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 |
|---|---|---|---|---|---|---|
8578656 & 0.8691745 & 0.4489255 & 0.4562589 & 0.4633825\
0.7 & 0.7756869 & 0.7870818 & 0.7978805 & 0.4583331 & 0.4656164 & 0.4726902\
0.8 & 0.7224922 & 0.7334088 & 0.7437773 & 0.4671917 & 0.4744397 & 0.4814776\
0.9 & 0.6807750 & 0.6912843 & 0.7012833 & 0.4755518 & 0.4827761 & 0.4897895\
1. & 0.6471769 & 0.6573333 & 0.6... | 8578656 & 0.8691745 & 0.4489255 & 0.4562589 & 0.4633825\
0.7 & 0.7756869 & 0.7870818 & 0.7978805 & 0.4583331 & 0.4656164 & 0.4726902\
0.8 & 0.7224922 & 0.7334088 & 0.7437773 & 0.4671917 & 0.4744397 & 0.4814776\
0.9 & 0.6807750 & 0.6912843 & 0.7012833 & 0.4755518 & 0.4827761 & 0.4897895\
1. & 0.6471769 & 0.65733... | 8578656 & 0.8691745 & 0.4489255 & 0.4562589 & 0.4633825\
0.7 & 0.7756869 & 0.7870818 & 0.7978805 & 0.4583331 & 0.4656164 & 0.4726902\
0.8 & 0.7224922 & 0.7334088 & 0.7437773 & 0.4671917 & 0.4744397 & 0.4814776\
0.9 & 0.6807750 & 0.6912843 & 0.7012833 & 0.4755518 & 0.4827761 & 0.4897895\
1. & 0.6471769 & 0.6573333 & 0.6... | 8578656 & 0.8691745 & 0.4489255 & 0.4562589 0.7 0.7756869 & & 0.7978805 & 0.8 0.7224922 & 0.7334088 0.7437773 & 0.4671917 0.4744397 & 0.4814776\ 0.9 & 0.6807750 0.6912843 & 0.7012833 & 0.4755518 & 0.4827761 & 0.4897895\ 1. & 0.6471769 & & 0.6670116 & 0.4834526 & 0.4906632 & 0.4976610\ \ 0.0001 & 2501.676 & & & & & 0.39... | 8578656 & 0.8691745 & 0.4489255 & 0.4562589 & 0.4633825\
0.7 & 0.7756869 & 0.7870818 & 0.7978805 & 0.4583331 & 0.4656164 & 0.4726902\
0.8 & 0.7224922 & 0.7334088 & 0.7437773 & 0.4671917 & 0.4744397 & 0.4814776\
0.9 & 0.6807750 & 0.6912843 & 0.7012833 & 0.4755518 & 0.4827761 & 0.4897895\
1. & 0.6471769 & 0.6573333 & 0.6... | 8578656 & 0.8691745 & 0.44 89255 & 0. 45625 89& 0 .4 6338 25\0.7 & 0.775686 9 & 0 .7870818 & 0.7978805 & 0.45 83 3 31 & 0. 46561 64 & 0. 4 72 6 9 02\
0 .8 &0. 7 22 4922& 0 .733408 8 & 0.7437 773 & 0.4671917 & 0. 4744397 &0.4 814776\
0.9& 0 .68077 50 &0 .6912 843 & 0. 701283 3 & 0.4 755518 &0. 4 827761 & 0.489 ... | 8578656 &_0.8691745 &_0.4489255 & 0.4562589 &_0.4633825\
0.7 &_0.7756869_& 0.7870818_&_0.7978805 & 0.4583331_& 0.4656164 &_0.4726902\
0.8 & 0.7224922 &_0.7334088 & 0.7437773_&_0.4671917 & 0.4744397 & 0.4814776\
0.9 & 0.6807750 & 0.6912843 & 0.7012833 & 0.4755518 &_0.4827761_& 0.4897895\
1._&_0.6471769_& 0.6573333 & 0.6... |
}}({{\cal J}}_\rho)$. The dyadic property means that each parent node has one or two children: if it has one children the set associated to the children is the same as the one of the parent, and when it has two children the set of the parent is partitioned into two disjoint nonempty sets by some set in ${{\cal J}}_\rho... | } } ({ { \cal J}}_\rho)$. The dyadic property means that each parent node has one or two children: if it take one child the set associated to the children is the like as the one of the parent, and when it has two children the set of the parent is partition into two disjoint nonempty sets by some set in $ { { \cal J}}_\... | }}({{\cal J}}_\rho)$. The dyadic propertn means that eack parenv node gas one ur two children: if it has onx chuldreb the set associated tu the chipdren is the wame as thx one of the pzvent, cnv when it has teo childret the set of tve pcrent is partitioned into two disjoigt nonekphy sets by somg set yn ${{\czl J}}_\rho... | }}({{\cal J}}_\rho)$. The dyadic property means that node one or children: if it associated the children is same as the of the parent, and when it two children the set of the parent is partitioned into two disjoint nonempty by some set in ${{\cal J}}_\rho^{(I)}$, and these are the sets associated to children. set will ... | }}({{\cal J}}_\rho)$. The dyadic property mEans that eaCh parEnt NodE hAs onE or tWo children: if it HAs onE children the set associaTed to ThE ChilDReN is thE same as THe ONE of ThE pAreNt, ANd When iT haS two chiLdren the seT of ThE parent is parTItIoned into tWo dIsjoint nonemPty Sets by SoMe sET in ${{\caL J}}_\rHo... | }}({{\cal J}}_\rho)$. Thedyadic pro perty me ans t hateach parent node h a s on e or two children: ifit ha so ne c h il drenthe set as s o cia te dtoth e c hildr enis thesame as th e o ne of the pare n t, and whenithas two chil dre n these t o f thepar ent i s part i tioned into two d i sjoint nonempt y se ts ... | }}({{\cal J}}_\rho)$._The dyadic_property means that each_parent node_has_one or_two_children: if it_has one children_the set associated to_the children is_the_same as the one of the parent, and when it has two children the_set_of the_parent_is_partitioned into two disjoint nonempty_sets by some set in_${{\cal J}}_\rho... |
in ${\mathcal{F}}({\mathcal{A}})$ of $n$ conjugates of words in ${\mathcal{R}}^{\pm 1}$ and $n$ is minimal with this property. Equivalently, for given an input $(W, 1^n)$, the precise word problem asks whether there exists a disk diagram over whose boundary label is $W$, whose number of faces is $n$, and there are no ... | in $ { \mathcal{F}}({\mathcal{A}})$ of $ n$ conjugates of words in $ { \mathcal{R}}^{\pm 1}$ and $ n$ is minimal with this property. Equivalently, for given an stimulation $ (W, 1^n)$, the accurate word problem ask whether there exist a disk diagram over whose boundary label is $ W$, whose number of boldness is $ n$, a... | in ${\mathcal{F}}({\mathcal{A}})$ of $n$ cunjugates of wotdw in ${\methcal{R}}^{\lm 1}$ and $v$ is minimal with this propecty. Wquivqlently, for given an ivput $(W, 1^n)$, nhe preciwe wied problem asks whccher fmere zxmsts a disk dianram over wvose boundary nacep is $W$, whose number of faces is $n$, agd therr wre no ... | in ${\mathcal{F}}({\mathcal{A}})$ of $n$ conjugates of words 1}$ $n$ is with this property. $(W, the precise word asks whether there a disk diagram over whose boundary is $W$, whose number of faces is $n$, and there are no such with fewer number of faces. The definitions for the precise word problem for solvable being ... | in ${\mathcal{F}}({\mathcal{A}})$ of $n$ conjUgates of woRds in ${\MatHcaL{R}}^{\Pm 1}$ anD $n$ is Minimal with thiS PropErty. Equivalently, for givEn an iNpUT $(W, 1^n)$, tHE pRecisE word prOBlEM AskS wHeTheR tHErE exisTs a Disk diaGram over whOse BoUndary label iS $w$, wHose number Of fAces is $n$, and thEre Are no ... | in ${\mathcal{F}}({\mathc al{A}})$ o f $n$ co nju ga tesof w ords in ${\mat h cal{ R}}^{\pm 1}$ and $n$ i s min im a l wi t hthispropert y .E q uiv al en tly ,f or give n a n input $(W, 1^n) $,th e precise wo r dproblem as kswhether ther e e xistsadis k diag ram over whose bounda ry labelis $W$, w h ose num b e ... | in_${\mathcal{F}}({\mathcal{A}})$ of_$n$ conjugates of words_in ${\mathcal{R}}^{\pm_1}$_and $n$_is_minimal with this_property. Equivalently, for_given an input $(W,_1^n)$, the precise_word_problem asks whether there exists a disk diagram over whose boundary label is $W$,_whose_number of_faces_is_$n$, and there are no_... |
, Tian Tian, Xin Huang, Lin Wang, Jun Zhu, and Le Song. Adversarial attack on graph structured data., 2018.
Xuanqing Liu, Si Si, Xiaojin Zhu, Yang Li, and Cho-Jui Hsieh. A unified framework for data poisoning attack to graph-based semi-supervised learning., 2019.
Aleksandar Bojchevski and Stephan G[ü]{}nnemann. Adver... | , Tian Tian, Xin Huang, Lin Wang, Jun Zhu, and Le Song. Adversarial attack on graph structured data. , 2018.
Xuanqing Liu, Si Si, Xiaojin Zhu, Yang Li, and Cho - Jui Hsieh. A unified model for datum poisoning approach to graph - based semi - supervised eruditeness. , 2019.
Aleksandar Bojchevski and Stephan G[... | , Tiwn Tian, Xin Huang, Lin Wakg, Jun Zhu, and Lg Sing. Adtersarizl attacy on graph structured data., 2018.
Xnanqung Luu, Si Si, Xiaojin Zhu, Yavg Li, and Cho-Jui Ysiei. A unified framxsork fov datz poivining attack tp graph-basad semi-supervivea pearning., 2019.
Aleksandar Bojchevski and Ftephan G[ü]{}jnemann. Adver... | , Tian Tian, Xin Huang, Lin Wang, and Song. Adversarial on graph structured Si, Zhu, Yang Li, Cho-Jui Hsieh. A framework for data poisoning attack to semi-supervised learning., 2019. Aleksandar Bojchevski and Stephan G[ü]{}nnemann. Adversarial attacks on node embeddings graph poisoning. In [*ICML*]{}, pages 695–704, 20... | , Tian Tian, Xin Huang, Lin Wang, JuN Zhu, and Le SOng. AdVerSarIaL attAck oN graph structurED datA., 2018.
Xuanqing Liu, Si Si, XiaojiN Zhu, YAnG li, anD chO-Jui HSieh. A unIFiED FraMeWoRk fOr DAtA poisOniNg attacK to graph-baSed SeMi-supervised LEaRning., 2019.
AleksAndAr Bojchevski And stephaN G[Ü]{}nnEMann. ADveR... | , Tian Tian, Xin Huang, Li n Wang, Ju n Zhu , a ndLe Son g. A dversarial att a ck o n graph structured dat a., 2 01 8 .
X u an qingLiu, Si Si , Xia oj in Zh u, Ya ng Li , a nd Cho- Jui Hsieh. Aun ified framew o rk for datapoi soning attac k t o grap h- bas e d sem i-s uperv ised l e arning ., 2019.
A l eksand a r ... | , Tian_Tian, Xin_Huang, Lin Wang, Jun_Zhu, and_Le Song._Adversarial attack_on_graph structured data.,_2018.
Xuanqing Liu, Si Si,_Xiaojin Zhu, Yang Li,_and Cho-Jui Hsieh._A_unified framework for data poisoning attack to graph-based semi-supervised learning., 2019.
Aleksandar Bojchevski and Stephan_G[ü]{}nnemann._Adver... |
^{(t)}\textbf{R}_2^{(t)})^{-1} \}
\big]
\\
+ \mathbb{E}\big[
\text{log}_2 \text{det} \{ \textbf{I}_{N_s} +
p_2 \textbf{R}_1^{(t)^H}\textbf{H}_{r,1}^{(t)}\bar{\textbf{F}}^{(t)}\textbf{H}_{2,r}^{(t-1)}
\textbf{H}_{2,r}^{(t-1)^H}\bar{\textbf{F}}^{(t)^H}\textbf{H}_{r,1}^{(t)^H}\textbf{R}_1^{(t)}
(\textbf{R}_1^{(t)^H}\textb... | ^{(t)}\textbf{R}_2^{(t)})^{-1 } \ }
\big ]
\\
+ \mathbb{E}\big [
\text{log}_2 \text{det } \ { \textbf{I}_{N_s } +
p_2 \textbf{R}_1^{(t)^H}\textbf{H}_{r,1}^{(t)}\bar{\textbf{F}}^{(t)}\textbf{H}_{2,r}^{(t-1) }
\textbf{H}_{2,r}^{(t-1)^H}\bar{\textbf{F}}^{(t)^H}\textbf{H}_{r,1}^{(t)^H}\textbf{R}_1^{(t) }
(\te... | ^{(t)}\tedtbf{R}_2^{(t)})^{-1} \}
\big]
\\
+ \mathbb{E}\big[
\texu{log}_2 \text{det} \{ \texjbd{I}_{N_s} +
p_2 \textbr{R}_1^{(t)^H}\textcf{H}_{r,1}^{(t)}\bar{\textbf{F}}^{(t)}\textbf{H}_{2,r}^{(t-1)}
\texvbf{H}_{2,e}^{(t-1)^H}\bae{\textbf{F}}^{(t)^H}\textbf{H}_{r,1}^{(t)^H}\tebtbf{R}_1^{(t)}
(\texnbf{R}_1^{(t)^H}\texrb... | ^{(t)}\textbf{R}_2^{(t)})^{-1} \} \big] \\ + \mathbb{E}\big[ \text{log}_2 \textbf{I}_{N_s} p_2 \textbf{R}_1^{(t)^H}\textbf{H}_{r,1}^{(t)}\bar{\textbf{F}}^{(t)}\textbf{H}_{2,r}^{(t-1)} (\textbf{R}_1^{(t)^H}\textbf{A}_1^{(t)}\textbf{R}_1^{(t)})^{-1} \} \big] + + \alpha^{(1)^{-2}}\sigma_{n,l}^2\textbf{I}_{N_s}$ and \textb... | ^{(t)}\textbf{R}_2^{(t)})^{-1} \}
\big]
\\
+ \mathbb{E}\big[
\texT{log}_2 \text{deT} \{ \textBf{I}_{n_s} +
p_2 \TeXtbf{r}_1^{(t)^H}\tExtbf{H}_{r,1}^{(t)}\bar{\texTBf{F}}^{(t)}\Textbf{H}_{2,r}^{(t-1)}
\textbf{H}_{2,r}^{(t-1)^H}\bar{\TextbF{F}}^{(T)^h}\texTBf{h}_{r,1}^{(t)^H}\tExtbf{R}_1^{(t)}
(\TExTBF{R}_1^{(t)^h}\tExTb... | ^{(t)}\textbf{R}_2^{(t)})^ {-1} \}
\b ig]
\ \
+ \m at hbb{ E}\b ig[
\text{log} _ 2 \t ext{det} \{ \textbf{I} _{N_s }+
p_2 \t extbf {R}_1^{ ( t) ^ H }\t ex tb f{H }_ { r, 1}^{( t)} \bar{\t extbf{F}}^ {(t )} \textbf{H}_{ 2 ,r }^{(t-1)}\te xtbf{H}_{2,r }^{ (t-1)^ H} \ba r {\tex tbf {F}}^ {(t)^H } \textb f{H}_{r,1 }^ { (... | ^{(t)}\textbf{R}_2^{(t)})^{-1} \}
\big]
\\
+_\mathbb{E}\big[
\text{log}_2 \text{det}_\{ \textbf{I}_{N_s} +
p_2 \textbf{R}_1^{(t)^H}\textbf{H}_{r,1}^{(t)}\bar{\textbf{F}}^{(t)}\textbf{H}_{2,r}^{(t-1)}
\textbf{H}_{2,r}^{(t-1)^H}\bar{\textbf{F}}^{(t)^H}\textbf{H}_{r,1}^{(t)^H}\textbf{R}_1^{(t)}
(\textbf{R}_1^{(t)^H}\textb... |
users. Moreover, users can add other users as trustworthy users if they like their reviews. The Epinions dataset is provided by [@tang-etal12b], and CiaoDVD is provided by [@guo2014etaf]. The statistics of the two datasets are shown in Table \[tb-dataset-stat\]. Note that although the datasets include other SIs such a... | users. Moreover, users can add early drug user as trustworthy users if they like their follow-up. The Epinions dataset is provide by [ @tang - etal12b ], and CiaoDVD is provided by [ @guo2014etaf ]. The statistics of the two datasets are show in Table \[tb - dataset - stat\ ]. notice that although the datasets in... | usfrs. Moreover, users can aad other users cw trusvworthy users iw they like their reviews. Thx Epunionw dataset is provided cy [@tang-etap12b], and CuaoDTD is provided bb [@fuo2014etaf]. The sfwtiscirs of the two dstasets ara shown in Tabne \[gb-bataset-stat\]. Note that although the dwtasets ijclude other SYs slcr a... | users. Moreover, users can add other users users they like reviews. The Epinions and is provided by The statistics of two datasets are shown in Table Note that although the datasets include other SIs such as the categories and of items, the only relation that exists between nodes of same type is relation. in experiment... | users. Moreover, users can add oTher users aS trusTwoRthY uSers If thEy like their revIEws. THe Epinions dataset is proVided By [@TAng-eTAl12B], and CIaoDVD iS PrOVIdeD bY [@gUo2014eTaF]. thE statIstIcs of thE two dataseTs aRe Shown in Table \[TB-dAtaset-stat\]. notE that althougH thE datasEtS inCLude oTheR SIs sUch a... | users. Moreover, users ca n add othe r use rsastr ustw orth y users if the y lik e their reviews. The E pinio ns data s et is p rovided by [ @ta ng -e tal 12 b ], andCia oDVD is providedby[@ guo2014etaf] . T he statist ics of the twodat asetsar e s h own i n T able\[tb-d a taset- stat\]. N ot e thata lthough t ... | users._Moreover, users_can add other users_as trustworthy_users_if they_like_their reviews. The_Epinions dataset is_provided by [@tang-etal12b], and CiaoDVD_is provided by [@guo2014etaf]._The_statistics of the two datasets are shown in Table \[tb-dataset-stat\]. Note that although the datasets_include_other SIs_such_a... |
where $d$ is the dimension of the underlying Hilbert space in which $\rho$ acts, and the reduced density matrix of a given subsystem $\alpha_k$ is obtained by tracing out all the other subsystems $\rho^{\alpha_j} = \mbox{Tr}_{\{\alpha_k \} \neq \alpha_j} [\rho]$. In the above problem, the subsystems considered corresp... | where $ d$ is the dimension of the underlying Hilbert space in which $ \rho$ acts, and the dilute concentration matrix of a given subsystem $ \alpha_k$ is obtained by trace out all the early subsystems $ \rho^{\alpha_j } = \mbox{Tr}_{\{\alpha_k \ } \neq \alpha_j } [ \rho]$. In the above problem, the subsystems consider... | whfre $d$ is the dimension on the underlying Hilberv space in whicf $\rho$ acts, and the reduced dxnsiry maugix of a given subsysgem $\alpha_n$ is obtqinev by tracing out all the other dubsvsvems $\rho^{\alpha_j} = \mbox{Tr}_{\{\alpva_k \} \neq \alpha_b} [\fhl]$. In the above problem, the subsysteis consodfred corresp... | where $d$ is the dimension of the space which $\rho$ and the reduced subsystem is obtained by out all the subsystems $\rho^{\alpha_j} = \mbox{Tr}_{\{\alpha_k \} \neq [\rho]$. In the above problem, the subsystems considered correspond to spin and parity, and $P$, for particles $A$ and $B$, i.e. $\{\alpha_k\} \equiv \{(S... | where $d$ is the dimension of the Underlying hilbeRt sPacE iN whiCh $\rhO$ acts, and the redUCed dEnsity matrix of a given suBsystEm $\ALpha_K$ Is ObtaiNed by trACiNG Out AlL tHe oThER sUbsysTemS $\rho^{\alpHa_j} = \mbox{Tr}_{\{\aLphA_k \} \Neq \alpha_j} [\rho]$. iN tHe above proBleM, the subsysteMs cOnsideReD coRResp... | where $d$ is the dimensio n of the u nderl yin g H il bert spa ce in which $\ r ho$acts, and the reduceddensi ty matr i xof agiven s u bs y s tem $ \a lph a_ k $is ob tai ned bytracing ou t a ll the other s u bs ystems $\r ho^ {\alpha_j} = \m box{Tr }_ {\{ \ alpha _k\} \n eq \al p ha_j}[\rho]$.In the ab o ve prob ... | where_$d$ is_the dimension of the_underlying Hilbert_space_in which_$\rho$_acts, and the_reduced density matrix_of a given subsystem_$\alpha_k$ is obtained_by_tracing out all the other subsystems $\rho^{\alpha_j} = \mbox{Tr}_{\{\alpha_k \} \neq \alpha_j} [\rho]$. In_the_above problem,_the_subsystems_considered corresp... |
ovae II. The selection effect and the frequencies per unit blue luminosity. A&A 273,383
Colgate S.A., McKee C. (1969) Early Supernova Luminosity. ApJ 157, 623
Della Valle M., Kissler-Patig M., Danziger J., Storm J. (1998) Globular cluster calibration of the peak brightness of the type Ia supernova 1992A and the value... | ovae II. The selection effect and the frequencies per unit aristocratic luminosity. A&A 273,383
Colgate S.A., McKee C. (1969) Early Supernova Luminosity. ApJ 157, 623
Della Valle M., Kissler - Patig M., Danziger J., Storm J. (1998) Globular bunch calibration of the peak brightness of the character Ia supernov... | ovaf II. The selection effecu and the frequeneues pec unit glue lumknosity. A&A 273,383
Colgate S.A., McKee C. (1969) Warly Supernova Luminosity. XpJ 157, 623
Dellw Valle M., Kiswoer-Patig M., Danzigev J., Sflrm L. (1998) Globular clusjer calibrathon of the pean crnghtness of the type Ia supernova 1992A wnd the vwlue... | ovae II. The selection effect and the unit luminosity. A&A Colgate S.A., McKee ApJ 623 Della Valle Kissler-Patig M., Danziger Storm J. (1998) Globular cluster calibration the peak brightness of the type Ia supernova 1992A and the value of MNRAS 299,267 Drenkhahn G., Richtler T. (1999) SN1994D in NGC4526: a normally bri... | ovae II. The selection effect aNd the frequEncieS peR unIt Blue LumiNosity. A&A 273,383
ColgatE s.A., Mckee C. (1969) Early Supernova LumiNositY. APj 157, 623
DelLA VAlle M., kissler-pAtIG m., DaNzIgEr J., stORm j. (1998) GlobUlaR clusteR calibratiOn oF tHe peak brightNEsS of the type ia sUpernova 1992A and The Value... | ovae II. The selection eff ect and th e fre que nci es per uni t blue luminos i ty.A&A 273,383
Colgate S .A.,Mc K ee C . ( 1969) EarlyS up e r nov aLu min os i ty . ApJ 15 7, 623
Della Val leM. , Kissler-Pa t ig M., Danzi ger J., Storm J . ( 1998)Gl obu l ar cl ust er ca librat i on ofthe peakbr i ghtnes s of ... | ovae II._The selection_effect and the frequencies_per unit_blue_luminosity. A&A 273,383
Colgate_S.A.,_McKee C. (1969)_Early Supernova Luminosity._ApJ 157, 623
Della Valle M.,_Kissler-Patig M., Danziger_J.,_Storm J. (1998) Globular cluster calibration of the peak brightness of the type Ia_supernova_1992A and_the_value... |
mathrm{0} + \frac{3}{2} t^\frac{2}{3}$. If we choose, as we did in the case of the Mobius strip example, to enforce continuity between points of the space-time manifold covered by different choices of tetrad, and further to conserve the momentum of the photon as measured in this way (respecting the signed volume of spa... | mathrm{0 } + \frac{3}{2 } t^\frac{2}{3}$. If we choose, as we did in the case of the Mobius strip example, to enforce continuity between period of the quad - time manifold covered by unlike choices of tetrad, and further to conserve the momentum of the photon as measure in this way (respecting the sign volume of space)... | matjrm{0} + \frac{3}{2} t^\frac{2}{3}$. If we chuose, as we did nb the rase of the Mobkus strip example, to enforce cintinyity between points of the spacv-time manufolv covered by difhsrent cmjicea of cevrad, and furthet to conserva the momentum ow che photon as measured in this way (rqspectimg the signed vojume jf sla... | mathrm{0} + \frac{3}{2} t^\frac{2}{3}$. If we choose, did the case the Mobius strip points the space-time manifold by different choices tetrad, and further to conserve the of the photon as measured in this way (respecting the signed volume of then there is a unique geodesic on the other side of the singularity connects... | mathrm{0} + \frac{3}{2} t^\frac{2}{3}$. If we choose, As we did in tHe casE of The moBius StriP example, to enfoRCe coNtinuity between points oF the sPaCE-timE MaNifolD covereD By DIFfeReNt ChoIcES oF tetrAd, aNd furthEr to conserVe tHe Momentum of thE PhOton as measUreD in this way (reSpeCting tHe SigNEd volUme Of spa... | mathrm{0} + \frac{3}{2} t^ \frac{2}{3 }$. I f w e c ho ose, aswe did in thec aseof the Mobius strip ex ample ,t o en f or ce co ntinuit y b e t wee npo int so fthe s pac e-timemanifold c ove re d by differe n tchoices of te trad, and fu rth er toco nse r ve th e m oment um oft he pho ton as me as u red in this wa y ... | mathrm{0} +_\frac{3}{2} t^\frac{2}{3}$._If we choose, as_we did_in_the case_of_the Mobius strip_example, to enforce_continuity between points of_the space-time manifold_covered_by different choices of tetrad, and further to conserve the momentum of the photon_as_measured in_this_way_(respecting the signed volume of_spa... |
[@gordon:salmon:smith:1993], where particles are propagated from the model transition, more sophisticated filters can readily be used in the CPF procedure. For instance, performance gains can be obtained with auxiliary particle filters [@pitt1999filtering; @johansen2008note], as illustrated in Section \[sec:numerics:h... | [ @gordon: salmon: smith:1993 ], where particles are propagated from the model transition, more advanced filter can readily be used in the CPF operation. For example, performance gains can be obtain with auxiliary particle filter [ @pitt1999filtering; @johansen2008note ], as illustrated in Section \[sec: numerics: hidd... | [@gogdon:salmon:smith:1993], where pavticles are propctated hrom ths model gransition, more sophisticatev fioters can readily be used iv the CPF proceduee. Fie instance, performance gzlns ccn be obtained wlth auxiliasy particle fintdrd [@pitt1999filtering; @johansen2008note], as illtstratec ln Section \[sec:gumegiss:h... | [@gordon:salmon:smith:1993], where particles are propagated from the more filters can be used in performance can be obtained auxiliary particle filters @johansen2008note], as illustrated in Section \[sec:numerics:hiddenar\]. presenting algorithms we focus on bootstrap particle filters for simplicity. When the transitio... | [@gordon:salmon:smith:1993], where parTicles are pRopagAteD frOm The mOdel Transition, more SOphiSticated filters can readIly be UsED in tHE CpF proCedure. FOR iNSTanCe, PeRfoRmANcE gainS caN be obtaIned with auXilIaRy particle fiLTeRs [@pitt1999filtEriNg; @johansen2008noTe], aS illusTrAteD In SecTioN \[sec:nUmericS:H... | [@gordon:salmon:smith:199 3], whereparti cle s a re pro paga ted from the m o deltransition, more sophi stica te d fil t er s can readil y b e use din th eC PF proc edu re. For instance, pe rf ormance gain s c an be obta ine d with auxil iar y part ic lef ilter s [ @pitt 1999fi l tering ; @johans en 2 008not ... | [@gordon:salmon:smith:1993],_where particles_are propagated from the_model transition,_more_sophisticated filters_can_readily be used_in the CPF_procedure. For instance, performance_gains can be_obtained_with auxiliary particle filters [@pitt1999filtering; @johansen2008note], as illustrated in Section \[sec:numerics:h... |
}$$ is the free-field part of the $\varphi$ hamiltonian, plus counterterms (some of which involve the $\chi$ field) that are necessary to remove infinities in this composite operator.
Eq. (\[n2\]) is a highly plausible definition of the heavy particle number density (at low temperatures). However, it does not correspo... | } $ $ is the free - field part of the $ \varphi$ hamiltonian, plus counterterms (some of which involve the $ \chi$ sphere) that are necessary to absent infinities in this composite operator.
Eq. (\[n2\ ]) is a highly plausible definition of the heavy atom number density (at low temperature). However, it does not c... | }$$ is the free-field part of tme $\varphi$ hamiltonian, pnus cohnterteros (some of which involve the $\cyi$ fitjd) that are necessxry to relove infunitmes in this composite opcxator.
Se. (\[n2\]) iv a highly plauxible defitition of the vexvv particle number density (at low tem[eraturrs). However, it dogs nou cjrreaio... | }$$ is the free-field part of the plus (some of involve the $\chi$ remove in this composite Eq. (\[n2\]) is highly plausible definition of the heavy number density (at low temperatures). However, it does not correspond in any obvious to how this number density would be determined experimentally. Standard methods all in... | }$$ is the free-field part of the $\vaRphi$ hamiltOnian, PluS coUnTertErms (Some of which invOLve tHe $\chi$ field) that are necesSary tO rEMove INfInitiEs in thiS CoMPOsiTe OpEraToR.
eq. (\[N2\]) is a hIghLy plausIble definiTioN oF the heavy parTIcLe number deNsiTy (at low tempeRatUres). HoWeVer, IT does Not CorreSpo... | }$$ is the free-field part of the $\ varph i$ham il toni an,plus counterte r ms ( some of which involvethe $ \c h i$ f i el d) th at aren ec e s sar yto re mo v einfin iti es in t his compos ite o perator.
Eq . ( \[n2\]) is ahighly plaus ibl e defi ni tio n of t heheavy parti c le num ber densi ty (at lo w tem... | }$$ is_the free-field_part of the $\varphi$_hamiltonian, plus_counterterms_(some of_which_involve the $\chi$_field) that are_necessary to remove infinities_in this composite_operator.
Eq. (\[n2\])_is a highly plausible definition of the heavy particle number density (at low temperatures)._However,_it does_not_correspo... |
in the upper-half complex plane. The Unruh modes annihilate the Minkowski vacuum state $$\begin{aligned}
\hat{c}_{m\omega} | 0_M \rangle = \hat{d}_{m\omega} | 0_M \rangle = 0\end{aligned}$$ as noted above.
Circuit model {#circuit}
=============
General formalism
-----------------
How are the states of a quantum fie... | in the upper - half complex plane. The Unruh modes annihilate the Minkowski void department of state $ $ \begin{aligned }
\hat{c}_{m\omega } | 0_M \rangle = \hat{d}_{m\omega } | 0_M \rangle = 0\end{aligned}$$ as noted above.
Circuit exemplar { # circumference }
= = = = = = = = = = = = =
General formalism
--... | in the upper-half complex puane. The Unruh modes ainihilafe the Mknkowski vacuum state $$\begin{apitned}
\hqt{c}_{m\omega} | 0_M \rangle = \hxt{d}_{m\omega} | 0_M \rangoe = 0\tnd{aligned}$$ as notxs above.
Gnrcuif modzl {#circuit}
=============
Generak formalisk
-----------------
How are the sdaged of a quantum fie... | in the upper-half complex plane. The Unruh the vacuum state \hat{c}_{m\omega} | 0_M \rangle 0\end{aligned}$$ as noted Circuit model {#circuit} General formalism ----------------- How are the of a quantum field affected by an object (such as a beamsplitter) that uniformly accelerated in the $R$ wedge? This is the questi... | in the upper-half complex planE. The Unruh mOdes aNniHilAtE the minkOwski vacuum staTE $$\begIn{aligned}
\hat{c}_{m\omega} | 0_M \raNgle = \hAt{D}_{M\omeGA} | 0_M \RanglE = 0\end{aliGNeD}$$ AS noTeD aBovE.
CIRcUit moDel {#Circuit}
=============
general forMalIsM
-----------------
How are the stATeS of a quantuM fiE... | in the upper-half complex plane. Th e Unr uhmod es ann ihil ate the Minkow s ki v acuum state $$\begin{a ligne d} \hat { c} _{m\o mega} | 0_ M \ra ng le =\h a t{ d}_{m \om ega} |0_M \rangl e = 0 \end{aligned } $$ as notedabo ve.
Circuit mo del {# ci rcu i t}
== === ===== ===
G e neralformalism
- - ------ - --... | in_the upper-half_complex plane. The Unruh_modes annihilate_the_Minkowski vacuum_state_$$\begin{aligned}
\hat{c}_{m\omega} | 0_M_\rangle = \hat{d}_{m\omega}_| 0_M \rangle =_0\end{aligned}$$ as noted_above.
Circuit_model {#circuit}
=============
General formalism
-----------------
How are the states of a quantum fie... |
focal plane by dividing the images by a master flat-field image. We adopted the flat-fielding method that was used by the Kilo Degree Survey (KiDS, [@deJong2015]): the master flat-field is achieved by first median-combining and normalizing 8 dome flat-fields and 8 twilight flat-fields, and then multiplying the average... | focal plane by dividing the images by a master bland - sphere image. We adopted the flat - fielding method acting that was used by the Kilo Degree Survey (KiDS, [ @deJong2015 ] ): the master flat - plain is achieved by first median - combining and normalizing 8 attic bland - fields and 8 twilight bland - fields, and th... | fofal plane by dividing tht images by a masjee flat-hield ijage. We xdopted the flat-fielding metiod rhat qas used by the Kilo Ddgree Surney (KiDS, [@eeJoig2015]): the master flef-field lf acgleved uy first median-gombining atd normalizing 8 aole flat-fields and 8 twilight flat-fiejds, and tjen multiplyind tht aderafv... | focal plane by dividing the images by flat-field We adopted flat-fielding method that Degree (KiDS, [@deJong2015]): the flat-field is achieved first median-combining and normalizing 8 dome and 8 twilight flat-fields, and then multiplying the averaged flat-fields with each other. spatial Fourier modes are corrected usin... | focal plane by dividing the imAges by a masTer flAt-fIelD iMage. we adOpted the flat-fiELdinG method that was used by thE Kilo deGRee SURvEy (KiDs, [@deJong2015]): THe MASteR fLaT-fiElD Is AchieVed By first Median-combIniNg And normaliziNG 8 dOme flat-fieLds And 8 twilight fLat-Fields, AnD thEN multIplYing tHe averAGe... | focal plane by dividing t he imagesby amas ter f lat- fiel d image. We ad o pted the flat-fielding met hod t ha t was us ed by the Ki l oD e gre eSu rve y( Ki DS, [ @de Jong201 5]): the m ast er flat-fieldi sachieved b y f irst median- com bining a ndn ormal izi ng 8dome f l at-fie lds and 8 t w ilight flat-fi ... | focal_plane by_dividing the images by_a master_flat-field_image. We_adopted_the flat-fielding method_that was used_by the Kilo Degree_Survey (KiDS, [@deJong2015]):_the_master flat-field is achieved by first median-combining and normalizing 8 dome flat-fields and 8_twilight_flat-fields, and_then_multiplying_the average... |
the unit Euclidean norm eigenvector associated with $\lambda_{n,j}$. Then the empirical probability transition matrix $P_n$ has the decomposition $$\begin{aligned}
P_{n}^t = \sum_{j=0}^{n-1} \lambda_{n,j}^t\, \psi_{n,j}\,\varphi_{n,j}^T,\end{aligned}$$ where $\psi_{n,j} = D_n^{-1/2}\,\phi_{n,j}\in{\mathbb{R}}^n$ and $... | the unit Euclidean norm eigenvector associated with $ \lambda_{n, j}$. Then the empirical probability transition matrix $ P_n$ take the decay $ $ \begin{aligned }
P_{n}^t = \sum_{j=0}^{n-1 } \lambda_{n, j}^t\, \psi_{n, j}\,\varphi_{n, j}^T,\end{aligned}$$ where $ \psi_{n, j } = D_n^{-1/2}\,\phi_{n, j}\in{\mathbb{R}}^... | thf unit Euclidean norm einenvector associcred wivh $\lambsa_{n,j}$. Thev the empirical probability vranwitiob matrix $P_n$ has the dezompositiln $$\begin{qligied}
P_{n}^t = \sum_{j=0}^{n-1} \lambda_{n,j}^t\, \ifi_{n,j}\,\bwrphn_{n,o}^T,\end{aligned}$$ whgre $\psi_{n,j} = D_t^{-1/2}\,\phi_{n,j}\in{\mathbb{S}}^n$ aud $... | the unit Euclidean norm eigenvector associated with the probability transition $P_n$ has the \lambda_{n,j}^t\, where $\psi_{n,j} = and $\varphi_{n,j} = so that $\psi_{n,j}=D_n^{-1} \varphi_{n,j}$ for each In particular, $\psi_{n,j}$ has unit $L^2(\mbox{diag}(D_n))$ norm, and $\varphi_{n,j}$ has unit $L^2(\mbox{diag}(D_... | the unit Euclidean norm eigenVector assoCiateD wiTh $\lAmBda_{n,J}$. TheN the empirical pRObabIlity transition matrix $P_N$ has tHe DEcomPOsItion $$\Begin{alIGnED}
p_{n}^t = \SuM_{j=0}^{N-1} \laMbDA_{n,J}^t\, \psi_{N,j}\,\vArphi_{n,j}^t,\end{aligneD}$$ whErE $\psi_{n,j} = D_n^{-1/2}\,\phi_{n,J}\In{\Mathbb{R}}^n$ anD $... | the unit Euclidean norm e igenvector asso cia ted w ith$\la mbda_{n,j}$. T h en t he empirical probabili ty tr an s itio n m atrix $P_n$h as t hede co mpo si t io n $$\ beg in{alig ned}
P_{n} ^t=\sum_{j=0}^{ n -1 } \lambda_ {n, j}^t\, \psi_ {n, j}\,\v ar phi _ {n,j} ^T, \end{ aligne d }$$ wh ere $\psi _{ n ,j} =D ... | the_unit Euclidean_norm eigenvector associated with_$\lambda_{n,j}$. Then_the_empirical probability_transition_matrix $P_n$ has_the decomposition $$\begin{aligned}
P_{n}^t_= \sum_{j=0}^{n-1} \lambda_{n,j}^t\, \psi_{n,j}\,\varphi_{n,j}^T,\end{aligned}$$_where $\psi_{n,j} =_D_n^{-1/2}\,\phi_{n,j}\in{\mathbb{R}}^n$_and $... |
10^{-2}$ & $7.5^{+2.3}_{-1.8}$ & 2.1 &$ 80\pm 17$\
$\geq 32$ & 1,993 & $25^{+9}_{-6}$ & 7.6 & $151 \pm 17$ & $4.1\times 10^{-3}$ & $13^{+5}_{-3}$ & 4.1 & $152 \pm 19$\
------------------------------------------------------------------------
$\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ & 1.8& $132 \pm 15$ & $8.6\times 10^... | 10^{-2}$ & $ 7.5^{+2.3}_{-1.8}$ & 2.1 & $ 80\pm 17$\
$ \geq 32 $ & 1,993 & $ 25^{+9}_{-6}$ & 7.6 & $ 151 \pm 17 $ & $ 4.1\times 10^{-3}$ & $ 13^{+5}_{-3}$ & 4.1 & $ 152 \pm 19$\
------------------------------------------------------------------------
$ \geq 8 $ & 36,928 & $ 6.6^{+2.0}_{-1.5}$ & 1.8 & $ 132 \pm ... | 10^{-2}$ & $7.5^{+2.3}_{-1.8}$ & 2.1 &$ 80\pm 17$\
$\geq 32$ & 1,993 & $25^{+9}_{-6}$ & 7.6 & $151 \pm 17$ & $4.1\times 10^{-3}$ & $13^{+5}_{-3}$ & 4.1 & $152 \pm 19$\
------------------------------------------------------------------------
$\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ & 1.8& $132 \pm 15$ & $8.6\tioes 10^... | 10^{-2}$ & $7.5^{+2.3}_{-1.8}$ & 2.1 &$ 80\pm 32$ 1,993 & & 7.6 & 10^{-3}$ $13^{+5}_{-3}$ & 4.1 $152 \pm 19$\ $\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ 1.8& $132 \pm 15$ & $8.6\times 10^{-4}$ & $6.0^{+1.0}_{-0.9}$ & 0.94 & $98 9$\ Ahlers, M. 2019,, 886, L18 Al Samarai, I. for the Pierre Auger 2016, ICRC2015]{}, Bonino, e... | 10^{-2}$ & $7.5^{+2.3}_{-1.8}$ & 2.1 &$ 80\pm 17$\
$\geq 32$ & 1,993 & $25^{+9}_{-6}$ & 7.6 & $151 \pm 17$ & $4.1\times 10^{-3}$ & $13^{+5}_{-3}$ & 4.1 & $152 \pm 19$\
------------------------------------------------------------------------
$\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ & 1.8& $132 \pm 15$ & $8.6\times 10^... | 10^{-2}$ & $7.5^{+2.3}_{- 1.8}$ & 2. 1 &$80\ pm17 $\
$ \geq 32$ & 1,993 & $25^ {+9}_{-6}$ & 7.6 & $15 1 \pm 1 7 $ &$ 4. 1\tim es 10^{ - 3} $ & $ 13 ^{ +5} _{ - 3} $ & 4 .1& $152\pm 19$\
--- -- ------------ - -- ---------- --- ------------ --- ------ -- --- - ----- --- ----
$\geq 8$ & 3 6,928 & $ 6. 6 ^{+2.0 } _{-1.5... | 10^{-2}$_& $7.5^{+2.3}_{-1.8}$_& 2.1 &$ 80\pm_17$\
$\geq 32$_&_1,993 &_$25^{+9}_{-6}$_& 7.6 &_$151 \pm 17$_& $4.1\times 10^{-3}$ &_$13^{+5}_{-3}$ & 4.1_&_$152 \pm 19$\
------------------------------------------------------------------------
$\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ & 1.8& $132 \pm 15$ &_$8.6\times_10^... |
E^s\oplus E^c\oplus E^u$, with non trivial extremal sub-bundles $E^s$ and $E^u$, and there exists $n\in {\mathbb{N}}$ such that $E^s$ and $E^u$ are uniformly contracted by $Df^n$ and $Df^{-n}$, respectively.
If the center bundle $E^c$ is trivial, then $f$ is called [*Anosov*]{}. For convenience, given a partially hype... | E^s\oplus E^c\oplus E^u$, with non trivial extremal sub - bundles $ E^s$ and $ E^u$, and there exists $ n\in { \mathbb{N}}$ such that $ E^s$ and $ E^u$ are uniformly compress by $ Df^n$ and $ Df^{-n}$, respectively.
If the plaza bundle $ E^c$ is trivial, then $ f$ is called [ * Anosov * ] { }. For appliance, given a... | E^s\oolus E^c\oplus E^u$, with non trivial extremco sub-bnndles $S^s$ and $E^j$, and there exists $n\in {\mathbu{N}}$ sych tyat $E^s$ and $E^u$ are unifurmly connracted bt $Df^i$ and $Df^{-n}$, respecvjvely.
If the cskter yuidle $E^c$ is trivlal, then $f$ hs called [*Anosmv*]{}. Flr convenience, given a partially hy[e... | E^s\oplus E^c\oplus E^u$, with non trivial extremal and and there $n\in {\mathbb{N}}$ such uniformly by $Df^n$ and respectively. If the bundle $E^c$ is trivial, then $f$ called [*Anosov*]{}. For convenience, given a partially hyperbolic diffeomorphism $f$, we consider its hyperbolic splitting $TM=E^s\oplus E^c\oplus E^... | E^s\oplus E^c\oplus E^u$, with non trIvial extreMal suB-buNdlEs $e^s$ anD $E^u$, aNd there exists $n\IN {\matHbb{N}}$ such that $E^s$ and $E^u$ are UnifoRmLY conTRaCted bY $Df^n$ and $dF^{-n}$, RESpeCtIvEly.
if THe CenteR buNdle $E^c$ iS trivial, thEn $f$ Is Called [*Anosov*]{}. fOr ConveniencE, giVen a partiallY hyPe... | E^s\oplus E^c\oplus E^u$,with non t rivia l e xtr em al s ub-b undles $E^s$ a n d $E ^u$, and there exists$n\in { \ math b b{ N}}$such th a t$ E ^s$ a nd $E ^u $ a re un ifo rmly co ntracted b y $ Df ^n$ and $Df^ { -n }$, respec tiv ely.
If the ce nter b un dle $E^c$ is triv ial, t h en $f$ is calle d[ *Anoso ... | E^s\oplus E^c\oplus_E^u$, with_non trivial extremal sub-bundles_$E^s$ and_$E^u$,_and there_exists_$n\in {\mathbb{N}}$ such_that $E^s$ and_$E^u$ are uniformly contracted_by $Df^n$ and_$Df^{-n}$,_respectively.
If the center bundle $E^c$ is trivial, then $f$ is called [*Anosov*]{}. For convenience,_given_a partially_hype... |
\sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where each $a_{n_i},\ 1 \leq i\leq
d_r,$ satisfies the following system of non-linear recurrence relations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} = a_{(n-1)_k}, {\hspace*{2em}}1\leq k\leq d_r \Bigg\}$$ with initial values $a_{A_k},\ 1\leq k\leq d_r,$ as obtained in (ii).
\... | \sum_{i=1}^{d_r}a_{n_i}x^{d_r - i}\right)$$ where each $ a_{n_i},\ 1 \leq i\leq
d_r,$ satisfies the following system of non - linear recurrence relative $ $ \Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j } = a_{(n-1)_k }, { \hspace*{2em}}1\leq k\leq d_r \Bigg\}$$ with initial value $ a_{A_k},\ 1\leq k\leq d_r,$ as obtained ... | \sum_{l=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where eagh $a_{n_i},\ 1 \leq i\leq
b_e,$ sativfies fhe folluwing system of non-linear rerurrwnce eelations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{v_i}a_{n_j} = a_{(n-1)_n}, {\hspace*{2wm}}1\lew k\leq d_r \Bmfg\}$$ with initizp vannes $a_{A_k},\ 1\leq k\lea d_r,$ as obdained in (ii).
\... | \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where each $a_{n_i},\ 1 \leq i\leq the system of recurrence relations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} \Bigg\}$$ initial values $a_{A_k},\ k\leq d_r,$ as in (ii). \(iv) For $n > the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr... | \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where eaCh $a_{n_i},\ 1 \leq i\lEq
d_r,$ sAtiSfiEs The fOlloWing system of noN-LineAr recurrence relations $$\BIgg\{\suM_{i+J=2K}(-1)^ja_{n_I}A_{n_J} = a_{(n-1)_k}, {\hSpace*{2em}}1\LEq K\LEq d_R \BIgG\}$$ wiTh INiTial vAluEs $a_{A_k},\ 1\leQ k\leq d_r,$ as oBtaInEd in (ii).
\... | \sum_{i=1}^{d_r}a_{n_i}x^{ d_r-i}\rig ht)$$ wh ere e ach$a_{ n_i},\ 1 \leqi \leq
d_r,$ satisfies the f ollow in g sys t em of n on-line a rr e cur re nc e r el a ti ons $ $\B igg\{\s um_{i+j=2k }(- 1) ^ja_{n_i}a_{ n _j } = a_{(n- 1)_ k}, {\hspace *{2 em}}1\ le q k \ leq d _r\Bigg \}$$ w i th ini tial valu es $a_{A... | \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where_each $a_{n_i},\_1 \leq i\leq
d_r,$ satisfies_the following_system_of non-linear_recurrence_relations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} =_a_{(n-1)_k}, {\hspace*{2em}}1\leq k\leq_d_r \Bigg\}$$ with initial_values $a_{A_k},\ 1\leq_k\leq_d_r,$ as obtained in (ii).
\... |
partition of $\Omega$, are two strong factors which hinder us from linking $f(\alpha)=d_H(\Omega_{\alpha(\beta)})$ with $d_H(G_{\beta})$ directly through, say, so simple a way as an equality. Still, we know:
1. For $\beta=\infty$ we have $\alpha(\beta)=\alpha_{\min}$.
2. For $\beta=2$ we have $\alpha(\beta)=\alpha_... | partition of $ \Omega$, are two strong factors which hinder us from connect $ f(\alpha)=d_H(\Omega_{\alpha(\beta)})$ with $ d_H(G_{\beta})$ immediately through, say, so simple a direction as an equality. even, we know:
1. For $ \beta=\infty$ we have $ \alpha(\beta)=\alpha_{\min}$.
2. For $ \beta=2 $ we hold $... | parhition of $\Omega$, are two rtrong factors cyich hmnder ua from lknking $f(\alpha)=d_H(\Omega_{\alpha(\bete)})$ wirh $d_H(T_{\beta})$ directly through, say, so spmple a wqy aw an equalivg. Still, we knka:
1. Fmc $\beta=\infty$ we mave $\alpha(\bata)=\alpha_{\min}$.
2. Fmr $\bzta=2$ we have $\alpha(\beta)=\alpha_... | partition of $\Omega$, are two strong factors us linking $f(\alpha)=d_H(\Omega_{\alpha(\beta)})$ $d_H(G_{\beta})$ directly through, as equality. Still, we 1. For $\beta=\infty$ have $\alpha(\beta)=\alpha_{\min}$. 2. For $\beta=2$ we $\alpha(\beta)=\alpha_{\max}$. 3. $f(\alpha_{\min})=0$. 4. $f(\alpha)$ is increasing fo... | partition of $\Omega$, are two strOng factors Which HinDer Us From LinkIng $f(\alpha)=d_H(\OmeGA_{\alpHa(\beta)})$ with $d_H(G_{\beta})$ direcTly thRoUGh, saY, So SimplE a way as AN eQUAliTy. stIll, We KNoW:
1. For $\bEta=\Infty$ we Have $\alpha(\bEta)=\AlPha_{\min}$.
2. For $\betA=2$ We Have $\alpha(\bEta)=\Alpha_... | partition of $\Omega$, are two stron g fac tor s w hi ch h inde r us from link i ng $ f(\alpha)=d_H(\Omega_{ \alph a( \ beta ) }) $ wit h $d_H( G _{ \ b eta }) $dir ec t ly thro ugh , say,so simplea w ay as an equal i ty . Still, w e k now:
1. Fo r $ \beta= \i nft y $ wehav e $\a lpha(\ b eta)=\ alpha_{\m in } $.
... | partition of_$\Omega$, are_two strong factors which_hinder us_from_linking $f(\alpha)=d_H(\Omega_{\alpha(\beta)})$_with_$d_H(G_{\beta})$ directly through,_say, so simple_a way as an_equality. Still, we_know:
1._ For $\beta=\infty$ we have $\alpha(\beta)=\alpha_{\min}$.
2. For $\beta=2$ we have $\alpha(\beta)=\alpha_... |
u_1-u_2|^\nu |\Theta_n^*(u)|du \\
& \le c \,n^{4r-\nu -1}
\int_{0}^{1/n} \left| \frac{\sin\frac{(n+1)u_2 \pi}{2}} {\sin \frac{u_2\pi}{2}} \right|^{2r}
du \le c \, n^{6 r - \nu -2}.\end{aligned}$$ upon using $|\sin n u / \sin u| \le n$. The other two cases can be handled similarly. As a result, we conclu... | u_1 - u_2|^\nu |\Theta_n^*(u)|du \\
& \le c \,n^{4r-\nu -1 }
\int_{0}^{1 / n } \left| \frac{\sin\frac{(n+1)u_2 \pi}{2 } } { \sin \frac{u_2\pi}{2 } } \right|^{2r }
du \le c \, n^{6 r - \nu -2}.\end{aligned}$$ upon using $ |\sin n u / \sin u| \le n$. The other two case can be cover similarly. As a ... | u_1-u_2|^\nk |\Theta_n^*(u)|du \\
& \le c \,n^{4v-\nu -1}
\int_{0}^{1/n} \left| \frac{\sii\frac{(n+1)u_2 \pi}{2}} {\sin \wrac{u_2\pi}{2}} \right|^{2r}
du \le r \, n^{6 r - \ny -2}.\end{aligned}$$ upon usine $|\sin n u / \sin u| \oe n$. Rhe other vso cases can gc hanblxd similarly. As a result, fe conclu... | u_1-u_2|^\nu |\Theta_n^*(u)|du \\ & \le c \,n^{4r-\nu \left| \pi}{2}} {\sin \right|^{2r} du \le \nu upon using $|\sin u / \sin \le n$. The other two cases be handled similarly. As a result, we conclude that $I_n^{r,p} \le c n^{6 - p -2}$. The case $p = 0$ gives the lower bound estimate The estimate over quantity $\lamb... | u_1-u_2|^\nu |\Theta_n^*(u)|du \\
& \le c \,n^{4r-\nu -1}
\int_{0}^{1/n} \lEft| \frac{\sin\Frac{(n+1)U_2 \pi}{2}} {\Sin \FrAc{u_2\pI}{2}} \rigHt|^{2r}
du \le c \, n^{6 r - \nu -2}.\enD{AligNed}$$ upon using $|\sin n u / \sin u| \lE n$. The OtHEr twO CaSes caN be handLEd SIMilArLy. as a ReSUlT, we coNclU... | u_1-u_2|^\nu |\Theta_n^*(u )|du \\
&\le c\, n^{4 r-\n u -1}
\int_ { 0}^{ 1/n} \left| \frac{\sin \frac {( n +1)u _ 2\pi}{ 2}} {\s i n\ f rac {u _2 \pi }{ 2 }} \rig ht| ^{2r}
du \le c \, n^{6 r - \n u -2}.\end {al igned}$$ upo n u sing $ |\ sin n u / \s in u| \le n $ . Theother two c a ses ca n be ha... | u_1-u_2|^\nu |\Theta_n^*(u)|du_\\
_ &_\le c_\,n^{4r-\nu_-1}
__\int_{0}^{1/n} \left| \frac{\sin\frac{(n+1)u_2_\pi}{2}} {\sin \frac{u_2\pi}{2}}_\right|^{2r}
_ du__ \le c \, n^{6 r - \nu -2}.\end{aligned}$$ upon using $|\sin n u_/_\sin u|_\le_n$._The other two cases can_be handled similarly. As a_result, we_conclu... |
[@Sayed2013intr] for definition and properties of the norm): $$\label{eq:spr}
\begin{split}
\rho \left({\boldsymbol{A}}_I^\top \left[{\boldsymbol{I}}_{LN} - \mu\, ({\boldsymbol{H}}_R + \eta \, {\boldsymbol{Q}})\right]\right) \leq \|{\boldsymbol{A}}_I^\top \left({\boldsymbol{I}}_{LN} - \mu\, ({\boldsymbol{H}}_R ... | [ @Sayed2013intr ] for definition and properties of the norm ): $ $ \label{eq: spr }
\begin{split }
\rho \left({\boldsymbol{A}}_I^\top \left[{\boldsymbol{I}}_{LN } - \mu\, ({ \boldsymbol{H}}_R + \eta \, { \boldsymbol{Q}})\right]\right) \leq \|{\boldsymbol{A}}_I^\top \left({\boldsymbol{I}}_{LN } - \mu\, ({ \... | [@Sayfd2013intr] for definition ana properties of the nocm): $$\labem{eq:spr}
\begin{split}
\rho \left({\bolddynbol{A}}_U^\top \left[{\boldsymbol{I}}_{LN} - \mu\, ({\bolddymbol{H}}_R + \ete \, {\boldsymbol{Q}})\rijgt]\right) \leq \|{\bkpdsykuol{A}}_I^\top \left({\bokdsymbol{I}}_{LT} - \mu\, ({\boldsymbml{F}}_R ... | [@Sayed2013intr] for definition and properties of the \begin{split} \left({\boldsymbol{A}}_I^\top \left[{\boldsymbol{I}}_{LN} \mu\, ({\boldsymbol{H}}_R + \left({\boldsymbol{I}}_{LN} \mu\, ({\boldsymbol{H}}_R + \, {\boldsymbol{Q}})\right)\|_{b,\infty} \end{split}$$ using norm inequalities and the fact ${\boldsymbol{A}}$... | [@Sayed2013intr] for definition and Properties Of the NorM): $$\laBeL{eq:sPr}
\beGin{split}
\rho \lefT({\BoldSymbol{A}}_I^\top \left[{\boldsymBol{I}}_{Ln} - \mU\, ({\BoldSYmBol{H}}_R + \Eta \, {\boldSYmBOL{Q}})\rIgHt]\RigHt) \LEq \|{\BoldsYmbOl{A}}_I^\top \Left({\boldsyMboL{I}}_{lN} - \mu\, ({\boldsymbOL{H}}_r ... | [@Sayed2013intr] for defin ition andprope rti esof the nor m): $$\label{e q :spr }
\begin{split}
\rh o\ left ( {\ bolds ymbol{A } }_ I ^ \to p\l eft [{ \ bo ldsym bol {I}}_{L N} - \mu\, ({ \b oldsymbol{H} } _R + \eta \, {\ boldsymbol{Q }}) \right ]\ rig h t) \l eq\|{\b oldsym b ol{A}} _I^\top \ le f t({\bo ... | [@Sayed2013intr] for_definition and_properties of the norm):_$$\label{eq:spr}
__ \begin{split}
__ \rho_\left({\boldsymbol{A}}_I^\top \left[{\boldsymbol{I}}_{LN} -_\mu\, ({\boldsymbol{H}}_R + \eta_\, {\boldsymbol{Q}})\right]\right) \leq_\|{\boldsymbol{A}}_I^\top_\left({\boldsymbol{I}}_{LN} - \mu\, ({\boldsymbol{H}}_R ... |
= F(\sqrt{\alpha_1\alpha_2})$, so ${\mathfrak{p}}$ ramifies in $E_i$ for an even number of $i$’s. In particular, ${\mathfrak{p}}$ is not totally ramified in $E/F$.
\[ram1\] Let $K$ be a non-biquadratic quartic CM field of type $\Phi$ with the real quadratic subfield $F$ and let $K^r$ be its reflex field with the quad... | = F(\sqrt{\alpha_1\alpha_2})$, so $ { \mathfrak{p}}$ ramifies in $ E_i$ for an even number of $ i$ ’s. In particular, $ { \mathfrak{p}}$ is not wholly branch in $ E / F$.
\[ram1\ ] Let $ K$ be a non - biquadratic quartic CM field of character $ \Phi$ with the real quadratic subfield $ F$ and get $ K^r$ be its reflex... | = F(\dqrt{\alpha_1\alpha_2})$, so ${\mathfrxk{p}}$ ramifies in $E_i$ for an evsn numbef of $i$’s. In particular, ${\mathfrek{p}}$ us nou totally ramified kn $E/F$.
\[ram1\] Pet $K$ be a nib-biquadratmd quartle CM rleld mh type $\Phi$ with the real xuadratic subfheud $F$ and let $K^r$ be its reflex field wyth the qkad... | = F(\sqrt{\alpha_1\alpha_2})$, so ${\mathfrak{p}}$ ramifies in $E_i$ even of $i$’s. particular, ${\mathfrak{p}}$ is \[ram1\] $K$ be a quartic CM field type $\Phi$ with the real quadratic $F$ and let $K^r$ be its reflex field with the quadratic subfield $F^r$. the following assertions hold. - If a prime $p$ is ramified ... | = F(\sqrt{\alpha_1\alpha_2})$, so ${\mathfrak{P}}$ ramifies iN $E_i$ foR an EveN nUmbeR of $i$’S. In particular, ${\mAThfrAk{p}}$ is not totally ramifieD in $E/F$.
\[RaM1\] let $K$ BE a Non-biQuadratIC qUARtiC Cm fIelD oF TyPe $\Phi$ WitH the reaL quadratic SubFiEld $F$ and let $K^r$ BE iTs reflex fiEld With the quad... | = F(\sqrt{\alpha_1\alpha_ 2})$, so $ {\mat hfr ak{ p} }$ r amif ies in $E_i$ f o r an even number of $i$’s. In p ar t icul a r, ${\m athfrak { p} } $ is n ot to ta l ly rami fie d in $E /F$.
\[ra m1\ ]Let $K$ be a no n-biquadra tic quartic CMfie ld ofty pe$ \Phi$ wi th th e real quadra tic subfi el d $F$ a ... | =_F(\sqrt{\alpha_1\alpha_2})$, so_${\mathfrak{p}}$ ramifies in $E_i$_for an_even_number of_$i$’s._In particular, ${\mathfrak{p}}$_is not totally_ramified in $E/F$.
\[ram1\] Let_$K$ be a_non-biquadratic_quartic CM field of type $\Phi$ with the real quadratic subfield $F$ and let_$K^r$_be its_reflex_field_with the quad... |
based dynamics with integrate-and-fire dynamics where the firing of the neuron is represented by the resetting of the membrane potential whenever it crosses a threshold. Our justification for using this approximation hinges on three considerations: 1) both the conductance-based and integrate-and-fire models are Type I ... | based dynamics with integrate - and - fire moral force where the ignition of the neuron is represented by the resetting of the membrane electric potential whenever it crosses a doorsill. Our justification for using this estimate hinge on three considerations: 1) both the conductance - establish and integrate - and - fi... | basfd dynamics with integraue-and-fire dynamics where the fjring of the neuron is represented bb thw restnting of the membrane potentiap whenevwr iu crosses a thresikld. Our justirlcatimi for using thix approximdtion hinges ot ghxee considerations: 1) both the conductwnce-basrd and integrate-wnd-fprq mosvlw are Type I ... | based dynamics with integrate-and-fire dynamics where the the is represented the resetting of crosses threshold. Our justification using this approximation on three considerations: 1) both the and integrate-and-fire models are Type I (in the sense of positive PRC), 2) action potentials (spike widths) are narrow compare... | based dynamics with integratE-and-fire dyNamicS whEre ThE firIng oF the neuron is rePReseNted by the resetting of thE membRaNE potENtIal whEnever iT CrOSSes A tHrEshOlD. ouR justIfiCation fOr using thiS apPrOximation hinGEs On three conSidErations: 1) both The ConducTaNce-BAsed aNd iNtegrAte-and-FIre modEls are TypE I ... | based dynamics with integr ate-and-fi re dy nam ics w here the firing of the neur on is represented by t he re se t ting of themembran e p o t ent ia lwhe ne v er it c ros ses a t hreshold.Our j ustification fo r using th isapproximatio n h ingeson th r ee co nsi derat ions:1 ) both the cond uc t ance-b a sed ... | based dynamics_with integrate-and-fire_dynamics where the firing_of the_neuron_is represented_by_the resetting of_the membrane potential_whenever it crosses a_threshold. Our justification_for_using this approximation hinges on three considerations: 1) both the conductance-based and integrate-and-fire models_are_Type I_... |
) =0\ne \b $ that partition the $(q+1)^2$ singular points in $\Omega_0^\perp$, sending $$\label{generators}
\begin{array}{llll}
\hspace{-5.5pt}u_s\col (0,\b,{\gamma },0)\mapsto
(0,\b,{\gamma }+ \b s ,0)
\vspace{2pt}
\\
j\col (0,\b,{\gamma },0)\mapsto (0,{\gamma }, \b,0) . \hspace{185pt}
\end{array}\vspace{-1pt}$$ An... | ) = 0\ne \b $ that partition the $ (q+1)^2 $ singular points in $ \Omega_0^\perp$, sending $ $ \label{generators }
\begin{array}{llll }
\hspace{-5.5pt}u_s\col (0,\b,{\gamma }, 0)\mapsto
(0,\b,{\gamma } + \b s, 0)
\vspace{2pt }
\\
j\col (0,\b,{\gamma }, 0)\mapsto (0,{\gamma }, \b,0) . \hspace{185pt }
\e... | ) =0\ne \b $ that partition the $(q+1)^2$ singular points in $\Omxga_0^\perp$, sending $$\label{generators}
\begin{array}{llpl}
\yspact{-5.5it}u_s\col (0,\b,{\gamma },0)\mapsto
(0,\b,{\gamma }+ \b s ,0)
\vspqce{2pu}
\\
j\col (0,\b,{\gamma },0)\mapsvk (0,{\gamma }, \b,0) . \gdpacz{185pv}
\end{array}\vspace{-1kt}$$ An... | ) =0\ne \b $ that partition the points $\Omega_0^\perp$, sending \begin{array}{llll} \hspace{-5.5pt}u_s\col (0,\b,{\gamma ,0) \\ j\col (0,\b,{\gamma (0,{\gamma }, \b,0) \hspace{185pt} \end{array}\vspace{-1pt}$$ An [*ordinary*]{} point is singular point in $\Omega_0^\perp$ of the form $\< (0,\b, {\gamma },0)\>$ such tha... | ) =0\ne \b $ that partition the $(q+1)^2$ singuLar points iN $\OmegA_0^\peRp$, sEnDing $$\LabeL{generators}
\begIN{arrAy}{llll}
\hspace{-5.5pt}u_s\col (0,\b,{\gaMma },0)\maPsTO
(0,\b,{\gaMMa }+ \B s ,0)
\vspAce{2pt}
\\
j\cOL (0,\b,{\GAMma },0)\MaPsTo (0,{\gAmMA }, \b,0) . \HspacE{185pt}
\End{arraY}\vspace{-1pt}$$ AN... | ) =0\ne \b $ that partitio n the $(q+ 1)^2$ si ngu la r po ints in $\Omega_0^ \ perp $, sending $$\label{ge nerat or s }
\b e gi n{arr ay}{lll l }\ h spa ce {- 5.5 pt } u_ s\col (0 ,\b,{\g amma },0)\ map st o
(0,\b,{\g a mm a }+ \b s ,0 )
\vspace{2p t}\\
j\c ol (0 , \b,{\ gam ma }, 0)\map s to (0, {\gamma } ,\ b... | ) =0\ne_\b $_that partition the $(q+1)^2$_singular points_in_$\Omega_0^\perp$, sending_$$\label{generators}
\begin{array}{llll}
\hspace{-5.5pt}u_s\col_(0,\b,{\gamma },0)\mapsto
(0,\b,{\gamma_}+ \b_s ,0)
\vspace{2pt}
\\
j\col (0,\b,{\gamma },0)\mapsto_(0,{\gamma }, \b,0)__. \hspace{185pt}
\end{array}\vspace{-1pt}$$ An... |
size scaling of the discontinuity and its effect on the scaling exponents, which have also previously been studied using a complementary approach with the addition of small, non-zero bending rigidity [@sharma_strain-controlled_2016]. Using these modified exponents, we test scaling relations recently predicted for fiber... | size scaling of the discontinuity and its effect on the scaling exponents, which have also previously been study use a complementary approach with the addition of little, non - zero bending rigidity [ @sharma_strain - controlled_2016 ]. use these modify exponents, we test scaling relations recently predicted for charac... | sizf scaling of the discontlnuity and its eydect oi the sdaling ebponents, which have also pretiouwly btvn studied using a cooplementagy approaxh wmth the addition of small, non-zsvo beudmng rigidity [@shsrma_strain-wontrolled_2016]. Usitg tkese modified exponents, we test scalyng relstlons recently kredibtqd fkg niber... | size scaling of the discontinuity and its the exponents, which also previously been with addition of small, bending rigidity [@sharma_strain-controlled_2016]. these modified exponents, we test scaling recently predicted for fiber networks [@shivers_scaling_2019]. ![ \[fig:1\] Rigidity phase diagram of central networks.... | size scaling of the discontinUity and its EffecT on The ScAlinG expOnents, which havE Also Previously been studied uSing a CoMPlemENtAry apProach wITh THE adDiTiOn oF sMAlL, non-zEro Bending Rigidity [@shArmA_sTrain-controlLEd_2016]. using these ModIfied exponenTs, wE test sCaLinG RelatIonS receNtly prEDicted For fiber... | size scaling of the discon tinuity an d its ef fec ton t he s caling exponen t s, w hich have also previou sly b ee n stu d ie d usi ng a co m pl e m ent ar yapp ro a ch with th e addit ion of sma ll, n on-zero bend i ng rigidity[@s harma_strain -co ntroll ed _20 1 6]. U sin g the se mod i fied e xponents, w e ... | size scaling_of the_discontinuity and its effect_on the_scaling_exponents, which_have_also previously been_studied using a_complementary approach with the_addition of small,_non-zero_bending rigidity [@sharma_strain-controlled_2016]. Using these modified exponents, we test scaling relations recently predicted for_fiber... |
right) \gg Y^{\sigma -1/2}, \quad t\in \cA_2.\end{aligned}$$
We refer the reader to either [@Bou Section 1] or [@Ivic Chapter 11] for details of the following reduction from Lemma \[lem:zerodensity1\] which makes use of Heath-Brown’s twelfth power moment estimate [@HB0].
\[lem:zerodensity2\] Let $Y\le T^{A}$ be some ... | right) \gg Y^{\sigma -1/2 }, \quad t\in \cA_2.\end{aligned}$$
We refer the reader to either [ @Bou Section 1 ] or [ @Ivic Chapter 11 ] for details of the following decrease from Lemma \[lem: zerodensity1\ ] which do use of Heath - Brown ’s twelfth power here and now appraisal [ @HB0 ].
\[lem: zerodens... | rigjt) \gg Y^{\sigma -1/2}, \quad t\in \cX_2.\end{aligned}$$
We rgfwr the readed to eitfer [@Bou Section 1] or [@Ivic Chaptec 11] foe detqils of the following feduction from Lenma \[ltm:zerodensity1\] whirg makes use or Heach-Urown’s twelfth kower moment estimate [@HB0].
\[lem:sefobensity2\] Let $Y\le T^{A}$ be some ... | right) \gg Y^{\sigma -1/2}, \quad t\in \cA_2.\end{aligned}$$ the to either Section 1] or of following reduction from \[lem:zerodensity1\] which makes of Heath-Brown’s twelfth power moment estimate \[lem:zerodensity2\] Let $Y\le T^{A}$ be some parameter. There exists some $N$ satisfying $$\begin{aligned} Y^{4/3}<N<Y^{2+... | right) \gg Y^{\sigma -1/2}, \quad t\in \cA_2.\end{Aligned}$$
We rEfer tHe rEadEr To eiTher [@bou Section 1] or [@IvIC ChaPter 11] for details of the folLowinG rEDuctIOn From LEmma \[lem:ZErODEnsItY1\] wHicH mAKeS use oF HeAth-BrowN’s twelfth pOweR mOment estimatE [@hB0].
\[Lem:zerodenSitY2\] Let $Y\le T^{A}$ be sOme ... | right) \gg Y^{\sigma -1/2} , \quad t\ in \c A_2 .\e nd {ali gned }$$
We refert he r eader to either [@BouSecti on 1] o r [ @Ivic Chapte r 1 1 ] fo rde tai ls of thefol lowingreductionfro mLemma \[lem: z er odensity1\ ] w hich makes u seof Hea th -Br o wn’stwe lfthpowerm omentestimate[@ H B0].
\ [lem:ze r o de nsi... | right) \gg_Y^{\sigma -1/2},_\quad t\in \cA_2.\end{aligned}$$
We refer_the reader_to_either [@Bou Section 1]_or [@Ivic_Chapter 11] for details_of the following_reduction from Lemma \[lem:zerodensity1\] which_makes use of_Heath-Brown’s_twelfth power moment estimate [@HB0].
\[lem:zerodensity2\] Let $Y\le T^{A}$ be some ... |
84 protocol. The essence of this protocol is that the communicators can freely select the message mode and the control mode. In BB84 protocol, when Alice and Bob want to transform $n$ bit message, it need about $4n$ qubit. On the other hand, comparing with the ‘ping-pong’ protocol, we use a single qubit to realize the ... | 84 protocol. The essence of this protocol is that the communicators can freely select the message mode and the restraint mood. In BB84 protocol, when Alice and Bob want to transform $ n$ bit message, it want about $ 4n$ qubit. On the other hired hand, comparing with the ‘ ping - pong ’ protocol, we practice a individua... | 84 prltocol. The essence of thls protocol is tkqt the commuhicators can freely select the messaje mide abd the control mode. In BB84 protobol, when Qlict and Bob want to transfovi $n$ glt mevwage, it need anout $4n$ qubid. On the other hxnb, comparing with the ‘ping-pong’ protocjl, we uxe a single qubij to gewlizs the ... | 84 protocol. The essence of this protocol the can freely the message mode BB84 when Alice and want to transform bit message, it need about $4n$ On the other hand, comparing with the ‘ping-pong’ protocol, we use a single to realize the deterministic secure direct communication instead of using entanglement. Also, there ... | 84 protocol. The essence of this pRotocol is tHat thE coMmuNiCatoRs caN freely select tHE mesSage mode and the control mOde. In bB84 PRotoCOl, When ALice and bOb WANt tO tRaNsfOrM $N$ bIt mesSagE, it need About $4n$ qubiT. On ThE other hand, coMPaRing with thE ‘piNg-pong’ protocOl, wE use a sInGle QUbit tO reAlize The ... | 84 protocol. The essence o f this pro tocol is th at the com municators can free ly select the messagemodean d the co ntrol mode.I nB B 84pr ot oco l, wh en Al ice and Bo b want totra ns form $n$ bit me ssage, itnee d about $4n$ qu bit. O nthe other ha nd, c ompari n g with the ‘pin g- p ong’ p r otocol, w eu... | 84 protocol._The essence_of this protocol is_that the_communicators_can freely_select_the message mode_and the control_mode. In BB84 protocol,_when Alice and_Bob_want to transform $n$ bit message, it need about $4n$ qubit. On the other_hand,_comparing with_the_‘ping-pong’_protocol, we use a single_qubit to realize the ... |
metry\_properties\]); note that if the noise $\xi(t)$ has time-reversal symmetry, the term $\propto \omega_1\omega_2(\omega_1+\omega_2)$ in the integrand in Eq. (\[eq:2nd\_moment\_weak\_noise\]) can be disregarded.
For weak noise $\xi(t)$, the leading contribution to the variance of $u$ comes from the second-order ter... | metry\_properties\ ]); note that if the noise $ \xi(t)$ has time - transposition isotropy, the term $ \propto \omega_1\omega_2(\omega_1+\omega_2)$ in the integrand in Eq. (\[eq:2nd\_moment\_weak\_noise\ ]) can be disregarded.
For fallible randomness $ \xi(t)$, the leading contribution to the variance of $ u$ comes... | metgy\_properties\]); note that in the noise $\xi(t)$ kqs timx-reverszl symmegry, the term $\propto \omega_1\omeja_2(\omwga_1+\omtya_2)$ in the integrand kn Eq. (\[eq:2nd\_loment\_weqk\_nomse\]) can be disrejzrded.
Fov weai noivx $\xi(t)$, the leadikg contribudion to the vasixnee of $u$ comes from the second-order tqr... | metry\_properties\]); note that if the noise $\xi(t)$ symmetry, term $\propto in the integrand disregarded. weak noise $\xi(t)$, leading contribution to variance of $u$ comes from the term $\propto \Xi_2$. To reveal a nonzero third-order noise correlator, in addition to variance of $u$ one should measure the third cumu... | metry\_properties\]); note that if The noise $\xi(T)$ has tIme-RevErSal sYmmeTry, the term $\propTO \omeGa_1\omega_2(\omega_1+\omega_2)$ in the IntegRaND in EQ. (\[Eq:2Nd\_momEnt\_weak\_NOiSE\]) Can Be DiSreGaRDeD.
For wEak Noise $\xi(T)$, the leadinG coNtRibution to thE VaRiance of $u$ cOmeS from the secoNd-oRder teR... | metry\_properties\]); note that if t he no ise $\ xi (t)$ has time-reversal symm etry, the term $\propt o \om eg a _1\o m eg a_2(\ omega_1 + \o m e ga_ 2) $inth e i ntegr and in Eq. (\[eq:2nd \_m om ent\_weak\_n o is e\]) can b e d isregarded.
Fo r weak n ois e $\xi (t) $, th e lead i ng con tribution t o th... | metry\_properties\]); note_that if_the noise $\xi(t)$ has_time-reversal symmetry,_the_term $\propto_\omega_1\omega_2(\omega_1+\omega_2)$_in the integrand_in Eq. (\[eq:2nd\_moment\_weak\_noise\]) can_be disregarded.
For weak noise_$\xi(t)$, the leading_contribution_to the variance of $u$ comes from the second-order ter... |
7 $\pm$ 2 [MeV/]{}$c^2$, i.e. compatible with the detector resolution. Therefore, only an upper limit is derived: the full width of the signal is smaller than 15 at 95 % C.L. There is no natural explanation of such a small value, neither for the D$^{*'}$ nor for higher orbital excitations [@pene].
Various checks were... | 7 $ \pm$ 2 [ MeV/]{}$c^2 $, i.e. compatible with the detector resolution. Therefore, only an upper terminus ad quem is derive: the full width of the signal is smaller than 15 at 95% C.L. There be no lifelike explanation of such a small value, neither for the D$^{*'}$ nor for higher orbital excitations [ @pene ]. ... | 7 $\pl$ 2 [MeV/]{}$c^2$, i.e. compatible witm the detector rgsilutioi. Thererore, onlh an upper limit is derived: vhe dull qidth of the signal is smaller nhan 15 at 95 % C.L. Uhere is no naturem explakction lf sbci a small value, neither fmr the D$^{*'}$ nor fmr hngher orbital excitations [@pene].
Various checks wfre... | 7 $\pm$ 2 [MeV/]{}$c^2$, i.e. compatible with resolution. only an limit is derived: signal smaller than 15 95 % C.L. is no natural explanation of such small value, neither for the D$^{*'}$ nor for higher orbital excitations [@pene]. Various were performed. Varying the background shape and the kinematical cuts has no ef... | 7 $\pm$ 2 [MeV/]{}$c^2$, i.e. compatible with the Detector reSolutIon. theReFore, Only An upper limit is DErivEd: the full width of the sigNal is SmALler THaN 15 at 95 % C.L. there is NO nATUraL eXpLanAtIOn Of sucH a sMall valUe, neither fOr tHe d$^{*'}$ nor for higheR OrBital excitAtiOns [@pene].
VarioUs cHecks wErE... | 7 $\pm$ 2 [MeV/]{}$c^2$,i.e. compa tible wi thth e de tect or resolution. Ther efore, only an upper l imitis deri v ed : the full w i dt h ofth esig na l i s sma lle r than15 at 95 % C. L. There is no na tural expl ana tion of such asmallva lue , neit her forthe D$ ^ {*'}$nor for h ig h er orb i tal exc i t at... | 7_$\pm$ 2 [MeV/]{}$c^2$,_i.e. compatible with the_detector resolution._Therefore,_only an_upper_limit is derived:_the full width_of the signal is_smaller than 15 at_95_% C.L. There is no natural explanation of such a small value, neither for_the_D$^{*'}$ nor_for_higher_orbital excitations [@pene].
Various checks were... |
in [@Sun; @LS], and they are closely related to orthogonal algebraic polynomials on the domain bounded by Steiner’s hyercycloid [@K; @LSX], much as Chebyshev polynomials arise from exponentials. In fact, the trigonometric functions arise from the exponentials by symmetry are called generalized cosine functions in [@LS... | in [ @Sun; @LS ], and they are closely related to orthogonal algebraic polynomials on the knowledge domain restrict by Steiner ’s hyercycloid [ @K; @LSX ], much as Chebyshev polynomials arise from exponential. In fact, the trigonometric functions arise from the exponentials by isotropy are address generalized cosine fu... | in [@Sun; @LS], and they are clorely related to orthogmnal amgebraic polynomials on the domain blubded vy Steiner’s hyercycloia [@K; @LSX], mlch as Chwbysiev polynomials edise from expkkenticlw. In fact, the jrigonometriw functions arhsd yrom the exponentials by symmetry arq callec heneralized cofine suncfpoks in [@LS... | in [@Sun; @LS], and they are closely orthogonal polynomials on domain bounded by as polynomials arise from In fact, the functions arise from the exponentials by are called generalized cosine functions in [@LSX; @Sun], and there are also generalized functions that are anti-symmetric. Our results on the hexagonal domain ... | in [@Sun; @LS], and they are closely rElated to orThogoNal AlgEbRaic PolyNomials on the doMAin bOunded by Steiner’s hyercyCloid [@k; @Lsx], mucH As chebyShev polYNoMIAls ArIsE frOm EXpOnentIalS. In fact, The trigonoMetRiC functions arISe From the expOneNtials by symmEtrY are caLlEd gENeralIzeD cosiNe funcTIons in [@lS... | in [@Sun; @LS], and theyare closel y rel ate d t oorth ogon al algebraic p o lyno mials on the domain bo unded b y Ste i ne r’s h yercycl o id [ @K; @ LS X], m u ch as C heb yshev p olynomials ar is e from expon e nt ials. In f act , the trigon ome tric f un cti o ns ar ise from the e x ponent ials by s ym m e... | in_[@Sun; @LS],_and they are closely_related to_orthogonal_algebraic polynomials_on_the domain bounded_by Steiner’s hyercycloid_[@K; @LSX], much as_Chebyshev polynomials arise_from_exponentials. In fact, the trigonometric functions arise from the exponentials by symmetry are called_generalized_cosine functions_in_[@LS... |
interface (GUI).
- A logic tier, based on an application server.
- A data tier, combining native XML databases with bulk data storage.
In Simflowny 2, the original Graphical User Interface (GUI) of version 1, written ad hoc for the original PDE family, has been completely redone to make it flexible enough to au... | interface (GUI).
- A logic tier, based on an application server.
- A datum grade, combine native XML databases with bulk data memory.
In Simflowny 2, the original Graphical User Interface (GUI) of version 1, written ad hoc for the original PDE class, has been completely redone to stool it flexible enough... | inherface (GUI).
- A logic titr, based on an apklucatioi served.
- A daga tier, combining native XML dqtabawes with bulk data stofage.
In Silflowny 2, the iriginal Gczphical User Jkterfccx (GUI) of versiok 1, written dd hoc for the ofiyinal PDE family, has been completely redone tl make it flexyble qnoufh to au... | interface (GUI). - A logic tier, based application - A tier, combining native storage. Simflowny 2, the Graphical User Interface of version 1, written ad hoc the original PDE family, has been completely redone to make it flexible enough automatically accommodate new families of equations, both of PDE nature or otherwis... | interface (GUI).
- A logic tier, basEd on an applIcatiOn sErvEr.
- a datA tieR, combining natiVE XML Databases with bulk data sToragE.
IN simfLOwNy 2, the OriginaL grAPHicAl usEr INtERfAce (GUi) of Version 1, Written ad hOc fOr The original Pde fAmily, has beEn cOmpletely redOne To make It FleXIble eNouGh to aU... | interface (GUI).
- A l ogic tier, base d o n a nappl icat ion server.
- Adata tier, combining n ative X M L da t ab aseswith bu l kd a tast or age .In Simf low ny 2, t he origina l G ra phical UserI nt erface (GU I)of version 1 , w ritten a d h o c for th e ori ginalP DE fam ily, hasbe e n comp l etely r e ... | interface_(GUI).
- _ A logic tier,_based on_an_application server.
-__ A data_tier, combining native_XML databases with bulk_data storage.
In Simflowny_2,_the original Graphical User Interface (GUI) of version 1, written ad hoc for the_original_PDE family,_has_been_completely redone to make it_flexible enough to au... |
a_{min}$ and $a_{max}$, and it decreases very quickly from infinity at $a_{min}$ to an almost vanishing value at some place. It remains almost unchanged until $a$ reaches a value near $a_0=1$, where it starts increasing to a finite value when $a$ gets around $a_0=1$, and then decreases again until $a$ arrives at a poin... | a_{min}$ and $ a_{max}$, and it decreases very quickly from infinity at $ a_{min}$ to an almost vanishing value at some home. It remain almost unchanged until $ a$ reaches a value near $ a_0=1 $, where it begin increasing to a finite value when $ a$ gets about $ a_0=1 $, and then decrease again until $ a$ arrives at a ... | a_{mij}$ and $a_{max}$, and it decreares very quickli drom iifinity at $a_{min}$ to an almost vanishing valux at some place. It remains almort unchanhed untio $a$ ceaches a value isar $a_0=1$, wmzre if staxtw increasing tp a finite value when $a$ cegs around $a_0=1$, and then decreases again tntil $a$ agrives at a poyn... | a_{min}$ and $a_{max}$, and it decreases very infinity $a_{min}$ to almost vanishing value almost until $a$ reaches value near $a_0=1$, it starts increasing to a finite when $a$ gets around $a_0=1$, and then decreases again until $a$ arrives at point near $a_{max}$, from which the integrand increases again very quickly... | a_{min}$ and $a_{max}$, and it decreases Very quicklY from InfIniTy At $a_{mIn}$ to An almost vanishINg vaLue at some place. It remainS almoSt UNchaNGeD untiL $a$ reachES a VALue NeAr $A_0=1$, whErE It StartS inCreasinG to a finite ValUe When $a$ gets aroUNd $A_0=1$, and then deCreAses again untIl $a$ ArriveS aT a pOIn... | a_{min}$ and $a_{max}$, an d it decre asesver y q ui ckly fro m infinity at$ a_{m in}$ to an almost vani shing v a luea tsomeplace.I tr e mai ns a lmo st un chang eduntil $ a$ reaches ava lue near $a_ 0 =1 $, where i t s tarts increa sin g to a f ini t e val uewhen$a$ ge t s arou nd $a_0=1 $, and th e n decre a s ... | a_{min}$ and_$a_{max}$, and_it decreases very quickly_from infinity_at_$a_{min}$ to_an_almost vanishing value_at some place._It remains almost unchanged_until $a$ reaches_a_value near $a_0=1$, where it starts increasing to a finite value when $a$ gets_around_$a_0=1$, and_then_decreases_again until $a$ arrives at_a poin... |
and every $G\in RC(Y)$, and for every $\p\in\SKAL((A,\rho,\BBBB),(B,\eta,\BBBB\ap))$ and for every bounded ultrafilter $u$ in $B$ (see \[boundcl\]) $$\label{psi1a}
\Psi_1^a(\p)(\s_u)=\s_{\p\inv(u)},$$ where $\s_{\p\inv(u)}$ is a cluster in $(A,C_\rho)$ (see \[Alexprn\], (\[sigmau\]) and \[uniqult\] for $C_\rho$ and $\... | and every $ G\in RC(Y)$, and for every $ \p\in\SKAL((A,\rho,\BBBB),(B,\eta,\BBBB\ap))$ and for every bounded ultrafilter $ u$ in $ B$ (see \[boundcl\ ]) $ $ \label{psi1a }
\Psi_1^a(\p)(\s_u)=\s_{\p\inv(u)},$$ where $ \s_{\p\inv(u)}$ is a cluster in $ (A, C_\rho)$ (see \[Alexprn\ ], (\[sigmau\ ]) and \[uniqult\ ] for ... | anf every $G\in RC(Y)$, and for tvery $\p\in\SKAL((A,\rho,\YVBB),(B,\ete,\BBBB\ap))$ and for every bounded ultrafilter $u$ ib $B$ (stv \[boundcl\]) $$\label{psi1a}
\Psk_1^a(\p)(\s_u)=\s_{\p\inn(u)},$$ where $\w_{\p\int(u)}$ is a cluster mh $(A,C_\rho)$ (see \[Amcxprn\], (\[wigmau\]) and \[uniault\] for $C_\sho$ and $\... | and every $G\in RC(Y)$, and for every for bounded ultrafilter in $B$ (see is cluster in $(A,C_\rho)$ \[Alexprn\], (\[sigmau\]) and for $C_\rho$ and $\s_u$, and note by \[conclustth\], any bounded cluster $\s$ in $(B,\eta,\BBBB\ap)$ can be written in the $\s_u$ for some bounded ultrafilter $u$ in $B$); then $\l^g: Id_{\... | and every $G\in RC(Y)$, and for every $\P\in\SKAL((A,\rhO,\BBBB),(b,\etA,\BBbB\Ap))$ anD for Every bounded ulTRafiLter $u$ in $B$ (see \[boundcl\]) $$\labeL{psi1a}
\psI_1^A(\p)(\s_u)=\S_{\P\iNv(u)},$$ whEre $\s_{\p\inV(U)}$ iS A CluStEr In $(A,c_\rHO)$ (sEe \[AleXprN\], (\[sigmau\]) And \[uniqult\] For $c_\rHo$ and $\... | and every $G\in RC(Y)$, a nd for eve ry $\ p\i n\S KA L((A ,\rh o,\BBBB),(B,\e t a,\B BBB\ap))$ and for ever y bou nd e d ul t ra filte r $u$ i n $ B $ (s ee \ [bo un d cl \]) $ $\l abel{ps i1a}
\Psi_ 1^a (\ p)(\s_u)=\s_ { \p \inv(u)},$ $ w here $\s_{\p \in v(u)}$ i s a clust erin $( A,C_\r h o)$ (s ee \[Alex pr n... | and_every $G\in_RC(Y)$, and for every_$\p\in\SKAL((A,\rho,\BBBB),(B,\eta,\BBBB\ap))$ and_for_every bounded_ultrafilter_$u$ in $B$_(see \[boundcl\]) $$\label{psi1a}
\Psi_1^a(\p)(\s_u)=\s_{\p\inv(u)},$$_where $\s_{\p\inv(u)}$ is a_cluster in $(A,C_\rho)$_(see_\[Alexprn\], (\[sigmau\]) and \[uniqult\] for $C_\rho$ and $\... |
^-=\min\{0,r\}$, and the paths $\gamma_+,\gamma_-$ satisfying $-\gamma_+\subseteq\gamma_-$ with, for some $\phi_+\in (\pi/3,\pi/2)$, $\phi_-\in (\pi/2,\pi-\phi_+)$. The Airy$_{2\to 1}$ process crosses over between the Airy$_2$ and the Airy$_1$ processes in the sense that ${\mathrm{Airy}_{2\to 1}}(t+\tau)$ converges to ... | ^-=\min\{0,r\}$, and the paths $ \gamma_+,\gamma_-$ satisfying $ -\gamma_+\subseteq\gamma_-$ with, for some $ \phi_+\in (\pi/3,\pi/2)$, $ \phi_-\in (\pi/2,\pi-\phi_+)$. The Airy$_{2\to 1}$ process crosses over between the Airy$_2 $ and the Airy$_1 $ summons in the common sense that $ { \mathrm{Airy}_{2\to 1}}(t+\tau)$ ... | ^-=\min\{0,g\}$, and the paths $\gamma_+,\gamoa_-$ satisfying $-\gcnma_+\subveteq\gzmma_-$ witf, for some $\phi_+\in (\pi/3,\pi/2)$, $\phi_-\in (\'i/2,\pi-\phi_+)$. Tye Airy$_{2\to 1}$ process crorses over between the Qiry$_2$ and tis Airy$_1$ ixocesacs in vhe sense that ${\kathrm{Airy}_{2\do 1}}(t+\tau)$ convercer co ... | ^-=\min\{0,r\}$, and the paths $\gamma_+,\gamma_-$ satisfying $-\gamma_+\subseteq\gamma_-$ some (\pi/3,\pi/2)$, $\phi_-\in The Airy$_{2\to 1}$ Airy$_2$ the Airy$_1$ processes the sense that 1}}(t+\tau)$ converges to $2^{1/3}{\mathrm{Airy}_1}(2^{-2/3}t)$ as $\tau\to\infty$ to ${\mathrm{Airy}_2}(1;t)$ (the Airy$_2$ proce... | ^-=\min\{0,r\}$, and the paths $\gamma_+,\gamma_-$ Satisfying $-\Gamma_+\SubSetEq\GammA_-$ witH, for some $\phi_+\in (\pI/3,\Pi/2)$, $\phI_-\in (\pi/2,\pi-\phi_+)$. The Airy$_{2\to 1}$ proCess cRoSSes oVEr BetweEn the AiRY$_2$ aND The aiRy$_1$ ProCeSSeS in thE seNse that ${\Mathrm{Airy}_{2\To 1}}(t+\TaU)$ converges to ... | ^-=\min\{0,r\}$, and the p aths $\gam ma_+, \ga mma _- $ sa tisf ying $-\gamma_ + \sub seteq\gamma_-$ with, f or so me $\ph i _+ \in ( \pi/3,\ p i/ 2 ) $,$\ ph i_- \i n ( \pi/2 ,\p i-\phi_ +)$. The A iry $_ {2\to 1}$ pr o ce ss crosses ov er between t heAiry$_ 2$ an d theAir y$_1$ proce s ses in the sens et hat $... | ^-=\min\{0,r\}$, and_the paths_$\gamma_+,\gamma_-$ satisfying $-\gamma_+\subseteq\gamma_-$ with,_for some_$\phi_+\in_(\pi/3,\pi/2)$, $\phi_-\in_(\pi/2,\pi-\phi_+)$._The Airy$_{2\to 1}$_process crosses over_between the Airy$_2$ and_the Airy$_1$ processes_in_the sense that ${\mathrm{Airy}_{2\to 1}}(t+\tau)$ converges to ... |
{j}\bar{l}}E_{ik}} \over {(i,k)}} \right ).
\label{6}$$ For $u=\iota$ or $v=\iota, \iota^2=0$ superalgebra $osp(m|2n)$ is contracted to inhomogeneous superalgebra, which is semidirect sum $ \{E_{i\bar{j}}\} \S (so(m) \bigoplus sp(2n)),$ with all anticommutators of the odd generators equal to zero $\{E_{i\bar{j}},E_{k\b... | { j}\bar{l}}E_{ik } } \over { (i, k) } } \right).
\label{6}$$ For $ u=\iota$ or $ v=\iota, \iota^2=0 $ superalgebra $ osp(m|2n)$ is contracted to inhomogeneous superalgebra, which is semidirect sum $ \{E_{i\bar{j}}\ } \S (so(m) \bigoplus sp(2n)),$ with all anticommutators of the odd generators adequate to zero $ \{E_... | {j}\bag{l}}E_{ik}} \over {(i,k)}} \right ).
\labeu{6}$$ For $u=\iota$ or $r=\uota, \imta^2=0$ suleralgebfa $osp(m|2n)$ is contracted to iniomoteneoys superalgebra, which ks semidigect sum $ \{E_{i\ber{j}}\} \S (so(m) \bigoplna sp(2n)),$ wlch alm antncimmutators of jhe odd genesators equal tm xexo $\{E_{i\bar{j}},E_{k\b... | {j}\bar{l}}E_{ik}} \over {(i,k)}} \right ). \label{6}$$ For $v=\iota, superalgebra $osp(m|2n)$ contracted to inhomogeneous $ \S (so(m) \bigoplus with all anticommutators the odd generators equal to zero \} = 0.$ Example: CK contractions of $osp(3|2)$ -------------------------------------- This superalgebra has $so(3)$ ... | {j}\bar{l}}E_{ik}} \over {(i,k)}} \right ).
\label{6}$$ FOr $u=\iota$ or $v=\Iota, \iOta^2=0$ SupErAlgeBra $oSp(m|2n)$ is contractED to iNhomogeneous superalgebRa, whiCh IS semIDiRect sUm $ \{E_{i\bar{J}}\} \s (sO(M) \BigOpLuS sp(2N)),$ wITh All anTicOmmutatOrs of the odD geNeRators equal tO ZeRo $\{E_{i\bar{j}},E_{k\B... | {j}\bar{l}}E_{ik}} \over { (i,k)}} \r ight).\la be l{6} $$ F or $u=\iota$ o r $v= \iota, \iota^2=0$ supe ralge br a $os p (m |2n)$ is con t ra c t edto i nho mo g en eoussup eralgeb ra, whichisse midirect sum $\{E_{i\bar {j} }\} \S (so(m ) \ bigopl us sp ( 2n)), $ w ith a ll ant i commut ators ofth e odd g e nera... | {j}\bar{l}}E_{ik}} \over_{(i,k)}} \right_).
\label{6}$$ For $u=\iota$ or_$v=\iota, \iota^2=0$_superalgebra_$osp(m|2n)$ is_contracted_to inhomogeneous superalgebra,_which is semidirect_sum $ \{E_{i\bar{j}}\} \S_(so(m) \bigoplus sp(2n)),$_with_all anticommutators of the odd generators equal to zero $\{E_{i\bar{j}},E_{k\b... |
^{[3]}$}
\put(197, 42){$S$}
\put(307, 94){$R^{[L]}$}
\put(309, 42){$S$}
\put(335, 79){\footnotesize $c_L$}
\put(360, 67){$Q$}
\put(363, 37){$Y$}
\put(29, 6){\footnotesize $s_1$}
\put(118, 6){\footnotesize $s_2$}
\put(207, 6){\footnotesize $s_3$}
\put(319, 6){\footnotesize $s_L$}
\end{overpic}$$ where structu... | ^{[3]}$ }
\put(197, 42){$S$ }
\put(307, 94){$R^{[L]}$ }
\put(309, 42){$S$ }
\put(335, 79){\footnotesize $ c_L$ }
\put(360, 67){$Q$ }
\put(363, 37){$Y$ }
\put(29, 6){\footnotesize $ s_1 $ }
\put(118, 6){\footnotesize $ s_2 $ }
\put(207, 6){\footnotesize $ s_3 $ }
\put(319, 6){\footnotesize ... | ^{[3]}$}
\puh(197, 42){$S$}
\put(307, 94){$R^{[L]}$}
\put(309, 42){$S$}
\put(335, 79){\foounotesize $c_L$}
\put(360, 67){$Q$}
\put(363, 37){$B$}
\put(29, 6){\fkotnoteskze $s_1$}
\put(118, 6){\footnotesize $s_2$}
\put(207, 6){\fiotnouvsize $s_3$}
\put(319, 6){\footnoteskze $s_L$}
\end{lverpic}$$ qhert structu... | ^{[3]}$} \put(197, 42){$S$} \put(307, 94){$R^{[L]}$} \put(309, 42){$S$} $c_L$} 67){$Q$} \put(363, \put(29, 6){\footnotesize $s_1$} $s_3$} 6){\footnotesize $s_L$} \end{overpic}$$ structure fragments $S$ Clebsh-Gordan coefficients. The tensor $Y$ can any random tensor; no algorithm based on a SU(2) invariant benchmark ca... | ^{[3]}$}
\put(197, 42){$S$}
\put(307, 94){$R^{[L]}$}
\put(309, 42){$S$}
\put(335, 79){\footnotesIze $c_L$}
\put(360, 67){$Q$}
\pUt(363, 37){$Y$}
\puT(29, 6){\foOtnOtEsizE $s_1$}
\puT(118, 6){\footnotesize $s_2$}
\PUt(207, 6){\foOtnotesize $s_3$}
\put(319, 6){\footnoteSize $s_l$}
\eND{oveRPiC}$$ wherE structU... | ^{[3]}$}
\put(197, 42){$S $}
\put(3 07, 9 4){ $R^ {[ L]}$ }
\ put(309, 42){$ S $}
\put(335, 79){\footnot esize $ c _L$}
\put( 360, 67 ) {$ Q $ }
\p ut (36 3, 37 ){$Y$ }
\put(29 , 6){\foot not es ize $s_1$}
\ pu t(118, 6){ \fo otnotesize $ s_2 $}
\p ut (20 7 , 6){ \fo otnot esize$ s_3$} \put(319 ,6 ){\foo t notesi... | ^{[3]}$}
\put(197,_42){$S$}
\put(307,_94){$R^{[L]}$}
\put(309, 42){$S$}
\put(335,_79){\footnotesize $c_L$}_
_\put(360, 67){$Q$}
_\put(363,_37){$Y$}
\put(29, 6){\footnotesize_$s_1$}
\put(118, 6){\footnotesize_$s_2$}
\put(207, 6){\footnotesize $s_3$}
_\put(319, 6){\footnotesize $s_L$}
\end{overpic}$$_where_structu... |
2000 *Phys. Rev. Lett.* [**85**]{} 3313
Cerf N J, Bourennane M, Karlsson A and Gisin N 2002 *Phys. Rev. Lett.* [**88**]{} 127902
Vaidman L, Aharonov Y and Albert D Z 1987 *Phys. Rev. Lett.* [**58**]{} 1385
Aharonov Y and Englert B-G 2001 *Z. Naturforsch. A* [**56**]{} 16
Aravind P K 2003 *Z. Naturforsch. A* [**58*... | 2000 * Phys. Rev. Lett. * [ * * 85 * * ] { } 3313
Cerf N J, Bourennane M, Karlsson A and Gisin N 2002 * Phys. Rev. Lett. * [ * * 88 * * ] { } 127902
Vaidman L, Aharonov Y and Albert D Z 1987 * Phys. Rev. Lett. * [ * * 58 * * ] { } 1385
Aharonov Y and Englert B - G 2001 * Z. Naturforsch. A * [ * * 56 * * ] { } ... | 2000 *Pjys. Rev. Lett.* [**85**]{} 3313
Cerf N J, Buurennane M, Karlsson A and Gjsin N 2002 *Ohys. Rev. Lett.* [**88**]{} 127902
Vaidman L, Ahaconoc Y abd Albert D Z 1987 *Phys. Rex. Lett.* [**58**]{} 1385
Ajaronov T anv Englert B-G 2001 *Z. Izturforsch. A* [**56**]{} 16
Wravnnv P K 2003 *Z. Naturfprsch. A* [**58*... | 2000 *Phys. Rev. Lett.* [**85**]{} 3313 Cerf Bourennane Karlsson A Gisin N 2002 Vaidman Aharonov Y and D Z 1987 Rev. Lett.* [**58**]{} 1385 Aharonov Y Englert B-G 2001 *Z. Naturforsch. A* [**56**]{} 16 Aravind P K 2003 *Z. A* [**58**]{} 2212 Hayashi A, Horibe M and Hashimoto T 2005 *Phys. Rev. [**71**]{} Lee Kim S and ... | 2000 *Phys. Rev. Lett.* [**85**]{} 3313
Cerf N J, BourennaNe M, KarlssoN A and gisIn N 2002 *phYs. ReV. LetT.* [**88**]{} 127902
Vaidman L, AharoNOv Y aNd Albert D Z 1987 *Phys. Rev. Lett.* [**58**]{} 1385
AHaronOv y And ENGlErt B-G 2001 *z. NaturfORsCH. a* [**56**]{} 16
ArAvInD P K 2003 *z. NATuRforsCh. A* [**58*... | 2000 *Phys. Rev. Lett.* [ **85**]{}3313
Ce rfNJ, B oure nnane M, Karls s on A and Gisin N 2002 *Phy s. Re v. Lett . *[**88 **]{} 1 2 79 0 2
V ai dm anL, Ah arono v Y and Al bert D Z 1 987 * Phys. Rev. L e tt .* [**58** ]{} 1385
Aharo nov Y and E ngl e rt B- G 2 001 * Z. Nat u rforsc h. A* [** 56 * *]{} 1 6
... | 2000_*Phys. Rev._Lett.* [**85**]{} 3313
Cerf N_J, Bourennane_M,_Karlsson A_and_Gisin N 2002_*Phys. Rev. Lett.*_[**88**]{} 127902
Vaidman L, Aharonov_Y and Albert_D_Z 1987 *Phys. Rev. Lett.* [**58**]{} 1385
Aharonov Y and Englert B-G 2001 *Z. Naturforsch._A*_[**56**]{} 16
Aravind_P_K_2003 *Z. Naturforsch. A* [**58*... |
f^N, g^N)$, then $\psi_N(i) \geq 1- \epsilon_N$ for every $N$ and $$\begin{aligned}
\lim_{N \to \infty}-\frac{1}{N}\log\phi_N(i) = D^*(i).\end{aligned}$$
See Section \[achievable\].
Note that selecting an experiment that minimizes $\mathscr{M}_i(u,\rho,s)$ over the set $\mathcal{U}$ is equivalent to selecting an expe... | f^N, g^N)$, then $ \psi_N(i) \geq 1- \epsilon_N$ for every $ N$ and $ $ \begin{aligned }
\lim_{N \to \infty}-\frac{1}{N}\log\phi_N(i) = D^*(i).\end{aligned}$$
See Section \[achievable\ ].
Note that selecting an experiment that minimize $ \mathscr{M}_i(u,\rho, s)$ over the set $ \mathcal{U}$ is equivalent to cho... | f^N, h^N)$, then $\psi_N(i) \geq 1- \epsilun_N$ for every $N$ and $$\bejin{alighed}
\lim_{N \go \infty}-\frac{1}{N}\log\phi_N(i) = D^*(i).\end{eligbed}$$
Set Section \[achievabld\].
Note than selectibg ai experiment thav minimidzs $\mafmscr{M}_n(u,\cho,s)$ over the sgt $\mathcal{U}$ hs equivalent do szlecting an expe... | f^N, g^N)$, then $\psi_N(i) \geq 1- \epsilon_N$ $N$ $$\begin{aligned} \lim_{N \infty}-\frac{1}{N}\log\phi_N(i) = D^*(i).\end{aligned}$$ selecting experiment that minimizes over the set is equivalent to selecting an experiment maximizes $(1-\mathscr{M}_i(u,\rho,s))/s$. When $s$ is small, this function can be approximate... | f^N, g^N)$, then $\psi_N(i) \geq 1- \epsilon_N$ fOr every $N$ anD $$\begiN{alIgnEd}
\Lim_{N \To \inFty}-\frac{1}{N}\log\phi_n(I) = D^*(i).\eNd{aligned}$$
See Section \[achIevabLe\].
nOte tHAt SelecTing an eXPeRIMenT tHaT miNiMIzEs $\matHscR{M}_i(u,\rho,S)$ over the seT $\maThCal{U}$ is equivaLEnT to selectiNg aN expe... | f^N, g^N)$, then $\psi_N(i ) \geq 1-\epsi lon _N$ f or e very $N$ and $$\be g in{a ligned}
\lim_{N \to \i nfty} -\ f rac{ 1 }{ N}\lo g\phi_N ( i) = D^ *( i) .\e nd { al igned }$$
See S ection \[a chi ev able\].
Not e t hat select ing an experime ntthat m in imi z es $\ mat hscr{ M}_i(u , \rho,s )$ over t he set... | f^N, g^N)$,_then $\psi_N(i)_\geq 1- \epsilon_N$ for_every $N$_and_$$\begin{aligned}
\lim_{N \to_\infty}-\frac{1}{N}\log\phi_N(i)_= D^*(i).\end{aligned}$$
See Section_\[achievable\].
Note that selecting_an experiment that minimizes_$\mathscr{M}_i(u,\rho,s)$ over the_set_$\mathcal{U}$ is equivalent to selecting an expe... |
and such that there is a point $z$ on the boundary of the ball that belongs to $\{u_1=u_2\}$ (this can be done by choosing any ball and enlarging the radius until the boundary touches $\{u_1=u_2\}$). Observe that $v< 0$ on $B_r(y)$ and it extends to $z$ with $v(z)=0$ and $Dv(z)=0$ (because $u_1(z)=u_2(z)$ and $Du_1(z)... | and such that there is a point $ z$ on the boundary of the ball that belong to $ \{u_1 = u_2\}$ (this can be do by choosing any ball and enlarging the spoke until the boundary touches $ \{u_1 = u_2\}$). Observe that $ v < 0 $ on $ B_r(y)$ and it widen to $ z$ with $ v(z)=0 $ and $ Dv(z)=0 $ (because $ u_1(z)=u_2(z)$ an... | anf such that there is a puint $z$ on the boundary of ths ball tfat belongs to $\{u_1=u_2\}$ (this can bx dobe by choosing any ball and enlarginh the raeius yntil the ukundary touchsd $\{u_1=u_2\}$). Ibserve that $v< 0$ on $B_r(y)$ atd it extends do $z$ with $v(z)=0$ and $Dv(z)=0$ (because $u_1(z)=u_2(z)$ and $Dt_1(z)... | and such that there is a point the of the that belongs to by any ball and the radius until boundary touches $\{u_1=u_2\}$). Observe that $v< on $B_r(y)$ and it extends to $z$ with $v(z)=0$ and $Dv(z)=0$ (because $u_1(z)=u_2(z)$ $Du_1(z) = Du_2(z)$ by the structural assumptions): this contradicts Hopf boundary point lem... | and such that there is a point $z$ On the boundAry of The BalL tHat bElonGs to $\{u_1=u_2\}$ (this can bE Done By choosing any ball and enLargiNg THe raDIuS untiL the bouNDaRY TouChEs $\{U_1=u_2\}$). OBsERvE that $V< 0$ on $b_r(y)$ and iT extends to $Z$ wiTh $V(z)=0$ and $Dv(z)=0$ (becaUSe $U_1(z)=u_2(z)$ and $Du_1(z)... | and such that there is apoint $z$on th e b oun da ry o f th e ball that be l ongs to $\{u_1=u_2\}$ (thi s can b e don e b y cho osing a n yb a llan denl ar g in g the ra dius un til the bo und ar y touches $\ { u_ 1=u_2\}$). Ob serve that $ v<0$ on$B _r( y )$ an d i t ext ends t o $z$ w ith $v(z) =0 $ and $ D ... | and_such that_there is a point_$z$ on_the_boundary of_the_ball that belongs_to $\{u_1=u_2\}$ (this_can be done by_choosing any ball_and_enlarging the radius until the boundary touches $\{u_1=u_2\}$). Observe that $v< 0$ on $B_r(y)$_and_it extends_to_$z$_with $v(z)=0$ and $Dv(z)=0$ (because_$u_1(z)=u_2(z)$ and $Du_1(z)... |
[ kms$^{-1}$]{} were found in the SE region, including the region of our R1, R2, R9 and R13 for which HETGS data imply velocities between $-2500$ and $-1000$[ kms$^{-1}$]{}. This discrepancy is not too surprising. Although @hwang01 argued that they can ignore the ionization effects on the Si He$\alpha$ centroid to a re... | [ kms$^{-1}$ ] { } were found in the SE region, including the region of our R1, R2, R9 and R13 for which HETGS datum entail velocities between $ -2500 $ and $ -1000 $ [ kms$^{-1}$ ] { }. This discrepancy is not too surprising. Although @hwang01 argue that they can neglect the ionization effects on the Si He$\alph... | [ kms$^{-1}$]{} aere found in the SE reglon, including thg eegion of oud R1, R2, R9 xnd R13 for which HETGS data ilpoy veoocities between $-2500$ and $-1000$[ yms$^{-1}$]{}. This fiscrepabcy ms not too surprmaing. Albkough @mwang01 ergued that thei can ignore the ionizatiot dfyects on the Si He$\alpha$ centroid to w re... | [ kms$^{-1}$]{} were found in the SE the of our R2, R9 and imply between $-2500$ and kms$^{-1}$]{}. This discrepancy not too surprising. Although @hwang01 argued they can ignore the ionization effects on the Si He$\alpha$ centroid to a approximation, they note that high spectral resolution measurements are more desirab... | [ kms$^{-1}$]{} were found in the SE region, Including tHe regIon Of oUr r1, R2, R9 aNd R13 fOr which HETGS daTA impLy velocities between $-2500$ and $-1000$[ Kms$^{-1}$]{}. ThIs DIscrEPaNcy is Not too sURpRISinG. ALtHouGh @HWaNg01 argUed That theY can ignore The IoNization effeCTs On the Si He$\aLphA$ centroid to a Re... | [ kms$^{-1}$]{} were found in the SE regi on, in cl udin g th e region of ou r R1, R2, R9 and R13 for wh ich H ET G S da t aimply veloci t ie s bet we en $- 25 0 0$ and$-1 000$[ k ms$^{-1}$] {}. T his discrepa n cy is not to o s urprising. A lth ough @ hw ang 0 1 arg ued that theyc an ign ore the i on i zat... | [ kms$^{-1}$]{} were found_in the_SE region, including the_region of_our_R1, R2,_R9_and R13 for_which HETGS data_imply velocities between $-2500$_and $-1000$[ kms$^{-1}$]{}. This_discrepancy_is not too surprising. Although @hwang01 argued that they can ignore the ionization effects_on_the Si_He$\alpha$_centroid_to a re... |
kian of $J_\nu$ and $Y_\nu$ obeys $$W\{J_\nu(z),Y_\nu(z)\} = 2/(\pi z)$$ (cf., e.g., [@NIST 10.5.2]) and also that the derivatives of $J_\nu$ and $Y_\nu$ obey $$J_0'(z)=-J_1(z), \quad Y_0'(z)=Y_1(z)$$ cf., e.g. [@NIST 10.6.3]. In asymptotic analysis of the solution of the stochastic equation, we will need information a... | kian of $ J_\nu$ and $ Y_\nu$ obeys $ $ W\{J_\nu(z),Y_\nu(z)\ } = 2/(\pi z)$$ (cf. , e.g., [ @NIST 10.5.2 ]) and also that the derivative instrument of $ J_\nu$ and $ Y_\nu$ obey $ $ J_0'(z)=-J_1(z), \quad Y_0'(z)=Y_1(z)$$ californium. , e.g. [ @NIST 10.6.3 ]. In asymptotic analysis of the solution of the stochasti... | kiaj of $J_\nu$ and $Y_\nu$ obeys $$W\{M_\nu(z),Y_\nu(z)\} = 2/(\pi z)$$ (cy., e.g., [@NISV 10.5.2]) and zlso thag the derivatives of $J_\nu$ and $Y_\bu$ obtj $$J_0'(z)=-J_1(z), \quad Y_0'(z)=Y_1(z)$$ cf., e.e. [@NIST 10.6.3]. In asymptoric enalysis of the solution of ths stoehestic equation, ee will nead information a... | kian of $J_\nu$ and $Y_\nu$ obeys $$W\{J_\nu(z),Y_\nu(z)\} z)$$ e.g., [@NIST and also that $Y_\nu$ $$J_0'(z)=-J_1(z), \quad Y_0'(z)=Y_1(z)$$ e.g. [@NIST 10.6.3]. asymptotic analysis of the solution of stochastic equation, we will need information about the asymptotic behaviour of $J_\nu(t)$ and as $t\to\infty$. The req... | kian of $J_\nu$ and $Y_\nu$ obeys $$W\{J_\nu(z),y_\nu(z)\} = 2/(\pi z)$$ (cf., e.G., [@NIST 10.5.2]) And AlsO tHat tHe deRivatives of $J_\nu$ ANd $Y_\nU$ obey $$J_0'(z)=-J_1(z), \quad Y_0'(z)=Y_1(z)$$ cf., e.g. [@NiST 10.6.3]. In AsYMptoTIc AnalySis of thE SoLUTioN oF tHe sToCHaStic eQuaTion, we wIll need infOrmAtIon a... | kian of $J_\nu$ and $Y_\nu $ obeys $$ W\{J_ \nu (z) ,Y _\nu (z)\ } = 2/(\pi z)$ $ (cf ., e.g., [@NIST 10.5.2 ]) an da lsot ha t the deriva t iv e s of $ J_ \nu $a nd $Y_\ nu$ obey $ $J_0'(z)=- J_1 (z ), \quad Y_0 ' (z )=Y_1(z)$$ cf ., e.g. [@NI ST10.6.3 ]. In asymp tot ic an alysis of the solution o f the s ... | kian of_$J_\nu$ and_$Y_\nu$ obeys $$W\{J_\nu(z),Y_\nu(z)\} =_2/(\pi z)$$_(cf.,_e.g., [@NIST 10.5.2])_and_also that the_derivatives of $J_\nu$_and $Y_\nu$ obey $$J_0'(z)=-J_1(z),_\quad Y_0'(z)=Y_1(z)$$ cf.,_e.g. [@NIST_10.6.3]. In asymptotic analysis of the solution of the stochastic equation, we will need_information_a... |
Au-coated sphere. Our measurements are performed by means of dynamic atomic force microscope (AFM) operated in the frequency-shift technique described in detail in Refs. [@27; @28]. We demonstrate significant increase in the gradient of the Casimir force in comparison with that between a Si plate covered with a SiO${}... | Au - coated sphere. Our measurements are performed by mean of active nuclear force microscope (AFM) operated in the frequency - shift proficiency described in detail in Refs. [ @27; @28 ]. We demonstrate meaning addition in the gradient of the Casimir force in comparison with that between a Si plate covered with a Si... | Au-foated sphere. Our measurtments are performed by keans kf dynamkc atomic force microscope (AHM) operattb in the frequency-shkft technpque desceibev in detail in Rxrs. [@27; @28]. We demonsfvate vmgnificant incrgase in the cradient of tha Zadimir force in comparison with that betweem w Si plate covgred eyth z SiO${}... | Au-coated sphere. Our measurements are performed by dynamic force microscope operated in the in [@27; @28]. We significant increase in gradient of the Casimir force in with that between a Si plate covered with a SiO${}_2$ film and an sphere, i.e., in the absence of graphene sheet. At short separations this increase up ... | Au-coated sphere. Our measuremEnts are perFormeD by MeaNs Of dyNamiC atomic force miCRoscOpe (AFM) operated in the freQuencY-sHIft tEChNique DescribED iN DEtaIl In refS. [@27; @28]. WE DeMonstRatE signifIcant increAse In The gradient oF ThE Casimir foRce In comparison WitH that bEtWeeN A Si plAte CoverEd with A siO${}... | Au-coated sphere. Our mea surementsare p erf orm ed bymean s of dynamic a t omic force microscope (AFM ) ope ra t ed i n t he fr equency - sh i f t t ec hn iqu ed es cribe d i n detai l in Refs. [@ 27 ; @28]. We d e mo nstrate si gni ficant incre ase in th egra d ientofthe C asimir forcein compar is o n with that... | Au-coated_sphere. Our_measurements are performed by_means of_dynamic_atomic force_microscope_(AFM) operated in_the frequency-shift technique_described in detail in_Refs. [@27; @28]. We_demonstrate_significant increase in the gradient of the Casimir force in comparison with that between_a_Si plate_covered_with_a SiO${}... |
beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big],\nonumber\\
&&\nonumber\\
{\cal L}_{b} \longrightarrow \tilde {\cal L}_{b} & = & \tilde {\cal L}_0 + \frac {\partial}
{\partial \bar \theta}\, \frac {\partial} {\partial \theta}\, \Big [ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu (h)} + \... | beta^{(h) } \cdot { \tilde { \bar \beta}}^{(h) } \Big],\nonumber\\
& & \nonumber\\
{ \cal L}_{b } \longrightarrow \tilde { \cal L}_{b } & = & \tilde { \cal L}_0 + \frac { \partial }
{ \partial \bar \theta}\, \frac { \partial } { \partial \theta}\, \Big [ \frac { i } { 2 } \, \tilde { \cal A}_\mu^{(h) }
\cdo... | betw^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Nig],\nonumber\\
&&\nonumywr\\
{\cal N}_{b} \lonfrightarfow \tilde {\cal L}_{b} & = & \tilde {\cap O}_0 + \frqc {\partial}
{\partial \bar \theta}\, \frwc {\partiql} {\pertial \theta}\, \Bmf [ \frac {i} {2} \, \tjpde {\eao A}_\mu^{(h)}
\cdot \tikde {\cal A}^{\mg (h)} + \... | beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big],\nonumber\\ &&\nonumber\\ \longrightarrow {\cal L}_{b} = & \tilde {\partial \theta}\, \frac {\partial} \theta}\, \Big [ {i} {2} \, \tilde {\cal A}_\mu^{(h)} \tilde {\cal A}^{\mu (h)} + \tilde {\cal F}^{(h)} \cdot {\tilde {\bar {\cal F}}}^{(h)} \frac {i}{2} \, \tilde \P... | beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big],\Nonumber\\
&&\noNumbeR\\
{\caL L}_{b} \LoNgriGhtaRrow \tilde {\cal L}_{b} & = & \TIlde {\Cal L}_0 + \frac {\partial}
{\partial \Bar \thEtA}\, \Frac {\PArTial} {\pArtial \tHEtA}\, \bIg [ \fRaC {i} {2} \, \TilDe {\CAl a}_\mu^{(h)}
\cDot \Tilde {\caL A}^{\mu (h)} + \... | beta^{(h)} \cdot {\tilde { \bar \beta }}^{( h)} \B ig ],\n onum ber\\
&&\nonum b er\\
{\cal L}_{b} \longrig htarr ow \til d e{\cal L}_{b} &= & \ ti ld e { \c a lL}_0+ \ frac {\ partial} {\p ar tial \bar \t h et a}\, \frac {\ partial} {\p art ial \ th eta } \, \ Big [ \f rac {i } {2} \ , \tilde{\ c al A}_ \ mu^{... | beta^{(h)} \cdot_{\tilde {\bar_\beta}}^{(h)} \Big],\nonumber\\
&&\nonumber\\
{\cal L}_{b} \longrightarrow_\tilde {\cal_L}_{b}_& =_&_\tilde {\cal L}_0_+ \frac {\partial}_
{\partial \bar \theta}\, \frac_{\partial} {\partial _\theta}\,_ \Big [ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu_(h)}_+ \... |
.eps){width="0.85\linewidth"}
Once the link is removed in a rewiring event, nodes can be left without any link, so that the phase of previously connected and synchronized oscillators will start to drift away one from each other due to their natural frequencies difference $\delta \omega$, loosing any information regard... | .eps){width="0.85\linewidth " }
Once the link is removed in a rewiring event, node can be leave without any link, so that the phase of previously connected and synchronized oscillator will start to drift away one from each early ascribable to their natural frequencies difference $ \delta \omega$, loosing any informa... | .eps){aidth="0.85\linewidth"}
Once the llnk is removed iu a rewmring ebent, nodds can be left without any lmnk, wo thqt the phase of previojsly connvcted and synrhronized oscillefors will stadb to brmft away one frpm each otver due to thehr nctural frequencies difference $\delta \jmega$, lpoding any inforiatipg refard... | .eps){width="0.85\linewidth"} Once the link is removed in event, can be without any link, previously and synchronized oscillators start to drift one from each other due to natural frequencies difference $\delta \omega$, loosing any information regarding their previous mutual synchronization their phase difference appro... | .eps){width="0.85\linewidth"}
Once the lInk is removEd in a RewIriNg EvenT, nodEs can be left witHOut aNy link, so that the phase of PreviOuSLy coNNeCted aNd synchROnIZEd oScIlLatOrS WiLl staRt tO drift aWay one from EacH oTher due to theIR nAtural freqUenCies differenCe $\dElta \omEgA$, loOSing aNy iNformAtion rEGard... | .eps){width="0.85\linewidt h"}
Oncethe l ink is r emov ed i n a rewiring e v ent, nodes can be left wit houtan y lin k ,so th at thep ha s e of p re vio us l yconne cte d and s ynchronize d o sc illators wil l s tart to dr ift away one fr omeach o th erd ue to th eir n atural freque ncies dif fe r ence $ \ de... | .eps){width="0.85\linewidth"}
Once the_link is_removed in a rewiring_event, nodes_can_be left_without_any link, so_that the phase_of previously connected and_synchronized oscillators will_start_to drift away one from each other due to their natural frequencies difference $\delta_\omega$,_loosing any_information_regard... |
to the dimension of the control distribution, the system will be called a *fully actuated nonholonomic system*.
A **solution of a fully actuated nonholonomic problem** is an admissible curve $\gamma:I\rightarrow\mathcal{D}$ such that $$\nabla_{\gamma(t)}^{\mathcal{G}^{\mathcal{D}}}\gamma(t)+grad_{\mathcal{G}^{\mathca... | to the dimension of the control distribution, the system will be called a * amply trip nonholonomic system *.
A * * solution of a fully actuated nonholonomic trouble * * is an admissible curve $ \gamma: I\rightarrow\mathcal{D}$ such that $ $ \nabla_{\gamma(t)}^{\mathcal{G}^{\mathcal{D}}}\gamma(t)+grad_{\mathcal{G}^{... | to the dimension of the coktrol distribution, the vystem will be called a *fully actuated noniolobomic system*.
A **solution of a fully acnuated nobholibomic problem** is ak admjdsibnx curve $\gamma:I\rlghtarrow\madhcal{D}$ such thdt $$\ncbla_{\gamma(t)}^{\mathcal{G}^{\mathcal{D}}}\gamma(t)+grad_{\iathcal{B}^{\mwthca... | to the dimension of the control distribution, will called a actuated nonholonomic system*. actuated problem** is an curve $\gamma:I\rightarrow\mathcal{D}$ such $$\nabla_{\gamma(t)}^{\mathcal{G}^{\mathcal{D}}}\gamma(t)+grad_{\mathcal{G}^{\mathcal{D}}}V(\tau_{\mathcal{D}}(\gamma(t)))\in \Gamma (\tau_D),$$ or, equivalentl... | to the dimension of the controL distributIon, thE sySteM wIll bE calLed a *fully actuaTEd noNholonomic system*.
A **solutIon of A fULly aCTuAted nOnholonOMiC PRobLeM** iS an AdMIsSible CurVe $\gamma:i\rightarroW\maThCal{D}$ such that $$\NAbLa_{\gamma(t)}^{\maThcAl{G}^{\mathcal{D}}}\gAmmA(t)+grad_{\MaThcAL{G}^{\matHca... | to the dimension of the c ontrol dis tribu tio n,th e sy stem will be calle d a * fully actuated nonholo nomic s y stem * .
A ** solutio n o f a f ul ly ac tu a te d non hol onomicproblem**isan admissiblec ur ve $\gamma :I\ rightarrow\m ath cal{D} $suc h that $$ \nabl a_{\ga m ma(t)} ^{\mathca l{ G }^{\ma t hc... | to_the dimension_of the control distribution,_the system_will_be called_a_*fully actuated nonholonomic_system*.
A **solution of_a fully actuated nonholonomic_problem** is an_admissible_curve $\gamma:I\rightarrow\mathcal{D}$ such that $$\nabla_{\gamma(t)}^{\mathcal{G}^{\mathcal{D}}}\gamma(t)+grad_{\mathcal{G}^{\mathca... |
usual one.
By Proposition \[prop:reflexible\], the characteristic ideals ${\rm char}_{R}(M)$ and ${\rm char}_{R}(N)$ are reflexive. Hence, by Lemma \[lem:inequality\] and the condition (G$_{1}$), we may assume that $R$ is a Gorenstein local ring with $\dim(R) \leq 1$.
By the horseshoe lemma, a projective resolution ... | usual one.
By Proposition \[prop: reflexible\ ], the characteristic ideals $ { \rm char}_{R}(M)$ and $ { \rm char}_{R}(N)$ are reflexive. Hence, by Lemma \[lem: inequality\ ] and the circumstance (G$_{1}$), we may wear that $ R$ is a Gorenstein local ring with $ \dim(R) \leq 1$.
By the horseshoe lemma, a proj... | uskal one.
By Proposition \[prok:reflexible\], the ckqractecistic jdeals ${\ro char}_{R}(M)$ and ${\rm char}_{R}(N)$ are rxflezive. Yence, by Lemma \[lem:inequxlity\] and the coneitiib (G$_{1}$), we may assume bkat $R$ ls a Yocenstein local ting with $\dik(R) \leq 1$.
By the vofszshoe lemma, a projective resolution ... | usual one. By Proposition \[prop:reflexible\], the characteristic char}_{R}(M)$ ${\rm char}_{R}(N)$ reflexive. Hence, by (G$_{1}$), may assume that is a Gorenstein ring with $\dim(R) \leq 1$. By horseshoe lemma, a projective resolution of $M$ can be built up inductively with $n$-th item in the resolution of $M$ equal t... | usual one.
By Proposition \[prop:Reflexible\], The chAraCteRiStic IdeaLs ${\rm char}_{R}(M)$ and ${\rM Char}_{r}(N)$ are reflexive. Hence, by LEmma \[lEm:INequALiTy\] and The condITiON (g$_{1}$), we MaY aSsuMe THaT $R$ is a gorEnstein Local ring wIth $\DiM(R) \leq 1$.
By the hoRSeShoe lemma, a ProJective resolUtiOn ... | usual one.
By Propositio n \[prop:r eflex ibl e\] ,thechar acteristic ide a ls $ {\rm char}_{R}(M)$ and ${\r mc har} _ {R }(N)$ are re f le x i ve. H en ce, b y L emma\[l em:ineq uality\] a ndth e condition( G$ _{1}$), we ma y assume tha t $ R$ isaGor e nstei n l ocalring w i th $\d im(R) \le q1 $.
By the hor ... | usual_one.
By Proposition \[prop:reflexible\],_the characteristic ideals ${\rm_char}_{R}(M)$ and_${\rm_char}_{R}(N)$ are_reflexive._Hence, by Lemma \[lem:inequality\]_and the condition_(G$_{1}$), we may assume_that $R$ is_a_Gorenstein local ring with $\dim(R) \leq 1$.
By the horseshoe lemma, a projective resolution ... |
the solutions to separations of multiples of the KI resolution only. Thus, at the 99.7 % (67 %) confidence level of rejecting binary models, it is very unlikely that more than one out of our five targets have a binary companion of 0.08 (0.05) brightness of the primary.]{} The masses that such flux limits represent dep... | the solutions to separations of multiples of the KI resolution merely. therefore, at the 99.7 % (67 %) confidence level of disapprove binary model, it is very unlikely that more than one out of our five targets hold a binary companion of 0.08 (0.05) brightness of the primary. ] { } The mass that such flux limits def... | thf solutions to separatioks of multiples of the NI resklution unly. Thus, at the 99.7 % (67 %) confidencx lecel od rejecting binary moddls, it is very unoikeot that morx than okz out lf obr five targets mave a binasy companion ox 0.08 (0.05) yrightness of the primary.]{} The masses that sicj flux limits teprexqnt svp... | the solutions to separations of multiples of resolution Thus, at 99.7 % (67 binary it is very that more than out of our five targets have binary companion of 0.08 (0.05) brightness of the primary.]{} The masses that such limits represent depend on both the luminosity of the primary and the assumed of system. on PMS evo... | the solutions to separations Of multipleS of thE KI ResOlUtioN onlY. Thus, at the 99.7 % (67 %) confIDencE level of rejecting binarY modeLs, IT is vERy UnlikEly that MOrE THan OnE oUt oF oUR fIve taRgeTs have a Binary compAniOn Of 0.08 (0.05) brightness OF tHe primary.]{} THe mAsses that sucH flUx limiTs RepREsent Dep... | the solutions to separati ons of mul tiple s o f t he KIreso lution only. T h us,at the 99.7 % (67 %) c onfid en c e le v el of r ejectin g b i n ary m od els ,i tis ve ryunlikel y that mor e t ha n one out of ou r five tar get s have a bin ary compa ni ono f 0.0 8 ( 0.05) brigh t ness o f the pri ma r y.]{}T he... | the_solutions to_separations of multiples of_the KI_resolution_only. Thus,_at_the 99.7 % (67 %)_confidence level of_rejecting binary models, it_is very unlikely_that_more than one out of our five targets have a binary companion of 0.08_(0.05)_brightness of_the_primary.]{}_The masses that such flux_limits represent dep... |
\omega\right\vert \ll\left\vert \delta_{i}\right\vert $ such that $\delta
_{1}\approx\delta_{2}$. In the tight-binding limit, the external potential corresponds to the Hamiltonian $$H_{\text{e}}\left( t\right) =\hbar\Omega\sum_{l=0,1\text{; }q=\text{L,R}}\cos\left( \varphi_{l,q}\mathbf{+}\omega t\right) b_{l,q}^{\... | \omega\right\vert \ll\left\vert \delta_{i}\right\vert $ such that $ \delta
_ { 1}\approx\delta_{2}$. In the tight - binding limit, the external potential corresponds to the Hamiltonian $ $ H_{\text{e}}\left ( t\right) = \hbar\Omega\sum_{l=0,1\text {; } q=\text{L, R}}\cos\left ( \varphi_{l, q}\mathbf{+}\omega t\ri... |
\omeha\right\vert \ll\left\vert \dtlta_{i}\right\vert $ socy that $\delta
_{1}\zpprox\deuta_{2}$. In the tight-binding limiv, thw exttgnal potential corresoonds to nhe Hamilronien $$H_{\text{e}}\left( t\cjght) =\hncr\Omefw\sum_{n=0,1\vext{; }q=\text{L,R}}\cos\keft( \varpvi_{l,q}\mathbf{+}\omegd g\rnght) b_{l,q}^{\... | \omega\right\vert \ll\left\vert \delta_{i}\right\vert $ such that $\delta the limit, the potential corresponds to }q=\text{L,R}}\cos\left( t\right) b_{l,q}^{\dag}b_{l,q}, \label{eq:He}$$ $\varphi_{l,\text{L/R}}=\mp\frac{k_{x}\lambda_{\text{s}}}{4}+\frac{lk_{y}\lambda_{\text{s}}}{2}$. The wavevectors and $k_{y}$ can be ... |
\omega\right\vert \ll\left\vert \dElta_{i}\right\Vert $ sUch ThaT $\dElta
_{1}\ApprOx\delta_{2}$. In the tiGHt-biNding limit, the external pOtentIaL CorrESpOnds tO the HamILtONIan $$h_{\tExT{e}}\lEfT( T\rIght) =\hBar\omega\suM_{l=0,1\text{; }q=\texT{L,R}}\CoS\left( \varphi_{l,Q}\MaThbf{+}\omega t\RigHt) b_{l,q}^{\... |
\omega\right\vert \ll\lef t\vert \de lta_{ i}\ rig ht \ver t $such that $\de l ta
_ {1}\approx\delta_{2}$. In t he tigh t -b indin g limit , t h e ex te rn alpo t en tialcor respond s to the H ami lt onian $$H_{\ t ex t{e}}\left ( t\right) =\ hba r\Omeg a\ sum _ {l=0, 1\t ext{; }q=\t e xt{L,R }}\cos\le ft ( \va... |
\omega\right\vert \ll\left\vert_\delta_{i}\right\vert $_such that $\delta
_{1}\approx\delta_{2}$. In_the tight-binding_limit,_the external_potential_corresponds to the_Hamiltonian $$H_{\text{e}}\left( _t\right) =\hbar\Omega\sum_{l=0,1\text{; }q=\text{L,R}}\cos\left(_ \varphi_{l,q}\mathbf{+}\omega t\right)__b_{l,q}^{\... |
Phys. Rep. 325, 83 (1999). X-G. Huang, Rep. Prog. Phys. 79, 076302 (2016).
J. Schwinger, Phys. Rev. 82, 664(1951). See for example, G. Dunne and T.M. Hall, Phys. Lett. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Pauli, Rev. Mod. Phys. 13, 203(1941). Q.-G. Lin, J. Phys. G. 25, 1793(1999). H.K. Lee and Y.S. Yoon,... | Phys. Rep. 325, 83 (1999). X - G. Huang, Rep. Prog. Phys. 79, 076302 (2016).
J. Schwinger, Phys. Rev. 82, 664(1951). See for example, G. Dunne and T.M. Hall, Phys. Lett. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Pauli, Rev. Mod. Phys. 13, 203(1941). Q.-G. Lin, J. Phys. G. 25, 1793(1999). H.K. Lee and Y.S. Yo... | Phjs. Rep. 325, 83 (1999). X-G. Huang, Rep. Pvog. Phys. 79, 076302 (2016).
J. Schcunger, 'hys. Reb. 82, 664(1951). See wor example, G. Dunne and T.M. Hell, Phys. Oett. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Oauli, Rev. Mod. Phyw. 13, 203(1941). W.-T. Lin, J. Phba. G. 25, 1793(1999). H.K. Lee zkd Y.S. Boon,... | Phys. Rep. 325, 83 (1999). X-G. Huang, Phys. 076302 (2016). Schwinger, Phys. Rev. G. and T.M. Hall, Lett. B419, 322(1998); Rev. D60, 065002(1999). W. Pauli, Rev. Phys. 13, 203(1941). Q.-G. Lin, J. Phys. G. 25, 1793(1999). H.K. Lee and Yoon, JHEP 0603, 078(2006). S.P. Gavrilov and D.M. Gitman Phys. Rev. D 87, (2013). Ki... | Phys. Rep. 325, 83 (1999). X-G. Huang, Rep. Prog. Phys. 79, 076302 (2016).
j. Schwinger, phys. REv. 82, 664(1951). SEe fOr ExamPle, G. dunne and T.M. Hall, pHys. LEtt. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Pauli, Rev. MOd. PhyS. 13, 203(1941). Q.-g. lin, J. pHyS. G. 25, 1793(1999). H.K. LEe and Y.S. yOoN,... | Phys. Rep. 325, 83 (1999) . X-G. Hua ng, R ep. Pr og . Ph ys.79, 076302 (20 1 6).
J. Schwinger, Phys. R ev. 8 2, 664( 1 95 1). S ee fore xa m p le, G .Dun ne an d T.M . H all, Ph ys. Lett.B41 9, 322(1998);P hy s. Rev. D6 0,065002(1999) . W . Paul i, Re v . Mod . P hys.13, 20 3 (1941) . Q.-G. L in , J. Ph y s. G. ... | Phys._Rep. 325,_83 (1999). X-G. Huang,_Rep. Prog._Phys._79, 076302_(2016).
J._Schwinger, Phys. Rev._82, 664(1951). See_for example, G. Dunne_and T.M. Hall,_Phys._Lett. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Pauli, Rev. Mod. Phys. 13, 203(1941)._Q.-G._Lin, J._Phys._G._25, 1793(1999). H.K. Lee and_Y.S. Yoon,... |
beta$ to ${\mathcal{C}}$ does preserve finite products; this is part of proof of [@Glicksberg Thm. 3]. In conclusion, [Theorem [\[thm:main\]]{}]{}(a) applies to the situation $$\xymatrix{
{\mathcal{C}}=\mathsf{PsLocComp} \ar@<0.5ex>[r]^-{F=\beta} & \mathsf{CompHaus}={\mathcal{D}}\,, \ar@<0.5ex>[l]^-{G}
}$$ and sh... | beta$ to $ { \mathcal{C}}$ does preserve finite products; this is part of proof of [ @Glicksberg Thm. 3 ]. In ending, [ Theorem [ \[thm: main\]]{}]{}(a) enforce to the situation $ $ \xymatrix {
{ \mathcal{C}}=\mathsf{PsLocComp } \ar@<0.5ex>[r]^-{F=\beta } & \mathsf{CompHaus}={\mathcal{D}}\, , \ar@<0.5ex>[l]^-... | betw$ to ${\mathcal{C}}$ does presevve finite produers; thiv is pzrt of pfoof of [@Glicksberg Thm. 3]. In coicluwion, [Ukeorem [\[thm:main\]]{}]{}(a) applids to the situatiin $$\xbmatrix{
{\mathcem{C}}=\mathsn{'sLocDlmp} \cr@<0.5xx>[r]^-{F=\beta} & \mathsn{CompHaus}={\madhcal{D}}\,, \ar@<0.5ex>[l]^-{G}
}$$ xnb sh... | beta$ to ${\mathcal{C}}$ does preserve finite products; part proof of Thm. 3]. In the $$\xymatrix{ {\mathcal{C}}=\mathsf{PsLocComp} \ar@<0.5ex>[r]^-{F=\beta} \mathsf{CompHaus}={\mathcal{D}}\,, \ar@<0.5ex>[l]^-{G} }$$ shows that every algebraic structure on pseudocompact and locally compact topological space $X$ ascends... | beta$ to ${\mathcal{C}}$ does preservE finite proDucts; ThiS is PaRt of ProoF of [@Glicksberg THM. 3]. In cOnclusion, [Theorem [\[thm:maiN\]]{}]{}(a) appLiES to tHE sItuatIon $$\xymaTRiX{
{\MAthCaL{C}}=\MatHsF{pslocCoMp} \aR@<0.5ex>[r]^-{F=\beTa} & \mathsf{CoMpHAuS}={\mathcal{D}}\,, \ar@<0.5eX>[L]^-{G}
}$$ And sh... | beta$ to ${\mathcal{C}}$ d oes preser ve fi nit e p ro duct s; t his is part of proo f of [@Glicksberg Thm. 3].In conc l us ion,[Theore m [ \ [ thm :m ai n\] ]{ } ]{ }(a)app lies to the situa tio n$$\xymatrix{ {\mathca l{C }}=\mathsf{P sLo cComp} \ ar@ < 0.5ex >[r ]^-{F =\beta } & \ma thsf{Comp Ha u s}={\m a th... | beta$ to_${\mathcal{C}}$ does_preserve finite products; this_is part_of_proof of_[@Glicksberg_Thm. 3]. In conclusion,_[Theorem [\[thm:main\]]{}]{}(a) applies to_the situation $$\xymatrix{
_ {\mathcal{C}}=\mathsf{PsLocComp}_\ar@<0.5ex>[r]^-{F=\beta}_& \mathsf{CompHaus}={\mathcal{D}}\,, \ar@<0.5ex>[l]^-{G}
}$$ and sh... |
, for instance. The displacement field is, then, $\Delta \mathbf{X}$=$\Delta \widetilde{\mathbf{X}}$ +$\Delta \mathbf{b}$. While the first moment of the displacement field can be free of noise if $\left\langle \mathbf{b} \right\rangle_t=0$, its second moment can be expressed as:
$$\begin{array}{ccl}
\left\langle (... | , for instance. The displacement field is, then, $ \Delta \mathbf{X}$=$\Delta \widetilde{\mathbf{X}}$ + $ \Delta \mathbf{b}$. While the first moment of the displacement sphere can be spare of noise if $ \left\langle \mathbf{b } \right\rangle_t=0 $, its second moment can be expressed as:
$ $ \begin{array}{ccl }
... | , fog instance. The displacemtnt field is, then, $\Delta \kathbf{S}$=$\Delta \wkdetilde{\mathbf{X}}$ +$\Delta \mathbf{u}$. Whule tye first moment of the displacelent fieod cen be free of nomae if $\lcyt\lanfpe \mctibf{b} \right\ranglg_t=0$, its secong moment can ba dx'ressed as:
$$\begin{array}{ccl}
\left\langlq (... | , for instance. The displacement field is, \mathbf{X}$=$\Delta +$\Delta \mathbf{b}$. the first moment be of noise if \mathbf{b} \right\rangle_t=0$, its moment can be expressed as: $$\begin{array}{ccl} (\Delta \mathbf{X})^2\right\rangle_t &=& \left\langle \widetilde{\mathbf{v}}^2\right\rangle_t \Delta t^2 + 2\left\langl... | , for instance. The displacemenT field is, thEn, $\DelTa \mAthBf{x}$=$\DelTa \wiDetilde{\mathbf{X}}$ +$\dElta \Mathbf{b}$. While the first moMent oF tHE disPLaCemenT field cAN bE FRee Of NoIse If $\LEfT\langLe \mAthbf{b} \rIght\rangle_T=0$, itS sEcond moment cAN bE expressed As:
$$\bEgin{array}{ccl}
\LefT\langlE (... | , for instance. The displa cement fie ld is , t hen ,$\De lta\mathbf{X}$=$\ D elta \widetilde{\mathbf{X} }$ +$ \D e lta\ ma thbf{ b}$. Wh i le t hefi rs t m om e nt of t hedisplac ement fiel d c an be free ofn oi se if $\le ft\ langle \math bf{ b} \ri gh t\r a ngle_ t=0 $, it s seco n d mome nt can be e x presse d ... | , for_instance. The_displacement field is, then,_$\Delta \mathbf{X}$=$\Delta_\widetilde{\mathbf{X}}$_+$\Delta \mathbf{b}$._While_the first moment_of the displacement_field can be free_of noise if_$\left\langle_\mathbf{b} \right\rangle_t=0$, its second moment can be expressed as:
$$\begin{array}{ccl}
\left\langle (... |
Pr[E(y_i,Z)]^2\Pr[Y=y_i],$$ since $Z$ and $Z'$ are i.i.d. Therefore, by Jensen’s inequality, we obtain $$\label{eqn:Jensen-1}
\Pr[E(Y,Z)]^2\leq \sum_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i]=\Pr[E(Y,Z)\cap E(Y,Z')].$$ We also have $$\Pr[E(Y,Z)\cap E(Y,Z')]=\sum_{i=1}^m\sum_{j=1}^m\Pr[E(Y,z_i)\cap E(Y,z_j)]\Pr[Z=z_i]\Pr[Z=z_j],$... | Pr[E(y_i, Z)]^2\Pr[Y = y_i],$$ since $ Z$ and $ Z'$ are i.i.d. Therefore, by Jensen ’s inequality, we obtain $ $ \label{eqn: Jensen-1 }
\Pr[E(Y, Z)]^2\leq \sum_{i=1}^n\Pr[E(y_i, Z)]^2\Pr[Y = y_i]=\Pr[E(Y, Z)\cap E(Y, Z')].$$ We also have $ $ \Pr[E(Y, Z)\cap E(Y, Z')]=\sum_{i=1}^m\sum_{j=1}^m\Pr[E(Y, z_i)\cap E(Y, z_j... | Pr[E(j_i,Z)]^2\Pr[Y=y_i],$$ since $Z$ and $Z'$ ave i.i.d. Therefore, by Jenven’s ihequalith, we obtain $$\label{eqn:Jensen-1}
\Pr[X(Y,Z)]^2\lwq \sun_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i]=\Pr[E(Y,Z)\cap E(Y,Z')].$$ We apso have $$\Pr[E(B,Z)\cap E(Y,Z')]=\sum_{i=1}^m\sum_{j=1}^m\Pr[E(Y,z_l)\eap E(G,d_j)]\Pr[Z=v_i]\'r[Z=z_j],$... | Pr[E(y_i,Z)]^2\Pr[Y=y_i],$$ since $Z$ and $Z'$ are i.i.d. Jensen’s we obtain \Pr[E(Y,Z)]^2\leq \sum_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i]=\Pr[E(Y,Z)\cap E(Y,Z')].$$ E(Y,z_j)]\Pr[Z=z_i]\Pr[Z=z_j],$$ $$\Pr[E(Y,Z)\cap E(Y,Z')\cap E(Y',Z)\cap \sum_{i=1}^n\sum_{j=1}^n \Pr[E(Y,z_i)\cap E(Y,z_j)]^2\Pr[Z=z_i]\Pr[Z=z_j].$$ again, by... | Pr[E(y_i,Z)]^2\Pr[Y=y_i],$$ since $Z$ and $Z'$ are i.I.d. ThereforE, by JeNseN’s iNeQualIty, wE obtain $$\label{eqN:jensEn-1}
\Pr[E(Y,Z)]^2\leq \sum_{i=1}^n\Pr[E(y_i,Z)]^2\PR[Y=y_i]=\PR[E(y,z)\cap e(y,Z')].$$ we alsO have $$\Pr[e(y,Z)\CAP E(Y,z')]=\sUm_{I=1}^m\sUm_{J=1}^M\PR[E(Y,z_i)\Cap e(Y,z_j)]\Pr[Z=Z_i]\Pr[Z=z_j],$... | Pr[E(y_i,Z)]^2\Pr[Y=y_i],$ $ since $Z $ and $Z '$ar e i. i.d. Therefore, by Jens en’s inequality, we ob tain$$ \ labe l {e qn:Je nsen-1} \P r [ E(Y ,Z )] ^2\ le q \ sum_{ i=1 }^n\Pr[ E(y_i,Z)]^ 2\P r[ Y=y_i]=\Pr[E ( Y, Z)\cap E(Y ,Z' )].$$ We als o h ave $$ \P r[E ( Y,Z)\ cap E(Y, Z')]=\ s um_{i= 1}^m\sum_ {j = 1}^... | Pr[E(y_i,Z)]^2\Pr[Y=y_i],$$ since_$Z$ and_$Z'$ are i.i.d. Therefore,_by Jensen’s_inequality,_we obtain_$$\label{eqn:Jensen-1}
\Pr[E(Y,Z)]^2\leq_\sum_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i]=\Pr[E(Y,Z)\cap E(Y,Z')].$$ We_also have $$\Pr[E(Y,Z)\cap_E(Y,Z')]=\sum_{i=1}^m\sum_{j=1}^m\Pr[E(Y,z_i)\cap E(Y,z_j)]\Pr[Z=z_i]\Pr[Z=z_j],$... |
As already mentioned, SN1992A is one of the best ever observed Ia SNe and therefore enters the zero point determination of the Ia Hubble diagram with a high weight. The GCS of NGC1380 has been analysed by Kissler-Patig et al. [@kissler:97] and Della Valle et al. [@della:98]. Interestingly, it turned out that the GCS c... | As already mentioned, SN1992A is one of the best ever observed Ia SNe and consequently insert the zero point decision of the Ia Hubble diagram with a gamey weight. The GCS of NGC1380 has been analysed by Kissler - Patig et al. [ @kissler:97 ] and Della Valle et al. [ @della:98 ]. Interestingly, it flex out that t... |
As wlready mentioned, SN1992A is one of the besj wver ouserved Ia SNe xnd therefore enters the zerl point determination of the Ka Hubble diagram witi a high weight. Vge GCS of NGC1380 mas bzei analysed by Klssler-Patig et al. [@kissler:97] dna Bella Valle et al. [@della:98]. Interestingly, yt turnrd out that the DCS b... | As already mentioned, SN1992A is one of ever Ia SNe therefore enters the Ia diagram with a weight. The GCS NGC1380 has been analysed by Kissler-Patig al. [@kissler:97] and Della Valle et al. [@della:98]. Interestingly, it turned out that GCS could be separated into an elongated metal-rich bulge and a spherical, metal-p... |
As already mentioned, SN1992A is onE of the best Ever oBseRveD IA SNe And tHerefore enters THe zeRo point determination of The Ia huBBle dIAgRam wiTh a high WEiGHT. ThE GcS Of NgC1380 HAs Been aNalYsed by KIssler-PatiG et Al. [@Kissler:97] and DeLLa valle et al. [@dEllA:98]. InterestingLy, iT turneD oUt tHAt the gCS C... |
As already mentioned, SN1 992A is on e ofthe be st eve r ob served Ia SNea nd t herefore enters the ze ro po in t det e rm inati on of t h eI a Hu bb le di ag r am with ahigh we ight. TheGCS o f NGC1380 ha s b een analys edby Kissler-P ati g et a l. [@ k issle r:9 7] an d Dell a Valle et al. [ @d e lla:98 ... |
As already_mentioned, SN1992A_is one of the_best ever_observed_Ia SNe_and_therefore enters the_zero point determination_of the Ia Hubble_diagram with a_high_weight. The GCS of NGC1380 has been analysed by Kissler-Patig et al. [@kissler:97] and Della Valle_et al._[@della:98]. Interestingly,_it_turned_out that the GCS c... |
V. Loo, L. Lanco, A. Lemaitre, I. Sagnes, O. Krebs, P. Voisin, and P. Senellart, Appl. Phys. Lett. [**97**]{}, 241110 (2010).
R. J. Barbour, P. A. Dalgarno, A. Curran, K. M. Nowak, H. J. Baker, D. R. Hall, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, J. Appl. Phys. [**110**]{}, 053107 (2011).
We assume that a s... | V. Loo, L. Lanco, A. Lemaitre, I. Sagnes, O. Krebs, P. Voisin, and P. Senellart, Appl. Phys. Lett. [ * * 97 * * ] { }, 241110 (2010).
R. J. Barbour, P. A. Dalgarno, A. Curran, K. M. Nowak, H. J. Baker, D. R. Hall, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, J. Appl. Phys. [ * * 110 * * ] { }, 053107 (2011).
... |
V. Llo, L. Lanco, A. Lemaitre, I. Ragnes, O. Krebs, K. Coisin, and P. Senellaft, Appl. Phys. Lett. [**97**]{}, 241110 (2010).
R. J. Barblue, P. A. Dalgarno, A. Curran, K. M. Nowak, H. U. Baker, E. R. Iall, N. G. Stoltz, '. M. Petroff, ans R. J. Xarburton, J. Appk. Phys. [**110**]{}, 053107 (2011).
Wa assume that d r... | V. Loo, L. Lanco, A. Lemaitre, I. Krebs, Voisin, and Senellart, Appl. Phys. J. P. A. Dalgarno, Curran, K. M. H. J. Baker, D. R. Hall, G. Stoltz, P. M. Petroff, and R. J. Warburton, J. Appl. Phys. [**110**]{}, (2011). We assume that a single layer of quantum dots is put inside cavity linear is $\mu$m as in Ref.. N. Perr... |
V. Loo, L. Lanco, A. Lemaitre, I. SagneS, O. Krebs, P. VoIsin, aNd P. senElLart, appl. phys. Lett. [**97**]{}, 241110 (2010).
R. J. BarbOUr, P. A. dalgarno, A. Curran, K. M. Nowak, h. J. BakEr, d. r. HalL, n. G. stoltZ, P. M. PetrOFf, AND R. J. waRbUrtOn, j. apPl. PhyS. [**110**]{}, 053107 (2011).
We Assume tHat a s... |
V. Loo, L. Lanco, A. Lema itre, I. S agnes , O . K re bs,P. V oisin, and P.S enel lart, Appl. Phys. Lett . [** 97 * *]{} , 2 41110 (2010) .
R . J. B ar bou r, P. A. D alg arno, A . Curran,K.M. Nowak, H. J . B aker, D. R . H all, N. G. S tol tz, P. M . P e troff , a nd R. J. Wa r burton , J. Appl .P hys. [ * ... |
V. Loo,_L. Lanco,_A. Lemaitre, I. Sagnes,_O. Krebs,_P._Voisin, and_P._Senellart, Appl. Phys._Lett. [**97**]{}, 241110_(2010).
R. J. Barbour, P._A. Dalgarno, A._Curran,_K. M. Nowak, H. J. Baker, D. R. Hall, N. G. Stoltz, P. M._Petroff,_and R._J._Warburton,_J. Appl. Phys. [**110**]{}, 053107_(2011).
We assume that a s... |
0 \to L(C_0) \to E \to F \to 0$$ such that $E$ is semi-stable (see the proof of Proposition \[prop:1\]). Thus $Z$ is not empty. Therefore, to compute the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$, it is enough to consider $Pic(M^0 \cup (Z^0 \cap M_0^{\mu}) )$, where $Z^0$ is the subscheme of $M_H(r;c_1,c_2)$ con... | 0 \to L(C_0) \to E \to F \to 0$$ such that $ E$ is semi - stable (see the proof of Proposition \[prop:1\ ]). Thus $ Z$ is not empty. consequently, to calculate the Picard group of $ \overline{M}_{H_x}(r;c_1,c_2)$, it is adequate to consider $ Pic(M^0 \cup (Z^0 \cap M_0^{\mu }) ) $, where $ Z^0 $ is the subscheme of $ M... |
0 \to L(C_0) \to E \to F \to 0$$ such tmat $E$ is semi-stayoe (see the pdoof of Oroposition \[prop:1\]). Thus $Z$ is nlt emptt. Therefore, to compute the Picagd group if $\oterline{M}_{H_x}(r;c_1,c_2)$, it is enounk to dlnsibec $Pic(M^0 \cup (Z^0 \cak M_0^{\mu}) )$, where $Z^0$ is the subswhdmz of $M_H(r;c_1,c_2)$ con... | 0 \to L(C_0) \to E \to F such $E$ is (see the proof is empty. Therefore, to the Picard group $\overline{M}_{H_x}(r;c_1,c_2)$, it is enough to consider \cup (Z^0 \cap M_0^{\mu}) )$, where $Z^0$ is the subscheme of $M_H(r;c_1,c_2)$ consisting stable sheaves which are defined by the exact sequence. We set $V_i=\cal O_X(-n... |
0 \to L(C_0) \to E \to F \to 0$$ such that $E$ is seMi-stable (seE the pRooF of prOposItioN \[prop:1\]). Thus $Z$ is noT EmptY. Therefore, to compute the picarD gROup oF $\OvErlinE{M}_{H_x}(r;c_1,c_2)$, IT iS ENouGh To ConSiDEr $pic(M^0 \cUp (Z^0 \Cap M_0^{\mu}) )$, wHere $Z^0$ is the SubScHeme of $M_H(r;c_1,c_2)$ cON... |
0 \to L(C_0) \to E \to F\to 0$$ su ch th at$E$ i s se mi-s table (see the proo f of Proposition \[pro p:1\] ). Thus $Z $ isnot emp t y. T her ef or e,to co mpute th e Picar d group of $\ ov erline{M}_{H _ x} (r;c_1,c_2 )$, it is enoug h t o cons id er$ Pic(M ^0\cup(Z^0 \ c ap M_0 ^{\mu}) ) $, where$ Z^0$ is t he ... |
0 \to_L(C_0) \to_E \to F \to_0$$ such_that_$E$ is_semi-stable_(see the proof_of Proposition \[prop:1\])._Thus $Z$ is not_empty. Therefore, to_compute_the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$, it is enough to consider $Pic(M^0 \cup (Z^0 \cap_M_0^{\mu})_)$, where_$Z^0$_is_the subscheme of $M_H(r;c_1,c_2)$ con... |
h$ and $\ell$ are odd integers).[@Matsumura] This suggests that some structural change takes place at $T=T_{\rm Q}$ from the tetragonal phase at high temperatures.[@Tanaka; @Hirota] A buckling of sheets of B and C atoms was proposed,[@Tanaka] and the non-resonant intensities by the buckling has recently been evaluated;... | h$ and $ \ell$ are odd integers).[@Matsumura ] This suggests that some structural change take home at $ T = T_{\rm Q}$ from the tetragonal phase at high temperatures.[@Tanaka; @Hirota ] A buckling of sheets of B and C atom was proposed,[@Tanaka ] and the non - resonant intensities by the buckling has recently been eval... | h$ ajd $\ell$ are odd integers).[@Mxtsumura] This sogtests vhat soje strucgural change takes place at $V=T_{\rm Q}$ frim the tetragonal phasd at high temperarurew.[@Ranaka; @Hirota] A bughlinf of vieets of B and G atoms was proposed,[@Tanakd] xnb the non-resonant intensities by the bucklimg has recently feen qvalhated;... | h$ and $\ell$ are odd integers).[@Matsumura] This some change takes at $T=T_{\rm Q}$ high @Hirota] A buckling sheets of B C atoms was proposed,[@Tanaka] and the intensities by the buckling has recently been evaluated; about $0.01$ ${\rm \AA}$ shift B and/or C atoms may be sufficient to give rise to such large It not in... | h$ and $\ell$ are odd integers).[@MatsUmura] This sUggesTs tHat SoMe stRuctUral change takeS PlacE at $T=T_{\rm Q}$ from the tetragoNal phAsE At hiGH tEmperAtures.[@TANaKA; @hirOtA] A BucKlINg Of sheEts Of B and C Atoms was prOpoSeD,[@Tanaka] and thE NoN-resonant iNteNsities by the BucKling hAs RecENtly bEen EvaluAted;... | h$ and $\ell$ are odd inte gers).[@Ma tsumu ra] Th is sug gest s that some st r uctu ral change takes place at $ T= T _{\r m Q }$ fr om thet et r a gon al p has ea thightem peratur es.[@Tanak a;@H irota] A buc k li ng of shee tsof B and C a tom s waspr opo s ed,[@ Tan aka]and th e non-r esonant i nt e nsitie s b... | h$ and_$\ell$ are_odd integers).[@Matsumura] This suggests_that some_structural_change takes_place_at $T=T_{\rm Q}$_from the tetragonal_phase at high temperatures.[@Tanaka;_@Hirota] A buckling_of_sheets of B and C atoms was proposed,[@Tanaka] and the non-resonant intensities by the_buckling_has recently_been_evaluated;... |
by @jordan04, mass segregation and stellar evolution effects are the leading causes of the blue/red GC size differences. In this scenario little change of relative GC sizes with galactocentric distances is expected.
By comparing the average half-light radii of the red and blue GC in NGC5866 we do not find any signifi... | by @jordan04, mass segregation and stellar evolution effects are the lead campaign of the blue / red GC size differences. In this scenario little variety of relative GC sizes with galactocentric distances is expect.
By comparing the average half - unaccented radii of the red and aristocratic GC in NGC5866 we do not ... | by @jordan04, mass segregation and stellar evolution effecfs are tfe leading causes of the blux/red GC suze differences. In thir scenaril little chaige of relative JD sizes with fwlaccorentric distancgs is expectad.
By comparing tfe average half-light radii of the red and blie GC in NGC5866 we qo npe fihd any signifi... | by @jordan04, mass segregation and stellar evolution the causes of blue/red GC size change relative GC sizes galactocentric distances is By comparing the average half-light radii the red and blue GC in NGC5866 we do not find any significant between the two subpopulations. Furthermore, the size versus galactocentric rad... | by @jordan04, mass segregation anD stellar evOlutiOn eFfeCtS are The lEading causes of THe blUe/red GC size differences. in thiS sCEnarIO lIttle Change oF ReLATivE Gc sIzeS wITh GalacTocEntric dIstances is ExpEcTed.
By compariNG tHe average hAlf-Light radii of The Red and BlUe Gc In NGC5866 We dO not fInd any SIgnifi... | by @jordan04, mass segreg ation andstell arevo lu tion eff ects are the l e adin g causes of the blue/r ed GC s i ze d i ff erenc es. Int hi s sce na ri o l it t le chan geof rela tive GC si zes w ith galactoc e nt ric distan ces is expected .
By com pa rin g theave ragehalf-l i ght ra dii of th er ed and blue G... | by_@jordan04, mass_segregation and stellar evolution_effects are_the_leading causes_of_the blue/red GC_size differences. In_this scenario little change_of relative GC_sizes_with galactocentric distances is expected.
By comparing the average half-light radii of the red and_blue_GC in_NGC5866_we_do not find any signifi... |
tilde{E}(\tilde{T})-\tilde{E}_{bp}$ for the parameter values $\tilde{\omega}_i$ presented in Fig. \[fig1\], \[fig2\]. \[fig3\]](fig3_2.eps "fig:"){width="80.00000%"}
The dependencies obtained enable us to find the heat capacity of a TI bipolaron gas: $C_v(\tilde{T})=d\tilde{E}/d\tilde{T}$. With the use of (20) $C_v(\t... | tilde{E}(\tilde{T})-\tilde{E}_{bp}$ for the parameter values $ \tilde{\omega}_i$ presented in Fig. \[fig1\ ], \[fig2\ ]. \[fig3\]](fig3_2.eps " fig:"){width="80.00000% " }
The dependencies prevail enable us to receive the heat capability of a TI bipolaron flatulence: $ C_v(\tilde{T})=d\tilde{E}/d\tilde{T}$. With t... | tilfe{E}(\tilde{T})-\tilde{E}_{bp}$ for tht parameter values $\tilde{\mmega}_i$ presentdd in Fig. \[fig1\], \[fig2\]. \[fig3\]](fig3_2.eps "fmg:"){wieth="80.00000%"}
Tht dependencies obtakned enabpe us to finv the heat capacmfy of a TI billlarmi gas: $C_v(\tilde{T})=d\jilde{E}/d\tilde{D}$. With the use ow (20) $C_v(\t... | tilde{E}(\tilde{T})-\tilde{E}_{bp}$ for the parameter values $\tilde{\omega}_i$ presented \[fig1\], \[fig3\]](fig3_2.eps "fig:"){width="80.00000%"} dependencies obtained enable capacity a TI bipolaron $C_v(\tilde{T})=d\tilde{E}/d\tilde{T}$. With the of (20) $C_v(\tilde{T})$ for $\tilde{T}\leq\tilde{T}_c$ is as: $$\labe... | tilde{E}(\tilde{T})-\tilde{E}_{bp}$ for thE parameter ValueS $\tiLde{\OmEga}_i$ PresEnted in Fig. \[fig1\], \[fIG2\]. \[fig3\]](Fig3_2.eps "fig:"){width="80.00000%"}
The depenDenciEs OBtaiNEd EnablE us to fiND tHE HeaT cApAciTy OF a tI bipOlaRon gas: $C_V(\tilde{T})=d\tiLde{e}/d\Tilde{T}$. With thE UsE of (20) $C_v(\t... | tilde{E}(\tilde{T})-\tilde {E}_{bp}$for t hepar am eter val ues $\tilde{\o m ega} _i$ presented in Fig.\[fig 1\ ] , \[ f ig 2\].\[fig3\ ] ]( f i g3_ 2. ep s " fi g :" ){wid th= "80.000 00%"}
The de pe ndencies obt a in ed enableusto find thehea t capa ci tyo f a T I b ipola ron ga s : $C_v (\tilde{T }) = d\tild e {E... | tilde{E}(\tilde{T})-\tilde{E}_{bp}$ for_the parameter_values $\tilde{\omega}_i$ presented in_Fig. \[fig1\], \[fig2\]._\[fig3\]](fig3_2.eps_"fig:"){width="80.00000%"}
The dependencies_obtained_enable us to_find the heat_capacity of a TI_bipolaron gas: $C_v(\tilde{T})=d\tilde{E}/d\tilde{T}$._With_the use of (20) $C_v(\t... |
We identify satellite galaxies in a suite of eight simulations of the formation of $L_*$ galaxies in the $\Lambda$CDM scenario. This series has been presented by Abadi, Navarro & Steinmetz (2006), and follow the same numerical scheme originally introduced by @steinmetzandnavarro02. The “primary” galaxies in these simu... | We identify satellite galaxies in a suite of eight simulations of the constitution of $ L_*$ galaxy in the $ \Lambda$CDM scenario. This series has been presented by Abadi, Navarro & Steinmetz (2006), and follow the like numerical scheme originally introduce by @steinmetzandnavarro02. The “ primary ” galaxies in these s... |
We ldentify satellite galaxles in a suite oy eight simulztions ow the formation of $L_*$ galaxied un tht $\Lambda$CDM scenariu. This segies has veen presented ug Abadi, Navardl & Scemnmetz (2006), and folkow the sake numerical swhdmz originally introduced by @steinmetzwndnavatrl02. The “primary” dalaqiqs ih these simu... | We identify satellite galaxies in a suite simulations the formation $L_*$ galaxies in has presented by Abadi, & Steinmetz (2006), follow the same numerical scheme originally by @steinmetzandnavarro02. The “primary” galaxies in these simulations have been analyzed in detail several recent papers, which the interested re... |
We identify satellite galaxiEs in a suite Of eigHt sImuLaTionS of tHe formation of $L_*$ GAlaxIes in the $\Lambda$CDM scenaRio. ThIs SErieS HaS been PresentED bY aBadI, NAvArrO & STEiNmetz (2006), And Follow tHe same numeRicAl Scheme originALlY introduceD by @SteinmetzandNavArro02. ThE “pRimARy” galAxiEs in tHese siMU... |
We identify satellite gal axies in a suit e o f e ig ht s imul ations of thef orma tion of $L_*$ galaxies in t he $\La m bd a$CDM scenar i o. T his s er ies h a sbeenpre sentedby Abadi,Nav ar ro & Steinme t z(2006), an d f ollow the sa menumeri ca l s c hemeori ginal ly int r oduced by @stei nm e tzandn a varro0... |
We identify_satellite galaxies_in a suite of_eight simulations_of_the formation_of_$L_*$ galaxies in_the $\Lambda$CDM scenario._This series has been_presented by Abadi,_Navarro_& Steinmetz (2006), and follow the same numerical scheme originally introduced by @steinmetzandnavarro02. The_“primary”_galaxies in_these_simu... |
\in \mathcal{D}_{m+1}(U)$ as the current $\p T(\omega):= T(d \omega)$. The **support** $supp \: T $ of a current $T$ is the complement of the union of all open sets $W$ such that $T (\omega) = 0$ for $\omega \in \mathcal{D}^n(U)$ with $supp \; \omega \subset W$. For any open $W \subset U$ and $T \in \mathcal{D}_{m}(U... | \in \mathcal{D}_{m+1}(U)$ as the current $ \p T(\omega):= T(d \omega)$. The * * support * * $ supp \: T $ of a current $ T$ is the complement of the union of all open set $ W$ such that $ metric ton (\omega) = 0 $ for $ \omega \in \mathcal{D}^n(U)$ with $ supp \; \omega \subset W$. For any open $ W \subset U$ and $ T ... | \in \mathcal{D}_{m+1}(U)$ as the currekt $\p T(\omega):= T(d \omega)$. Thx **suppodt** $supp \: T $ of a current $T$ is the colpoemenu of the union of aul open svts $W$ sucy thet $T (\omega) = 0$ for $\omega \ik \matggal{D}^n(B)$ xith $supp \; \omegs \subset W$. For any open $F \ruyset U$ and $T \in \mathcal{D}_{m}(U... | \in \mathcal{D}_{m+1}(U)$ as the current $\p T(\omega):= The $supp \: $ of a of union of all sets $W$ such $T (\omega) = 0$ for $\omega \mathcal{D}^n(U)$ with $supp \; \omega \subset W$. For any open $W \subset U$ $T \in \mathcal{D}_{m}(U)$ we write $T \llcorner W$ for the current in $\mathcal{D}_{m}(W)$ get **restrict... | \in \mathcal{D}_{m+1}(U)$ as the current $\p t(\omega):= T(d \omEga)$. ThE **suPpoRt** $Supp \: t $ of a Current $T$ is the cOMpleMent of the union of all opeN sets $w$ sUCh thAT $T (\Omega) = 0$ For $\omegA \In \MAThcAl{d}^n(u)$ wiTh $SUpP \; \omegA \suBset W$. FoR any open $W \sUbsEt u$ and $T \in \mathcAL{D}_{M}(U... | \in \mathcal{D}_{m+1}(U)$ as the cu rrent $\ p T (\ omeg a):= T(d \omega)$. The**support** $supp \: T $ of a curr e nt $T$is thec om p l eme nt o f t he un ion o f a ll open sets $W$suc hthat $T (\om e ga ) = 0$ for $\ omega \in \m ath cal{D} ^n (U) $ with $s upp \ ; \ome g a \sub set W$. F or any op e n $W... | \in_\mathcal{D}_{m+1}(U)$ as_the current $\p T(\omega):=_T(d \omega)$._The_**support** $supp_\:_T $ of_a current $T$_is the complement of_the union of_all_open sets $W$ such that $T (\omega) = 0$ for $\omega \in \mathcal{D}^n(U)$ with_$supp_\; \omega_\subset_W$._For any open $W \subset_U$ and $T \in_\mathcal{D}_{m}(U... |
-by-name, the pioneering paper [@pag41] by Plotkin already verifies completeness of the $\lambda$-calculus in Thm. 6 (p. 153). The proof is by simulation relations. It is extended by de Groote to the call-by-name $\lambda\mu$-calculus [@owr18]. Fujita establishes completeness by the inverse translation [@cmv29][@eiq93]... | -by - name, the pioneering paper [ @pag41 ] by Plotkin already verifies completeness of the $ \lambda$-calculus in Thm. 6 (p. 153). The proof is by model relative. It is extended by de Groote to the call - by - name $ \lambda\mu$-calculus [ @owr18 ]. Fujita establish completeness by the inverse translation [ @cmv... | -by-nwme, the pioneering paper [@pag41] by Plotkin alreadb verifjes compueteness of the $\lambda$-calculns ib Thm. 6 (p. 153). The proof is by simjlation rvlations. Ut iw extended ug de Groote to bhe cclo-by-name $\lambda\ku$-calculus [@owr18]. Fujita esdaclnshes completeness by the inverse trwnslatipn [@cmv29][@eiq93]... | -by-name, the pioneering paper [@pag41] by Plotkin completeness the $\lambda$-calculus Thm. 6 (p. simulation It is extended de Groote to call-by-name $\lambda\mu$-calculus [@owr18]. Fujita establishes completeness the inverse translation [@cmv29][@eiq93]. He manages to deal with reduction using an idea to ours. Also Ho... | -by-name, the pioneering paper [@pAg41] by PlotkiN alreAdy VerIfIes cOmplEteness of the $\laMBda$-cAlculus in Thm. 6 (p. 153). The proof iS by siMuLAtioN ReLatioNs. It is eXTeNDEd bY dE GRooTe TO tHe calL-by-Name $\lamBda\mu$-calcuLus [@OwR18]. Fujita estabLIsHes completEneSs by the inverSe tRanslaTiOn [@cMV29][@eiq93]... | -by-name, the pioneering p aper [@pag 41] b y P lot ki n al read y verifies com p lete ness of the $\lambda$- calcu lu s inT hm . 6 ( p. 153) . T h e pr oo fisby si mulat ion relati ons. It is ex te nded by de G r oo te to thecal l-by-name $\ lam bda\mu $- cal c ulus[@o wr18] . Fuji t a esta blishes c om p letene ... | -by-name, the_pioneering paper_[@pag41] by Plotkin already_verifies completeness_of_the $\lambda$-calculus_in_Thm. 6 (p. 153). The_proof is by_simulation relations. It is_extended by de Groote_to_the call-by-name $\lambda\mu$-calculus [@owr18]. Fujita establishes completeness by the inverse translation [@cmv29][@eiq93]... |
Metelmann and A. A. Clerk, Quantum-limited amplification via reservoir engineering, Phys. Rev. Lett. **112**, 133904 (2014).
L. Mercier de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, M. A. Sillanpää, Realization of directional amplification in a microwave optomechanical device, arXiv:1811.06036.
A. Nunnenkamp, V. S... | Metelmann and A. A. Clerk, Quantum - limited amplification via reservoir engineering, Phys. Rev. Lett. * * 112 * *, 133904 (2014).
L. Mercier de Lépinay, E. Damskägg, C. F. Ockeloen - Korppi, M. A. Sillanpää, Realization of directional amplification in a microwave optomechanical device, arXiv:1811.06036.
A. Nunne... | Mehelmann and A. A. Clerk, Quxntum-limited amkluficatmon via reservokr engineering, Phys. Rev. Lett. **112**, 133904 (2014).
L. Meecier de Lépinay, E. Damsyägg, C. F. Obkeloen-Koeppi, N. A. Sillan'ää, Realization kn dirzcvional amplificstion in a microwave optmmdckanical device, arXiv:1811.06036.
A. Nunnenkamp, V. S... | Metelmann and A. A. Clerk, Quantum-limited amplification engineering, Rev. Lett. 133904 (2014). L. C. Ockeloen-Korppi, M. A. Realization of directional in a microwave optomechanical device, arXiv:1811.06036. Nunnenkamp, V. Sudhir, A. K. Feofanov, A. Roulet, and T. J. Kippenberg, Quantum-limited and parametric instabili... | Metelmann and A. A. Clerk, QuantuM-limited amPlifiCatIon ViA resErvoIr engineering, PHYs. ReV. Lett. **112**, 133904 (2014).
L. Mercier de Lépinay, e. DamsKäGG, C. F. OCKeLoen-KOrppi, M. A. sIlLANpäÄ, REaLizAtIOn Of dirEctIonal amPlificatioN in A mIcrowave optoMEcHanical devIce, ArXiv:1811.06036.
A. NunnenKamP, V. S... | Metelmann and A. A. Clerk , Quantum- limit edamp li fica tion via reservoir engi neering, Phys. Rev. Le tt. * *1 1 2**, 13 3904(2014).
L . Mer ci er de L é pi nay,E.Damskäg g, C. F. O cke lo en-Korppi, M . A . Sillanpä ä,Realizationofdirect io nal ampli fic ation in am icrowa ve optome ch a nicald evice,a r Xi v... | Metelmann_and A._A. Clerk, Quantum-limited amplification_via reservoir_engineering,_Phys. Rev._Lett._**112**, 133904 (2014).
L._Mercier de Lépinay,_E. Damskägg, C. F._Ockeloen-Korppi, M. A._Sillanpää,_Realization of directional amplification in a microwave optomechanical device, arXiv:1811.06036.
A. Nunnenkamp, V. S... |
ta- hypernuclei. For anti-flavour (positive strangeness, beauty or negative charm) the same formula as above holds, but with certain changes for the hyperfine splitting constants, $c_F \to c_{\bar F}$ and $\bar c_F \to \bar c_{\bar F}$ in the last term $\Delta M_{HFS} $. $c_{\bar F}$ ($\bar c_{\bar F}$) is obtained fro... | ta- hypernuclei. For anti - flavour (positive strangeness, beauty or negative appeal) the like formula as above holds, but with sealed changes for the hyperfine splitting constant, $ c_F \to c_{\bar F}$ and $ \bar c_F \to \bar c_{\bar F}$ in the last term $ \Delta M_{HFS } $. $ c_{\bar F}$ ($ \bar c_{\bar F}$) is obtai... | ta- jypernuclei. For anti-flavuur (positive sttabgenesv, beaufy or neeative charm) the same formule as abovt holds, but with ceftain chajges for the yyperfine splittinn conabants, $r_F \to c_{\bar F}$ anc $\bar c_F \tm \bar c_{\bar F}$ it ghz last term $\Delta M_{HFS} $. $c_{\bar F}$ ($\bar c_{\far F}$) ix lbtained fro... | ta- hypernuclei. For anti-flavour (positive strangeness, beauty charm) same formula above holds, but hyperfine constants, $c_F \to F}$ and $\bar \to \bar c_{\bar F}$ in the term $\Delta M_{HFS} $. $c_{\bar F}$ ($\bar c_{\bar F}$) is obtained from $c_F$ c_F$) by means of substitution $\mu\to -\mu$: $$c_{\bar F} =1-{\The... | ta- hypernuclei. For anti-flavoUr (positive StranGenEss, BeAuty Or neGative charm) the SAme fOrmula as above holds, but wIth ceRtAIn chANgEs for The hypeRFiNE SplItTiNg cOnSTaNts, $c_F \To c_{\Bar F}$ and $\Bar c_F \to \bar C_{\baR F}$ In the last terM $\deLta M_{HFS} $. $c_{\baR F}$ ($\bAr c_{\bar F}$) is obtAinEd fro... | ta- hypernuclei. For anti- flavour (p ositi vestr an gene ss,beauty or nega t ivecharm) the same formul a asab o ve h o ld s, bu t withc er t a inch an ges f o rthe h ype rfine s plitting c ons ta nts, $c_F \t o c _{\bar F}$ an d $\bar c_F\to \barc_ {\b a r F}$ in thelast t e rm $\D elta M_{H FS } $. $c _ {\bar ... | ta- hypernuclei._For anti-flavour_(positive strangeness, beauty or_negative charm)_the_same formula_as_above holds, but_with certain changes_for the hyperfine splitting_constants, $c_F \to_c_{\bar_F}$ and $\bar c_F \to \bar c_{\bar F}$ in the last term $\Delta M_{HFS}_$._$c_{\bar F}$_($\bar_c_{\bar_F}$) is obtained fro... |
s in Mathematics Vol. 227, Springer-Verlag, New York, 2004.
M. Reineke, Quivers, desingularizations and canonical bases. Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), 325–344, Progr. Math., 210, Birkhäuser Boston, Boston, MA, 2003. [math.AG/0104284]{}
C. M. Ringel, Representations of $K$-species and bi... | s in Mathematics Vol. 227, Springer - Verlag, New York, 2004.
M. Reineke, Quivers, desingularizations and canonical bases. Studies in memory of Issai Schur (Chevaleret / Rehovot, 2000), 325–344, Progr. Math. , 210, Birkhäuser Boston, Boston, MA, 2003. [ math. AG/0104284 ] { }
C. M. Ringel, Representations of $ K$... | s ij Mathematics Vol. 227, Sprinner-Verlag, New Yotk, 2004.
M. Reiieke, Qujvers, deringularizations and canonicel bqses. Wtudies in memory of Irsai Schug (Chevaleeet/Rthovot, 2000), 325–344, Progr. Mavg., 210, Birkmäbser Glstou, Uoston, MA, 2003. [math.SG/0104284]{}
C. M. Ringal, Representathovs of $K$-species and bi... | s in Mathematics Vol. 227, Springer-Verlag, New M. Quivers, desingularizations canonical bases. Studies (Chevaleret/Rehovot, 325–344, Progr. Math., Birkhäuser Boston, Boston, 2003. [math.AG/0104284]{} C. M. Ringel, Representations $K$-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302. C. M. Ringel, algebras a... | s in Mathematics Vol. 227, Springer-verlag, New YOrk, 2004.
M. REinEke, quIverS, desIngularizationS And cAnonical bases. Studies in MemorY oF issaI scHur (ChEvalereT/reHOVot, 2000), 325–344, prOgR. MaTh., 210, bIrKhäusEr BOston, BoSton, MA, 2003. [math.aG/0104284]{}
C. m. RIngel, RepreseNTaTions of $K$-spEciEs and bi... | s in Mathematics Vol. 227, Springer- Verla g,New Y ork, 200 4.
M. Reineke , Qui vers, desingularizatio ns an dc anon i ca l bas es. Stu d ie s inme mo ryof Is sai S chu r (Chev aleret/Reh ovo t, 2000), 325– 3 44 , Progr. M ath ., 210, Birk häu ser Bo st on, Bosto n,MA, 2 003. [ m ath.AG /0104284] {}
C. M. Ringel... | s in_Mathematics Vol._227, Springer-Verlag, New York,_2004.
M. Reineke,_Quivers,_desingularizations and_canonical_bases. Studies in_memory of Issai_Schur (Chevaleret/Rehovot, 2000), 325–344,_Progr. Math., 210,_Birkhäuser_Boston, Boston, MA, 2003. [math.AG/0104284]{}
C. M. Ringel, Representations of $K$-species and bi... |
uklm\], we obtain that $\p(c^*)\le\p\cuk(a^*)$. Since $\p(d)\le\p\cuk(b)$, we get that $(\p\cuk(a^*))^*\le (\p(c^*))^*\lle
\p(d)\le\p\cuk(b)$. Therefore, $(\p\cuk(a^*))^*\lle\p\cuk(b)$. So, (DLC3) is fulfilled.
For verifying (DLC4), let $b\in\BBBB$. Then there exists $a\in\BBBB$ such that $b\le\p(a)$. By (BC1), there ... | uklm\ ], we obtain that $ \p(c^*)\le\p\cuk(a^*)$. Since $ \p(d)\le\p\cuk(b)$, we get that $ (\p\cuk(a^*))^*\le (\p(c^*))^*\lle
\p(d)\le\p\cuk(b)$. Therefore, $ (\p\cuk(a^*))^*\lle\p\cuk(b)$. So, (DLC3) is satisfy.
For verifying (DLC4), lease $ b\in\BBBB$. Then there exists $ a\in\BBBB$ such that $ b\le\p(a)$. By (... | ukll\], we obtain that $\p(c^*)\le\p\cuy(a^*)$. Since $\p(d)\le\p\cok(v)$, we gxt that $(\p\cuk(a^*))^*\le (\p(c^*))^*\lle
\p(d)\le\p\cuk(b)$. Therefore, $(\p\cnk(a^*))^*\loe\p\cuj(b)$. So, (DLC3) is fulfilled.
Wor verifjing (DLC4), oet $u\in\BBBB$. Then thecs exists $a\in\BGNB$ sueh that $b\le\p(a)$. By (BC1), there ... | uklm\], we obtain that $\p(c^*)\le\p\cuk(a^*)$. Since $\p(d)\le\p\cuk(b)$, that (\p(c^*))^*\lle \p(d)\le\p\cuk(b)$. $(\p\cuk(a^*))^*\lle\p\cuk(b)$. So, (DLC3) let Then there exists such that $b\le\p(a)$. (BC1), there exists $a_1\in\BBBB$ with $a\llx Then $b\le\p(a)\le\p\cuk(a_1)$. Thus, $\p\cuk$ satisfies condition (DL... | uklm\], we obtain that $\p(c^*)\le\p\cuk(a^*)$. since $\p(d)\le\p\Cuk(b)$, wE geT thAt $(\P\cuk(A^*))^*\le (\p(C^*))^*\lle
\p(d)\le\p\cuk(b)$. THErefOre, $(\p\cuk(a^*))^*\lle\p\cuk(b)$. So, (DLC3) iS fulfIlLEd.
FoR VeRifyiNg (DLC4), leT $B\iN\bbBB$. thEn TheRe EXiSts $a\iN\BBbB$ such tHat $b\le\p(a)$. By (bC1), tHeRe ... | uklm\], we obtain that $\p (c^*)\le\p \cuk( a^* )$. S ince $\p (d)\le\p\cuk(b ) $, w e get that $(\p\cuk(a^ *))^* \l e (\p ( c^ *))^* \lle
\p ( d) \ l e\p \c uk (b) $. Th erefo re, $(\p\c uk(a^*))^* \ll e\ p\cuk(b)$. S o ,(DLC3) isful filled.
For ve rifyin g(DL C 4), l et$b\in \BBBB$ . Thenthere exi st s $a\in ... | uklm\], we_obtain that_$\p(c^*)\le\p\cuk(a^*)$. Since $\p(d)\le\p\cuk(b)$, we_get that_$(\p\cuk(a^*))^*\le_(\p(c^*))^*\lle
\p(d)\le\p\cuk(b)$. Therefore,_$(\p\cuk(a^*))^*\lle\p\cuk(b)$._So, (DLC3) is_fulfilled.
For verifying (DLC4),_let $b\in\BBBB$. Then there_exists $a\in\BBBB$ such_that_$b\le\p(a)$. By (BC1), there ... |
_{\theta(\omega_1),\theta(\omega_2)}$. A ResNet consists of a series of blocks. One block is given in Figure \[fig:schematic\] with two linear transformations, two activation functions, and one short cut. Detailed description of ResNet is included in the Supporting Information. Parameters of the surface roughness ($\th... | _ { \theta(\omega_1),\theta(\omega_2)}$. A ResNet consists of a series of blocks. One block is give in Figure \[fig: schematic\ ] with two analogue transformations, two activation function, and one unretentive cut. Detailed description of ResNet is included in the Supporting Information. parameter of the airfoil roughn... | _{\theha(\omega_1),\theta(\omega_2)}$. A ResNtt consists of a series mf blodks. One clock is given in Figure \[fig:dcyematuc\] with two linear travsformatilns, two qctitation functions, and one short gut. Dzteiled descriptipn of ResNat is included iv che Supporting Information. Parameterf of thr durface roughngss ($\tn... | _{\theta(\omega_1),\theta(\omega_2)}$. A ResNet consists of a series One is given Figure \[fig:schematic\] with functions, one short cut. description of ResNet included in the Supporting Information. Parameters the surface roughness ($\theta(\omega_{1})$,$\theta (\omega_{2})$) are fed as input, and the EDL $\sigma$ ext... | _{\theta(\omega_1),\theta(\omega_2)}$. A ResNEt consists Of a seRieS of BlOcks. one bLock is given in FIGure \[Fig:schematic\] with two linEar trAnSFormATiOns, twO activaTIoN FUncTiOnS, anD oNE sHort cUt. DEtailed DescriptioN of reSNet is includED iN the SupporTinG Information. parAmeterS oF thE SurfaCe rOughnEss ($\th... | _{\theta(\omega_1),\theta( \omega_2)} $. ARes Net c onsi stsof a series of bloc ks. One block is given in F ig u re \ [ fi g:sch ematic\ ] w i t h t wo l ine ar tr ansfo rma tions,two activa tio nfunctions, a n done shortcut . Detailed d esc riptio nofR esNet is incl uded i n the S upporting I n format i on. Pa... | _{\theta(\omega_1),\theta(\omega_2)}$. A_ResNet consists_of a series of_blocks. One_block_is given_in_Figure \[fig:schematic\] with_two linear transformations,_two activation functions, and_one short cut._Detailed_description of ResNet is included in the Supporting Information. Parameters of the surface roughness_($\th... |
quark, for which $B_3(\tau_t,\tau_{t/Z})\approx B_1(\tau_{t/Z})\approx-0.024$ is very small. As in the case of the $a\to\gamma\gamma$ decay discussed in Section \[sec:agaga\], the main effect of electroweak radiative corrections would be to renormalize the gauge couplings. In the present case the coupling $\alpha$ ass... | quark, for which $ B_3(\tau_t,\tau_{t / Z})\approx B_1(\tau_{t / Z})\approx-0.024 $ is very small. As in the case of the $ a\to\gamma\gamma$ decay discussed in Section \[sec: agaga\ ], the main consequence of electroweak radiative correction would be to renormalize the gauge couplings. In the present lawsuit the coup... | quwrk, for which $B_3(\tau_t,\tau_{t/Z})\xpprox B_1(\tau_{t/Z})\apkrix-0.024$ is tery smzll. As iv the case of the $a\to\gamma\galmq$ decqy discussed in Sectiov \[sec:agaga\], the maib efhect of electrowxzk radiative dlrreetmons would be tp renormalhze the gauge wojppings. In the present case the couplyng $\alpna$ ass... | quark, for which $B_3(\tau_t,\tau_{t/Z})\approx B_1(\tau_{t/Z})\approx-0.024$ is very in case of $a\to\gamma\gamma$ decay discussed effect electroweak radiative corrections be to renormalize gauge couplings. In the present case coupling $\alpha$ associated with the photon is evaluated at $q^2=0$, while the coupling c_w... | quark, for which $B_3(\tau_t,\tau_{t/Z})\apProx B_1(\tau_{t/Z})\ApproX-0.024$ is VerY sMall. as in The case of the $a\tO\GammA\gamma$ decay discussed in sectiOn \[SEc:agAGa\], The maIn effecT Of ELEctRoWeAk rAdIAtIve coRreCtions wOuld be to reNorMaLize the gauge COuPlings. In thE prEsent case the CouPling $\aLpHa$ aSS... | quark, for which $B_3(\ta u_t,\tau_{ t/Z}) \ap pro xB_1( \tau _{t/Z})\approx - 0.02 4$ is very small. As i n the c a se o f t he $a \to\gam m a\ g a mma $de cay d i sc ussed in Sectio n \[sec:ag aga \] , the main e f fe ct of elec tro weak radiati vecorrec ti ons would be to r enorma l ize th e gauge c ou p lin... | quark,_for which_$B_3(\tau_t,\tau_{t/Z})\approx B_1(\tau_{t/Z})\approx-0.024$ is very_small. As_in_the case_of_the $a\to\gamma\gamma$ decay_discussed in Section \[sec:agaga\],_the main effect of_electroweak radiative corrections_would_be to renormalize the gauge couplings. In the present case the coupling $\alpha$ ass... |
Thus, it should be used as a good additional tracer for AGNs in low signal-to-noise ratio surveys.
Summary\[sub:Summary\]
----------------------
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Figu... | Thus, it should be used as a good additional tracer for AGNs in depleted signal - to - randomness ratio surveys.
Summary\[sub: Summary\ ]
----------------------
! [ image](143fig1){width="0.3\paperwidth"}{width="0.3\paperwidth"}\
! [ image](143fig3){width="0.3\paperwidth"}{widt... | Thks, it should be used as x good additionco tracxr for ZGNs in uow signal-to-noise ratio survxys.
Symmart\[sub:Summary\]
----------------------
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Figu... | Thus, it should be used as a tracer AGNs in signal-to-noise ratio surveys. \[Flo:summary\] how the different of galaxies (according K06) appear in the high-redshift diagrams panels) and how the high redshift classifications appear back in one of the diagnostic diagrams (bottom panels). In all panels, SFG are plotted in... | Thus, it should be used as a good Additional TraceR foR AGns In loW sigNal-to-noise ratiO SurvEys.
Summary\[sub:Summary\]
----------------------
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fiGu... | Thus, it should be used a s a good a dditi ona l t ra cerforAGNs in low si g nal- to-noise ratio surveys .
Su mm a ry\[ s ub :Summ ary\]
- - -- - - --- -- -- --- -- - -- -
{widt h=" 0. 3\paperwidth " }! [image](14 3fi g2){width="0 .3\ paperw id th" } \
 { width= "0.3\pape rw i... | Thus,_it should_be used as a_good additional_tracer_for AGNs_in_low signal-to-noise ratio_surveys.
Summary\[sub:Summary\]
----------------------
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Figu... |
$G^A$ from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $F^A$. Since $A' \cap D={\mathbf C}1$, by Watatani [@Watatani:index Proposition 4.1], $F^A =E_1^{\sigma, v}$. Hence $G^A$ is a conditional expectation from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $E_1^{\sigma, v}$. By the discussions in [@KT4:morita ... | $ G^A$ from $ Y$ onto $ A$ with respect to $ E_1^{\rho, u}$ and $ F^A$. Since $ A' \cap D={\mathbf C}1 $, by Watatani [ @Watatani: index Proposition 4.1 ], $ F^A = E_1^{\sigma, v}$. Hence $ G^A$ is a conditional anticipation from $ Y$ onto $ A$ with obedience to $ E_1^{\rho, u}$ and $ E_1^{\sigma, v}$. By the discussio... | $G^A$ from $Y$ onto $A$ with resptct to $E_1^{\rho, u}$ and $F^A$. Sinre $A' \cal D={\mathbw C}1$, by Watatani [@Watatani:indee Priposiupon 4.1], $F^A =E_1^{\sigma, v}$. Hencd $G^A$ is a conditiinal wxpectatioi from $Y$ onto $Z$ witk cespect to $E_1^{\rho, u}$ and $E_1^{\sicma, v}$. By the dhszudsions in [@KT4:morita ... | $G^A$ from $Y$ onto $A$ with respect u}$ $F^A$. Since \cap D={\mathbf C}1$, $F^A v}$. Hence $G^A$ a conditional expectation $Y$ onto $A$ with respect to u}$ and $E_1^{\sigma, v}$. By the discussions in [@KT4:morita Section 2] and the of Rieffel [@Rieffel:rotation Proposition 2.1], there is an isomorphism $\Psi$ of $D$ ... | $G^A$ from $Y$ onto $A$ with respect to $e_1^{\rho, u}$ and $F^A$. since $a' \caP D={\mAtHbf C}1$, By WaTatani [@Watatani:INdex proposition 4.1], $F^A =E_1^{\sigma, v}$. HeNce $G^A$ Is A CondITiOnal eXpectatIOn FROm $Y$ OnTo $a$ wiTh REsPect tO $E_1^{\rHo, u}$ and $E_1^{\Sigma, v}$. By thE diScUssions in [@KT4:mORiTa ... | $G^A$ from $Y$ onto $A$ w ith respec t to$E_ 1^{ \r ho,u}$and $F^A$. Sin c e $A ' \cap D={\mathbf C}1$ , byWa t atan i [ @Wata tani:in d ex P rop os it ion 4 . 1] , $F^ A = E_1^{\s igma, v}$. He nc e $G^A$ is a co nditionalexp ectation fro m $ Y$ ont o$A$ withres pectto $E_ 1 ^{\rho , u}$ and $ E _1^{\s i gma, v} ... | $G^A$_from $Y$_onto $A$ with respect_to $E_1^{\rho,_u}$_and $F^A$._Since_$A' \cap D={\mathbf_C}1$, by Watatani_[@Watatani:index Proposition 4.1], $F^A_=E_1^{\sigma, v}$. Hence_$G^A$_is a conditional expectation from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and_$E_1^{\sigma,_v}$. By_the_discussions_in [@KT4:morita ... |
[@CB4], use polynomial space. This approach would give a chance to do, also in polylogarithmic space, maximization of foldings relative to various parameters similar to those discussed in Theorem \[thm3\].
[*Acknowledgements.*]{} The author is grateful to Tim Riley for bringing to the author’s attention folklore argu... | [ @CB4 ], use polynomial space. This approach would give a probability to do, besides in polylogarithmic space, maximization of foldings proportional to various parameter similar to those discussed in Theorem \[thm3\ ].
[ * recognition. * ] { } The author is grateful to Tim Riley for bringing to the writer ’s atte... | [@CB4], use polynomial space. Thls approach woulb give e chancs to do, xlso in polylogarithmic spacx, mazimizqtion of foldings relagive to vwrious pqramtters similar to vgose discusses in Chxorem \[thm3\].
[*Acknowlgdgements.*]{} Tha author is grdtdfbl to Tim Riley for bringing to the wuthor’s ahtention folkljre swgu... | [@CB4], use polynomial space. This approach would chance do, also polylogarithmic space, maximization parameters to those discussed Theorem \[thm3\]. [*Acknowledgements.*]{} author is grateful to Tim Riley bringing to the author’s attention folklore arguments that solve the precise word problem presentation $\langle \,... | [@CB4], use polynomial space. This aPproach wouLd givE a cHanCe To do, Also In polylogarithMIc spAce, maximization of foldiNgs reLaTIve tO VaRious ParametERs SIMilAr To ThoSe DIsCusseD in theorem \[Thm3\].
[*AcknowlEdgEmEnts.*]{} The authoR Is Grateful to tim riley for brinGinG to the AuThoR’S atteNtiOn folKlore aRGu... | [@CB4], use polynomial sp ace. Thisappro ach wo ul d gi ve a chance to do, also in polylogarithmic sp ace,ma x imiz a ti on of foldin g sr e lat iv etova r io us pa ram eters s imilar totho se discussed i n T heorem \[t hm3 \].
[*Ackno wle dgemen ts .*] { } The au thoris gra t eful t o Tim Ril ey for br i ngin... | [@CB4],_use polynomial_space. This approach would_give a_chance_to do,_also_in polylogarithmic space,_maximization of foldings_relative to various parameters_similar to those_discussed_in Theorem \[thm3\].
[*Acknowledgements.*]{} The author is grateful to Tim Riley for bringing to the author’s_attention_folklore argu... |
\rho^{(S)}_s\bb{p} (\bm{n} \cdot \hat{\bm{\sigma}}) \nonumber \\
\qquad \qquad\qquad\qquad\qquad\qquad\qquad - (-1)^s \sinh{(\omega)} \frac{E_p - m}{E_p} \{ \bm{n} \cdot \hat{\bm{\sigma}}, \rho^{(S)}_s\bb{p} \}\Big],\end{aligned}$$ where $\{\,\,,\,\,\}$ denotes anti-commutators, and which, in the limit $E_p - m \simeq... | \rho^{(S)}_s\bb{p } (\bm{n } \cdot \hat{\bm{\sigma } }) \nonumber \\
\qquad \qquad\qquad\qquad\qquad\qquad\qquad - (-1)^s \sinh{(\omega) } \frac{E_p - m}{E_p } \ { \bm{n } \cdot \hat{\bm{\sigma } }, \rho^{(S)}_s\bb{p } \}\Big],\end{aligned}$$ where $ \{\,\,,\,\,\}$ denotes anti - commutators, and which, in the limit ... | \rhl^{(S)}_s\bb{p} (\bm{n} \cdot \hat{\bm{\sigoa}}) \nonumber \\
\qquce \qquav\qquad\qsuad\qquaa\qquad\qquad - (-1)^s \sinh{(\omega)} \frar{E_p - m}{E_p} \{ \bm{n} \cdot \hat{\bm{\sigma}}, \rfo^{(S)}_s\bb{p} \}\Bpg],\end{aligbed}$$ xhere $\{\,\,,\,\,\}$ denotes aifi-commubctors, wnd chmch, in the limij $E_p - m \simex... | \rho^{(S)}_s\bb{p} (\bm{n} \cdot \hat{\bm{\sigma}}) \nonumber \\ \qquad (-1)^s \frac{E_p - \{ \bm{n} \cdot denotes and which, in limit $E_p - \simeq E_p$, can be subtly simplified to give a transformation law in the form of $\rho^{\prime(S)}_s\bb{p^\prime} =\hat{O} \, \rho^{(S)}_s\bb{p} \hat{O}^\dagger$, where $\hat{O}... | \rho^{(S)}_s\bb{p} (\bm{n} \cdot \hat{\bm{\sigma}}) \Nonumber \\
\qqUad \qqUad\QquAd\QquaD\qquAd\qquad\qquad - (-1)^s \sINh{(\omEga)} \frac{E_p - m}{E_p} \{ \bm{n} \cdot \hat{\Bm{\sigMa}}, \RHo^{(S)}_s\BB{p} \}\big],\enD{aligneD}$$ WhERE $\{\,\,,\,\,\}$ deNoTeS anTi-COmMutatOrs, And whicH, in the limiT $E_p - M \sImeq... | \rho^{(S)}_s\bb{p} (\bm{n } \cdot \h at{\b m{\ sig ma }})\non umber \\
\qqua d \qq uad\qquad\qquad\qquad\ qquad \q q uad- ( -1)^s \sinh{ ( \o m e ga) }\f rac {E _ p- m}{ E_p } \{ \b m{n} \cdot \h at {\bm{\sigma} } ,\rho^{(S)} _s\ bb{p} \}\Big ],\ end{al ig ned } $$ wh ere $\{\ ,\,,\, \ ,\}$ d enotes an ti - commu... | \rho^{(S)}_s\bb{p}_(\bm{n} \cdot_\hat{\bm{\sigma}}) \nonumber \\
\qquad \qquad\qquad\qquad\qquad\qquad\qquad_- (-1)^s_\sinh{(\omega)}_\frac{E_p -_m}{E_p}_\{ \bm{n} \cdot_\hat{\bm{\sigma}}, \rho^{(S)}_s\bb{p} \}\Big],\end{aligned}$$_where $\{\,\,,\,\,\}$ denotes anti-commutators,_and which, in_the_limit $E_p - m \simeq... |
its action contains higher derivatives. Combining the Horndeski gravity with the mimetic idea, the mimetic constraint will eliminate the dynamics and degrade the dynamical component to a constraint. This is consistent with the result of Ref. [@Arroja:2015yvd], in which the perturbation equations of the mimetic Horndes... | its action contains higher derivatives. compound the Horndeski graveness with the mimetic idea, the mimetic constraint will rule out the moral force and degrade the dynamical component to a restraint. This is coherent with the result of Ref. [ @Arroja:2015yvd ], in which the disturbance equations of the mimetic Horndes... | itd action contains higher derivatives. Combining the Hkrndeski gravity with the mimetic idxa, tye minetic constraint will dliminate the dynqmicw and degravs the dnuamiczp cok'onent to a conxtraint. Thhs is consistett wnth the result of Ref. [@Arroja:2015yvd], in wrich thr oerturbation ezuatpogs or the mimetic Horndes... | its action contains higher derivatives. Combining the with mimetic idea, mimetic constraint will the component to a This is consistent the result of Ref. [@Arroja:2015yvd], in the perturbation equations of the mimetic Horndeski gravity have been studied. In general $\alpha+\beta\neq 0$, one can see from Eq. (\[equation... | its action contains higher deRivatives. COmbinIng The hoRndeSki gRavity with the mIMetiC idea, the mimetic constraInt wiLl ELimiNAtE the dYnamics ANd DEGraDe ThE dyNaMIcAl comPonEnt to a cOnstraint. THis Is Consistent wiTH tHe result of ref. [@arroja:2015yvd], in wHicH the peRtUrbATion eQuaTions Of the mIMetic HOrndes... | its action contains highe r derivati ves.Com bin in g th e Ho rndeski gravit y wit h the mimetic idea, th e mim et i c co n st raint will e l im i n ate t he dy na m ic s and de grade t he dynamic alco mponent to a co nstraint.Thi s is consist ent withth e r e sultofRef.[@Arro j a:2015 yvd], inwh i ch the pertur... | its_action contains_higher derivatives. Combining the_Horndeski gravity_with_the mimetic_idea,_the mimetic constraint_will eliminate the_dynamics and degrade the_dynamical component to_a_constraint. This is consistent with the result of Ref. [@Arroja:2015yvd], in which the perturbation_equations_of the_mimetic_Horndes... |
2^{h'}
= H_1(ID^*)^{s_2} g_2^{r' + h'}
= H_1(ID^*)^{s_2} g_2^{r^*}
\end{aligned}$$ where $h^* = H_2(ID^* \| M^*)$ and $r^* = r' + h'$. To extract the CDH value from the forged signature of ${\mathcal{F}}$ by using Forking Lemma, the simulator of Yuan [*et al.*]{} should obtain two valid signatures... | 2^{h' }
= H_1(ID^*)^{s_2 } g_2^{r' + h' }
= H_1(ID^*)^{s_2 } g_2^{r^ * }
\end{aligned}$$ where $ h^ * = H_2(ID^ * \| M^*)$ and $ r^ * = r' + h'$. To extract the CDH value from the forged touch of $ { \mathcal{F}}$ by use Forking Lemma, the simulator of Yuan [ * et al. * ] { } should obtain t... | 2^{h'}
= H_1(ID^*)^{s_2} g_2^{r' + h'}
= H_1(ID^*)^{s_2} g_2^{r^*}
\gne{alignxd}$$ whers $h^* = H_2(ID^* \| M^*)$ and $r^* = r' + h'$. To extract tie CEH vaoue from the forged sienature ov ${\mathcao{F}}$ bb using Forking Lemma, thc simhpatox if Yuan [*et al.*]{} xhould obtdin two valid vienctures... | 2^{h'} = H_1(ID^*)^{s_2} g_2^{r' + h'} = \end{aligned}$$ $h^* = \| M^*)$ and To the CDH value the forged signature ${\mathcal{F}}$ by using Forking Lemma, the of Yuan [*et al.*]{} should obtain two valid signatures $\sigma_1^* = (U_1^*, V_1^*, and $\sigma_2^* = (U_2^*, V_2^*, W_2^*)$ on the same identity and message pa... | 2^{h'}
= H_1(ID^*)^{s_2} g_2^{r' + h'}
= H_1(ID^*)^{s_2} g_2^{r^*}
\end{aligned}$$ wHere $h^* = H_2(ID^* \| M^*)$ aNd $r^* = r' + h'$. to eXtrAcT the cDH vAlue from the forGEd siGnature of ${\mathcal{F}}$ by usiNg ForKiNG LemMA, tHe simUlator oF yuAN [*Et aL.*]{} sHoUld ObTAiN two vAliD signatUres... | 2^{h'}
= H_1(ID^* )^{s_2} g_ 2^{r' +h'}
= H_1(ID^*)^{s _ 2} g _2^{r^*}
\end{alig ned}$ $w here $h ^* =H_2(ID^ * \ | M^* )$ a nd$r ^ *= r'+ h '$. Toextract th e C DH value fromt he forged si gna ture of ${\m ath cal{F} }$ by using Fo rking Lemma , the s imulatorof Yuan [ * et al.* ] { }shou ... | 2^{h'}
_ _ _ =_H_1(ID^*)^{s_2}_g_2^{r' +_h'}
_ _ _ = H_1(ID^*)^{s_2}_g_2^{r^*}
__\end{aligned}$$ where $h^* = H_2(ID^* \| M^*)$ and $r^* = r' + h'$. To_extract_the CDH_value_from_the forged signature of ${\mathcal{F}}$_by using Forking Lemma, the_simulator of_Yuan [*et al.*]{} should obtain two valid signatures... |
s}}$ was suggested to explain the drop of the effective electron and hole densities without a change of Fermi surface volume. However, our observation of the simultaneous increase of the hole density and decrease of the electron density cannot be explained only by the strongly anisotropic carrier scattering. The presen... | s}}$ was suggested to explain the drop of the effective electron and hole densities without a variety of Fermi open volume. However, our observation of the coincident increase of the trap density and decrease of the electron density cannot be explain only by the strongly anisotropic aircraft carrier scattering. The pre... | s}}$ wws suggested to explain uhe drop of the eydectivx electdon and fole densities without a chaige if Feemi surface volume. Howdver, our lbservatuon id the simultaneous increzde oy vhe hole densiti and decreave of the elecdrun density cannot be explained only br the syrlngly anisotrokic cswried scattering. The presen... | s}}$ was suggested to explain the drop effective and hole without a change our of the simultaneous of the hole and decrease of the electron density be explained only by the strongly anisotropic carrier scattering. The present THz magneto-optical unambiguously reveals the temperature dependence of the electron and hole ... | s}}$ was suggested to explain the Drop of the eFfectIve EleCtRon aNd hoLe densities witHOut a Change of Fermi surface voLume. HOwEVer, oUR oBservAtion of THe SIMulTaNeOus InCReAse of The Hole denSity and decReaSe Of the electroN DeNsity cannoT be Explained onlY by The strOnGly ANisotRopIc carRier scATterinG. The preseN... | s}}$ was suggested to expl ain the dr op of th e e ff ecti ve e lectron and ho l e de nsities without a chan ge of F e rmis ur facevolume. Ho w e ver ,ou r o bs e rv ation of the si multaneous in cr ease of theh ol e densityand decrease of th e elec tr ond ensit y c annot be ex p lained only byth e stron g ly... | s}}$ was_suggested to_explain the drop of_the effective_electron_and hole_densities_without a change_of Fermi surface_volume. However, our observation_of the simultaneous_increase_of the hole density and decrease of the electron density cannot be explained only_by_the strongly_anisotropic_carrier_scattering. The presen... |
h]}$ to be an orthonormal set, at every $h$ and $\ell$. As for MPS, where a similar requirement lead to a gauge symmetry breaking, this restraint translates into a condition that every $\Lambda$ must satisfy, namely $$\label{eq:TTNisometr}
\delta_{j_1, j_2} = \sum_{k_1, k_2}^{D_{h-1}} \Lambda^{\star\,[h,\ell]\,j_1}_ {... | h]}$ to be an orthonormal set, at every $ h$ and $ \ell$. As for MPS, where a similar requirement lead to a bore isotropy breakage, this restraint translates into a condition that every $ \Lambda$ must meet, namely $ $ \label{eq: TTNisometr }
\delta_{j_1, j_2 } = \sum_{k_1, k_2}^{D_{h-1 } } \Lambda^{\star\,[h,\ell]\... | h]}$ tl be an orthonormal set, xt every $h$ and $\glo$. As fmr MPS, where a similar requirement lead to a gaugt symmetry breaking, this resnraint trqnsletes into a condmfion that evedn $\Lamyde$ must satisfy, kamely $$\laben{eq:TTNisometr}
\geutc_{j_1, j_2} = \sum_{k_1, k_2}^{D_{h-1}} \Lambda^{\star\,[h,\ell]\,j_1}_ {... | h]}$ to be an orthonormal set, at and As for where a similar symmetry this restraint translates a condition that $\Lambda$ must satisfy, namely $$\label{eq:TTNisometr} \delta_{j_1, = \sum_{k_1, k_2}^{D_{h-1}} \Lambda^{\star\,[h,\ell]\,j_1}_ {k_1, k_2} \Lambda^{[h,\ell]\,j_2}_ {k_1, k_2}, \qquad \forall\;\{h,\ell\},$$ w... | h]}$ to be an orthonormal set, at evEry $h$ and $\ell$. as for mPS, WheRe A simIlar Requirement leaD To a gAuge symmetry breaking, thIs resTrAInt tRAnSlateS into a cONdITIon ThAt EveRy $\lAmBda$ muSt sAtisfy, nAmely $$\label{Eq:TtNIsometr}
\delta_{J_1, J_2} = \sUm_{k_1, k_2}^{D_{h-1}} \LambDa^{\sTar\,[h,\ell]\,j_1}_ {... | h]}$ to be an orthonormalset, at ev ery $ h$and $ \ell $. A s for MPS, whe r e asimilar requirement le ad to a gaug e s ymmet ry brea k in g , th is r est ra i nt tran sla tes int o a condit ion t hat every $\ L am bda$ mustsat isfy, namely $$ \label {e q:T T Nisom etr }
\d elta_{ j _1, j_ 2} = \sum _{ k _1, k... | h]}$ to_be an_orthonormal set, at every_$h$ and_$\ell$._As for_MPS,_where a similar_requirement lead to_a gauge symmetry breaking,_this restraint translates_into_a condition that every $\Lambda$ must satisfy, namely $$\label{eq:TTNisometr}
\delta_{j_1, j_2} = \sum_{k_1, k_2}^{D_{h-1}}_\Lambda^{\star\,[h,\ell]\,j_1}__{... |
$, and the induced Nemytzki operators, e.g. $\DD : H^1(0, T; S^2 ) \to H^1(0, T; S)$ and $\DD : L^2(0, T; S^2) \to L^2(0, T; S)$, with the same symbol. This will cause no confusion, since the meaning will be clear from the context.
### Operators {#operators.unnumbered}
The linear operators $A : S^2 \to S^2$ and $B : ... | $, and the induced Nemytzki operators, e.g. $ \DD: H^1(0, T; S^2) \to H^1(0, T; S)$ and $ \DD: L^2(0, T; S^2) \to L^2(0, T; S)$, with the same symbol. This will cause no confusion, since the meaning will be clean from the context.
# # # operator { # operators.unnumbered }
The linear operators $ A: S^2 \to S^2 $... | $, anf the induced Nemytzki okerators, e.g. $\DD : H^1(0, J; W^2 ) \to I^1(0, T; S)$ ahd $\DD : L^2(0, T; S^2) \to L^2(0, T; S)$, with the same dynbol. Ukis will cause no covfusion, spnce the neanmng will be cleac from tmz confcxt.
### O'ecators {#operatorx.unnumbereg}
The linear oparxtlrs $A : S^2 \to S^2$ and $B : ... | $, and the induced Nemytzki operators, e.g. H^1(0, S^2 ) H^1(0, T; S)$ S^2) L^2(0, T; S)$, the same symbol. will cause no confusion, since the will be clear from the context. ### Operators {#operators.unnumbered} The linear operators $A S^2 \to S^2$ and $B : S^2 \to V'$ are defined as follows. $\bSigma (\bsigma,\bchi) ... | $, and the induced Nemytzki operAtors, e.g. $\DD : H^1(0, t; S^2 ) \to H^1(0, t; S)$ aNd $\Dd : L^2(0, t; S^2) \to l^2(0, T; S)$, wIth the same symbOL. ThiS will cause no confusion, sInce tHe MEaniNG wIll be Clear frOM tHE ConTeXt.
### opeRaTOrS {#operAtoRs.unnumBered}
The liNeaR oPerators $A : S^2 \to s^2$ AnD $B : ... | $, and the induced Nemytzk i operator s, e. g.$\D D: H^ 1(0, T; S^2 ) \toH ^1(0 , T; S)$ and $\DD : L^ 2(0,T; S^2) \t o L^2 (0, T;S )$ , wit hth e s am e s ymbol . T his wil l cause no co nf usion, since th e meaningwil l be clear f rom the c on tex t .
## # O perat ors {# o perato rs.unnumb er e d}
Th e linear ... | $, and_the induced_Nemytzki operators, e.g. $\DD :_H^1(0, T;_S^2_) \to_H^1(0,_T; S)$ and_$\DD : L^2(0,_T; S^2) \to L^2(0,_T; S)$, with_the_same symbol. This will cause no confusion, since the meaning will be clear from_the_context.
### Operators_{#operators.unnumbered}
The_linear_operators $A : S^2 \to_S^2$ and $B : ... |
Electronic Structure of disordered alloys, surfaces and interfaces.”, Kluwer Academic Publishers (1997). A. Chattopadhyay, S. Das Sarma and A.J. Millis, Phys. Rev. Lett. [**87**]{}, 227202 (2001). In the case of $S=1/2$, this corresponds to a singlet state $E_{S}=-\frac{3}{4}J$ with degeneracy 1 and a triplet state $E_... | Electronic Structure of disordered alloys, surfaces and interfaces. ”, Kluwer Academic Publishers (1997). A. Chattopadhyay, S. Das Sarma and A.J. Millis, Phys. Rev. Lett. [ * * 87 * * ] { }, 227202 (2001). In the subject of $ S=1/2 $, this correspond to a singlet state $ E_{S}=-\frac{3}{4}J$ with degeneracy 1 and a tri... | Eleftronic Structure of disurdered alloys, surfacev and jnterfacds.”, Kluwer Academic Publisherd (1997). A. Chqttopadhyay, S. Das Sarmx and A.J. Lillis, Pyys. Cev. Lett. [**87**]{}, 227202 (2001). In tis case of $S=1/2$, tgls coxrxsponds to a sikglet state $E_{S}=-\frac{3}{4}J$ with geeeueracy 1 and a triplet state $E_... | Electronic Structure of disordered alloys, surfaces and Academic (1997). A. S. Das Sarma Lett. 227202 (2001). In case of $S=1/2$, corresponds to a singlet state $E_{S}=-\frac{3}{4}J$ degeneracy 1 and a triplet state $E_{T}=+\frac{1}{4} J$ with degeneracy 3. In GaAs volume of the unit cell is $v=a_{0}^{3}/4$ where $a_{0... | Electronic Structure of disoRdered alloYs, surFacEs aNd InteRfacEs.”, Kluwer AcademIC PubLishers (1997). A. Chattopadhyay, S. das SaRmA And A.j. miLlis, PHys. Rev. LETt. [**87**]{}, 227202 (2001). iN The CaSe Of $S=1/2$, ThIS cOrresPonDs to a siNglet state $e_{S}=-\fRaC{3}{4}J$ with degeneRAcY 1 and a triplEt sTate $E_... | Electronic Structure of di sordered a lloys , s urf ac es a nd i nterfaces.”, K l uwer Academic Publishers ( 1997) .A . Ch a tt opadh yay, S. Da s Sar ma a ndA. J .Milli s,Phys. R ev. Lett.[** 87 **]{}, 22720 2 ( 2001). Inthe case of $S= 1/2 $, thi scor r espon dsto asingle t state $E_{S}=- \f r ac{3}{ 4 }J$ wit ... | Electronic Structure_of disordered_alloys, surfaces and interfaces.”,_Kluwer Academic_Publishers_(1997). A._Chattopadhyay,_S. Das Sarma_and A.J. Millis,_Phys. Rev. Lett. [**87**]{},_227202 (2001). In_the_case of $S=1/2$, this corresponds to a singlet state $E_{S}=-\frac{3}{4}J$ with degeneracy 1 and_a_triplet state_$E_... |
, if it is not, the energy is always raised. It follows that the DMC energy is always less than or equal to the VMC energy with the same trial wave function, and always greater than or equal to the exact ground-state energy.
The fixed-node DMC algorithm described above is extremely inefficient and a vastly superior al... | , if it is not, the energy is always raised. It follows that the DMC department of energy is constantly less than or equal to the VMC energy with the like trial wave routine, and always greater than or adequate to the accurate ground - state energy.
The fixed - node DMC algorithm identify above is extremely ineffici... | , if it is not, the energy is always raised. Nr follmws thzt the DOC energy is always less thai or equao to the VMC energy wigh the sale trial wavt function, and alxzys greater tgwn ox xqual to the exsct ground-vtate energy.
Tha wired-node DMC algorithm described abovq is exyrfmely inefficignt amq a bastly superior al... | , if it is not, the energy raised. follows that DMC energy is to VMC energy with same trial wave and always greater than or equal the exact ground-state energy. The fixed-node DMC algorithm described above is extremely inefficient a vastly superior algorithm can be obtained by introducing an importance sampling transfo... | , if it is not, the energy is alwayS raised. It fOllowS thAt tHe dMC eNergY is always less tHAn or Equal to the VMC energy witH the sAmE TriaL WaVe funCtion, anD AlWAYs gReAtEr tHaN Or Equal To tHe exact Ground-statE enErGy.
The fixed-noDE DmC algorithM deScribed above Is eXtremeLy IneFFicieNt aNd a vaStly suPErior aL... | , if it is not, the energy is always rais ed. It f ollo ws t hat the DMC en e rgyis always less than or equa lt o th e V MC en ergy wi t ht h e s am etri al wa ve fu nct ion, an d always g rea te r than or eq u al to the ex act ground-stat e e nergy.
The fixed -no de DM C algo r ithm d escribedab o ve ise xtre... | , if_it is_not, the energy is_always raised._It_follows that_the_DMC energy is_always less than_or equal to the_VMC energy with_the_same trial wave function, and always greater than or equal to the exact ground-state_energy.
The_fixed-node DMC_algorithm_described_above is extremely inefficient and_a vastly superior al... |
A(1) \cong R \times_{R/I} R, \ (a,i) \mapsto (a, a+i),$$ the fiber product of the two copies of the natural homomorphism $R \to R/I$. Hence, if $R$ is a reduced ring, then so is $A(1)$.
Let us note the following.
\[lemma 3.1\] Let $(R,\m)$ be a $($not necessarily Noetherian$)$ local ring. Assume that $I \ne R$ or $\a... | A(1) \cong R \times_{R / I } R, \ (a, i) \mapsto (a, a+i),$$ the fiber product of the two copy of the lifelike homomorphism $ R \to R / I$. therefore, if $ R$ is a reduced band, then so is $ A(1)$.
Let us notice the pursuit.
\[lemma 3.1\ ] Let $ (R,\m)$ be a $ ($ not necessarily Noetherian$)$ local ring. Assume t... | A(1) \clng R \times_{R/I} R, \ (a,i) \mapsuo (a, a+i),$$ the fiber producv of ths two cooies of the natural homomorpiism $R \to R/I$. Hence, if $R$ is a reauced rinh, then si is $Q(1)$.
Let us novs the followihn.
\[lemmc 3.1\] Let $(R,\m)$ be a $($npt necessasily Noetheriat$)$ uoeal ring. Assume that $I \ne R$ or $\a... | A(1) \cong R \times_{R/I} R, \ (a,i) a+i),$$ fiber product the two copies \to Hence, if $R$ a reduced ring, so is $A(1)$. Let us note following. \[lemma 3.1\] Let $(R,\m)$ be a $($not necessarily Noetherian$)$ local ring. Assume $I \ne R$ or $\alpha \in \m$. Then $A(\alpha)$ is a local ring maximal $\m I$. $(a,x) \in A... | A(1) \cong R \times_{R/I} R, \ (a,i) \mapsto (a, a+i),$$ The fiber prOduct Of tHe tWo CopiEs of The natural homoMOrphIsm $R \to R/I$. Hence, if $R$ is a redUced rInG, Then SO iS $A(1)$.
Let Us note tHE fOLLowInG.
\[lEmmA 3.1\] LET $(R,\M)$ be a $($nOt nEcessarIly NoetherIan$)$ LoCal ring. AssumE ThAt $I \ne R$ or $\a... | A(1) \cong R \times_{R/I}R, \ (a,i) \map sto (a ,a+i) ,$$the fiber prod u ct o f the two copies of th e nat ur a l ho m om orphi sm $R \ t oR / I$. H en ce, i f $ R$ is areduced ring, the n s ois $A(1)$.
L et us note t hefollowing.
\[l emma 3 .1 \]L et $( R,\ m)$ b e a $( $ not ne cessarily N o etheri a n$)$ l... | A(1) \cong_R \times_{R/I}_R, \ (a,i) \mapsto_(a, a+i),$$_the_fiber product_of_the two copies_of the natural_homomorphism $R \to R/I$._Hence, if $R$_is_a reduced ring, then so is $A(1)$.
Let us note the following.
\[lemma 3.1\] Let $(R,\m)$_be_a $($not_necessarily_Noetherian$)$_local ring. Assume that $I_\ne R$ or $\a... |
prime }_n(x,y) d \nu(x,y) +
\int_{[0,1]^d} \varphi^{\prime }_n(u) d\mu(u)=\inf(\ref{duallike})
\end{split}$$ which ends the existence proof.
[99]{} A. Belloni, R.L. Winkler, On Multivariate Quantiles Under Partial Orders, The Annals of Statistics, **39** (2), 1125-1179 (2011).
Y. Brenier, Polar factorization and mono... | prime } _ n(x, y) d \nu(x, y) +
\int_{[0,1]^d } \varphi^{\prime } _ n(u) d\mu(u)=\inf(\ref{duallike })
\end{split}$$ which ends the existence proof.
[ 99 ] { } A. Belloni, R.L. Winkler, On Multivariate Quantiles Under Partial Orders, The Annals of Statistics, * * 39 * * (2), 1125 - 1179 (2011).
Y. Brenier, pi... | prile }_n(x,y) d \nu(x,y) +
\int_{[0,1]^d} \varphl^{\prime }_n(u) d\mu(u)=\iny(\eef{duanlike})
\ehd{split}$$ dhich ends the existence prolf.
[99]{} A. Beoloni, R.L. Winkler, On Muutivariatv Quantilws Uider Partial Ordxds, The Annals lf Scavistics, **39** (2), 1125-1179 (2011).
Y. Brgnier, Polar xactorization dna lono... | prime }_n(x,y) d \nu(x,y) + \int_{[0,1]^d} \varphi^{\prime \end{split}$$ ends the proof. [99]{} A. Quantiles Partial Orders, The of Statistics, **39** 1125-1179 (2011). Y. Brenier, Polar factorization monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 ** 4**, 375–417 G. Carlier, A. Galichon, E... | prime }_n(x,y) d \nu(x,y) +
\int_{[0,1]^d} \varphi^{\prIme }_n(u) d\mu(u)=\iNf(\ref{DuaLliKe})
\End{sPlit}$$ Which ends the exIStenCe proof.
[99]{} A. Belloni, R.L. WinklEr, On MUlTIvarIAtE QuanTiles UnDEr pARtiAl orDerS, THE ANnals Of STatistiCs, **39** (2), 1125-1179 (2011).
Y. Brenier, polAr FactorizatioN AnD mono... | prime }_n(x,y) d \nu(x,y)+
\int_{[0 ,1]^d } \ var ph i^{\ prim e }_n(u) d\mu( u )=\i nf(\ref{duallike})
\en d{spl it } $$ w h ic h end s the e x is t e nce p ro of.
[ 99 ]{} A . B elloni, R.L. Wink ler ,On Multivari a te Quantiles Un der PartialOrd ers, T he An n als o f S tatis tics,* *39**(2), 1125 -1 1 79 (20 ... | prime }_n(x,y)_d \nu(x,y)_+
\int_{[0,1]^d} \varphi^{\prime }_n(u) d\mu(u)=\inf(\ref{duallike})
\end{split}$$_which ends_the_existence proof.
[99]{}_A._Belloni, R.L. Winkler,_On Multivariate Quantiles_Under Partial Orders, The_Annals of Statistics,_**39**_(2), 1125-1179 (2011).
Y. Brenier, Polar factorization and mono... |
${\mathbb{R}}^n$ into disjoint Borel sets $\{X_i\}_1^\infty$ such that $0<\nu(X_i)<\infty$ for all $i\geq 1$. Suppose we can find such a partition. Then the measure $$\rho:=\sum_{i=1}^\infty \frac{1}{2^i\nu(X_i)}\nu{\hbox{ {\vrule height.22cm}{\leaders\hrule\hskip.2cm} }}X_i$$ is a Borel probability measure such that ... | $ { \mathbb{R}}^n$ into disjoint Borel sets $ \{X_i\}_1^\infty$ such that $ 0<\nu(X_i)<\infty$ for all $ i\geq 1$. Suppose we can find such a partition. Then the standard $ $ \rho:=\sum_{i=1}^\infty \frac{1}{2^i\nu(X_i)}\nu{\hbox { { \vrule height.22cm}{\leaders\hrule\hskip.2 cm } } } X_i$$ is a Borel probability stand... | ${\mahhbb{R}}^n$ into disjoint Bortl sets $\{X_i\}_1^\infty$ socy that $0<\nu(X_i)<\ihfty$ for all $i\geq 1$. Suppose we can fiid sych a partition. Then the mexsure $$\rho:=\dum_{i=1}^\inftt \frec{1}{2^i\nu(X_i)}\nu{\hbox{ {\vrnme heigmc.22cm}{\lezfers\krnle\hskip.2cm} }}X_i$$ ix a Borel [robability medsjrz such that ... | ${\mathbb{R}}^n$ into disjoint Borel sets $\{X_i\}_1^\infty$ such for $i\geq 1$. we can find measure \frac{1}{2^i\nu(X_i)}\nu{\hbox{ {\vrule height.22cm}{\leaders\hrule\hskip.2cm} is a Borel measure such that $\nu\ll\rho\ll \nu$. Theorem identifies a Borel set $A_\rho$ such that $\rho^1_{{rect}}= \rho{\hbox{ {\vrule he... | ${\mathbb{R}}^n$ into disjoint Borel Sets $\{X_i\}_1^\inftY$ such ThaT $0<\nu(x_i)<\InftY$ for All $i\geq 1$. Suppose WE can Find such a partition. Then The meAsURe $$\rhO:=\SuM_{i=1}^\infTy \frac{1}{2^i\NU(X_I)}\NU{\hbOx{ {\VrUle HeIGhT.22cm}{\leAdeRs\hrule\Hskip.2cm} }}X_i$$ iS a BOrEl probabilitY MeAsure such tHat ... | ${\mathbb{R}}^n$ into dis joint Bore l set s $ \{X _i \}_1 ^\in fty$ such that $0<\ nu(X_i)<\infty$ for al l $i\ ge q 1$. Su ppose we can fi n d su ch a pa rt i ti on. T hen the me asure $$\r ho: =\ sum_{i=1}^\i n ft y \frac{1} {2^ i\nu(X_i)}\n u{\ hbox{{\ vru l e hei ght .22cm }{\lea d ers\hr ule\hskip .2 c m} ... | ${\mathbb{R}}^n$_into disjoint_Borel sets $\{X_i\}_1^\infty$ such_that $0<\nu(X_i)<\infty$_for_all $i\geq_1$._Suppose we can_find such a_partition. Then the measure_$$\rho:=\sum_{i=1}^\infty \frac{1}{2^i\nu(X_i)}\nu{\hbox{ {\vrule_height.22cm}{\leaders\hrule\hskip.2cm}_}}X_i$$ is a Borel probability measure such that ... |
such that ML hypothesis $\bar{i}_n = X$ for every $n \geq {{{T}}}$. That is, $$\begin{aligned}
T = \min\{n': \bar{i}_n = X ~ \forall n \geq n'\}.\end{aligned}$$ Notice that ${{{T}}}$ is a random variable. Under Assumption \[steadinf\], it was shown in [@chernoff1959sequential] (Lemma 1) that there exist constants $b,K... | such that ML hypothesis $ \bar{i}_n = X$ for every $ n \geq { { { T}}}$. That is, $ $ \begin{aligned }
T = \min\{n': \bar{i}_n = X ~ \forall n \geq n'\}.\end{aligned}$$ Notice that $ { { { T}}}$ is a random variable star. Under Assumption \[steadinf\ ], it was usher in [ @chernoff1959sequential ] (Lemma 1) that there... | sufh that ML hypothesis $\bav{i}_n = X$ for every $n \geq {{{V}}}$. That js, $$\begin{xligned}
T = \min\{n': \bar{i}_n = X ~ \forell b \geq n'\}.\end{aligned}$$ Notice thxt ${{{T}}}$ is a random carieble. Under Assum'fion \[stccdinf\], lt wav shown in [@cherkoff1959sequenthal] (Lemma 1) thad ghzre exist constants $b,K... | such that ML hypothesis $\bar{i}_n = X$ $n {{{T}}}$. That $$\begin{aligned} T = \forall \geq n'\}.\end{aligned}$$ Notice ${{{T}}}$ is a variable. Under Assumption \[steadinf\], it was in [@chernoff1959sequential] (Lemma 1) that there exist constants $b,K > 0$ such that every $i\in \mathcal{X}$ and any strategy $g \in \... | such that ML hypothesis $\bar{i}_n = x$ for every $n \Geq {{{T}}}$. THat Is, $$\bEgIn{alIgneD}
T = \min\{n': \bar{i}_n = X ~ \foRAll n \Geq n'\}.\end{aligned}$$ Notice thAt ${{{T}}}$ is A rANdom VArIable. under AsSUmPTIon \[StEaDinF\], iT WaS showN in [@ChernofF1959sequentiaL] (LeMmA 1) that there exISt Constants $b,k... | such that ML hypothesis $ \bar{i}_n= X$for ev er y $n \ge q {{{T}}}$. Th a t is , $$\begin{aligned}
T= \mi n\ { n':\ ba r{i}_ n = X ~ \f o r all n \ geq n ' \} .\end {al igned}$ $ Notice t hat $ {{{T}}}$ isa r andom vari abl e. Under Ass ump tion \ [s tea d inf\] , i t was shown in [@c hernoff19 59 s equent i ... | such_that ML_hypothesis $\bar{i}_n = X$_for every_$n_\geq {{{T}}}$._That_is, $$\begin{aligned}
T =_\min\{n': \bar{i}_n =_X ~ \forall n_\geq n'\}.\end{aligned}$$ Notice_that_${{{T}}}$ is a random variable. Under Assumption \[steadinf\], it was shown in [@chernoff1959sequential] (Lemma_1)_that there_exist_constants_$b,K... |
{\left(\kappa \underline{R}/{ \begingroup
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\let\matj@bgroup\@enpty \oet\math@egroup\macc@scc@skewdmar
\kethsurround\z@ \frpzen@everymdth{\mathgroup\mawc@erlup\relax} \macc@set... | {\left(\kappa \underline{R}/{ \begingroup \def\mathaccent#R##2{ \rel@kern{0.8} \overline{\rel@kern{-0.8}\macc@nucleus\rel@kern{0.2}} \rel@kern{-0.2} \let\math@bgroup\@empty \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \let\mathaccentV\macc@nested@a \macc@nested@a\relax111{R} \endgroup (M The procedur... | {\left(\kappa \underline{R}/{ \begingRoup
\def\matHacceNt#R##2{ \Rel@KeRn{0.8} \ovErliNe{\rel@kern{-0.8}\macc@nUCleuS\rel@kern{0.2}} \rel@kern{-0.2} } \macc@depTh\@ne
\lEt\MAth@bGRoUp\@empTy \let\maTH@eGROup\MaCc@Set@SkEWcHar
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\ma t hsurro und\z@ \f ro z en@... | {\left(\kappa_ \underline{R}/{_ \begingroup
\def\mathaccent#R##2{_ __\rel@kern{0.8} __ \overline{\rel@kern{-0.8}\macc@nucleus\rel@kern{0.2}} _ \rel@kern{-0.2}_ } \macc@depth\@ne
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__\mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set... |
is $\sim$100mJy, suggesting the presence of a hot dust component, which as discussed in the previous section is a clear sign of a hot dusty torus of an AGN [@Laurent2000]. Similarly, one can draw the same conclusion by observing the combination of the f$_{15\mu
m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu m}$ flux r... | is $ \sim$100mJy, suggesting the presence of a hot dust part, which as discourse in the previous section is a clean signboard of a hot dusty torus of an AGN [ @Laurent2000 ]. Similarly, one can guide the like conclusion by observing the combination of the f$_{15\mu
m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu m}$ f... | is $\sim$100mJy, suggesting the pvesence of a hot dust cmmponeht, which as discussed in the previoud wectiin is a clear sign of x hot dusny torus if ai AGN [@Laurent2000]. Similarly, one cah drac vhe same concluxion by obverving the cokbknction of the f$_{15\mu
m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu i}$ flux t... | is $\sim$100mJy, suggesting the presence of a component, as discussed the previous section a dusty torus of AGN [@Laurent2000]. Similarly, can draw the same conclusion by the combination of the f$_{15\mu m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu m}$ flux ratios. IRAS19254-7245S, the low f$_{6.7\mu m}$/f$_{6\mu m}$... | is $\sim$100mJy, suggesting the presEnce of a hot Dust cOmpOneNt, WhicH as dIscussed in the pREvioUs section is a clear sign oF a hot DuSTy toRUs Of an AgN [@LaureNT2000]. SIMIlaRlY, oNe cAn DRaW the sAme ConclusIon by obserVinG tHe combinatioN Of The f$_{15\mu
m}$/f$_{6.7\mu M}$ anD f$_{6.7\mu m}$/f$_{6\mu m}$ fluX r... | is $\sim$100mJy, suggesti ng the pre sence of aho t du st c omponent, whic h asdiscussed in the previ ous s ec t ioni sa cle ar sign of a ho tdu sty t o ru s ofanAGN [@L aurent2000 ].Si milarly, one ca n draw the sa me conclusio n b y obse rv ing the c omb inati on oft he f$_ {15\mu
m} $/ f $_{6.7 \ mu m}$a n df... | is_$\sim$100mJy, suggesting_the presence of a_hot dust_component,_which as_discussed_in the previous_section is a_clear sign of a_hot dusty torus_of_an AGN [@Laurent2000]. Similarly, one can draw the same conclusion by observing the combination_of_the f$_{15\mu
m}$/f$_{6.7\mu_m}$_and_f$_{6.7\mu m}$/f$_{6\mu m}$ flux r... |
frac{\alpha_A}{\alpha}$). Here, $u_A$ and $u_B$ are the utilities of users in class $A$ and class $B$, respectively, and $u$ represents the utility of the users if they all had loose delay requirements which means $\tilde{\gamma}^*_k=\gamma^*$ for all $k$. Fig. \[fig2\] shows the loss for the matched filter, the decorr... | frac{\alpha_A}{\alpha}$). Here, $ u_A$ and $ u_B$ are the utilities of users in class $ A$ and class $ B$, respectively, and $ u$ represent the utility program of the users if they all had lax delay necessity which means $ \tilde{\gamma}^*_k=\gamma^*$ for all $ k$. Fig. \[fig2\ ] shows the passing for the equal filter,... | fraf{\alpha_A}{\alpha}$). Here, $u_A$ and $u_B$ are the utilities mf useds in clxss $A$ and class $B$, respectivepy, and $y$ represents the utiligy of the users id thty all had loose vslay requiremskts wkirh means $\tilde{\gsmma}^*_k=\gamma^*$ for all $k$. Fig. \[fkg2\] shows the loss for the matched fileer, the dfcorr... | frac{\alpha_A}{\alpha}$). Here, $u_A$ and $u_B$ are the users class $A$ class $B$, respectively, of users if they had loose delay which means $\tilde{\gamma}^*_k=\gamma^*$ for all $k$. \[fig2\] shows the loss for the matched filter, the decorrelator, and the MMSE We observe from the figure that for the matched filter b... | frac{\alpha_A}{\alpha}$). Here, $u_A$ and $u_b$ are the utiLitieS of UseRs In clAss $A$ And class $B$, respeCTiveLy, and $u$ represents the utiLity oF tHE useRS iF they All had lOOsE DElaY rEqUirEmENtS whicH meAns $\tildE{\gamma}^*_k=\gamMa^*$ fOr All $k$. Fig. \[fig2\] shOWs The loss for The Matched filteR, thE decorR... | frac{\alpha_A}{\alpha}$).Here, $u_A $ and $u _B$ a re t he u tilities of us e rs i n class $A$ and class$B$,re s pect i ve ly, a nd $u$r ep r e sen ts t heut i li ty of th e users if they a llha d loose dela y r equirement s w hich means $ \ti lde{\g am ma} ^ *_k=\ gam ma^*$ for a l l $k$. Fig. \[f ig 2 \] sho ... | frac{\alpha_A}{\alpha}$). Here,_$u_A$ and_$u_B$ are the utilities_of users_in_class $A$_and_class $B$, respectively,_and $u$ represents_the utility of the_users if they_all_had loose delay requirements which means $\tilde{\gamma}^*_k=\gamma^*$ for all $k$. Fig. \[fig2\] shows the_loss_for the_matched_filter,_the decorr... |
good model of citation influence could suggest a model of hypertext link importance. This could improve measures of the importance of web pages, such as PageRank [@Qi:2007:MSD:1244408.1244418].
Improved recommender systems: Researchers often need help identifying relevant work that they should read. Filtering out les... | good model of citation influence could suggest a exemplar of hypertext liaison importance. This could improve measures of the importance of web page, such as PageRank [ @Qi:2007: MSD:1244408.1244418 ].
Improved recommender systems: research worker often need help identify relevant work that they should read. trick... | gold model of citation infuuence could suytest a model of hypeftext link importance. This cluod imkgove measures of the kmportancv of web pagew, such as PefeRank [@Ql:2007:ISD:1244408.1244418].
Ijiroveb cecommender sysjems: Researcvers often neeg fepp identifying relevant work that trey shoilf read. Filterigg olt les... | good model of citation influence could suggest of link importance. could improve measures pages, as PageRank [@Qi:2007:MSD:1244408.1244418]. recommender systems: Researchers need help identifying relevant work that should read. Filtering out less relevant citations might help paper recommender systems [@MEET:MEET145047... | good model of citation influeNce could suGgest A moDel Of HypeRtexT link importancE. this Could improve measures of The imPoRTancE Of Web paGes, such AS PAGERaNk [@qi:2007:mSD:1244408.1244418].
imPRoVed reComMender sYstems: ReseArcHeRs often need hELp IdentifyinG reLevant work thAt tHey shoUlD reAD. FiltEriNg out Les... | good model of citation in fluence co uld s ugg est a mod el o f hypertext li n k im portance. This could i mprov em easu r es of t he impo r ta n c e o fwe b p ag e s, such as PageRa nk [@Qi:20 07: MS D:1244408.12 4 44 18].
Impr ove d recommende r s ystems :Res e arche rsoften needh elp id entifying r e levan... | good_model of_citation influence could suggest_a model_of_hypertext link_importance._This could improve_measures of the_importance of web pages,_such as PageRank [@Qi:2007:MSD:1244408.1244418].
Improved_recommender_systems: Researchers often need help identifying relevant work that they should read. Filtering out_les... |
=(\mu_1,\mu_2,\dots)$, $$\mu \le \lambda \qquad \Leftrightarrow \qquad
\mu_1 + \cdots+\mu_i \le \lambda_1+ \cdots +\lambda_i \quad \text{for all $i \ge 1$}.$$ Let ${\mathbb{C}}[\bx^{\pm 1}] = {\mathbb{C}}[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ be the ring of all Laurent polynomials in $n$ variables $\bx=(x_1,\dots,x_n)$. T... | =( \mu_1,\mu_2,\dots)$, $ $ \mu \le \lambda \qquad \Leftrightarrow \qquad
\mu_1 + \cdots+\mu_i \le \lambda_1 + \cdots + \lambda_i \quad \text{for all $ i \ge 1$}.$$ Let $ { \mathbb{C}}[\bx^{\pm 1 } ] = { \mathbb{C}}[x_1^{\pm 1 }, \dots, x_n^{\pm 1}]$ be the ring of all Laurent polynomials in $ n$ variables $ \bx=(x_1... | =(\mu_1,\mk_2,\dots)$, $$\mu \le \lambda \qquad \Leftrightarrow \qquad
\ku_1 + \cdkts+\mu_i \ld \lambda_1+ \cdots +\lambda_i \quad \vext{dor aol $i \ge 1$}.$$ Let ${\mathbb{C}}[\bx^{\om 1}] = {\mathhb{C}}[x_1^{\pm 1}, \eots, z_n^{\pm 1}]$ be tis ring of all Paurznv polynomials ik $n$ variablas $\bx=(x_1,\dots,x_n)$. T... | =(\mu_1,\mu_2,\dots)$, $$\mu \le \lambda \qquad \Leftrightarrow \qquad \cdots+\mu_i \lambda_1+ \cdots \quad \text{for all 1}] {\mathbb{C}}[x_1^{\pm 1}, \dots, 1}]$ be the of all Laurent polynomials in $n$ $\bx=(x_1,\dots,x_n)$. The Weyl group $W={\mathbb{Z}}_2 \wr {\mathfrak{S}}_n = {\mathbb{Z}}_2^n \rtimes {\mathfrak{... | =(\mu_1,\mu_2,\dots)$, $$\mu \le \lambda \qquad \LeFtrightarrOw \qquAd
\mU_1 + \cdOtS+\mu_i \Le \laMbda_1+ \cdots +\lambdA_I \quaD \text{for all $i \ge 1$}.$$ Let ${\mathbB{C}}[\bx^{\pM 1}] = {\mAThbb{c}}[X_1^{\pM 1}, \dots, X_n^{\pm 1}]$ be tHE rING of AlL LAurEnT PoLynomIalS in $n$ varIables $\bx=(x_1,\dOts,X_n)$. t... | =(\mu_1,\mu_2,\dots)$, $$\ mu \le \la mbda\qq uad \ Left righ tarrow \qquad\mu_ 1 + \cdots+\mu_i \le \ lambd a_ 1 + \c d ot s +\l ambda_i \q u a d \ te xt {fo ra ll $i \ ge1$}.$$Let ${\mat hbb {C }}[\bx^{\pm1 }] = {\mathb b{C }}[x_1^{\pm1}, \dots ,x_n ^ {\pm1}] $ bethe ri n g of a ll Lauren tp olynom i als in$ n $v... | =(\mu_1,\mu_2,\dots)$, $$\mu_\le \lambda_\qquad \Leftrightarrow \qquad
\mu_1_+ \cdots+\mu_i_\le_\lambda_1+ \cdots_+\lambda_i_\quad \text{for all_$i \ge 1$}.$$_Let ${\mathbb{C}}[\bx^{\pm 1}] =_{\mathbb{C}}[x_1^{\pm 1}, \dots,_x_n^{\pm_1}]$ be the ring of all Laurent polynomials in $n$ variables $\bx=(x_1,\dots,x_n)$. T... |
,25)[$E_6$]{} (132, 3)[$\alpha_2$]{} ( 61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (121,43)[$\alpha_4$]{} (151,43)[$\alpha_5$]{} (181,43)[$\alpha_6$]{} (125, 5) ( 65,35) ( 95,35) (125,35) (155,35) (185,35) (125,35)[(0,-1)[30]{}]{} ( 65,35)[(1,0)[30]{}]{} ( 95,35)[(1,0)[30]{}]{} (125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30... | , 25)[$E_6 $ ] { } (132, 3)[$\alpha_2 $ ] { } (61,43)[$\alpha_1 $ ] { } (91,43)[$\alpha_3 $ ] { } (121,43)[$\alpha_4 $ ] { } (151,43)[$\alpha_5 $ ] { } (181,43)[$\alpha_6 $ ] { } (125, 5) (65,35) (95,35) (125,35) (155,35) (185,35) (125,35)[(0,-1)[30 ] { } ] { } (65,35)[(1,0)[30 ] { } ] { } (95,35)[(1,0)[30 ] { } ] { } ... | ,25)[$E_6$]{} (132, 3)[$\wlpha_2$]{} ( 61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (121,43)[$\alphx_4$]{} (151,43)[$\alpha_5$]{} (181,43)[$\alpha_6$]{} (125, 5) ( 65,35) ( 95,35) (125,35) (155,35) (185,35) (125,35)[(0,-1)[30]{}]{} ( 65,35)[(1,0)[30]{}]{} ( 95,35)[(1,0)[30]{}]{} (125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30... | ,25)[$E_6$]{} (132, 3)[$\alpha_2$]{} ( 61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (181,43)[$\alpha_6$]{} 5) ( ( 95,35) (125,35) ( (125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30]{}]{} Again, $p=2$, we do have to worry about the precise of a Chevalley basis in the underlying Lie algebra. Further note that $G({{\mathbb{F}}}_2... | ,25)[$E_6$]{} (132, 3)[$\alpha_2$]{} ( 61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (121,43)[$\alpha_4$]{} (151,43)[$\alphA_5$]{} (181,43)[$\alpha_6$]{} (125, 5) ( 65,35) ( 95,35) (125,35) (155,35) (185,35) (125,35)[(0,-1)[30]{}]{} ( 65,35)[(1,0)[30]{}]{} ( 95,35)[(1,0)[30]{}]{} (125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30... | ,25)[$E_6$]{} (132, 3)[$\a lpha_2$]{} ( 61 ,43 )[$ \a lpha _1$] {} ( 91,43)[$\ a lpha _3$]{} (121,43)[$\alph a_4$] {} (151 , 43 )[$\a lpha_5$ ] {} ( 181 ,4 3) [$\ al p ha _6$]{ } ( 125, 5) ( 65,35)( 9 5, 35) (125,35) (1 55,35) (18 5,3 5) (125,35)[ (0, -1)[30 ]{ }]{ } ( 65 ,35 )[(1, 0)[30] { }]{} ( 95,35)[( 1, 0 )[3... | ,25)[$E_6$]{} (132,_3)[$\alpha_2$]{} (_61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (121,43)[$\alpha_4$]{}_(151,43)[$\alpha_5$]{} (181,43)[$\alpha_6$]{}_(125,_5) (_65,35)_( 95,35) (125,35)_(155,35) (185,35) (125,35)[(0,-1)[30]{}]{}_( 65,35)[(1,0)[30]{}]{} ( 95,35)[(1,0)[30]{}]{}_(125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30... |
] symmetry which can be extended to all sectors of the Lagrangian with an invisible axion solving the strong-CP problem [@Pal95];
5. Spontaneous CP violation in the electroweak sector [@Epele95]. There are several natural sources of explicit and spontaneous CP violations [@vicente];
6. The quark mass hierarchy [@au... | ] symmetry which can be extended to all sectors of the Lagrangian with an invisible axion clear the impregnable - CP problem [ @Pal95 ];
5. Spontaneous CP violation in the electroweak sector [ @Epele95 ]. There be several natural informant of explicit and spontaneous CP misdemeanor [ @vicente ];
6. The ... | ] sylmetry which can be extekded to all sectors of vhe Lagdangian dith an invisible axion solvmng rhe sugong-CP problem [@Pal95];
5. Spuntaneous CP violqtioi in the electroxsak secbjr [@Epspe95]. Tkece are several katural sousces of explicht aud spontaneous CP violations [@vicente];
6. The qusrn mass hierarcry [@au... | ] symmetry which can be extended to of Lagrangian with invisible axion solving Spontaneous violation in the sector [@Epele95]. There several natural sources of explicit and CP violations [@vicente]; 6. The quark mass hierarchy [@austr]; 7. Although the leptoquark-bilepton do not conserve each generation lepton number $... | ] symmetry which can be extendeD to all sectOrs of The lagRaNgiaN witH an invisible axIOn soLving the strong-CP probleM [@Pal95];
5. SPoNTaneOUs cP vioLation iN ThE ELecTrOwEak SeCToR [@EpelE95]. ThEre are sEveral natuRal SoUrces of expliCIt And spontanEouS CP violationS [@viCente];
6. THe QuaRK mass HieRarchY [@au... | ] symmetry which can be ex tended toall s ect ors o f th e La grangian witha n in visible axion solvingthe s tr o ng-C P p roble m [@Pal 9 5] ;
5. Sp ont an e ou s CPvio lationin the ele ctr ow eak sector [ @ Ep ele95]. Th ere are several na turalso urc e s ofexp licit and s p ontane ous CP vi ol a tions[ @vicen... | ] symmetry_which can_be extended to all_sectors of_the_Lagrangian with_an_invisible axion solving_the strong-CP problem [@Pal95];
5._ Spontaneous CP violation_in the electroweak_sector [@Epele95]._There are several natural sources of explicit and spontaneous CP violations [@vicente];
6. The quark_mass_hierarchy [@au... |
P. F. Harrison, D. H. Perkins and W. G. Scott, Phys.Lett. B530, 167 (2002). R. G. Moorhouse, Phys. Rev. D77,053008 (2008)
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abstract: 'We performed terahertz magneto-optical spectroscopy of FeSe thin film to elucidate the charge carrier dynamics. The measured diagonal (longitudinal) and off-diagonal (Hall) conducti... | P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B530, 167 (2002). R. G. Moorhouse, Phys. Rev. D77,053008 (2008)
---
abstract:' We performed terahertz magneto - optical spectroscopy of FeSe thin film to clarify the cathexis carrier dynamics. The measured aslant (longitudinal) and off - diagonal (Hall) co... | P. V. Harrison, D. H. Perkins akd W. G. Scott, Phys.Lett. B530, 167 (2002). R. G. Moorhoure, Phys. Rev. D77,053008 (2008)
---
abstract: 'We pxrfoemed uvrahertz magneto-opticxl spectrlscopy od FeWw thin film to elugndate bhe ckacge carrier dynsmics. The keasured diagotau (pongitudinal) and off-diagonal (Hall) cjnducti... | P. F. Harrison, D. H. Perkins and Scott, B530, 167 R. G. Moorhouse, abstract: performed terahertz magneto-optical of FeSe thin to elucidate the charge carrier dynamics. measured diagonal (longitudinal) and off-diagonal (Hall) conductivity spectra are well reproduced by two-carrier model, from which the carrier densitie... | P. F. Harrison, D. H. Perkins and W. G. SCott, Phys.LeTt. B530, 167 (2002). R. G. mooRhoUsE, PhyS. Rev. d77,053008 (2008)
---
abstract: 'We perFOrmeD terahertz magneto-opticAl speCtROscoPY oF FeSe Thin filM To ELUciDaTe The ChARgE carrIer DynamicS. The measurEd dIaGonal (longituDInAl) and off-diAgoNal (Hall) conduCti... | P. F. Harrison, D. H. Per kins and W . G.Sco tt, P hys. Lett . B530, 167 (2 0 02). R. G. Moorhouse, Phys . Rev .D 77,0 5 30 08 (2 008)
- - -a b str ac t: 'W ep er forme d t erahert z magneto- opt ic al spectrosc o py of FeSe t hin film to elu cid ate th echa r ge ca rri er dy namics . The m easured d ia g onal ... | P._F. Harrison,_D. H. Perkins and_W. G._Scott,_Phys.Lett. B530,_167_(2002). R. G._Moorhouse, Phys. Rev._D77,053008 (2008)
---
abstract: 'We_performed terahertz magneto-optical_spectroscopy_of FeSe thin film to elucidate the charge carrier dynamics. The measured diagonal (longitudinal)_and_off-diagonal (Hall)_conducti... |
{T}^H_{_m}$. Furthermore, because the complex codimension of $\mathcal{Z}^H_{m}\setminus \mathcal{Z}_{m}$ is as least one in $\mathcal{Z}^H_{_m}$, the complex codimension of $\mathcal{T}^H_{_m}\backslash\mathcal{T}_m$ is also as least one in $\mathcal{T}^H_{_m}$.
First, it is not hard to see that the restriction map $... | { T}^H_{_m}$. Furthermore, because the complex codimension of $ \mathcal{Z}^H_{m}\setminus \mathcal{Z}_{m}$ is as least one in $ \mathcal{Z}^H_{_m}$, the complex codimension of $ \mathcal{T}^H_{_m}\backslash\mathcal{T}_m$ is also as least one in $ \mathcal{T}^H_{_m}$.
First, it is not unvoiced to examine that the re... | {T}^H_{_m}$. Furthermore, because the complex codimeuwion oh $\mathczl{Z}^H_{m}\setoinus \mathcal{Z}_{m}$ is as least lnw in $\nathcal{Z}^H_{_m}$, the complex codimenspon of $\marhcao{R}^H_{_m}\backslash\mathcal{T}_m$ is wlso es least one in $\mathcal{T}^H_{_k}$.
First, it is nmt hcrd to see that the restriction map $... | {T}^H_{_m}$. Furthermore, because the complex codimension of is least one $\mathcal{Z}^H_{_m}$, the complex as one in $\mathcal{T}^H_{_m}$. it is not to see that the restriction map is holomorphic. Indeed, we know that $i_m:\,\mathcal{T}\rightarrow \mathcal{T}_m$ is the lifting of $i\circ and $\pi^H_{_m}|_{\mathcal{T}_... | {T}^H_{_m}$. Furthermore, because the cOmplex codiMensiOn oF $\maThCal{Z}^h_{m}\seTminus \mathcal{Z}_{M}$ Is as Least one in $\mathcal{Z}^H_{_m}$, thE compLeX CodiMEnSion oF $\mathcaL{t}^H_{_M}\BAckSlAsH\maThCAl{t}_m$ is aLso As least One in $\mathcAl{T}^h_{_m}$.
first, it is not HArD to see that The Restriction mAp $... | {T}^H_{_m}$. Furthermore,because th e com ple x c od imen sion of $\mathcal{ Z }^H_ {m}\setminus \mathcal{ Z}_{m }$ is a s l eastone in$ \m a t hca l{ Z} ^H_ {_ m }$ , the co mplex c odimension of $ \mathcal{T}^ H _{ _m}\backsl ash \mathcal{T}_ m$is als oasl eastone in $ \mathc a l{T}^H _{_m}$.
Fi r st, it is not... | {T}^H_{_m}$. Furthermore,_because the_complex codimension of $\mathcal{Z}^H_{m}\setminus_\mathcal{Z}_{m}$ is_as_least one_in_$\mathcal{Z}^H_{_m}$, the complex_codimension of $\mathcal{T}^H_{_m}\backslash\mathcal{T}_m$_is also as least_one in $\mathcal{T}^H_{_m}$.
First,_it_is not hard to see that the restriction map $... |
-qc\]]{}]{}. G. Compere, “[Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions]{},” [[ arXiv:0708.3153 \[hep-th\]]{}]{}.
R. M. Wald, “Black Hole Entropy is the Noether Charge,” [[ Phys. Rev.]{} [**D48**]{} (1993) 3427]{}, [[... | -qc\ ] ] { } ] { }. G. Compere, “ [ Symmetries and conservation laws in Lagrangian gauge theories with application to the automobile mechanic of black holes and to gravity in three dimensions ] { }, ” [ [ arXiv:0708.3153 \[hep - th\ ] ] { } ] { }.
R. M. Wald, “ Black Hole Entropy is the Noether Charge, ” [ [ Phys.... | -qc\]]{}]{}. H. Compere, “[Symmetries and gonservation laws in Lajrangiah gauge gheories with applications tl rhe mtbhanics of black holer and to hravity un tiree dimensions]{},” [[ arXiv:0708.3153 \[hc'-th\]]{}]{}.
R. J. Walb, “Ulack Hole Entrppy is the Noether Charga,” [[ Pkys. Rev.]{} [**D48**]{} (1993) 3427]{}, [[... | -qc\]]{}]{}. G. Compere, “[Symmetries and conservation laws gauge with applications the mechanics of in dimensions]{},” [[ arXiv:0708.3153 R. M. Wald, Hole Entropy is the Noether Charge,” Phys. Rev.]{} [**D48**]{} (1993) 3427]{}, [[ \[arXiv:gr-qc/9307038\]]{}]{}. V. Iyer and R. M. Wald, Properties of Noether Charge and... | -qc\]]{}]{}. G. Compere, “[Symmetries and coNservation Laws iN LaGraNgIan gAuge Theories with apPLicaTions to the mechanics of bLack hOlES and TO gRavitY in threE DiMENsiOnS]{},” [[ aRXiV:0708.3153 \[hEP-tH\]]{}]{}.
R. M. WaLd, “BLack HolE Entropy is The noEther Charge,” [[ PHYs. rev.]{} [**D48**]{} (1993) 3427]{}, [[... | -qc\]]{}]{}. G. Compere, “ [Symmetrie s and co nse rv atio n la ws in Lagrangi a n ga uge theories with appl icati on s tot he mech anics o f b l a ckho le s a nd to grav ity in thr ee dimensi ons ]{ },” [[ arXiv : 07 08.3153 \[ hep -th\]]{}]{}.
R . M. W al d,“ Black Ho le En tropyi s theNoether C ha r ge,” [ ... | -qc\]]{}]{}. G. Compere,_“[Symmetries and_conservation laws in Lagrangian_gauge theories_with_applications to_the_mechanics of black_holes and to_gravity in three dimensions]{},”_[[ arXiv:0708.3153 \[hep-th\]]{}]{}.
R._M._Wald, “Black Hole Entropy is the Noether Charge,” [[ Phys. Rev.]{} [**D48**]{} (1993) 3427]{},_[[... |
n$ is diagonalizing this matrix, which is possible since ${\mathbb{K}}_n[t^{\pm1}]$ is a PID. Since $c\neq1$ in $\pi_1({{\mathcal U}})/\pi_1({{\mathcal U}})^\prime$, it follows that $c \notin
\pi_1({{\mathcal U}})^{(n)}_r$ for all $n \geq 1$. Therefore $c\neq1$ in $\Gamma_n$ for all $n \geq 0$. Hence $1-c\neq0$ in ${\m... | n$ is diagonalizing this matrix, which is possible since $ { \mathbb{K}}_n[t^{\pm1}]$ is a PID. Since $ c\neq1 $ in $ \pi_1({{\mathcal U}})/\pi_1({{\mathcal U}})^\prime$, it follows that $ c \notin
\pi_1({{\mathcal U}})^{(n)}_r$ for all $ newton \geq 1$. consequently $ c\neq1 $ in $ \Gamma_n$ for all $ n \geq 0$. Hen... | n$ id diagonalizing this matvix, which is possible smnce ${\mafhbb{K}}_n[t^{\po1}]$ is a PID. Since $c\neq1$ in $\pi_1({{\methcql U}})/\pu_1({{\mathcal U}})^\prime$, it foluows that $c \notin
\pi_1({{\mauhcal U}})^{(n)}_r$ for all $n \geq 1$. Bkerefkve $c\nzq1$ in $\Gamma_n$ for all $n \geq 0$. Hence $1-c\neq0$ it ${\o... | n$ is diagonalizing this matrix, which is ${\mathbb{K}}_n[t^{\pm1}]$ a PID. $c\neq1$ in $\pi_1({{\mathcal $c \pi_1({{\mathcal U}})^{(n)}_r$ for $n \geq 1$. $c\neq1$ in $\Gamma_n$ for all $n 0$. Hence $1-c\neq0$ in ${\mathbb{Z}}\Gamma_n$ and is therefore invertible in ${\mathbb{K}}_n[t^{\pm1}]$. This allows to multiply ... | n$ is diagonalizing this matriX, which is poSsiblE siNce ${\MaThbb{k}}_n[t^{\pM1}]$ is a PID. Since $c\nEQ1$ in $\pI_1({{\mathcal U}})/\pi_1({{\mathcal U}})^\priMe$, it fOlLOws tHAt $C \notiN
\pi_1({{\mathCAl u}})^{(N)}_R$ foR aLl $N \geQ 1$. THErEfore $C\neQ1$ in $\GammA_n$ for all $n \gEq 0$. HEnCe $1-c\neq0$ in ${\m... | n$ is diagonalizing this m atrix, whi ch is po ssi bl e si nce${\mathbb{K}}_ n [t^{ \pm1}]$ is a PID. Sinc e $c\ ne q 1$ i n $ \pi_1 ({{\mat h ca l U}} )/ \p i_1 ({ { \m athca l U }})^\pr ime$, it f oll ow s that $c \n o ti n
\pi_1({{ \ma thcal U}})^{ (n) }_r$ f or al l $n \ geq 1$.Theref o re $c\ neq1$ in$\ G amm... | n$ is_diagonalizing this_matrix, which is possible_since ${\mathbb{K}}_n[t^{\pm1}]$_is_a PID._Since_$c\neq1$ in $\pi_1({{\mathcal_U}})/\pi_1({{\mathcal U}})^\prime$, it_follows that $c \notin
\pi_1({{\mathcal_U}})^{(n)}_r$ for all_$n_\geq 1$. Therefore $c\neq1$ in $\Gamma_n$ for all $n \geq 0$. Hence $1-c\neq0$ in_${\m... |
Spurious Correlations
=======================================
There are enormous differences in the size of academic journals, and these differences swamp the patterns that Davis was seeking in his analysis. The JCR indexes journals that range in size from tiny (*Astronomy and Astrophysics Review* has published 13 ar... | Spurious Correlations
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
There are enormous differences in the size of academic journals, and these differences deluge the design that Davis was seeking in his analysis. The JCR index journal that range in size from tiny (* Astronomy and As... | Spkrious Correlations
=======================================
There are enormous dndferenres in fhe size of academic journals, and thxse eiffeeences swamp the pattefns that Favis waw setking in his analbais. The JCR ihfexev journals that range in vize from tiny (*Artxonomy and Astrophysics Review* has ptblishec 13 ar... | Spurious Correlations ======================================= There are enormous differences size academic journals, these differences swamp seeking his analysis. The indexes journals that in size from tiny (*Astronomy and Review* has published 13 articles over the previous five years) to huge (*The of Biological Chemi... | Spurious Correlations
=======================================
There Are enormouS diffEreNceS iN the Size Of academic jourNAls, aNd these differences swamP the pAtTErns THaT DaviS was seeKInG IN hiS aNaLysIs. tHe jCR inDexEs journAls that ranGe iN sIze from tiny (*ASTrOnomy and AsTroPhysics RevieW* haS publiShEd 13 aR... | Spurious Correlations
=== ========== ===== === === == ==== ==== =====
There a r e en ormous differences inthe s iz e ofa ca demic journa l s, a ndth es e d if f er ences sw amp the patternstha tDavis was se e ki ng in hisana lysis. The J CRindexe sjou r nalstha t ran ge ins ize fr om tiny ( *A s tronom y and As ... | Spurious_Correlations
=======================================
There are_enormous differences in the_size of_academic_journals, and_these_differences swamp the_patterns that Davis_was seeking in his_analysis. The JCR_indexes_journals that range in size from tiny (*Astronomy and Astrophysics Review* has published 13_ar... |
the normalizer of $T$ in $G$. An element $\sigma\in
Aut(\mathfrak{g},\mathfrak{t})$ induces a symmetry of $\Phi$, and we denote the resulting group homomorphism by $$\tau: Aut(\mathfrak{g},\mathfrak{t}){\longrightarrow}Aut(\Phi), \ \sigma\mapsto (\sigma^{-1}_{|\mathfrak{t}})^*.$$
Recall also that the **Weyl group** o... | the normalizer of $ T$ in $ G$. An element $ \sigma\in
Aut(\mathfrak{g},\mathfrak{t})$ induces a symmetry of $ \Phi$, and we denote the result group homomorphism by $ $ \tau: Aut(\mathfrak{g},\mathfrak{t}){\longrightarrow}Aut(\Phi), \ \sigma\mapsto (\sigma^{-1}_{|\mathfrak{t}})^*.$$
Recall besides that the * * Wey... | thf normalizer of $T$ in $G$. Ak element $\sigma\iu
Qut(\matifrak{g},\mzthfrak{t})$ induces a symmetry of $\Phi$, aid ww denite the resulting grouo homomorihism by $$\rau: Eut(\mathfrak{g},\mathhdak{t}){\lonnxightzvrow}Abt(\'hi), \ \sigma\mapstp (\sigma^{-1}_{|\matvfrak{t}})^*.$$
Recall ansu chat the **Weyl group** o... | the normalizer of $T$ in $G$. An Aut(\mathfrak{g},\mathfrak{t})$ a symmetry $\Phi$, and we by Aut(\mathfrak{g},\mathfrak{t}){\longrightarrow}Aut(\Phi), \ \sigma\mapsto Recall also that **Weyl group** of $\Phi$, denoted by Aut(\Phi)$, is the group generated by the symmetries $s_{\alpha}$ (defined in (\[EQ\_symmetries\])... | the normalizer of $T$ in $G$. An elemEnt $\sigma\in
aut(\maThfRak{G},\mAthfRak{t})$ Induces a symmetRY of $\PHi$, and we denote the resultIng grOuP HomoMOrPhism By $$\tau: AuT(\MaTHFraK{g},\MaThfRaK{T}){\lOngriGhtArrow}AuT(\Phi), \ \sigma\mApsTo (\Sigma^{-1}_{|\mathfraK{T}})^*.$$
REcall also tHat The **Weyl group** O... | the normalizer of $T$ in$G$. An el ement $\ sig ma \inAut( \mathfrak{g},\ m athf rak{t})$ induces a sym metry o f $\P h i$ , and we den o te t here su lti ng gr oup h omo morphis m by $$\ta u:Au t(\mathfrak{ g }, \mathfrak{ t}) {\longrighta rro w}Aut( \P hi) , \ \s igm a\map sto (\ s igma^{ -1}_{|\ma th f rak{t} ... | the_normalizer of_$T$ in $G$. An_element $\sigma\in
Aut(\mathfrak{g},\mathfrak{t})$_induces_a symmetry_of_$\Phi$, and we_denote the resulting_group homomorphism by $$\tau:_Aut(\mathfrak{g},\mathfrak{t}){\longrightarrow}Aut(\Phi), \ \sigma\mapsto_(\sigma^{-1}_{|\mathfrak{t}})^*.$$
Recall_also that the **Weyl group** o... |
.1497$ $0.0016$ $88.7 $ $1.1 $ $1.6$ $1.57$
2013-09-07 987 Rozhen 2.0m $6542.50530$ $0.00018$ $5.97 $ $0.09 $ $0.1535$ $0.0012$ $88.3 $ $0.8 $ $0.9$ $0.67$
2013-09-07 987 Torun 0.6m $6542.50522$ $0.00052$ $5.89 $ $0.20 $ $0... | .1497 $ $ 0.0016 $ $ 88.7 $ $ 1.1 $ $ 1.6 $ $ 1.57 $
2013 - 09 - 07 987 Rozhen 2.0 m $ 6542.50530 $ $ 0.00018 $ $ 5.97 $ $ 0.09 $ $ 0.1535 $ $ 0.0012 $ $ 88.3 $ $ 0.8 $ $ 0.9 $ $ 0.67 $
2013 - 09 - 07 987 Torun 0.6 m... | .1497$ $0.0016$ $88.7 $ $1.1 $ $1.6$ $1.57$
2013-09-07 987 Tozhen 2.0m $6542.50530$ $0.00018$ $5.97 $ $0.09 $ $0.1535$ $0.0012$ $88.3 $ $0.8 $ $0.9$ $0.67$
2013-09-07 987 Torun 0.6m $6542.50522$ $0.00052$ $5.89 $ $0.20 $ $0... | .1497$ $0.0016$ $88.7 $ $1.1 $ $1.6$ 987 2.0m $6542.50530$ $5.97 $ $0.09 $0.8 $0.9$ $0.67$ 2013-09-07 Torun 0.6m $6542.50522$ $5.89 $ $0.20 $ $0.1515$ $0.0029$ $ $1.4 $ $3.5$ $2.33$ 2013-10-06 1001 OSN 1.5m $6572.60532$ $0.00018$ $6.11 $ $ $0.1465$ $0.0013$ $89.2 $ $0.7 $ $0.9$ $0.63$ 2013-11-01 1013 Rozhen 2.0m $0.000... | .1497$ $0.0016$ $88.7 $ $1.1 $ $1.6$ $1.57$
2013-09-07 987 Rozhen 2.0m $6542.50530$ $0.00018$ $5.97 $ $0.09 $ $0.1535$ $0.0012$ $88.3 $ $0.8 $ $0.9$ $0.67$
2013-09-07 987 Torun 0.6m $6542.50522$ $0.00052$ $5.89 $ $0.20 $ $0... | .1497$ $0.0016$ $8 8.7 $ $1. 1$ $1. 6$ $1.57$
2 013- 09-07 987 Rozhen 2.0m $ 65 42.50 530$ $ 0. 0 0 018 $ $5. 97 $ $0. 09$ $0 .1535$ $0.0012$ $88.3$ $0.8 $ $ 0.9 $ $0 .6 7$ 2013 -09 -07 987 Toru n 0.6m $654 2 .50522$ $ 0.00 052$ $5.89 $ $ 0. 2 0 $ ... | .1497$ _ _ _ _$0.0016$_ __ _ _$88.7 $ _$1.1 $ __ $1.6$ $1.57$
2013-09-07 987 _Rozhen_2.0m ___ $6542.50530$ _ $0.00018$ $5.97_$ _ $0.09 $ $0.1535$ __ _ $0.0012$ _ $88.3_$ _$0.8_$_ $0.9$_ $0.67$
2013-09-07 _ 987 _Torun 0.6m _ $6542.50522$ $0.00052$_ $5.89 $ _ $0.20_$ $0... |
the only deviation from regularity (see below).
For [NLTT 11748]{}, I recognize that $K_2$ is the radial velocity that was measured since the heavier object is the fainter one. So, $q\approx
0.15/0.71=0.21$, $K_2=271\,{\ensuremath{{\rm km\,s}^{-1}}}$, and $P=5.64\,$hr, which give ${\ensuremath{\Delta t_{\rm LT}}}=4.6... | the only deviation from regularity (see below).
For [ NLTT 11748 ] { }, I accredit that $ K_2 $ is the radial speed that was measured since the heavier aim is the fainter one. So, $ q\approx
0.15/0.71=0.21 $, $ K_2=271\,{\ensuremath{{\rm km\,s}^{-1}}}$, and $ P=5.64\,$hr, which grant $ { \ensuremath{\Delta t_{\r... | thf only deviation from renularity (see below).
For [NNTT 11748]{}, I decognizd that $K_2$ is the radial velocmty rhat qas measured since the heavier lbject iw tht fainter one. So, $q\approx
0.15/0.71=0.21$, $K_2=271\,{\ensurslath{{\xm km\,s}^{-1}}}$, and $P=5.64\,$hr, wmich give ${\etsuremath{\Delta t_{\fm LT}}}=4.6... | the only deviation from regularity (see below). 11748]{}, recognize that is the radial the object is the one. So, $q\approx $K_2=271\,{\ensuremath{{\rm km\,s}^{-1}}}$, and $P=5.64\,$hr, which give t_{\rm LT}}}=4.6\,$s. @sks+10 measured individual eclipse times to $\sim 10$s, making it hard detect an effect like this. H... | the only deviation from regulArity (see beLow).
FoR [NLtT 11748]{}, I ReCognIze tHat $K_2$ is the radiaL VeloCity that was measured sinCe the HeAVier OBjEct is The fainTEr ONE. So, $Q\aPpRox
0.15/0.71=0.21$, $k_2=271\,{\eNSuRematH{{\rm Km\,s}^{-1}}}$, and $P=5.64\,$Hr, which givE ${\enSuRemath{\Delta t_{\RM Lt}}}=4.6... | the only deviation from r egularity(seebel ow) .
For [NL TT 11748]{}, I reco gnize that $K_2$ is th e rad ia l vel o ci ty th at wasm ea s u red s in ceth e h eavie r o bject i s the fain ter o ne. So, $q\a p pr ox
0.15/0. 71= 0.21$, $K_2= 271 \,{\en su rem a th{{\ rmkm\,s }^{-1} } }$, an d $P=5.64 \, $ hr, wh i ... | the_only deviation_from regularity (see below).
For_[NLTT 11748]{}, I_recognize_that $K_2$_is_the radial velocity_that was measured_since the heavier object_is the fainter_one._So, $q\approx
0.15/0.71=0.21$, $K_2=271\,{\ensuremath{{\rm km\,s}^{-1}}}$, and $P=5.64\,$hr, which give ${\ensuremath{\Delta t_{\rm LT}}}=4.6... |
mathrm{bulge}}-M_B<0.4$ [@Sim86]. We ignore galaxies at the centers of simulated clusters since we have omitted the two central Coma galaxies from the analysis in this paper. Likewise, we exclude from the comparison galaxies that have been stripped-off their entire halo, because the only Coma galaxy that possibly lacks... | mathrm{bulge}}-M_B<0.4 $ [ @Sim86 ]. We ignore galaxies at the centers of simulated bunch since we have exclude the two central Coma galaxies from the psychoanalysis in this paper. besides, we exclude from the comparison galaxy that have been stripped - off their entire aura, because the only Coma galaxy that possibly ... | matjrm{bulge}}-M_B<0.4$ [@Sim86]. We ignore galaxies at thg xenterv of sjmulated clusters since we have omitved rhe tqo central Coma galaxids from tje analywis mn this paper. Likewise, wc excmmde fxon the comparispn galaxiev that have bean scripped-off their entire halo, because the onky Coma galaxy trat kosfiblg lacks... | mathrm{bulge}}-M_B<0.4$ [@Sim86]. We ignore galaxies at the simulated since we omitted the two analysis this paper. Likewise, exclude from the galaxies that have been stripped-off their halo, because the only Coma galaxy that possibly lacks dark matter inside $3\, has been excluded from the analysis in this paper as we... | mathrm{bulge}}-M_B<0.4$ [@Sim86]. We ignore gAlaxies at tHe cenTerS of SiMulaTed cLusters since we HAve oMitted the two central ComA galaXiES froM ThE analYsis in tHIs PAPer. liKeWisE, wE ExClude FroM the comParison galAxiEs That have been STrIpped-off thEir Entire halo, beCauSe the oNlY CoMA galaXy tHat poSsibly LAcks... | mathrm{bulge}}-M_B<0.4$ [@ Sim86]. We igno regal ax iesat t he centers ofs imul ated clusters since we have o m itte d t he tw o centr a lC o maga la xie sf ro m the an alysisin this pa per .Likewise, we ex clude from th e comparison ga laxies t hat havebee n str ipped- o ff the ir entire h a lo, be c ause th ... | mathrm{bulge}}-M_B<0.4$ [@Sim86]._We ignore_galaxies at the centers_of simulated_clusters_since we_have_omitted the two_central Coma galaxies_from the analysis in_this paper. Likewise,_we_exclude from the comparison galaxies that have been stripped-off their entire halo, because the_only_Coma galaxy_that_possibly_lacks... |
}
\frac{x}{\vartheta} = \frac1{\alpha^2} \int_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})-\theta}{\alpha}\right\}d\mathfrak{t}$$ depends on the details of the pattern $\mathfrak{u}(\mathfrak{t})$ in the vicinity of its maximum $\theta$. Nevertheless, these details seem not to be too ... | }
\frac{x}{\vartheta } = \frac1{\alpha^2 } \int_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})-\theta}{\alpha}\right\}d\mathfrak{t}$$ depends on the details of the pattern $ \mathfrak{u}(\mathfrak{t})$ in the vicinity of its maximum $ \theta$. however, these detail seem not to be too... | }
\fraf{x}{\vartheta} = \frac1{\alpha^2} \ikt_{\mathfrak{t}\in \majhvb{P}_\theva}\exp\lert\{\frac{\maghfrak{u}(\mathfrak{t})-\theta}{\alpha}\rijht\}d\nathfeak{t}$$ depends on the degails of nhe patteen $\methfrak{u}(\mathfrak{v})$ in the vicinjby of mts maximum $\theja$. Neverthelass, these detahlr deem not to be too ... | } \frac{x}{\vartheta} = \frac1{\alpha^2} \int_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})-\theta}{\alpha}\right\}d\mathfrak{t}$$ depends details the pattern in the vicinity these seem not to too essential; they mainly some cofactors of order unity, also Ref. [@we1]. To justify the lat... | }
\frac{x}{\vartheta} = \frac1{\alpha^2} \int_{\Mathfrak{t}\iN \mathBb{P}_\TheTa}\Exp\lEft\{\fRac{\mathfrak{u}(\maTHfraK{t})-\theta}{\alpha}\right\}d\mathFrak{t}$$ DePEnds ON tHe detAils of tHE pATTerN $\mAtHfrAk{U}(\MaThfraK{t})$ iN the vicInity of its MaxImUm $\theta$. NeverTHeLess, these dEtaIls seem not to Be tOo ... | }
\frac{x}{\vartheta} = \f rac1{\alph a^2} \i nt_ {\ math frak {t}\in \mathbb { P}_\ theta}\exp\left\{\frac {\mat hf r ak{u } (\ mathf rak{t}) - \t h e ta} {\ al pha }\ r ig ht\}d \ma thfrak{ t}$$ depen dson the details of the patte rn$\mathfrak{u }(\ mathfr ak {t} ) $ inthe vici nity o f its m aximum $\ th e ta$. ... | }
\frac{x}{\vartheta} =_\frac1{\alpha^2} _\int_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})-\theta}{\alpha}\right\}d\mathfrak{t}$$ depends on_the details_of_the pattern_$\mathfrak{u}(\mathfrak{t})$_in the vicinity_of its maximum_$\theta$. Nevertheless, these details_seem not to_be_too ... |
$, may be systematically affected by the echelle-order blaze residuals imprinted on the continuum in the UVES (slit-mode) observations. Differences in model atmospheres and hydrogen line-broadening theory used may also explain part of the offset (see Paper I).
![Comparison between our spectroscopic $T_{\rm eff}$-scale... | $, may be systematically affected by the echelle - order blaze residuals imprint on the continuum in the UVES (slit - mood) observations. Differences in model standard atmosphere and hydrogen line - broadening hypothesis used may also explain region of the offset (see Paper I).
! [ Comparison between our spectroscopi... | $, maj be systematically affegted by the echelle-ordec blaze residuaus imprinted on the continuul un tht UVES (slit-mode) obsdrvations. Differebces un model avjosphercf ans hydxojen line-broadenlng theory gsed may also axolcin part of the offset (see Paper I).
![Coiparisom hetween our spgctroxsopid $T_{\rm eff}$-scale... | $, may be systematically affected by the residuals on the in the UVES atmospheres hydrogen line-broadening theory may also explain of the offset (see Paper I). between our spectroscopic $T_{\rm eff}$-scale, the photometric values obtained from $v-y$ and $V-I$, the results from six other studies. []{data-label="fig:othe... | $, may be systematically affectEd by the echElle-oRdeR blAzE resIduaLs imprinted on tHE conTinuum in the UVES (slit-modE) obseRvATionS. diFfereNces in mODeL ATmoSpHeRes AnD HyDrogeN liNe-broadEning theorY usEd May also explaIN pArt of the ofFseT (see Paper I).
![CoMpaRison bEtWeeN Our spEctRoscoPic $T_{\rm EFf}$-scalE... | $, may be systematically a ffected by theech ell e- orde r bl aze residualsi mpri nted on the continuumin th eU VES( sl it-mo de) obs e rv a t ion s. D iff er e nc es in mo del atm ospheres a ndhy drogen line- b ro adening th eor y used may a lso expla in pa r t ofthe offs et (se e Paper I).
![C om p arison be... | $, may_be systematically_affected by the echelle-order_blaze residuals_imprinted_on the_continuum_in the UVES_(slit-mode) observations. Differences_in model atmospheres and_hydrogen line-broadening theory_used_may also explain part of the offset (see Paper I).
![Comparison between our spectroscopic $T_{\rm_eff}$-scale... |
. The Clarendon Press, Oxford University Press, New York, 1995.
G.E. Murphy, A new construction of Young’s seminormal representation of the symmetric group, [*J. Algebra*]{} [**69**]{} (1981), 287–297.
V. Reiner, D. Stanton, and P. Webb, Springer’s regular elements over arbitrary fields, [*Math. Proc. Cambridge Philo... | . The Clarendon Press, Oxford University Press, New York, 1995.
G.E. Murphy, A new construction of Young ’s seminormal representation of the symmetric group, [ * J. Algebra * ] { } [ * * 69 * * ] { } (1981), 287–297.
V. Reiner, D. Stanton, and P. Webb, Springer ’s even chemical element over arbitrary fields, [ * ... | . Thf Clarendon Press, Oxford University Press, New Bork, 1995.
G.E. Murphy, X new construction of Young’s swminoemal representation of the symmvtric groyp, [*J. Qlgebra*]{} [**69**]{} (1981), 287–297.
T. Reiner, D. Stahbon, aud P. Webb, Springgr’s regular alements over drcicrary fields, [*Math. Proc. Cambridge Phijo... | . The Clarendon Press, Oxford University Press, 1995. Murphy, A construction of Young’s group, Algebra*]{} [**69**]{} (1981), V. Reiner, D. and P. Webb, Springer’s regular elements arbitrary fields, [*Math. Proc. Cambridge Philos. Soc.*]{} [**141**]{} (2006), 209–229. V. Reiner, D. and D. White, The cyclic sieving phen... | . The Clarendon Press, Oxford UnIversity PrEss, NeW YoRk, 1995.
G.e. MUrphY, A neW construction oF younG’s seminormal representaTion oF tHE symMEtRic grOup, [*J. AlgEBrA*]{} [**69**]{} (1981), 287–297.
v. reiNeR, D. staNtON, aNd P. WeBb, SPringer’S regular elEmeNtS over arbitraRY fIelds, [*Math. PRoc. cambridge PhiLo... | . The Clarendon Press, Oxf ord Univer sityPre ss, N ew Y ork, 1995.
G.E. M u rphy , A new construction o f You ng ’ s se m in ormal repres e nt a t ion o fthe s y mm etric gr oup, [* J. Algebra *]{ }[**69**]{} ( 1 98 1), 287–29 7.
V. Reiner,D.Stanto n, an d P. W ebb , Spr inger’ s regul ar elemen ts over a r ... | . The_Clarendon Press,_Oxford University Press, New_York, 1995.
G.E._Murphy,_A new_construction_of Young’s seminormal_representation of the_symmetric group, [*J. Algebra*]{}_[**69**]{} (1981), 287–297.
V._Reiner,_D. Stanton, and P. Webb, Springer’s regular elements over arbitrary fields, [*Math. Proc. Cambridge_Philo... |
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