[auc\_compare\]: tSVD [@dhillon], tKatz [@dunlavy], and a degree-based model where the probability of a link is approximated as $\mathbb P(A_{ij}=1)=1-\exp(-d^\mathrm{out}_id^\mathrm{in}_j)$ where $d^\mathrm{out}_i$ and $d^\mathrm{in... | Table \[auc\_table\ ] are compared with other common link prediction method in Table \[auc\_compare\ ]: tSVD [ @dhillon ], tKatz [ @dunlavy ], and a academic degree - based model where the probability of a liaison is estimate as $ \mathbb P(A_{ij}=1)=1-\exp(-d^\mathrm{out}_id^\mathrm{in}_j)$ where $ d^\mathrm{out}_... | Tahle \[auc\_table\] are compared with other common linn predjction mdthods in Table \[auc\_compare\]: tSTD [@dyillob], tKatz [@dunlavy], and a aegree-basvd model qhert the probability of a likh is wpprmeimated as $\mathnb P(A_{ij}=1)=1-\exp(-d^\kathrm{out}_id^\matvro{iu}_j)$ where $d^\mathrm{out}_i$ and $d^\mathrm{in... | Table \[auc\_table\] are compared with other common methods Table \[auc\_compare\]: [@dhillon], tKatz [@dunlavy], the of a link approximated as $\mathbb where $d^\mathrm{out}_i$ and $d^\mathrm{in}_j$ are the and in-degree of each node. Overall, when compared to the results in Table the PMF models achieve impressive imp... | Table \[auc\_table\] are compared wIth other coMmon lInk PreDiCtioN metHods in Table \[auc\_COmpaRe\]: tSVD [@dhillon], tKatz [@dunlAvy], anD a DEgreE-BaSed moDel wherE ThE PRobAbIlIty Of A LiNk is aPprOximateD as $\mathbb P(a_{ij}=1)=1-\ExP(-d^\mathrm{out}_iD^\MaThrm{in}_j)$ wheRe $d^\Mathrm{out}_i$ anD $d^\mAthrm{iN... | Table \[auc\_table\] arecompared w ith o the r c om monlink prediction me t hods in Table \[auc\_compa re\]: t S VD [ @ dh illon ], tKat z [ @ d unl av y] , a nd adegre e-b ased mo del wherethe p robability o f a link is a ppr oximated as$\m athbbP( A_{ i j}=1) =1- \exp( -d^\ma t hrm{ou t}_id^\ma th r m{in}_ j ... | Table \[auc\_table\]_are compared_with other common link_prediction methods_in_Table \[auc\_compare\]: tSVD_[@dhillon],_tKatz [@dunlavy], and_a degree-based model_where the probability of_a link is_approximated_as $\mathbb P(A_{ij}=1)=1-\exp(-d^\mathrm{out}_id^\mathrm{in}_j)$ where $d^\mathrm{out}_i$ and $d^\mathrm{in... |
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