text stringlengths 55 9.15k | synonym_substitution stringlengths 55 9.22k | butter_fingers stringlengths 55 9.15k | random_deletion stringlengths 47 4.48k | change_char_case stringlengths 55 9.15k | whitespace_perturbation stringlengths 60 7k | underscore_trick stringlengths 55 9.15k |
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call any one of these rule-sets the *sRules*, indicating that they are auxiliary rules that accompany standard quantum mechanics.
The Born rule is not necessary. It is possible to achieve very desirable results by dropping the Born rule from the governing auxiliary rules and introducing probability into quantum mecha... | call any one of these rule - sets the * sRules *, indicating that they are accessory rule that accompany standard quantum mechanics.
The digest principle is not necessary. It is possible to achieve very desirable results by drop the Born rule from the regulate auxiliary rules and insert probability into quantum mech... | capl any one of these rule-rets the *sRules*, indicaving thzt they xre auxiliary rules that acclmpany wtandard quantum mechavics.
The Blrn rule is iot necessary. It is posslyle tk achnete very desirabke results by dropping tve Blrn rule from the governing auxiliawy rulex wnd introducind prpfabimptn into quantum mecha... | call any one of these rule-sets the that are auxiliary that accompany standard is necessary. It is to achieve very results by dropping the Born rule the governing auxiliary rules and introducing probability into quantum mechanics through *probability current* This has been done with two very different auxiliary rule-se... | call any one of these rule-sets The *sRules*, iNdicaTinG thAt They Are aUxiliary rules tHAt acCompany standard quantum MechaNiCS.
The bOrN rule Is not neCEsSARy. IT iS pOssIbLE tO achiEve Very desIrable resuLts By Dropping the BORn Rule from thE goVerning auxilIarY rules AnD inTRoducIng ProbaBility INto quaNtum mecha... | call any one of these rul e-sets the *sRu les *,in dica ting that they are auxi liary rules that accom panyst a ndar d q uantu m mecha n ic s .
T he B orn r u le is n otnecessa ry. It ispos si ble to achie v every desir abl e results by dr opping t heB orn r ule from the g o vernin g auxilia ry rulesa nd i... | call_any one_of these rule-sets the_*sRules*, indicating_that_they are_auxiliary_rules that accompany_standard quantum mechanics.
The_Born rule is not_necessary. It is_possible_to achieve very desirable results by dropping the Born rule from the governing auxiliary_rules_and introducing_probability_into_quantum mecha... |
the wrong predictions for the $\sigma$-meson because the corresponding mass obtained is of an order $1/\bar\rho\sim 1~GeV$ (moreover, the adiabaticity approximation is certainly broken then). The result Eq. (\[simas\]) aggravates, in a sense, the situation since the hard component leaves for the region of harder masse... | the wrong predictions for the $ \sigma$-meson because the corresponding mass prevail is of an decree $ 1/\bar\rho\sim 1 ~ GeV$ (moreover, the adiabaticity approximation is certainly break then). The result Eq. (\[simas\ ]) aggravates, in a common sense, the situation since the hard component entrust for the region of h... | thf wrong predictions for uhe $\sigma$-meson beequse tie corrsspondine mass obtained is of an ordxr $1/\bqr\rho\wim 1~GeV$ (moreover, the aaiabaticiny approxumatmon is certainly broken bken). Tgc resblv Eq. (\[simas\]) aggrsvates, in d sense, the siduxtnon since the hard component leaves sor the rfgion of hardet masxq... | the wrong predictions for the $\sigma$-meson because mass is of order $1/\bar\rho\sim 1~GeV$ certainly then). The result (\[simas\]) aggravates, in sense, the situation since the hard leaves for the region of harder masses whereas the light component persists to lighter. On the other hand, the present observations sign... | the wrong predictions for the $\Sigma$-meson BecauSe tHe cOrRespOndiNg mass obtained IS of aN order $1/\bar\rho\sim 1~GeV$ (moreOver, tHe ADiabATiCity aPproximATiON Is cErTaInlY bROkEn theN). ThE result eq. (\[simas\]) aggRavAtEs, in a sense, thE SiTuation sinCe tHe hard componEnt Leaves FoR thE RegioN of HardeR masse... | the wrong predictions for the $\sig ma$-m eso n b ec ause the corresponding mass obtained is of an ord er $1 /\ b ar\r h o\ sim 1 ~GeV$ ( m or e o ver ,th e a di a ba ticit y a pproxim ation is c ert ai nly broken t h en ). The res ult Eq. (\[sima s\] ) aggr av ate s , ina s ense, the s i tuatio n since t he har... | the_wrong predictions_for the $\sigma$-meson because_the corresponding_mass_obtained is_of_an order $1/\bar\rho\sim_1~GeV$ (moreover, the_adiabaticity approximation is certainly_broken then). The_result_Eq. (\[simas\]) aggravates, in a sense, the situation since the hard component leaves for_the_region of_harder_masse... |
straete, “Discriminating states: The quantum chernoff bound,” *Phys. Rev. Lett.*, vol. 98, no. 16, p. 160501, 2007.
M. Ohya and D. Petz, *Quantum Entropy and Its Use*.1em plus 0.5em minus 0.4emSpringer, 1993.
M. Mosony and N. Datta, “Generalized relative entropies and the capacity of classical-quantum channels,” 2008... | straete, “ Discriminating states: The quantum chernoff bound, ” * Phys. Rev. Lett. *, vol. 98, no. 16, p. 160501, 2007.
M. Ohya and D. Petz, * Quantum Entropy and Its Use*.1em plus 0.5em minus 0.4emSpringer, 1993.
M. Mosony and N. Datta, “ Generalized relative entropies and the capability of authorita... | strwete, “Discriminating statts: The quantum chgrboff bmund,” *Pgys. Rev. Uett.*, vol. 98, no. 16, p. 160501, 2007.
M. Ohya and D. Pevz, *Qyantun Entropy and Its Use*.1eo plus 0.5em minus 0.4enSprmnger, 1993.
M. Mosony anv N. Datta, “Generzpizeb celative entroples and the capacity of cnarsncal-quantum channels,” 2008... | straete, “Discriminating states: The quantum chernoff bound,” Lett.*, 98, no. p. 160501, 2007. *Quantum and Its Use*.1em 0.5em minus 0.4emSpringer, M. Mosony and N. Datta, “Generalized entropies and the capacity of classical-quantum channels,” 2008. \[Online\]. Available: <http://arxiv.org/abs/0810.3478> R. Bhatia, Ana... | straete, “Discriminating statEs: The quantUm cheRnoFf bOuNd,” *PhYs. ReV. Lett.*, vol. 98, no. 16, p. 160501, 2007.
M. OhYA and d. Petz, *Quantum Entropy and its UsE*.1eM Plus 0.5EM mInus 0.4eMSpringER, 1993.
M. mOSonY aNd n. DaTtA, “geNeralIzeD relatiVe entropieS anD tHe capacity of CLaSsical-quanTum Channels,” 2008... | straete, “Discriminating s tates: The quan tum ch er noff bou nd,” *Phys. Re v . Le tt.*, vol. 98, no. 16, p. 1 60 5 01,2 00 7.
M . Ohyaa nd D . P et z, *Q ua n tu m Ent rop y and I ts Use*.1e m p lu s 0.5em minu s 0 .4emSpring er, 1993.
M. M oso ny and N . D a tta,“Ge neral ized r e lative entropie sa nd th... | straete, “Discriminating_states: The_quantum chernoff bound,” *Phys._Rev. Lett.*,_vol. 98,_no. 16, p._160501,_2007.
M. Ohya and D. Petz,_*Quantum Entropy and_Its Use*.1em plus 0.5em_minus 0.4emSpringer, 1993.
M. Mosony_and_N. Datta, “Generalized relative entropies and the capacity of classical-quantum channels,” 2008... |
of cascade (obtaining $v^{+}$, $v^{-}$, $A^{+}$, $A^{-}$)if it is clear from the context which cascade we have on mind. If $v \in V \setminus
\max V$, then the set $v^+$ (if infinite) may be endowed with an order of the type $\omega$, and then by $(v_n)_{n \in \omega}$ we denote the sequence of elements of $v^+$, and ... | of cascade (obtaining $ v^{+}$, $ v^{-}$, $ A^{+}$, $ A^{-}$)if it is clear from the context which cascade we get on judgment. If $ v \in V \setminus
\max V$, then the set $ v^+$ (if countless) may be endowed with an order of the type $ \omega$, and then by $ (v_n)_{n \in \omega}$ we announce the succession of elemen... | of cascade (obtaining $v^{+}$, $v^{-}$, $A^{+}$, $A^{-}$)if it is cleat drom tie contsxt whicf cascade we have on mind. If $v \in V \setminus
\max V$, then thd set $v^+$ (iv infinire) mey be endowed wivg an order of bhe tvpx $\omega$, and thek by $(v_n)_{n \in \omega}$ we denode tke sequence of elements of $v^+$, and ... | of cascade (obtaining $v^{+}$, $v^{-}$, $A^{+}$, $A^{-}$)if clear the context cascade we have V \max V$, then set $v^+$ (if may be endowed with an order the type $\omega$, and then by $(v_n)_{n \in \omega}$ we denote the sequence elements of $v^+$, and by $v_{nV}$ - the $n$-th element of $v^{+V}$. The of \in ($r_V(v)$ ... | of cascade (obtaining $v^{+}$, $v^{-}$, $A^{+}$, $A^{-}$)if iT is clear frOm the ConTexT wHich CascAde we have on minD. if $v \iN V \setminus
\max V$, then the sEt $v^+$ (if InFInitE) MaY be enDowed wiTH aN ORdeR oF tHe tYpE $\OmEga$, anD thEn by $(v_n)_{n \In \omega}$ we dEnoTe The sequence oF ElEments of $v^+$, aNd ... | of cascade (obtaining $v^ {+}$, $v^{ -}$,$A^ {+} $, $A^ {-}$ )if it is clea r fro m the context which ca scade w e hav e o n min d. If $ v \ i n V\s et min us \m ax V$ , t hen the set $v^+$ (i finfinite) ma y b e endowedwit h an order o f t he typ e$\o m ega$, an d the n by $ ( v_n)_{ n \in \om eg a }$ wed ... | of_cascade (obtaining_$v^{+}$, $v^{-}$, $A^{+}$, $A^{-}$)if_it is_clear_from the_context_which cascade we_have on mind._If $v \in V_\setminus
\max V$, then_the_set $v^+$ (if infinite) may be endowed with an order of the type $\omega$,_and_then by_$(v_n)_{n_\in_\omega}$ we denote the sequence_of elements of $v^+$, and_... |
Colbert E. J. M., Heckman, T. M. Ptak, A. F. et al. 2004, ApJ 602, 231 Collin, S., Kawaguchi, T.: 2004, A&A 426, 797 Combes F.: 2001, in Starburst-AGN Connection, ed. I. Aretxaga et al., World Scientific Combes F.: 2003, in “Active Galactic Nuclei: from Central Engine to Host Galaxy” ASP Conf. Ser., 2003, ed. S. Colli... | Colbert E. J. M., Heckman, T. M. Ptak, A. F. et al. 2004, ApJ 602, 231 Collin, S., Kawaguchi, T.: 2004, A&A 426, 797 Combes F.: 2001, in Starburst - AGN Connection, ed. I. Aretxaga et al. , World Scientific Combes F.: 2003, in “ Active Galactic Nuclei: from Central Engine to Host Galaxy ” ASP Conf. Ser. , 2003, ed. S. ... | Copbert E. J. M., Heckman, T. M. Ktak, A. F. et al. 2004, AkJ 602, 231 Colnin, S., Iawaguchk, T.: 2004, A&A 426, 797 Combes F.: 2001, in Starbnrst-QGN Cinnection, ed. I. Aretxagx et al., Wlrld Sciwntihic Combes F.: 2003, in “Active Nclactjg Nucnxi: from Central Engine to Host Galaxy” AVP Clnf. Ser., 2003, ed. S. Colli... | Colbert E. J. M., Heckman, T. M. F. al. 2004, 602, 231 Collin, 426, Combes F.: 2001, Starburst-AGN Connection, ed. Aretxaga et al., World Scientific Combes 2003, in “Active Galactic Nuclei: from Central Engine to Host Galaxy” ASP Conf. 2003, ed. S. Collin, F. Combes and I. Shlosman, p. 411 Cowie, L. Songaila, Hu, M., J... | Colbert E. J. M., Heckman, T. M. Ptak, A. F. Et al. 2004, ApJ 602, 231 ColLin, S., KAwaGucHi, t.: 2004, A&A 426, 797 COmbeS F.: 2001, in Starburst-Agn ConNection, ed. I. Aretxaga et al., world scIEntiFIc combeS F.: 2003, in “ActIVe gALacTiC NUclEi: FRoM CentRal engine tO Host GalaxY” ASp COnf. Ser., 2003, ed. S. ColLI... | Colbert E. J. M., Heckman , T. M. Pt ak, A . F . e tal.2004 , ApJ 602, 231 Coll in, S., Kawaguchi, T.: 2004 ,A &A 4 2 6, 797CombesF .: 2 001 ,in St ar b ur st-AG N C onnecti on, ed. I. Ar et xaga et al., Wo rld Scient ifi c Combes F.: 20 03, in “ Act i ve Ga lac tic N uclei: from C entral En gi n e to H o st Gal... | Colbert_E. J._M., Heckman, T. M._Ptak, A._F._et al._2004,_ApJ 602, 231_Collin, S., Kawaguchi,_T.: 2004, A&A 426,_797 Combes F.:_2001,_in Starburst-AGN Connection, ed. I. Aretxaga et al., World Scientific Combes F.: 2003, in_“Active_Galactic Nuclei:_from_Central_Engine to Host Galaxy” ASP_Conf. Ser., 2003, ed. S._Colli... |
i+j}\right\rangle}_{N}k\underline
x}\beta_{ij\underline{{\left\langle{k+x}\right\rangle}_M}},\nonumber\\
\alpha^p_{ijk}\beta_{i{\left\langle{j+k}\right\rangle}_{N}\underline{{\left\langle{x+1}\right\rangle}_M}}
\beta_{jk\underline{{\left\langle{x+1}\right\rangle}_M}}&=\beta_{{\left\langle{i+j}\right\rangle}_{N}k
\under... | i+j}\right\rangle}_{N}k\underline
x}\beta_{ij\underline{{\left\langle{k+x}\right\rangle}_M}},\nonumber\\
\alpha^p_{ijk}\beta_{i{\left\langle{j+k}\right\rangle}_{N}\underline{{\left\langle{x+1}\right\rangle}_M } }
\beta_{jk\underline{{\left\langle{x+1}\right\rangle}_M}}&=\beta_{{\left\langle{i+j}\right\rangle}_{N}... | i+j}\rlght\rangle}_{N}k\underline
x}\beua_{ij\underline{{\left\langle{k+e}\right\rzngle}_M}},\novumber\\
\alpha^p_{ijk}\beta_{i{\left\langpe{h+k}\rigyt\rangle}_{N}\underline{{\left\uangle{x+1}\rihht\ranglw}_M}}
\beua_{jk\underline{{\left\langle{x+1}\rlyht\rahnle}_M}}&=\bzte_{{\left\langle{i+j}\rinht\rangle}_{N}k
\gnder... | i+j}\right\rangle}_{N}k\underline x}\beta_{ij\underline{{\left\langle{k+x}\right\rangle}_M}},\nonumber\\ \alpha^p_{ijk}\beta_{i{\left\langle{j+k}\right\rangle}_{N}\underline{{\left\langle{x+1}\right\rangle}_M}} \beta_{jk\underline{{\left\langle{x+1}\right\rangle}_M}}&=\beta_{{\left\langle{i+j}\right\rangle}_{N}k \under... | i+j}\right\rangle}_{N}k\underline
x}\Beta_{ij\undeRline{{\LefT\laNgLe{k+x}\RighT\rangle}_M}},\nonumbER\\
\alpHa^p_{ijk}\beta_{i{\left\langle{j+K}\righT\rANgle}_{n}\UnDerliNe{{\left\lANgLE{X+1}\riGhT\rAngLe}_m}}
\BeTa_{jk\uNdeRline{{\leFt\langle{x+1}\rIghT\rAngle}_M}}&=\beta_{{\leFT\lAngle{i+j}\rigHt\rAngle}_{N}k
\under... | i+j}\right\rangle}_{N}k\un derline
x} \beta _{i j\u nd erli ne{{ \left\langle{k + x}\r ight\rangle}_M}},\nonu mber\ \\ alph a ^p _{ijk }\beta_ { i{ \ l eft \l an gle {j + k} \righ t\r angle}_ {N}\underl ine {{ \left\langle { x+ 1}\right\r ang le}_M}}
\bet a_{ jk\und er lin e {{\le ft\ langl e{x+1} \ right\ rangle}_M }}... | i+j}\right\rangle}_{N}k\underline
x}\beta_{ij\underline{{\left\langle{k+x}\right\rangle}_M}},\nonumber\\
\alpha^p_{ijk}\beta_{i{\left\langle{j+k}\right\rangle}_{N}\underline{{\left\langle{x+1}\right\rangle}_M}}
\beta_{jk\underline{{\left\langle{x+1}\right\rangle}_M}}&=\beta_{{\left\langle{i+j}\right\rangle}_{N}k
\under... |
onomarev, and V. V. Zhytnikov, Gen. Relativ. Grav. [**21**]{}, 1107 (1989). [doi: [10.1007/BF00763457](http://dx.doi.org/10.1007/BF00763457)]{} P. Baekler, F. W. Hehl, and J. M. Nester, Phys. Rev. D [**83**]{}, 024001 (2011). [doi: [10.1103/PhysRevD.83.024001](http://dx.doi.org/10.1103/PhysRevD.83.024001)]{} D. Diakono... | onomarev, and V. V. Zhytnikov, Gen. Relativ. Grav. [ * * 21 * * ] { }, 1107 (1989). [ doi: [ 10.1007 / BF00763457](http://dx.doi.org/10.1007 / BF00763457) ] { } P. Baekler, F. W. Hehl, and J. M. Nester, Phys. Rev. D [ * * 83 * * ] { }, 024001 (2011). [ doi: [ 10.1103 / PhysRevD.83.024001](http://dx.doi.org/10... | onolarev, and V. V. Zhytnikov, Nen. Relativ. Grav. [**21**]{}, 1107 (1989). [doi: [10.1007/BF00763457](ittp://dx.dki.org/10.1007/BF00763457)]{} O. Baekler, F. W. Hehl, and J. M. Nxstee, Phyw. Rev. D [**83**]{}, 024001 (2011). [doi: [10.1103/PhysRevD.83.024001](hgtp://dx.doi.ogg/10.1103/PhysRevE.83.024001)]{} D. Viakono... | onomarev, and V. V. Zhytnikov, Gen. Relativ. 1107 [doi: [10.1007/BF00763457](http://dx.doi.org/10.1007/BF00763457)]{} Baekler, F. W. Phys. D [**83**]{}, 024001 [doi: [10.1103/PhysRevD.83.024001](http://dx.doi.org/10.1103/PhysRevD.83.024001)]{} D. A. G. Tumanov, and A. A. Phys. Rev. D [**84**]{}, 124042 (2011). \[arXiv:... | onomarev, and V. V. Zhytnikov, Gen. relativ. GraV. [**21**]{}, 1107 (1989). [doi: [10.1007/Bf00763457](htTp://dX.dOi.orG/10.1007/BF00763457)]{} P. baekler, F. W. Hehl, aND J. M. NEster, Phys. Rev. D [**83**]{}, 024001 (2011). [doi: [10.1103/PhysReVD.83.024001](httP://dX.Doi.oRG/10.1103/PHysReVD.83.024001)]{} D. DiakONo... | onomarev, and V. V. Zhytni kov, Gen.Relat iv. Gr av . [* *21* *]{}, 1107 (19 8 9).[doi: [10.1007/BF00763 457]( ht t p:// d x. doi.o rg/10.1 0 07 / B F00 76 34 57) ]{ } P . Bae kle r, F. W . Hehl, an d J .M. Nester, P h ys . Rev. D [ **8 3**]{}, 0240 01(2011) .[do i : [10 .11 03/Ph ysRevD . 83.024 001](http :/ / dx.do... | onomarev, and_V. V._Zhytnikov, Gen. Relativ. Grav. [**21**]{}, 1107 (1989)._[doi: [10.1007/BF00763457](http://dx.doi.org/10.1007/BF00763457)]{}_P._Baekler, F._W._Hehl, and J._M. Nester, Phys. Rev. D_[**83**]{}, 024001 (2011). [doi:_[10.1103/PhysRevD.83.024001](http://dx.doi.org/10.1103/PhysRevD.83.024001)]{} D. Diakono... |
ite] are constructed in such a way that $S_c$ is a good quantum number in the rotated-electron picture. The subspaces spanned by states whose number $2S_c$ is constant play an important role. For hole concentrations $0\leq x<1$ and maximum spin density $m=(1-x)$ there is a fully polarized vacuum, which remains invarian... | ite ] are constructed in such a way that $ S_c$ is a good quantum number in the rotate - electron movie. The subspaces spanned by states whose phone number $ 2S_c$ is constant play an significant role. For hole concentration $ 0\leq x<1 $ and maximum spin concentration $ m=(1 - x)$ there is a fully polarized vacuum, wh... | ite] are constructed in such a way that $S_c$ nw a gomd quahtum numcer in the rotated-electron pmctuee. Tht subspaces spanned by stated whose bumbtr $2S_c$ is constant play an imporfwnt xooe. For hole cokcentrationv $0\leq x<1$ and mafioul spin density $m=(1-x)$ there is a fully [olarizrd vacuum, which temaimf inbarian... | ite] are constructed in such a way is good quantum in the rotated-electron states number $2S_c$ is play an important For hole concentrations $0\leq x<1$ and spin density $m=(1-x)$ there is a fully polarized vacuum, which remains invariant under electron - rotated-electron unitary transformation, $$\vert 0_{\eta s}\rang... | ite] are constructed in such a wAy that $S_c$ is A good QuaNtuM nUmbeR in tHe rotated-electROn piCture. The subspaces spannEd by sTaTEs whOSe NumbeR $2S_c$ is coNStANT plAy An ImpOrTAnT role. for Hole conCentrationS $0\leQ x<1$ And maximum spIN dEnsity $m=(1-x)$ thEre Is a fully polaRizEd vacuUm, WhiCH remaIns InvarIan... | ite] are constructed in su ch a way t hat $ S_c $ i sa go od q uantum numberi n th e rotated-electron pic ture. T h e su b sp acesspanned by s tat es w hos en um ber $ 2S_ c$ is c onstant pl ayan important r o le . For hole co ncentrations $0 \leq x <1 $ a n d max imu m spi n dens i ty $m= (1-x)$ th er e is af ul... | ite] are_constructed in_such a way that_$S_c$ is_a_good quantum_number_in the rotated-electron_picture. The subspaces_spanned by states whose_number $2S_c$ is_constant_play an important role. For hole concentrations $0\leq x<1$ and maximum spin density $m=(1-x)$_there_is a_fully_polarized_vacuum, which remains invarian... |
practical implementation.
At the moment, the largest LPN instances with errors in ’s range $\tau \in [0.1, 0.15]$ are solved with variants of the low-memory algorithms <span style="font-variant:small-caps;">Pooled Gauss</span> and <span style="font-variant:small-caps;">Well-Pooled Gauss</span> of Esser, Kübler, May [... | practical implementation.
At the moment, the largest licensed practical nurse case with errors in ’s range $ \tau \in [ 0.1, 0.15]$ are clear with form of the low - memory algorithms < couple style="font - variant: small - caps;">Pooled Gauss</span > and < span style="font - variant: small - caps;">Well - pool Gauss... | prwctical implementation.
At the moment, the largesv LPN ihstances with errors in ’s range $\tau \mn [0.1, 0.15]$ are wolved with variants ow the low-lemory aogormthms <span style="hknt-variant:smamp-capv;">'ooled Gauss</spak> and <span vtyle="font-variatt:rmcll-caps;">Well-Pooled Gauss</span> of Esser, Kübler, Kaj [... | practical implementation. At the moment, the largest with in ’s $\tau \in [0.1, of low-memory algorithms <span Gauss</span> and <span Gauss</span> of Esser, Kübler, May [@C:EssKubMay17]. show that <span style="font-variant:small-caps;">Pooled Gauss</span> solves LSN for $\tau \leq 0.292$ faster than period finding algo... | practical implementation.
At The moment, tHe larGesT LPn iNstaNces With errors in ’s rANge $\tAu \in [0.1, 0.15]$ are solved with variaNts of ThE Low-mEMoRy algOrithms <SPaN STylE="fOnT-vaRiANt:Small-CapS;">Pooled gauss</span> aNd <sPaN style="font-vaRIaNt:small-capS;">WeLl-Pooled GausS</spAn> of EsSeR, KüBLer, MaY [... | practical implementation.
At the m oment , t hela rges t LP N instances wi t h er rors in ’s range $\tau \in[0 . 1, 0 . 15 ]$ ar e solve d w i t h v ar ia nts o f t he lo w-m emory a lgorithms<sp an style="font - va riant:smal l-c aps;">Pooled Ga uss</s pa n>a nd <s pan styl e="fon t -varia nt:small- ca p s;"... | practical_implementation.
At the_moment, the largest LPN_instances with_errors_in ’s_range_$\tau \in [0.1,_0.15]$ are solved_with variants of the_low-memory algorithms <span_style="font-variant:small-caps;">Pooled_Gauss</span> and <span style="font-variant:small-caps;">Well-Pooled Gauss</span> of Esser, Kübler, May [... |
isation, and overall $U(1)'$ neutrality implies they can even be much larger than the galactic escape velocity. It has been pointed out [@foot14] that this effect can give rise to observable keV electron recoils in direct detection experiments. It might even be possible to explain the DAMA annual modulation signal [@da... | isation, and overall $ U(1)'$ neutrality implies they can even be much larger than the astronomic evasion velocity. It has been pointed out [ @foot14 ] that this consequence can hold rise to observable keV electron recoil in lineal detection experiments. It might even be possible to explain the DAMA annual modulation s... | isahion, and overall $U(1)'$ neutrxlity implies tkwy can even ge much uarger than the galactic escepe celocuty. It has been pointea out [@foon14] that thus ehfect can give rmae to onfervznle kzV electron recolls in direwt detection efpdrnments. It might even be possible to qxplain tje DAMA annual modllwtioh signal [@da... | isation, and overall $U(1)'$ neutrality implies they be larger than galactic escape velocity. [@foot14] this effect can rise to observable electron recoils in direct detection experiments. might even be possible to explain the DAMA annual modulation signal [@dama1a; @dama1b; @dama2b; @dama2c] in this manner, since the ... | isation, and overall $U(1)'$ neutralIty implies They cAn eVen Be Much LargEr than the galacTIc esCape velocity. It has been pOinteD oUT [@fooT14] ThAt thiS effect CAn GIVe rIsE tO obSeRVaBle kev elEctron rEcoils in diRecT dEtection expeRImEnts. It mighT evEn be possible To eXplain ThE DAma annuAl mOdulaTion siGNal [@da... | isation, and overall $U(1) '$ neutral ity i mpl ies t heycaneven be much l a rger than the galactic esc ape v el o city . I t has been p o in t e d o ut [ @fo ot 1 4] that th is effe ct can giv e r is e to observa b le keV elect ron recoils indir ect de te cti o n exp eri ments . It m i ght ev en be pos si b l... | isation, and_overall $U(1)'$_neutrality implies they can_even be_much_larger than_the_galactic escape velocity._It has been_pointed out [@foot14] that_this effect can_give_rise to observable keV electron recoils in direct detection experiments. It might even be_possible_to explain_the_DAMA_annual modulation signal [@da... |
k})\\
h(\mathbf{x})
\end{array}\right.\label{eq:Relation I and h}$$
Equation (\[eq:Relation I and h\]) allows to postulate an equivalent condition which constitutes an upper limit for the expected value of the potential energy given the expected value for differential entropies
$$\left\langle V\right\rangle <\left\{ ... | k})\\
h(\mathbf{x })
\end{array}\right.\label{eq: Relation I and h}$$
Equation (\[eq: Relation I and h\ ]) allows to postulate an equivalent condition which constitutes an upper terminus ad quem for the expect value of the potential energy give the ask value for differential entropies
$ $ \left\langle V\right... | k})\\
h(\mwthbf{x})
\end{array}\right.\label{tq:Relation I and k}$$
Wquatimn (\[eq:Rslation K and h\]) allows to postulate en ewuivaoent condition which cunstituted an uppwr lmmit for the expxdted value of bhe pmvential energy niven the efpected value xof bifferential entropies
$$\left\langle V\ridht\rangke <\left\{ ... | k})\\ h(\mathbf{x}) \end{array}\right.\label{eq:Relation I and h}$$ Equation and allows to an equivalent condition for expected value of potential energy given expected value for differential entropies $$\left\langle <\left\{ \begin{array}{l} \left\langle h(t_{k})\right\rangle \\ \left\langle h(\mathbf{x})\right\rangle... | k})\\
h(\mathbf{x})
\end{array}\right.\labEl{eq:RelatiOn I anD h}$$
EQuaTiOn (\[eq:relaTion I and h\]) allowS To poStulate an equivalent conDitioN wHIch cONsTitutEs an uppER lIMIt fOr ThE exPeCTeD valuE of The poteNtial energY giVeN the expected VAlUe for diffeRenTial entropieS
$$\leFt\langLe v\riGHt\ranGle <\Left\{ ... | k})\\
h(\mathbf{x})
\end{a rray}\righ t.\la bel {eq :R elat ionI and h}$$
Eq u atio n (\[eq:Relation I and h\]) a l lows to post ulate a n e q u iva le nt co nd i ti on wh ich consti tutes an u ppe rlimit for th e e xpected va lue of the pote nti al ene rg y g i ven t heexpec ted va l ue for differen ti a l e... | k})\\
h(\mathbf{x})
\end{array}\right.\label{eq:Relation I_and h}$$
Equation_(\[eq:Relation I and h\])_allows to_postulate_an equivalent_condition_which constitutes an_upper limit for_the expected value of_the potential energy_given_the expected value for differential entropies
$$\left\langle V\right\rangle <\left\{ ... |
A_1}\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)},$$ which again yields for this case.
An argument identical to that for gives $$\|\nabla_{A_1}\Phi\times a_1\|_{L^{r'}(X)}
\leq
z\|\nabla_{A_1}\Phi\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)}.$$ proving. This completes Step \[step:Lrprime\_estimate\_FA1\_times\_a1\_and\_nablaA1Phi\_... | A_1}\|_{L^p(X) } \|a_1\|_{W_{A_1}^{1,p}(X)},$$ which again yields for this case.
An argument identical to that for gives $ $ \|\nabla_{A_1}\Phi\times a_1\|_{L^{r'}(X) }
\leq
z\|\nabla_{A_1}\Phi\|_{L^p(X) } \|a_1\|_{W_{A_1}^{1,p}(X)}.$$ proving. This dispatch Step \[step: Lrprime\_estimate\_FA1\_times\_a1\_and\_n... | A_1}\|_{L^p(D)} \|a_1\|_{W_{A_1}^{1,p}(X)},$$ which again yielas for this casg.
Ab argukent isentical to that for gives $$\|\nabla_{A_1}\Phi\vimew a_1\|_{L^{r'}(Z)}
\leq
z\|\nabla_{A_1}\Phi\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(B)}.$$ proving. This conpleues Step \[step:Lrprmje\_estimate\_FA1\_tjles\_a1\_cnv\_nablaA1Phi\_... | A_1}\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)},$$ which again yields for this argument to that gives $$\|\nabla_{A_1}\Phi\times a_1\|_{L^{r'}(X)} completes \[step:Lrprime\_estimate\_FA1\_times\_a1\_and\_nablaA1Phi\_times\_a1\]. \[step:Lrprime\_estimate\_Phi\_times\_nablaA1Phi\_and\_two\_similar\_terms\] We that $$\begin{al... | A_1}\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)},$$ which again yields For this casE.
An arGumEnt IdEntiCal tO that for gives $$\|\nABla_{A_1}\phi\times a_1\|_{L^{r'}(X)}
\leq
z\|\nabla_{A_1}\phi\|_{L^p(x)} \|a_1\|_{w_{a_1}^{1,p}(X)}.$$ pROvIng. ThIs complETeS sTep \[StEp:lrpRiME\_eStimaTe\_Fa1\_times\_a1\_And\_nablaA1PHi\_... | A_1}\|_{L^p(X)} \|a_1\|_{W _{A_1}^{1, p}(X) },$ $ w hi ch a gain yields for th i s ca se.
An argument ident icalto that fo r giv es $$\| \ na b l a_{ A_ 1} \Ph i\ t im es a_ 1\| _{L^{r' }(X)}
\leq
z\ |\ nabla_{A_1}\ P hi \|_{L^p(X) } \ |a_1\|_{W_{A _1} ^{1,p} (X )}. $ $ pro vin g. Th is com p letesStep \[st ep : Lrp... | A_1}\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)},$$_which again_yields for this case.
An_argument identical_to_that for_gives_$$\|\nabla_{A_1}\Phi\times a_1\|_{L^{r'}(X)}
\leq
z\|\nabla_{A_1}\Phi\|_{L^p(X)} \|a_1\|_{W_{A_1}^{1,p}(X)}.$$_proving. This completes_Step \[step:Lrprime\_estimate\_FA1\_times\_a1\_and\_nablaA1Phi\_... |
*$ stands for the convolution product), where $\theta$ is a radially symmetrical function, such that $\int_{\mathbb R ^3} \theta (\textbf{x})d\textbf{x}=1$ and $ \theta_\epsilon (\textbf{x})= \frac{1}{\epsilon^3}\theta (\textbf{x}/\epsilon)$[@RobVar08].
Numerical study
===============
We would like now to study the s... | * $ stands for the convolution product), where $ \theta$ is a radially symmetrical affair, such that $ \int_{\mathbb radius ^3 } \theta (\textbf{x})d\textbf{x}=1 $ and $ \theta_\epsilon (\textbf{x})= \frac{1}{\epsilon^3}\theta (\textbf{x}/\epsilon)$[@RobVar08 ].
Numerical study
= = = = = = = = = = = = = = =
We ... | *$ stwnds for the convolution product), where $\jhwta$ is a radjally syometrical function, such that $\ibt_{\matybb R ^3} \theta (\textbf{x})d\tdxtbf{x}=1$ anf $ \theta_\wpsioin (\textbf{x})= \frac{1}{\epsljon^3}\tgcta (\tzxvbf{x}/\epsilon)$[@RobVsr08].
Numerican study
===============
We woulg uine now to study the s... | *$ stands for the convolution product), where a symmetrical function, that $\int_{\mathbb R \theta_\epsilon \frac{1}{\epsilon^3}\theta (\textbf{x}/\epsilon)$[@RobVar08]. Numerical =============== We would now to study the statistical properties the velocity field defined by Eq. (\[eq:K41Field\]). Analytical formulas ar... | *$ stands for the convolution prOduct), where $\Theta$ Is a RadIaLly sYmmeTrical function, SUch tHat $\int_{\mathbb R ^3} \theta (\textBf{x})d\tExTBf{x}=1$ aND $ \tHeta_\ePsilon (\tEXtBF{X})= \frAc{1}{\EpSilOn^3}\THeTa (\texTbf{X}/\epsiloN)$[@RobVar08].
NumEriCaL study
===============
We woulD LiKe now to stuDy tHe s... | *$ stands for the convolut ion produc t), w her e $ \t heta $ is a radially sy m metr ical function, such th at $\ in t _{\m a th bb R^3} \th e ta ( \te xt bf {x} )d \ te xtbf{ x}= 1$ and$ \theta_\ eps il on (\textbf{ x }) = \frac{1} {\e psilon^3}\th eta (\tex tb f{x } /\eps ilo n)$[@ RobVar 0 8].
N umericalst u dy
... | *$ stands_for the_convolution product), where $\theta$_is a_radially_symmetrical function,_such_that $\int_{\mathbb R_^3} \theta (\textbf{x})d\textbf{x}=1$_and $ \theta_\epsilon (\textbf{x})=_\frac{1}{\epsilon^3}\theta (\textbf{x}/\epsilon)$[@RobVar08].
Numerical study
===============
We_would_like now to study the s... |
$)$.$ Let $A$ be the operator in $l_{q}\left( N\right) $ defined by$$\text{ }A=\left[ a_{mj}\right] \text{, }a_{mj}=g_{m}2^{sj},\text{ }m,j=1,2,...,N,\text{ }D\left( A\right) =\text{ }l_{q}^{s}\left( N\right) =$$
$$\left\{ \text{ }u=\left\{ u_{j}\right\},\text{ }j=1,2,...N,\left\Vert
u\right\Vert _{l_{q}^{s}\left( N\... | $) $ .$ Let $ A$ be the operator in $ l_{q}\left (N\right) $ defined by$$\text { } A=\left [ a_{mj}\right ] \text {, } a_{mj}=g_{m}2^{sj},\text { } m, j=1,2,... ,N,\text { } D\left (A\right) = \text { } l_{q}^{s}\left (N\right) = $ $
$ $ \left\ { \text { } u=\left\ { u_{j}\right\},\text { } j=1,2,... N,\left\Vert
... | $)$.$ Lft $A$ be the operator in $u_{q}\left( N\right) $ bwfined by$$\texf{ }A=\left[ x_{mj}\right] \text{, }a_{mj}=g_{m}2^{sj},\text{ }m,j=1,2,...,I,\texr{ }D\ledt( A\right) =\text{ }l_{q}^{s}\left( N\right) =$$
$$\lvft\{ \text{ }y=\lefu\{ u_{j}\right\},\text{ }j=1,2,...N,\lxrt\Vert
u\vnght\Vsvt _{l_{q}^{v}\oeft( N\... | $)$.$ Let $A$ be the operator in $ by$$\text{ }A=\left[ \text{, }a_{mj}=g_{m}2^{sj},\text{ }m,j=1,2,...,N,\text{ =$$ \text{ }u=\left\{ u_{j}\right\},\text{ u\right\Vert _{l_{q}^{s}\left( N\right) \sum\limits_{j=1}^{N}2^{sj}u_{j}^{q}\right) ^{\frac{1}{q}}<\infty \right\}.$$ From Theorem 3.1 obtain the following result *... | $)$.$ Let $A$ be the operator in $l_{q}\left( n\right) $ defiNed by$$\TexT{ }A=\lEfT[ a_{mj}\RighT] \text{, }a_{mj}=g_{m}2^{sj},\teXT{ }m,j=1,2,...,N,\Text{ }D\left( A\right) =\text{ }l_{q}^{s}\Left( N\RiGHt) =$$
$$\leFT\{ \tExt{ }u=\lEft\{ u_{j}\riGHt\},\TEXt{ }j=1,2,...n,\lEfT\VeRt
U\RiGht\VeRt _{l_{Q}^{s}\left( N\... | $)$.$ Let $A$ be the oper ator in $l _{q}\ lef t(N\ righ t) $ defined by$$\ t ext{ }A=\left[ a_{mj}\righ t] \t ex t {, } a _{ mj}=g _{m}2^{ s j} , \ tex t{ } m,j =1 , 2, ...,N ,\t ext{ }D \left( A\r igh t) =\text{ }l_ { q} ^{s}\left( N\ right) =$$
$$\ left\{ \ tex t { }u= \le ft\{u_{j}\ r ight\} ,\text{ } j= 1 ,... | $)$.$_Let $A$_be the operator in_$l_{q}\left( N\right)_$_defined by$$\text{_}A=\left[_a_{mj}\right] \text{, }a_{mj}=g_{m}2^{sj},\text{_}m,j=1,2,...,N,\text{ }D\left( A\right)_=\text{ }l_{q}^{s}\left( N\right) =$$
$$\left\{_\text{ }u=\left\{ u_{j}\right\},\text{_}j=1,2,...N,\left\Vert
u\right\Vert__{l_{q}^{s}\left( N\... |
biggr) \cdot
\frac{|x|_{r,i_{r}(s)}}{|x|_{L}} \cdot|x|_{L} \\
%
&&\qquad \ge\frac{a}{b^{3}} \cdot\bigl(1+ai_{r}(s)_{N}\bigr)^{-1} \cdot\frac
{1}{2} \cdot
|x|_{L}. %\ge\xxx N^{5},\end{aligned}$$ Because of $|x|_{2}\le|x|/2$, (\[eq7.5.1\]), $i_{r}(s)_{N}\le N$, $|x|>N^{6}$ and $aN \ge1$, this is at most $C_{29}N^{5}$, wh... | biggr) \cdot
\frac{|x|_{r, i_{r}(s)}}{|x|_{L } } \cdot|x|_{L } \\
%
& & \qquad \ge\frac{a}{b^{3 } } \cdot\bigl(1+ai_{r}(s)_{N}\bigr)^{-1 } \cdot\frac
{ 1}{2 } \cdot
|x|_{L }. % \ge\xxx N^{5},\end{aligned}$$ Because of $ |x|_{2}\le|x|/2 $, (\[eq7.5.1\ ]), $ i_{r}(s)_{N}\le N$, $ |x|>N^{6}$ and $ aN \ge1 $, this... | bighr) \cdot
\frac{|x|_{r,i_{r}(s)}}{|x|_{L}} \cdot|x|_{U} \\
%
&&\qquad \ge\frac{a}{y^{3}} \cdot\bmgl(1+ai_{r}(s)_{H}\bigr)^{-1} \cdut\frac
{1}{2} \cdot
|x|_{L}. %\ge\xxx N^{5},\end{aligied}$$ Vecauwe of $|x|_{2}\le|x|/2$, (\[eq7.5.1\]), $i_{r}(s)_{N}\le N$, $|x|>N^{6}$ and $aJ \ge1$, thiw is qt most $C_{29}N^{5}$, wh... | biggr) \cdot \frac{|x|_{r,i_{r}(s)}}{|x|_{L}} \cdot|x|_{L} \\ % &&\qquad \cdot\frac \cdot |x|_{L}. N^{5},\end{aligned}$$ Because of and \ge1$, this is most $C_{29}N^{5}$, where 0$ does not depend on $N$, or $\omega$. It follows from (\[eq41.2.6\]) and the succeeding inequalities that $$\label{eq41.3.2} |x|_{L}-|\tilde{... | biggr) \cdot
\frac{|x|_{r,i_{r}(s)}}{|x|_{L}} \cdot|x|_{l} \\
%
&&\qquad \ge\frAc{a}{b^{3}} \cDot\BigL(1+aI_{r}(s)_{N}\Bigr)^{-1} \Cdot\frac
{1}{2} \cdot
|x|_{L}. %\GE\xxx n^{5},\end{aligned}$$ Because of $|x|_{2}\lE|x|/2$, (\[eq7.5.1\]), $i_{R}(s)_{n}\Le N$, $|x|>n^{6}$ AnD $aN \ge1$, This is aT MoST $c_{29}N^{5}$, wH... | biggr) \cdot
\frac{|x|_{r, i_{r}(s)}} {|x|_ {L} } \ cd ot|x |_{L } \\
%
&&\qqua d \ge \frac{a}{b^{3}} \cdot\ bigl( 1+ a i_{r } (s )_{N} \bigr)^ { -1 } \cd ot \f rac
{ 1 }{ 2} \c dot
|x|_{L }. %\ge\xx x N ^{ 5},\end{alig n ed }$$ Becaus e o f $|x|_{2}\l e|x |/2$,(\ [eq 7 .5.1\ ]), $i_{ r}(s)_ { N}\leN$, $|x|> N^ { 6... | biggr) \cdot
\frac{|x|_{r,i_{r}(s)}}{|x|_{L}}_\cdot|x|_{L} \\
%
&&\qquad_\ge\frac{a}{b^{3}} \cdot\bigl(1+ai_{r}(s)_{N}\bigr)^{-1} \cdot\frac
{1}{2} \cdot
|x|_{L}._%\ge\xxx N^{5},\end{aligned}$$_Because_of $|x|_{2}\le|x|/2$,_(\[eq7.5.1\]),_$i_{r}(s)_{N}\le N$, $|x|>N^{6}$_and $aN \ge1$,_this is at most_$C_{29}N^{5}$, wh... |
] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q>3$, case $q=3$ can be checked by GAP. If $r=p=2$ then $s=3$ and $|x^G|>\frac{1}{4}q^{6\cdot3}.$ Represent $x$ as $(\lambda, j) \in F_{q^n}^{\times} \rtimes \mathbb{Z}_n$ such that $(\lambda, j)^2=(\lambda^{q^{(n-j)}+1}... | ] we obtain $ (q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $ q>3 $, case $ q=3 $ can be checked by GAP. If $ r = p=2 $ then $ s=3 $ and $ |x^G|>\frac{1}{4}q^{6\cdot3}.$ Represent $ x$ as $ (\lambda, j) \in F_{q^n}^{\times } \rtimes \mathbb{Z}_n$ such that $ (\lambda, j)^2=(\lamb... | ] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\ovevline{x}^{\overline{G}}| \le |x^G|$ fmr $q>3$, czse $q=3$ cav be checked by GAP. If $r=p=2$ thxn $s=3$ and $|z^G|>\frac{1}{4}q^{6\cdot3}.$ Represent $b$ as $(\lambfa, j) \in D_{q^n}^{\tmmes} \rtimes \mathug{Z}_n$ sucm thaf $(\lamyde, j)^2=(\lambda^{q^{(n-j)}+1}... | ] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q=3$ be checked GAP. If $r=p=2$ $x$ $(\lambda, j) \in \rtimes \mathbb{Z}_n$ such $(\lambda, j)^2=(\lambda^{q^{(n-j)}+1}, 2j)=1$. Then $j=3$ and so $\lambda = \theta^{m(q^3-1)}$, where $m=1, \ldots, (q^3 +1).$ Thus $|x^G ... | ] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overLine{x}^{\overlIne{G}}| \lE |x^G|$ For $Q>3$, cAse $q=3$ Can bE checked by GAP. IF $R=p=2$ thEn $s=3$ and $|x^G|>\frac{1}{4}q^{6\cdot3}.$ RepreSent $x$ As $(\LAmbdA, J) \iN F_{q^n}^{\tImes} \rtiMEs \MAThbB{Z}_N$ sUch ThAT $(\lAmbda, J)^2=(\laMbda^{q^{(n-j)}+1}... | ] we obtain $(q^n-(-1)^n)^ 3n^3<\frac {1}{6 }q^ {6\ cd ot4} <|\o verline{x}^{\o v erli ne{G}}| \le |x^G|$ for $q>3 $, case $q =3$ c an be c h ec k e d b yGA P.If $r =p=2$ th en $s=3 $ and $|x^ G|> \f rac{1}{4}q^{ 6 \c dot3}.$ Re pre sent $x$ as$(\ lambda ,j)\ in F_ {q^ n}^{\ times} \rtime s \mathbb {Z } _n$ su c h th... | ] we_obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}|_\le |x^G|$ for $q>3$,_case $q=3$_can_be checked_by_GAP. If $r=p=2$_then $s=3$ and_$|x^G|>\frac{1}{4}q^{6\cdot3}.$ Represent $x$ as_$(\lambda, j) \in_F_{q^n}^{\times}_\rtimes \mathbb{Z}_n$ such that $(\lambda, j)^2=(\lambda^{q^{(n-j)}+1}... |
CuO$_2$ plane and the relative weight of the $p_x$ and $p_y$ orbitals. Additional complications like multiple scattering of the photoelectrons, Bragg scattering from the BiO surface layer etc. do not seem to have a very strong impact on the intensity variations, not even at the relatively low photon energy of $22\;eV$... | CuO$_2 $ plane and the relative weight of the $ p_x$ and $ p_y$ orbitals. Additional complication like multiple dispersion of the photoelectrons, Bragg scattering from the BiO surface level etc. do not seem to have a very strong impingement on the intensity variations, not even at the relatively abject photon energy of... | CuL$_2$ plane and the relative weight of the $k_x$ and $p_b$ orbitzls. Addigional complications like muptuple wcattering of the photuelectrond, Bragg wcatuering from the BmK surfagz laysv etc. vo not seem to mave a very strong impact ov che intensity variations, not even at the rekahively low phojon emqrgy of $22\;eV$... | CuO$_2$ plane and the relative weight of and orbitals. Additional like multiple scattering from BiO surface layer do not seem have a very strong impact on intensity variations, not even at the relatively low photon energy of $22\;eV$. Despite simplicity we therefore believe that our theory has some merit, particularly ... | CuO$_2$ plane and the relative weiGht of the $p_x$ And $p_y$ OrbItaLs. addiTionAl complicationS Like Multiple scattering of thE photOeLEctrONs, bragg ScatterINg FROm tHe bio suRfACe Layer Etc. Do not seEm to have a vEry StRong impact on THe Intensity vAriAtions, not eveN at The relAtIveLY low pHotOn eneRgy of $22\;ev$... | CuO$_2$ plane and the rel ative weig ht of th e $ p_ x$ a nd $ p_y$ orbitals. Addi tional complications l ike m ul t iple sc atter ing oft he p hot oe le ctr on s ,Bragg sc atterin g from the Bi Osurface laye r e tc. do not se em to have a ve ry str on g i m pactonthe i ntensi t y vari ations, n ot even a t the... | CuO$_2$_plane and_the relative weight of_the $p_x$_and_$p_y$ orbitals._Additional_complications like multiple_scattering of the_photoelectrons, Bragg scattering from_the BiO surface_layer_etc. do not seem to have a very strong impact on the intensity variations,_not_even at_the_relatively_low photon energy of $22\;eV$... |
of 678 keV. As has been discussed, impurities of Kr below 10 ppb can be reached [@xenon10], making negligible the contamination produced by those radioactive decays.
Neutrons from target components
-------------------------------
One of the most important sources of background comes from neutrons produced by radioac... | of 678 keV. As has been discussed, impurities of Kr below 10 ppb can be reached [ @xenon10 ], making negligible the contaminant produce by those radioactive decays.
Neutrons from prey part
-------------------------------
One of the most important informant of background comes from neutron produce by radio... | of 678 keV. As has been discusstd, impurities of Kr belox 10 ppb czn be rexched [@xenon10], making negligible tye cobtamination produced bh those rwdioactice dtcays.
Neutrons from target compohcnts
-------------------------------
Oue of the most ikportant smurces of backcruuud comes from neutrons produced by rwdioac... | of 678 keV. As has been discussed, Kr 10 ppb be reached [@xenon10], by radioactive decays. Neutrons target components ------------------------------- of the most important sources of comes from neutrons produced by radioactive contamination of the materials constituting the detector To minimize their rate, the use of c... | of 678 keV. As has been discussed, imPurities of kr belOw 10 pPb cAn Be reAcheD [@xenon10], making neGLigiBle the contamination proDuced By THose RAdIoactIve decaYS.
NEUTroNs FrOm tArGEt CompoNenTs
-------------------------------
One of The most impOrtAnT sources of baCKgRound comes FroM neutrons proDucEd by raDiOac... | of 678 keV. As has been d iscussed,impur iti esof Krbelo w 10 ppb can b e rea ched [@xenon10], makin g neg li g ible th e con taminat i on p rod uc ed by t h os e rad ioa ctive d ecays.
Ne utr on s from targe t c omponents--- ------------ --- ------ -- --- - -
On e o f the mosti mporta nt source so f back g ro... | of_678 keV. As_has been discussed, impurities_of Kr_below_10 ppb can_be_reached [@xenon10], making negligible_the contamination produced_by those radioactive decays.
Neutrons_from target components
-------------------------------
One_of_the most important sources of background comes from neutrons produced by radioac... |
Neufeld & Hollenbach 1994, 1996), and thus deep observations are needed to detect them.
Young protostars exhibit a positive or almost flat ($>-0.1$) spectral index (defined as $\alpha$, where $S_\nu\propto \nu^a$) between e.g. 3.6 cm and 6 cm (André et al.1987; Anglada et al. 1998; Beltrán et al. 2001). This behaviou... | Neufeld & Hollenbach 1994, 1996), and thus deep observations are needed to detect them.
Young protostars parade a incontrovertible or almost flat ($ > -0.1 $) spectral exponent (defined as $ \alpha$, where $ S_\nu\propto \nu^a$) between e.g. 3.6 cm and 6 centimeter (André et al.1987; Anglada et al. 1998; Beltrán et ... | Nekfeld & Hollenbach 1994, 1996), and uhus deep observajiins arx needes to detdct them.
Young protostars exhmbit a powitive or almost flat ($>-0.1$) spectral index (dwfintd as $\alpha$, where $S_\nu\propbj \nu^z$) betcexn e.g. 3.6 cm and 6 gm (André et dl.1987; Anglada et dl. 1998; Yeltrán et al. 2001). This behaviou... | Neufeld & Hollenbach 1994, 1996), and thus are to detect Young protostars exhibit ($>-0.1$) index (defined as where $S_\nu\propto \nu^a$) e.g. 3.6 cm and 6 cm et al.1987; Anglada et al. 1998; Beltrán et al. 2001). This behaviour of radio spectral index is characteristic of free-free thermal emission from ionised gas. T... | Neufeld & Hollenbach 1994, 1996), and thus dEep observaTions Are NeeDeD to dEtecT them.
Young protOStarS exhibit a positive or almOst flAt ($>-0.1$) SPectRAl Index (Defined AS $\aLPHa$, wHeRe $s_\nu\PrOPtO \nu^a$) bEtwEen e.g. 3.6 cm And 6 cm (André Et aL.1987; ANglada et al. 1998; BeLTrÁn et al. 2001). This BehAviou... | Neufeld & Hollenbach 1994 , 1996), a nd th usdee pobse rvat ions are neede d todetect them.
Young pr otost ar s exh i bi t a p ositive or a lmo st f lat ( $ >- 0.1$) sp ectralindex (def ine das $\alpha$, wh ere $S_\nu \pr opto \nu^a$) be tweene. g.3 .6 cm an d 6 c m (And r é et a l.1987; A ng l ada et al. 199 ... | Neufeld_& Hollenbach_1994, 1996), and thus_deep observations_are_needed to_detect_them.
Young protostars exhibit_a positive or_almost flat ($>-0.1$) spectral_index (defined as_$\alpha$,_where $S_\nu\propto \nu^a$) between e.g. 3.6 cm and 6 cm (André et al.1987; Anglada_et_al. 1998;_Beltrán_et_al. 2001). This behaviou... |
OTWY Example 2.2]).
Let ${\mathfrak{q}}= (a)$ and set $f = (a,0) \in I$. Then, $I^2 = fI$, since $I^2 = (a^2) \times (aR + \alpha R) = (a^2) \times aR=fI$. Note that $I/fA = [(a) \oplus R]/[(a) \oplus (a)] \cong R/(a)$ and $A/I = [R \oplus R]/[(a) \oplus R] \cong R/(a)$ as $R$-modules. We then have $\ell_A(A/I) =\ell_... | OTWY Example 2.2 ]).
Let $ { \mathfrak{q}}= (a)$ and set $ f = (a,0) \in I$. Then, $ I^2 = fI$, since $ I^2 = (a^2) \times (aR + \alpha R) = (a^2) \times aR = fI$. Note that $ I / fA = [ (a) \oplus R]/[(a) \oplus (a) ] \cong R/(a)$ and $ deoxyadenosine monophosphate / I = [ R \oplus R]/[(a) \oplus radius ] \cong R/(... | OTWJ Example 2.2]).
Let ${\mathfrak{q}}= (x)$ and set $f = (a,0) \nb I$. Thxn, $I^2 = fJ$, since $K^2 = (a^2) \times (aR + \alpha R) = (a^2) \tilew aR=fU$. Note that $I/fA = [(a) \opljs R]/[(a) \oplls (a)] \cong R/(a)$ end $A/I = [R \oplus C]/[(z) \oplus R] \conf R/(a)$ cs $R$-modules. We tmen have $\eln_A(A/I) =\ell_... | OTWY Example 2.2]). Let ${\mathfrak{q}}= (a)$ and = \in I$. $I^2 = fI$, (aR \alpha R) = \times aR=fI$. Note $I/fA = [(a) \oplus R]/[(a) \oplus \cong R/(a)$ and $A/I = [R \oplus R]/[(a) \oplus R] \cong R/(a)$ as We then have $\ell_A(A/I) =\ell_A(I/fA) = \ell_R(R/(a))$. Hence, $A/I \cong I/fA$ as an because is cyclic Thu... | OTWY Example 2.2]).
Let ${\mathfrak{q}}= (a)$ aNd set $f = (a,0) \in I$. then, $I^2 = FI$, sIncE $I^2 = (A^2) \timEs (aR + \Alpha R) = (a^2) \times aR=Fi$. NotE that $I/fA = [(a) \oplus R]/[(a) \oplus (a)] \Cong R/(A)$ aND $A/I = [R \OPlUs R]/[(a) \oPlus R] \coNG R/(A)$ AS $R$-mOdUlEs. WE tHEn Have $\eLl_A(a/I) =\ell_... | OTWY Example 2.2]).
Let $ {\mathfrak {q}}= (a )$an d se t $f = (a,0) \in I $ . Th en, $I^2 = fI$, since$I^2=( a^2) \t imes(aR + \ a lp h a R) = ( a^2 )\ ti mes a R=f I$. Not e that $I/ fA=[(a) \oplusR ]/ [(a) \oplu s ( a)] \cong R/ (a) $ and$A /I= [R \ opl us R] /[(a)\ oplusR] \congR/ ( a)$ as $R$-mod u l es . W... | OTWY Example_2.2]).
Let ${\mathfrak{q}}=_(a)$ and set $f_= (a,0)_\in_I$. Then,_$I^2_= fI$, since_$I^2 = (a^2)_\times (aR + \alpha_R) = (a^2)_\times_aR=fI$. Note that $I/fA = [(a) \oplus R]/[(a) \oplus (a)] \cong R/(a)$ and $A/I_=_[R \oplus_R]/[(a)_\oplus_R] \cong R/(a)$ as $R$-modules._We then have $\ell_A(A/I) =\ell_... |
them. In Section \[sec:Hipp\] we compare our spectrophotometric parallaxes to Hipparcos parallaxes and ask how these comparisons are affected by neglecting extinction. In Section \[sec:all\] we analyse our distances to the generality of stars, using kinematic correction factors to test for systematic biases in distanc... | them. In Section \[sec: Hipp\ ] we compare our spectrophotometric parallaxes to Hipparcos parallaxes and ask how these comparisons are involve by neglect extinction. In Section \[sec: all\ ] we analyse our distances to the generality of ace, using kinematic correction agent to quiz for systematic bias in distances ... | thfm. In Section \[sec:Hipp\] we gompare our specjriphotoketric parallabes to Hipparcos parallaxes end qsk hiw these comparisons afe affectvd by negoectmng extinction. Ii Sectiok \[fec:amp\] we enalyse our disjances to tha generality ox rtcrs, using kinematic correction factows to trsh for systematyc bpafes jn distanc... | them. In Section \[sec:Hipp\] we compare our to parallaxes and how these comparisons In \[sec:all\] we analyse distances to the of stars, using kinematic correction factors test for systematic biases in distances as functions of surface gravity or effective and to modify distance pdfs (Section \[sec:SBA\]). In Section ... | them. In Section \[sec:Hipp\] we comPare our speCtropHotOmeTrIc paRallAxes to HipparcoS ParaLlaxes and ask how these coMpariSoNS are AFfEcted By negleCTiNG ExtInCtIon. in sEcTion \[sEc:aLl\] we anaLyse our disTanCeS to the generaLItY of stars, usIng Kinematic corRecTion faCtOrs TO test For SysteMatic bIAses in Distanc... | them. In Section \[sec:Hi pp\] we co mpare ou r s pe ctro phot ometric parall a xesto Hipparcos parallaxe s and a s k ho w t hesecompari s on s are a ff ect ed by negl ect ing ext inction. I n S ec tion \[sec:a l l\ ] we analy seour distance s t o thege ner a lityofstars , usin g kinem atic corr ec t ion fa c to... | them._In Section \[sec:Hipp\]_we compare our spectrophotometric_parallaxes to_Hipparcos_parallaxes and_ask_how these comparisons_are affected by_neglecting extinction. In Section \[sec:all\]_we analyse our_distances_to the generality of stars, using kinematic correction factors to test for systematic biases_in_distanc... |
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abstract: 'Guidelines and consistency rules of UML are used to control the degrees of freedom provided by the language to ... | ! [ image](figure9.eps){width="160 mm " }
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abstract:' guidepost and consistency rules of UML are used to see the degree of f... | {width="160mm"}
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abstract: 'Guideliner and condistency rults of UML are usev to conbxol tgc degxexs of freedom ptovided by tve language to ... | {width="160mm"} {width="160mm"} {width="160mm"} {width="160mm"} {width="160mm"} --- abstract: consistency of UML used to control by language to prevent Guidelines are used specific domains (e.g., avionics) to re... | {width="160mm"}
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AbstRAcT: 'GuidElines aND cONSisTeNcY ruLeS Of uML arE usEd to conTrol the degReeS oF freedom provIDeD by the langUagE to ... | {widt h="160mm"}
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ab str act: 'Guidel ine s andco nsi s tency ru les o f UMLa re use d to cont ro l ... | {width="160mm"}
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abstract:_'Guidelines and_consistency rules of UML_are used_to_control the_degrees_of freedom provided_by the language_to ... |
- 'E. Kontizas'
- 'A. Vallenari'
date: 'Received date / accepted'
title: 'A semi-empirical library of galaxy spectra for Gaia classification based on SDSS data and PÉGASE models'
---
[This paper is the third in a series implementing a classification system for Gaia observations of unresolved galaxies. The system makes... | -' E. Kontizas'
-' A. Vallenari'
date:' Received date / accepted'
title:' A semi - empirical library of galaxy spectra for Gaia categorization free-base on SDSS data and PÉGASE models'
---
[ This newspaper is the third in a serial implementing a classification system for Gaia notice of unresolved galaxies. T... | - 'E. Nontizas'
- 'A. Vallenari'
date: 'Received date / acceptxd'
title: 'A semi-eopirical library of galaxy s'ectea foe Gaia classification cased on DDSS datq anv PÉGASE models'
---
[Thma paper is ths thixd in a series ikplementinc a classificadiun system for Gaia observations of unwesolvec halaxies. The sistem iakea... | - 'E. Kontizas' - 'A. Vallenari' date: / title: 'A library of galaxy on data and PÉGASE --- [This paper the third in a series implementing classification system for Gaia observations of unresolved galaxies. The system makes use of galaxy spectra in order to determine spectral classes and estimate intrinsic astrophysica... | - 'E. Kontizas'
- 'A. Vallenari'
date: 'ReCeived date / AccepTed'
TitLe: 'a semI-empIrical library oF GalaXy spectra for Gaia classiFicatIoN BaseD On sDSS dAta and PégAse ModElS'
---
[THis PaPEr Is the ThiRd in a seRies implemEntInG a classificaTIoN system for gaiA observationS of UnresoLvEd gALaxieS. ThE systEm makeS... | - 'E. Kontizas'
- 'A. Vall enari'
dat e: 'R ece ive ddate / a ccepted'
title : 'Asemi-empirical library of g al a xy s p ec tra f or Gaia cl a s sif ic at ion b a se d onSDS S dataand PÉGASE mo de ls'
---
[Th i spaper is t hethird in a s eri es imp le men t ing a cl assif icatio n syste m for Gai ao bserva t io... | - 'E._Kontizas'
- 'A._Vallenari'
date: 'Received date /_accepted'
title: 'A_semi-empirical_library of_galaxy_spectra for Gaia_classification based on_SDSS data and PÉGASE_models'
---
[This paper is_the_third in a series implementing a classification system for Gaia observations of unresolved galaxies._The_system makes... |
.399249 0.399850 0.399970 0.400000 2.996339E-5
$0.5$ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5
$0.6$ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5
$0.7$ 0.688388 0.697700 0.699541 0.699908 0.700000 9.176289E-5
... | .399249 0.399850 0.399970 0.400000 2.996339E-5
$ 0.5 $ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5
$ 0.6 $ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5
$ 0.7 $ 0.688388 0.697700 0.699541 0.699908 ... | .399249 0.399850 0.399970 0.400000 2.996339E-5
$0.5$ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5
$0.6$ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5
$0.7$ 0.688388 0.697700 0.699541 0.699908 0.700000 9.176289E-5
... | .399249 0.399850 0.399970 0.400000 2.996339E-5 $0.5$ 0.494075 0.499953 4.681780E-5 $0.6$ 0.598310 0.599662 0.599932 0.699541 0.700000 9.176289E-5 $0.8$ 0.796996 0.799400 0.799880 1.198535E-4 $0.9$ 0.880804 0.896198 0.899241 0.899848 1.516896E-4 $1.0$ 0.976302 0.995307 0.999063 0.999812 1.000000 1.872712E-4 ------- ----... | .399249 0.399850 0.399970 0.400000 2.996339E-5
$0.5$ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5
$0.6$ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5
$0.7$ 0.688388 0.697700 0.699541 0.699908 0.700000 9.176289E-5
... | .399249 0.399850 0.399 970 0 .4000 00 2.9 9633 9E-5
$0.5$ 0.4 94075 0.498826 0.4 99765 0.49 9 95 3 0.500 0 00 4. 681 78 0 E- 5
$0. 6$ 0. 591468 0 .59 83 10 0.59966 2 0.599932 0.600000 6.74 17 63E - 5
$0. 7$ 0.6883 8 8 0. 697700 0. 6 99541 0.6999 0 8 0.700000 9... | .399249 _ 0.399850_ 0.399970 _ __ 0.400000__ _ 2.996339E-5
_ $0.5$ _ 0.494075 __0.498826 0.499765 0.499953 0.500000 __ ___4.681780E-5
$0.6$ _ 0.591468 0.598310_ _0.599662 0.599932 __0.600000 _ 6.741763E-5
$0.7$ _ 0.688388 0.697700 _ 0.699541 __0.699908_ _ 0.700000 _ 9.176289E-5
... |
given by $$\begin{aligned}
e^{-\lambda}(2\nu^{\prime\prime} + \nu^{\prime^2} - \lambda^{\prime}\nu^{\prime} - 2\frac{\nu^{\prime}}{r}) \nonumber \\
- 2(n-3)(\frac{e^{-\lambda}\lambda^{\prime}}{r}
+ 2\frac{e^{-\lambda}}{r^2} - \frac{2}{r^2}) = 0 \label{isotropy}. \end{aligned}$$ Let us rewrite this equation in a form... | given by $ $ \begin{aligned }
e^{-\lambda}(2\nu^{\prime\prime } + \nu^{\prime^2 } - \lambda^{\prime}\nu^{\prime } - 2\frac{\nu^{\prime}}{r }) \nonumber \\
- 2(n-3)(\frac{e^{-\lambda}\lambda^{\prime}}{r }
+ 2\frac{e^{-\lambda}}{r^2 } - \frac{2}{r^2 }) = 0 \label{isotropy }. \end{aligned}$$ Let us rewrite this eq... | gigen by $$\begin{aligned}
e^{-\lambaa}(2\nu^{\prime\prime} + \nu^{\primx^2} - \lambsa^{\prime}\nj^{\prime} - 2\frac{\nu^{\prime}}{r}) \nonumbec \\
- 2(n-3)(\drac{e^{-\oambda}\lambda^{\prime}}{r}
+ 2\frxc{e^{-\lambda}}{g^2} - \frac{2}{r^2}) = 0 \lebel{isotropy}. \end{emigned}$$ Let us vewrice this equation in a form... | given by $$\begin{aligned} e^{-\lambda}(2\nu^{\prime\prime} + \nu^{\prime^2} - 2\frac{\nu^{\prime}}{r}) \\ - + 2\frac{e^{-\lambda}}{r^2} - Let rewrite this equation a form that yields the universal character of the interior solution for all $n\geq4$, $$\begin{aligned} e^{-\lambda}(2\nu^{\prime\prime} + \nu^{\prime^2} -... | given by $$\begin{aligned}
e^{-\lambdA}(2\nu^{\prime\prIme} + \nu^{\PriMe^2} - \lAmBda^{\pRime}\Nu^{\prime} - 2\frac{\nu^{\pRIme}}{r}) \Nonumber \\
- 2(n-3)(\frac{e^{-\lambda}\laMbda^{\pRiME}}{r}
+ 2\frAC{e^{-\LambdA}}{r^2} - \frac{2}{r^2}) = 0 \LAbEL{IsoTrOpY}. \enD{aLIgNed}$$ LeT us Rewrite This equatiOn iN a Form... | given by $$\begin{aligned }
e^{-\lam bda}( 2\n u^{ \p rime \pri me} + \nu^{\pr i me^2 } - \lambda^{\prime}\n u^{\p ri m e} - 2\ frac{ \nu^{\p r im e } }{r }) \ non um b er \\
- 2( n-3)(\f rac{e^{-\l amb da }\lambda^{\p r im e}}{r}
+2\f rac{e^{-\lam bda }}{r^2 }- \ f rac{2 }{r ^2})= 0 \ l abel{i sotropy}. \ e nd{... | given_by $$\begin{aligned}
e^{-\lambda}(2\nu^{\prime\prime}_+ \nu^{\prime^2} - \lambda^{\prime}\nu^{\prime}_- 2\frac{\nu^{\prime}}{r})_\nonumber_\\
- 2(n-3)(\frac{e^{-\lambda}\lambda^{\prime}}{r}_
+_2\frac{e^{-\lambda}}{r^2} - \frac{2}{r^2})_= 0 _\label{isotropy}. \end{aligned}$$ Let us_rewrite this equation_in_a form... |
infty(A,\Phi))\|_{W^{-1, p}_{A_1}(X)} \quad\text{(by \eqref{eq:gauged_LS})}
\\
&\leq Z'C_1\|\hat\sM(A,\Phi)\|_{W^{-1, p}_{A_1}(X)}.\end{aligned}$$ But $\hat\sM(A,\Phi) = \Pi_{u_\infty(A_\infty, \Phi_\infty)}\sM$ and because the projection, $$\Pi_{u_\infty(A_\infty, \Phi_\infty)}:W^{k, p}_{A_1}(X; \Lambda^1 \otimes \ad ... | infty(A,\Phi))\|_{W^{-1, p}_{A_1}(X) } \quad\text{(by \eqref{eq: gauged_LS }) }
\\
& \leq Z'C_1\|\hat\sM(A,\Phi)\|_{W^{-1, p}_{A_1}(X)}.\end{aligned}$$ But $ \hat\sM(A,\Phi) = \Pi_{u_\infty(A_\infty, \Phi_\infty)}\sM$ and because the projection, $ $ \Pi_{u_\infty(A_\infty, \Phi_\infty)}:W^{k, p}_{A_1}(X; \Lambda^1 ... | infhy(A,\Phi))\|_{W^{-1, p}_{A_1}(X)} \quad\text{(by \edref{eq:gauged_LS})}
\\
&\lgq Z'C_1\|\hat\vM(A,\Phi)\|_{S^{-1, p}_{A_1}(X)}.\end{xligned}$$ But $\hat\sM(A,\Phi) = \Pi_{u_\inhty(A_\unfty, \Phi_\infty)}\sM$ and becausd the prouection, $$\Pi_{u_\iifty(A_\infty, \Phi_\inhfy)}:W^{k, p}_{A_1}(W; \Lamgfa^1 \ocines \ad ... | infty(A,\Phi))\|_{W^{-1, p}_{A_1}(X)} \quad\text{(by \eqref{eq:gauged_LS})} \\ &\leq Z'C_1\|\hat\sM(A,\Phi)\|_{W^{-1, $\hat\sM(A,\Phi) \Pi_{u_\infty(A_\infty, \Phi_\infty)}\sM$ because the projection, \ad \oplus E) \to p}_{A_1}(X; \Lambda^1 \otimes P \oplus E),$$ is bounded with one (for any $k\in\ZZ$ and $1<p<\infty$)... | infty(A,\Phi))\|_{W^{-1, p}_{A_1}(X)} \quad\text{(by \eqRef{eq:gaugeD_LS})}
\\
&\leQ Z'C_1\|\Hat\SM(a,\Phi)\|_{w^{-1, p}_{A_1}(X)}.\End{aligned}$$ But $\hAT\sM(A,\phi) = \Pi_{u_\infty(A_\infty, \Phi_\inFty)}\sM$ AnD BecaUSe The prOjectioN, $$\pi_{U_\INftY(A_\InFty, \phI_\InFty)}:W^{k, P}_{A_1}(X; \lambda^1 \oTimes \ad ... | infty(A,\Phi))\|_{W^{-1, p }_{A_1}(X) } \qu ad\ tex t{ (by\eqr ef{eq:gauged_L S })}\\
&\leq Z'C_1\|\hat\s M(A,\ Ph i )\|_ { W^ {-1,p}_{A_1 } (X ) } .\e nd {a lig ne d }$ $ But $\ hat\sM( A,\Phi) =\Pi _{ u_\infty(A_\ i nf ty, \Phi_\ inf ty)}\sM$ and be causeth e p r oject ion , $$\ Pi_{u_ \ infty( A_\infty, \ P hi_\i... | infty(A,\Phi))\|_{W^{-1, p}_{A_1}(X)}_\quad\text{(by \eqref{eq:gauged_LS})}
\\
&\leq_Z'C_1\|\hat\sM(A,\Phi)\|_{W^{-1, p}_{A_1}(X)}.\end{aligned}$$ But $\hat\sM(A,\Phi)_= \Pi_{u_\infty(A_\infty,_\Phi_\infty)}\sM$_and because_the_projection, $$\Pi_{u_\infty(A_\infty, \Phi_\infty)}:W^{k,_p}_{A_1}(X; \Lambda^1 \otimes_\ad ... |
D space as $$\hat c'(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c'^{-1}=-(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We set $$c'= e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\ha... | D space as $ $ \hat c'(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c'^{-1}=-(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We set $ $ c'= e^{\gamma_1\gamma_2\frac{2\pi}{2n}... | D soace as $$\hat c'(\begin{smallmxtrix}\gamma_1\\\gamma_2\gne{smallkatrix})\gat c'^{-1}=-(\begkn{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frec{2\pi}{2b}\\-\sin\feac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smaulmatrix})(\bvgin{smallnatrmx}\gamma_1\\\gamma_2\end{smallmatrlr}).$$ We act $$c'= z^{\gemma_1\gamma_2\frac{2\pi}{2k}}\ha... | D space as $$\hat c'(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c'^{-1}=-(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat{c}',$$ s... | D space as $$\hat c'(\begin{smallmatRix}\gamma_1\\\gaMma_2\enD{smAllMaTrix})\Hat c'^{-1}=-(\Begin{smallmatrIX}\cos\Frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frAc{2\pi}{2n}&\CoS\Frac{2\PI}{2n}\End{smAllmatrIX})(\bEGIn{sMaLlMatRiX}\GaMma_1\\\gaMma_2\End{smalLmatrix}).$$ We sEt $$c'= E^{\gAmma_1\gamma_2\fraC{2\Pi}{2N}}\ha... | D space as $$\hat c'(\begi n{smallmat rix}\ gam ma_ 1\ \\ga mma_ 2\end{smallmat r ix}) \hat c'^{-1}=-(\begin{ small ma t rix} \ co s\fra c{2\pi} { 2n } & \si n\ fr ac{ 2\ p i} {2n}\ \-\ sin\fra c{2\pi}{2n }&\ co s\frac{2\pi} { 2n }\end{smal lma trix})(\begi n{s mallma tr ix} \ gamma _1\ \\gam ma_2\e n d{smal lmatrix}) ... | D space_as $$\hat_c'(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c'^{-1}=-(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We set_$$c'= e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\ha... |
}^{3/2}B_{1}^{1/2}(2+vB_{1})v -
\nonumber\\
&& 2v^2B_{-1}B_{1}^2\left (\exp [\frac{B_{-1}}{v}]\Gamma \left (0,
\frac{B_{-1}}{v} + \sqrt{\frac{B_{-1}}{B_{1}}} \right )
+ \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2})\right )\nonumber\\
&& -2vB_{-1}B_{1}(\exp [\frac{B_{-1}}{v}]\Gamma\left (0, \frac{B_{-1}}{v}
+ \sqrt{\frac{B_{-1}}... | } ^{3/2}B_{1}^{1/2}(2+vB_{1})v -
\nonumber\\
& & 2v^2B_{-1}B_{1}^2\left (\exp [ \frac{B_{-1}}{v}]\Gamma \left (0,
\frac{B_{-1}}{v } + \sqrt{\frac{B_{-1}}{B_{1 } } } \right)
+ \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2})\right) \nonumber\\
& & -2vB_{-1}B_{1}(\exp [ \frac{B_{-1}}{v}]\Gamma\left (0, \frac{B_{-1}}{v }
+... | }^{3/2}B_{1}^{1/2}(2+vB_{1})g -
\nonumber\\
&& 2v^2B_{-1}B_{1}^2\left (\exp [\fvac{B_{-1}}{v}]\Gamma \left (0,
\frac{B_{-1}}{v} + \sqrt{\rrac{B_{-1}}{B_{1}}} \rkght )
+ \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2})\right )\nonumbec\\
&& -2vB_{-1}V_{1}(\exp [\drac{B_{-1}}{v}]\Gamma\left (0, \frac{B_{-1}}{x}
+ \sqrt{\frab{B_{-1}}... | }^{3/2}B_{1}^{1/2}(2+vB_{1})v - \nonumber\\ && 2v^2B_{-1}B_{1}^2\left (\exp [\frac{B_{-1}}{v}]\Gamma \frac{B_{-1}}{v} \sqrt{\frac{B_{-1}}{B_{1}}} \right + \Gamma (0, (0, + \sqrt{\frac{B_{-1}}{B_{1}}} \right) \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) && - \exp[vB_{1}]\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2} + - v^2B_{-1}\exp [B_{-1... | }^{3/2}B_{1}^{1/2}(2+vB_{1})v -
\nonumber\\
&& 2v^2B_{-1}B_{1}^2\left (\exp [\fraC{B_{-1}}{v}]\Gamma \leFt (0,
\fraC{B_{-1}}{v} + \SqrT{\fRac{B_{-1}}{b_{1}}} \rigHt )
+ \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2})\right )\NOnumBer\\
&& -2vB_{-1}B_{1}(\exp [\frac{B_{-1}}{v}]\Gamma\leFt (0, \fraC{B_{-1}}{V}
+ \Sqrt{\FRaC{B_{-1}}... | }^{3/2}B_{1}^{1/2}(2+vB_{1 })v -
\non umber \\&&2v ^2B_ {-1} B_{1}^2\left ( \ exp[\frac{B_{-1}}{v}]\Gam ma \l ef t (0,
\ frac{ B_{-1}} { v} + \s qr t{ \fr ac { B_ {-1}} {B_ {1}}} \ right )
+\Ga mm a (0, B_{-1} ^ {1 /2}B_{1}^{ 3/2 })\right )\n onu mber\\
& & - 2 vB_{- 1}B _{1}( \exp [ \ frac{B _{-1}}{v} ]\ G amma\l ... | }^{3/2}B_{1}^{1/2}(2+vB_{1})v -
\nonumber\\
&&_2v^2B_{-1}B_{1}^2\left (\exp_[\frac{B_{-1}}{v}]\Gamma \left (0,
\frac{B_{-1}}{v}_+ \sqrt{\frac{B_{-1}}{B_{1}}}_\right_)
+ \Gamma_(0,_B_{-1}^{1/2}B_{1}^{3/2})\right )\nonumber\\
&& -2vB_{-1}B_{1}(\exp_[\frac{B_{-1}}{v}]\Gamma\left (0, \frac{B_{-1}}{v}
+_\sqrt{\frac{B_{-1}}... |
known process that was found in 1D kinetic simulations [@araneda2008proton; @MatteiniEA2010]. Here we investigate the origin and its possible relation to the strong perpendicular heating that we observe in our simulations.
To explore the plasma heating and the proton beam formation, we have performed the same numeric... | known process that was found in 1D kinetic simulations [ @araneda2008proton; @MatteiniEA2010 ]. Here we investigate the beginning and its potential relation to the strong vertical heating that we respect in our simulations.
To research the plasma heating and the proton beam formation, we have perform the same nume... | knlwn process that was foukd in 1D kinetic simulatmons [@araheda2008protun; @MatteiniEA2010]. Here we investmgatw the origin and its possibue relatiln to thw stcong perpendiculed heatiky thaf we muserve in our slmulations.
Tm explore the [lxsla heating and the proton beam formwtion, wr jave performed the fame numeric... | known process that was found in 1D [@araneda2008proton; Here we the origin and strong heating that we in our simulations. explore the plasma heating and the beam formation, we have performed the same numerical experiment that consists of a initial perturbation with different plasma beta. Although the monochromatic case... | known process that was found iN 1D kinetic sImulaTioNs [@aRaNeda2008ProtOn; @MatteiniEA2010]. HeRE we iNvestigate the origin and Its poSsIBle rELaTion tO the strONg PERpeNdIcUlaR hEAtIng thAt wE observE in our simuLatIoNs.
To explore tHE pLasma heatiNg aNd the proton bEam FormatIoN, we HAve peRfoRmed tHe same NUmeric... | known process that was fo und in 1Dkinet icsim ul atio ns [ @araneda2008pr o ton; @MatteiniEA2010]. Her e wein v esti g at e the origin an d its p os sib le re latio n t o the s trong perp end ic ular heating th at we obse rve in our simu lat ions.
T o e x plore th e pla sma he a ting a nd the pr ot o n beam form... | known_process that_was found in 1D_kinetic simulations [@araneda2008proton;_@MatteiniEA2010]._Here we_investigate_the origin and_its possible relation_to the strong perpendicular_heating that we_observe_in our simulations.
To explore the plasma heating and the proton beam formation, we have_performed_the same_numeric... |
h s.), the following Hamiltonian was set up $$\label{eqn:H} H=\frac{(\hat{I}_{\pi}+\hat{I}_{\nu})^{2}}{2J_{intr}}+
\frac{(\hat{I}_{\pi}-\hat{I}_{\nu})^{2}}{2J_{intr}}+\frac{1}{2}C\theta^{2},
\label{eq:hamiltonian}$$ in which the restoring force constant $C$ is related to the symmetry energy constant in the semi-empiric... | h s.), the following Hamiltonian was set up $ $ \label{eqn: H } H=\frac{(\hat{I}_{\pi}+\hat{I}_{\nu})^{2}}{2J_{intr}}+
\frac{(\hat{I}_{\pi}-\hat{I}_{\nu})^{2}}{2J_{intr}}+\frac{1}{2}C\theta^{2 },
\label{eq: hamiltonian}$$ in which the restoring force changeless $ C$ is relate to the symmetry energy constant in the ... | h s.), the following Hamiltonixn was set up $$\lcvel{eqn:I} H=\frac{(\gat{I}_{\pi}+\hag{I}_{\nu})^{2}}{2J_{intr}}+
\frac{(\hat{I}_{\pi}-\hat{I}_{\nu})^{2}}{2J_{invr}}+\frqc{1}{2}C\thtna^{2},
\label{eq:hamiltonian}$$ kn which nhe restoeing dorce consvznt $C$ is relafcd to vhe symmetry engrgy constand in the semi-ekpkrnc... | h s.), the following Hamiltonian was set H=\frac{(\hat{I}_{\pi}+\hat{I}_{\nu})^{2}}{2J_{intr}}+ \label{eq:hamiltonian}$$ in the restoring force the energy constant in semi-empirical mass formula. known properties of deformed nuclei: the of inertia $J_{intr}$ of the ground-state band, the symmetry energy and the $B(E2;0... | h s.), the following Hamiltonian Was set up $$\laBel{eqN:H} H=\FraC{(\hAt{I}_{\pI}+\hat{i}_{\nu})^{2}}{2J_{intr}}+
\frac{(\haT{i}_{\pi}-\hAt{I}_{\nu})^{2}}{2J_{intr}}+\frac{1}{2}C\theta^{2},
\laBel{eq:HaMIltoNIaN}$$ in whIch the rEStORIng FoRcE coNsTAnT $C$ is rElaTed to thE symmetry eNerGy Constant in thE SeMi-empiric... | h s.), the following Hamil tonian was setup$$\ la bel{ eqn: H} H=\frac{(\h a t{I} _{\pi}+\hat{I}_{\nu})^ {2}}{ 2J _ {int r }} +
\fr ac{(\ha t {I } _ {\p i} -\ hat {I } _{ \nu}) ^{2 }}{2J_{ intr}}+\fr ac{ 1} {2}C\theta^{ 2 },
\label{eq :ha miltonian}$$ in which t her estor ing forc e cons t ant $C $ is rela te d ... | h s.),_the following_Hamiltonian was set up_$$\label{eqn:H} H=\frac{(\hat{I}_{\pi}+\hat{I}_{\nu})^{2}}{2J_{intr}}+
\frac{(\hat{I}_{\pi}-\hat{I}_{\nu})^{2}}{2J_{intr}}+\frac{1}{2}C\theta^{2},
\label{eq:hamiltonian}$$_in_which the_restoring_force constant $C$_is related to_the symmetry energy constant_in the semi-empiric... |
delta}^{\sigma'}(j)}
{|i-j|}\,,$$ $$\label{}
C_{\alpha \sigma,\beta \sigma'}
^{\gamma \sigma,\delta \sigma'}=\sum_{i}
{\phi_{\alpha}^{\sigma*}(i)\phi_{\gamma}^{\sigma}(i)
\phi_{\beta}^{\sigma'*}(i)\phi_{\delta}^{\sigma'}(i)}\,.$$ Omitting the off-diagonal matrix elements ($(\alpha,\beta)\not=
(\gamma,\delta)$) in Eq.(7... | delta}^{\sigma'}(j) }
{ |i - j|}\,,$$ $ $ \label { }
C_{\alpha \sigma,\beta \sigma' }
^{\gamma \sigma,\delta \sigma'}=\sum_{i }
{ \phi_{\alpha}^{\sigma*}(i)\phi_{\gamma}^{\sigma}(i)
\phi_{\beta}^{\sigma'*}(i)\phi_{\delta}^{\sigma'}(i)}\,.$$ Omitting the off - diagonal matrix elements ($ (\alpha,\beta)\not=
... | delha}^{\sigma'}(j)}
{|i-j|}\,,$$ $$\label{}
C_{\alpha \slgma,\beta \sigma'}
^{\gamma \sigka,\deltz \sigma'}=\sjm_{i}
{\phi_{\alpha}^{\sigma*}(i)\phi_{\gamma}^{\sigla}(u)
\phi_{\btna}^{\sigma'*}(i)\phi_{\delta}^{\sigma'}(k)}\,.$$ Omittinh the ofd-diajonal matrix elements ($(\alika,\betz)\kot=
(\gakna,\delta)$) in Eq.(7... | delta}^{\sigma'}(j)} {|i-j|}\,,$$ $$\label{} C_{\alpha \sigma,\beta \sigma'} ^{\gamma {\phi_{\alpha}^{\sigma*}(i)\phi_{\gamma}^{\sigma}(i) Omitting the matrix elements ($(\alpha,\beta)\not= Hartree-Fock The reason for this will be subsequently. The basic parameters of our framework are the electron number $n$, the long... | delta}^{\sigma'}(j)}
{|i-j|}\,,$$ $$\label{}
C_{\alpha \sIgma,\beta \siGma'}
^{\gaMma \SigMa,\DeltA \sigMa'}=\sum_{i}
{\phi_{\alpha}^{\SIgma*}(I)\phi_{\gamma}^{\sigma}(i)
\phi_{\beta}^{\Sigma'*}(I)\pHI_{\delTA}^{\sIgma'}(i)}\,.$$ omittinG ThE OFf-dIaGoNal MaTRiX elemEntS ($(\alpha,\bEta)\not=
(\gammA,\deLtA)$) in Eq.(7... | delta}^{\sigma'}(j)}
{|i-j |}\,,$$ $$ \labe l{}
C_ {\ alph a \s igma,\beta \si g ma'}
^{\gamma \sigma,\delt a \si gm a '}=\ s um _{i}{\phi_{ \ al p h a}^ {\ si gma *} ( i) \phi_ {\g amma}^{ \sigma}(i)
\p hi _{\beta}^{\s i gm a'*}(i)\ph i_{ \delta}^{\si gma '}(i)} \, .$$ Omitt ing theoff-di a gonalmatrix el em e nts... | delta}^{\sigma'}(j)}
{|i-j|}\,,$$ $$\label{}
C_{\alpha_\sigma,\beta \sigma'}
^{\gamma_\sigma,\delta \sigma'}=\sum_{i}
{\phi_{\alpha}^{\sigma*}(i)\phi_{\gamma}^{\sigma}(i)
\phi_{\beta}^{\sigma'*}(i)\phi_{\delta}^{\sigma'}(i)}\,.$$ Omitting the_off-diagonal matrix_elements_($(\alpha,\beta)\not=
(\gamma,\delta)$) in_Eq.(7... |
In Ref. [@Trifunovic2019], a complete classification of these intrinsic corner or hinge modes was derived and a higher-order bulk-boundary correspondence between these high codimension boundary modes and the topological bulk was obtained. These were accomplished by considering a $K$ subgroup series for a $d$-dimensiona... | In Ref. [ @Trifunovic2019 ], a complete classification of these intrinsic corner or hinge modes was derive and a eminent - order bulk - boundary correspondence between these gamey codimension boundary modes and the topological majority was obtained. These were accomplished by considering a $ K$ subgroup serial for a ... | In Gef. [@Trifunovic2019], a complete classification of theve intdinsic curner or hinge modes was dermved and q higher-order bulk-bounaary corrvspondencw beuween these high rkdimensljn bkmndarv nodes and the jopological tulk was obtaitea. Chese were accomplished by considerigg a $K$ xuhgroup series sor s $d$-dijvnwiona... | In Ref. [@Trifunovic2019], a complete classification of corner hinge modes derived and a high boundary modes and topological bulk was These were accomplished by considering a subgroup series for a $d$-dimensional crystal, $$K^{(d)} \subseteq \dots \subseteq K'' \subseteq K' K,$$ where $K\equiv K^{(0)}$ is the $K$ group... | In Ref. [@Trifunovic2019], a complete cLassificatIon of TheSe iNtRinsIc coRner or hinge modES was Derived and a higher-order Bulk-bOuNDary COrRespoNdence bETwEEN thEsE hIgh CoDImEnsioN boUndary mOdes and the TopOlOgical bulk waS ObTained. ThesE weRe accomplishEd bY consiDeRinG A $K$ subGroUp serIes for A $D$-dimenSiona... | In Ref. [@Trifunovic2019], a complet e cla ssi fic at ionof t hese intrinsic corn er or hinge modes wasderiv ed anda h igher -orderb ul k - bou nd ar y c or r es ponde nce betwee n these hi ghco dimension bo u nd ary modesand the topolog ica l bulk w aso btain ed. Thes e were accomp lished by c o nsider i ng a $... | In Ref. [@Trifunovic2019],_a complete_classification of these intrinsic_corner or_hinge_modes was_derived_and a higher-order_bulk-boundary correspondence between_these high codimension boundary_modes and the_topological_bulk was obtained. These were accomplished by considering a $K$ subgroup series for a_$d$-dimensiona... |
to the tensor product since here a tensor factor means an application of the Rota-Baxter operator $P_A$: $u_0\ot u_1= u_0P_A(u_1)$. Thus $$d_A(u_0\ot u_1)=d_A(u_0)P_A(u_1)+u_0\,d_A(P_A(u_1)) +\lambda d_A(u_0)d_A(P_A(u_1))= d(u_0)\ot u_1+u_0u_1+\lambda d(u_0)u_1.$$
$($[@GGZ; @GK3]$)$ Let $Y$ be a set with a set map $d... | to the tensor product since here a tensor factor means an application of the Rota - Baxter hustler $ P_A$: $ u_0\ot u_1= u_0P_A(u_1)$. therefore $ $ d_A(u_0\ot u_1)=d_A(u_0)P_A(u_1)+u_0\,d_A(P_A(u_1) ) + \lambda d_A(u_0)d_A(P_A(u_1))= d(u_0)\ot u_1+u_0u_1+\lambda d(u_0)u_1.$$
$ ($ [ @GGZ; @GK3]$)$ permit $ Y$ be a s... | to the tensor product sinct here a tensor fcxtor mxans an applicagion of the Rota-Baxter operavor $P_A$: $u_0\ou u_1= u_0P_A(u_1)$. Thus $$d_A(u_0\ot u_1)=d_A(u_0)P_A(u_1)+u_0\,f_A(P_A(u_1)) +\lanbda e_A(u_0)d_A(P_A(u_1))= d(n_0)\kt u_1+u_0u_1+\lambda s(m_0)u_1.$$
$($[@GGZ; @JK3]$)$ Let $Y$ be a sgt with a sed map $d... | to the tensor product since here a means application of Rota-Baxter operator $P_A$: u_1)=d_A(u_0)P_A(u_1)+u_0\,d_A(P_A(u_1)) d_A(u_0)d_A(P_A(u_1))= d(u_0)\ot u_1+u_0u_1+\lambda $($[@GGZ; @GK3]$)$ Let be a set with a set $d_0\colon Y\to \bfk[Y]$ and let $(\bfk[Y],d)$ be the commutative differential algebra of weight in ... | to the tensor product since heRe a tensor fActor MeaNs aN aPpliCatiOn of the Rota-BaxTEr opErator $P_A$: $u_0\ot u_1= u_0P_A(u_1)$. Thus $$d_A(U_0\ot u_1)=d_a(u_0)p_a(u_1)+u_0\,d_a(p_A(U_1)) +\lambDa d_A(u_0)d_A(p_a(u_1))= D(U_0)\Ot u_1+U_0u_1+\LaMbdA d(U_0)U_1.$$
$($[@GgZ; @GK3]$)$ LEt $Y$ Be a set wIth a set map $D... | to the tensor product sin ce here atenso r f act or mea ns a n applicationo f th e Rota-Baxter operator $P_A $: $u_0 \ ot u_1= u_0P_A ( u_ 1 ) $.Th us $$ d_ A (u _0\ot u_ 1)=d_A( u_0)P_A(u_ 1)+ u_ 0\,d_A(P_A(u _ 1) ) +\lambda d_ A(u_0)d_A(P_ A(u _1))=d( u_0 ) \ot u _1+ u_0u_ 1+\lam b da d(u _0)u_1.$$
$ ($[@GG ... | to_the tensor_product since here a_tensor factor_means_an application_of_the Rota-Baxter operator_$P_A$: $u_0\ot u_1=_u_0P_A(u_1)$. Thus $$d_A(u_0\ot u_1)=d_A(u_0)P_A(u_1)+u_0\,d_A(P_A(u_1))_+\lambda d_A(u_0)d_A(P_A(u_1))= d(u_0)\ot_u_1+u_0u_1+\lambda_d(u_0)u_1.$$
$($[@GGZ; @GK3]$)$ Let $Y$ be a set with a set map $d... |
the field equation with the potential (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\varphi^{2}+U\left( \varphi\right) -\mathcal{G}\right] }}d\varphi=0.$$
Obviously, for an $n$-term polynomial potential, once a potential in the duality family is solved, the other ... | the field equation with the potential (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left [ \frac{1}{2}m^{2}\varphi^{2}+U\left ( \varphi\right) -\mathcal{G}\right ] } } d\varphi=0.$$
Obviously, for an $ n$-term polynomial potential, once a electric potential in the dichotomy class i... | thf field equation with tht potential (\[varphnwqa\]):$$\bete_{\mu}y^{\mu}+\iht\frac{\sqft{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\varphi^{2}+U\pedt( \vqrphi\right) -\mathcal{G}\rieht] }}d\varihi=0.$$
Obviouwly, hor an $n$-term polbhomial ijtenflal, oucx a potential ik the dualidy family is smlxeb, the other ... | the field equation with the potential (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ -\mathcal{G}\right] Obviously, for $n$-term polynomial potential, duality is solved, the $n$ potentials are solved. The sine-Gordon equation \[sine-Gordon\] ======================================== sine... | the field equation with the poTential (\[varPhieqA\]):$$\beTa_{\mU}y^{\Mu}+\inT\fraC{\sqrt{-\beta^{2}}}{\sqrt{2\lEFt[ \frAc{1}{2}m^{2}\varphi^{2}+U\left( \varphi\riGht) -\maThCAl{G}\rIGhT] }}d\varPhi=0.$$
ObviOUsLY, For An $N$-tErm PoLYnOmial PotEntial, oNce a potentIal In The duality faMIlY is solved, tHe oTher ... | the field equation with t he potenti al (\ [va rph ie qa\] ):$$ \beta_{\mu}y^{ \ mu}+ \int\frac{\sqrt{-\beta ^{2}} }{ \ sqrt { 2\ left[ \frac { 1} { 2 }m^ {2 }\ var ph i ^{ 2}+U\ lef t( \va rphi\right ) -\ mathcal{G}\r i gh t] }}d\va rph i=0.$$
Obvi ous ly, fo ran$ n$-te rmpolyn omialp otenti al, onceap otenti a ... | the_field equation_with the potential (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[_ \frac{1}{2}m^{2}\varphi^{2}+U\left(__\varphi\right) _-\mathcal{G}\right]_ }}d\varphi=0.$$
Obviously, for_an $n$-term polynomial_potential, once a potential_in the duality_family_is solved, the other ... |
]{}*]{}.The experimental analysis of the inertial solution employs a single Ytterbium ion $^{171}$Yb$^+$, trapped in a six needle Paul trap schematically shown in Fig. \[ExpSche\]. The two-level-system used in our study is encoded in the hyperfine energy levels of $^{171}$Yb$^+$ represented as ${{\left| 0 \right>} \equ... | ] { } * ] { } .The experimental analysis of the inertial solution employs a single Ytterbium ion $ ^{171}$Yb$^+$, trapped in a six acerate leaf Paul ambush schematically shown in Fig. \[ExpSche\ ]. The two - level - system use in our study is encoded in the hyperfine department of energy levels of $ ^{171}$Yb$^+$ rep... | ]{}*]{}.The experimental analysis on the inertial solution emplogs a sinele Ytterbium ion $^{171}$Yb$^+$, trapped ib a sux needle Paul trap scfematicalpy shown in Hig. \[ExpSche\]. The txk-level-snftem msed nn our study is gncoded in tve hyperfine etefgv levels of $^{171}$Yb$^+$ represented as ${{\left| 0 \wight>} \eau... | ]{}*]{}.The experimental analysis of the inertial solution single ion $^{171}$Yb$^+$, in a six in \[ExpSche\]. The two-level-system in our study encoded in the hyperfine energy levels $^{171}$Yb$^+$ represented as ${{\left| 0 \right>} \equiv \,^{2}S_{1/2}\, {\left| F=0,m_{F}=0 \right>}}$ and ${{\left| \right>} \equiv \... | ]{}*]{}.The experimental analysis of The inertiaL soluTioN emPlOys a SingLe Ytterbium ion $^{171}$yB$^+$, traPped in a six needle Paul trAp schEmATicaLLy Shown In Fig. \[ExPscHE\]. the TwO-lEveL-sYStEm useD in Our studY is encoded In tHe Hyperfine eneRGy Levels of $^{171}$Yb$^+$ RepResented as ${{\leFt| 0 \rIght>} \eqU... | ]{}*]{}.The experimental a nalysis of theine rti al sol utio n employs a si n gleYtterbium ion $^{171}$ Yb$^+ $, trap p ed in a six ne e dl e Pau ltr apsc h em atica lly shownin Fig. \[ Exp Sc he\]. The tw o -l evel-syste m u sed in our s tud y is e nc ode d in t hehyper fine e n ergy l evels of$^ { 171}$Y b $^+$... | ]{}*]{}.The experimental_analysis of_the inertial solution employs_a single_Ytterbium_ion $^{171}$Yb$^+$,_trapped_in a six_needle Paul trap_schematically shown in Fig. \[ExpSche\]._The two-level-system used_in_our study is encoded in the hyperfine energy levels of $^{171}$Yb$^+$ represented as ${{\left|_0_\right>} \equ... |
is indeed an element of $\mathbb{Z}\left[\zeta_k \right].$ The result now follows by equivalence (\[modified condition\]).$ \qed$
Divisibility Results
====================
We use Theorem \[criterion\], in conjunction with the evaluations of the sums $K(\chi)$ given in Section $2,$ to obtain new results concerning th... | is indeed an element of $ \mathbb{Z}\left[\zeta_k \right].$ The result now follows by equality (\[modified condition\]).$ \qed$
Divisibility result
= = = = = = = = = = = = = = = = = = = =
We use Theorem \[criterion\ ], in conjunction with the evaluations of the sum $ K(\chi)$ given in Section $ 2,$ to prevail m... | is indeed an element of $\mauhbb{Z}\left[\zeta_k \riyyt].$ The resulf now foulows by equivalence (\[modifiev cobditiin\]).$ \qed$
Divisibility Resjlts
====================
We usv Theorem \[criuerion\], in conjuncvjon witm the cvaluctmons of the sumx $K(\chi)$ givan in Section $2,$ tu lbtain new results concerning th... | is indeed an element of $\mathbb{Z}\left[\zeta_k \right].$ now by equivalence condition\]).$ \qed$ Divisibility \[criterion\], conjunction with the of the sums given in Section $2,$ to obtain results concerning the divisors of $\text{gcd}(S_2(x),x^{q-1}+1).$ We first apply the evaluations of the Jacobi sums given in Co... | is indeed an element of $\mathbb{z}\left[\zeta_k \Right].$ the ResUlT now FollOws by equivalenCE (\[modIfied condition\]).$ \qed$
DivisIbiliTy rEsulTS
====================
WE use THeorem \[cRItERIon\], In CoNjuNcTIoN with The EvaluatIons of the sUms $k(\cHi)$ given in SecTIoN $2,$ to obtain nEw rEsults concerNinG th... | is indeed an element of $ \mathbb{Z} \left [\z eta _k \ri ght] .$ The resultn ow f ollows by equivalence(\[mo di f iedc on ditio n\]).$\ qe d $
D iv is ibi li t yResul ts======= ========== ===
We use Theor e m\[criterio n\] , in conjunc tio n with t hee valua tio ns of the s u ms $K( \chi)$ gi ve n in Se c tion... | is_indeed an_element of $\mathbb{Z}\left[\zeta_k \right].$_The result_now_follows by_equivalence_(\[modified condition\]).$ \qed$
Divisibility_Results
====================
We use Theorem_\[criterion\], in conjunction with_the evaluations of_the_sums $K(\chi)$ given in Section $2,$ to obtain new results concerning th... |
times that of the FFE, as indicated in Fig.\[fig2\]. Therefore, a saving in the number of steps $N_{s}$ provided by the ESSFM compared to the SSFM (as experimentally demonstrated in the next section) translates into a proportional saving in terms of latency, and an almost proportional saving in terms of complexity and... | times that of the FFE, as indicated in Fig.\[fig2\ ]. Therefore, a saving in the numeral of step $ N_{s}$ provided by the ESSFM compared to the SSFM (as experimentally prove in the next section) translate into a proportional saving in terms of rotational latency, and an almost proportional saving in term of complexity ... | tiles that of the FFE, as ikdicated in Fig.\[fnt2\]. Therxfore, a saving kn the number of steps $N_{s}$ prlvuded vy the ESSFM compared go the SSVM (as experinwntally demonstratcb in fme nert section) transkates into a proportionan raring in terms of latency, and an almoft propprhional saving yn ttrmf of bonplexity and... | times that of the FFE, as indicated Therefore, saving in number of steps compared the SSFM (as demonstrated in the section) translates into a proportional saving terms of latency, and an almost proportional saving in terms of complexity and consumption. Experimental results ==================== The experimental setup e... | times that of the FFE, as indicaTed in Fig.\[fiG2\]. TherEfoRe, a SaVing In thE number of steps $n_{S}$ proVided by the ESSFM compareD to thE SsfM (as EXpErimeNtally dEMoNSTraTeD iN thE nEXt SectiOn) tRanslatEs into a proPorTiOnal saving in TErMs of latencY, anD an almost proPorTional SaVinG In terMs oF compLexity ANd... | times that of the FFE, as indicated in F ig. \[f ig 2\]. The refore, a savi n g in the number of steps $ N_{s} $p rovi d ed by t he ESSF M c o m par ed t o t he SS FM (a s e xperime ntally dem ons tr ated in then ex t section) tr anslates int o a propo rt ion a l sav ing in t erms o f laten cy, and a na lmo... | times_that of_the FFE, as indicated_in Fig.\[fig2\]._Therefore,_a saving_in_the number of_steps $N_{s}$ provided_by the ESSFM compared_to the SSFM_(as_experimentally demonstrated in the next section) translates into a proportional saving in terms of_latency,_and an_almost_proportional_saving in terms of complexity_and... |
theory of Kohno connections [@rif65; @rif67]. If a connection $\mathcal{D}=d+A$ of the form \[berkohno\] satisfies $\mathcal{D}^{2}=0$, then the following relations hold
$$\begin{split}
\left[ B_{ij},B_{ik}+B_{jk}\right] = \left[ B_{ij}+B_{ik},B_{jk}\right] =0, \hspace{1cm} & \mathrm{for} \ i<j<k, \\
\left[ B_{ij}... | theory of Kohno connections [ @rif65; @rif67 ]. If a connection $ \mathcal{D}=d+A$ of the form \[berkohno\ ] satisfies $ \mathcal{D}^{2}=0 $, then the follow relative hold
$ $ \begin{split }
\left [ B_{ij},B_{ik}+B_{jk}\right ] = \left [ B_{ij}+B_{ik},B_{jk}\right ] = 0, \hspace{1 cm } & \mathrm{for } \ i < j <... | thfory of Kohno connectionr [@rif65; @rif67]. If a einnectmon $\matgcal{D}=d+A$ uf the form \[berkohno\] satisfixs $\mqthcao{D}^{2}=0$, then the following felations hold
$$\begun{spout}
\left[ B_{mn},B_{ik}+B_{jk}\vnght] = \peft[ U_{ij}+B_{ik},B_{jk}\right] =0, \hspace{1cm} & \mathrm{for} \ i<j<n, \\
\lzft[ B_{ij}... | theory of Kohno connections [@rif65; @rif67]. If $\mathcal{D}=d+A$ the form satisfies $\mathcal{D}^{2}=0$, then \left[ = \left[ B_{ij}+B_{ik},B_{jk}\right] \hspace{1cm} & \mathrm{for} i<j<k, \\ \left[ B_{ij},B_{kl}\right] =0, \hspace{5cm} \mathrm{for \ distinct \ i,j,k,l.} \end{split}$$ The above equations are called t... | theory of Kohno connections [@rIf65; @rif67]. If a coNnectIon $\MatHcAl{D}=d+a$ of tHe form \[berkohno\] SAtisFies $\mathcal{D}^{2}=0$, then the folLowinG rELatiONs Hold
$$\bEgin{splIT}
\lEFT[ B_{iJ},B_{Ik}+b_{jk}\RiGHt] = \Left[ B_{Ij}+B_{Ik},B_{jk}\riGht] =0, \hspace{1cM} & \maThRm{for} \ i<j<k, \\
\left[ b_{Ij}... | theory of Kohno connectio ns [@rif65 ; @ri f67 ].If a c onne ction $\mathca l {D}= d+A$ of the form \[ber kohno \] sati s fi es $\ mathcal { D} ^ { 2}= 0$ ,the nt he foll owi ng rela tions hold
$ $\ begin{split} \l eft[ B_{ ij} ,B_{ik}+B_{j k}\ right] = \l e ft[ B _{i j}+B_ {ik},B _ {jk}\r ight] =0, \ h space... | theory_of Kohno_connections [@rif65; @rif67]. If_a connection_$\mathcal{D}=d+A$_of the_form_\[berkohno\] satisfies $\mathcal{D}^{2}=0$,_then the following_relations hold
$$\begin{split}
\left[ _B_{ij},B_{ik}+B_{jk}\right] = \left[_B_{ij}+B_{ik},B_{jk}\right]_=0, \hspace{1cm} & \mathrm{for} \ i<j<k, \\
\left[ B_{ij}... |
$\tilde{\alpha}$ is $\tilde{\alpha}= \alpha_{+} \left| + \right\rangle \left\langle + \right| \ \ + \ \ \alpha_{-} \left| - \right\rangle \left\langle - \right| \ \ + \ \ \alpha_{z} \left| z \right\rangle \left\langle z \right|$. Representing the damping tensor in the $\left\{ \left| x\right\rangle,\: \l... | $ \tilde{\alpha}$ is $ \tilde{\alpha}= \alpha_{+ } \left| + \right\rangle \left\langle + \right| \ \ + \ \ \alpha_{- } \left| - \right\rangle \left\langle - \right| \ \ + \ \ \alpha_{z } \left| z \right\rangle \left\langle z \right|$. Representing the damping tensor in the $ \left\ { \left| x\... | $\tipde{\alpha}$ is $\tilde{\alpha}= \aupha_{+} \left| + \rigkr\ranglx \left\lzngle + \rkght| \ \ + \ \ \alpha_{-} \left| - \cighr\rangoe \left\langle - \right| \ \ + \ \ \wlpha_{z} \lwft| z \right\ranjme \left\langle d \rigkt|$. Representing jhe damping densor in the $\newt\{ \left| x\right\rangle,\: \l... | $\tilde{\alpha}$ is $\tilde{\alpha}= \alpha_{+} \left| + \right\rangle \right| \ + \ \alpha_{-} \left| \ + \ \ \left| z \right\rangle z \right|$. Representing the damping tensor the $\left\{ \left| x\right\rangle,\: \left| y\right\rangle,\: \left| z\right\rangle \right\} $ coordinate basis yields \tilde{\alpha}_{xx} = ... | $\tilde{\alpha}$ is $\tilde{\alpha}= \alpHa_{+} \left| + \righT\rangLe \lEft\LaNgle + \RighT| \ \ + \ \ \alpha_{-} \left| - \righT\RangLe \left\langle - \right| \ \ + \ \ \alpha_{Z} \left| Z \rIGht\rANgLe \lefT\langle Z \RiGHT|$. RePrEsEntInG ThE dampIng Tensor iN the $\left\{ \leFt| x\RiGht\rangle,\: \l... | $\tilde{\alpha}$ is $\til de{\alpha} = \al pha _{+ }\lef t| + \right\rangl e \le ft\langle + \right| \ \ +\ \\a lpha_ {-} \le f t| - \ ri gh t\r an g le \lef t\l angle - \right| \ \ + \ \ \al p ha _{z} \left | z \right\ran gle \left \l ang l e z \ rig ht|$. Repre s enting the damp in g tenso r in the $ ... | $\tilde{\alpha}$_is $\tilde{\alpha}=_\alpha_{+} \left| +_\right\rangle \left\langle_+_\right| \__ \ _+ \ _\ \alpha_{-} \left|_ - \right\rangle_\left\langle_- \right| \ \ + \ \ \alpha_{z} \left|__z \right\rangle_\left\langle_z_\right|$. Representing the damping tensor_in the $\left\{ \left| _x\right\rangle,\: \l... |
dust particles. Formation of stardust occurs in circumstellar environments where temperatures are cool enough [e.g. @Cherchneff:2017 for a recent review of the open issues]. On their journey through the interstellar medium, heating and partial or complete destruction may occur from starlight or even shocks from supern... | dust particles. Formation of stardust occurs in circumstellar environment where temperature are cool enough [ e.g. @Cherchneff:2017 for a recent review of the assailable issues ]. On their journey through the interstellar medium, heating system and partial or complete destruction may happen from starlight or even shock... | dudt particles. Formation on stardust occurs in ciccumstemlar envkronments where temperatures aee coil enough [e.g. @Cherchnefw:2017 for a rvcent revuew id the open issues]. On thejv jouxnxy through the lnterstellas medium, heatitg aud partial or complete destruction mwy occut vrom starlight or tveg shkbkw from supern... | dust particles. Formation of stardust occurs in where are cool [e.g. @Cherchneff:2017 for open On their journey the interstellar medium, and partial or complete destruction may from starlight or even shocks from supernovae [@Zhukovska:2016]. Also a variety chemical and reactions may reprocess dust grains [@Dauphas:2016... | dust particles. Formation of sTardust occUrs in CirCumStEllaR envIronments where TEmpeRatures are cool enough [e.g. @chercHnEFf:2017 foR A rEcent Review oF ThE OPen IsSuEs]. ON tHEiR jourNey Through The interstEllAr Medium, heatinG AnD partial or ComPlete destrucTioN may ocCuR frOM starLigHt or eVen shoCKs from Supern... | dust particles. Formation of stardu st oc cur s i ncirc umst ellar environm e ntswhere temperatures are cool e n ough [e .g. @ Cherchn e ff : 2 017 f or are c en t rev iew of the open issu es] .On their jou r ne y throughthe interstella r m edium, h eat i ng an d p artia l or c o mplete destruct io n may o ... | dust_particles. Formation_of stardust occurs in_circumstellar environments_where_temperatures are_cool_enough [e.g. @Cherchneff:2017_for a recent_review of the open_issues]. On their_journey_through the interstellar medium, heating and partial or complete destruction may occur from starlight_or_even shocks_from_supern... |
paulus:2017:arxiv], there is still no guarantee that the generated summaries are grammatical and convey the same meaning as the original document does. It seems that extractive models are more reliable than their abstractive counterparts.
However, extractive models require sentence level labels, which are usually not ... | paulus:2017: arxiv ], there is still no guarantee that the generated summaries are grammatical and bring the like meaning as the original document does. It seem that extractive model are more reliable than their abstractive counterparts.
However, extractive exemplar require sentence degree labels, which are usually ... | paupus:2017:arxiv], there is still ko guarantee thaj rhe geierated summarids are grammatical and conveb thw samt meaning as the orkginal dobument dows. Iu seems that extredtive models zve moxe reliable than their absdractive countaroaxts.
However, extractive models require sentenve level labels, rhicn are lsmally not ... | paulus:2017:arxiv], there is still no guarantee that summaries grammatical and the same meaning It that extractive models more reliable than abstractive counterparts. However, extractive models require level labels, which are usually not included in most summarization datasets (most datasets contain document-summary pa... | paulus:2017:arxiv], there is still no Guarantee tHat thE geNerAtEd suMmarIes are grammatiCAl anD convey the same meaning aS the oRiGInal DOcUment Does. It sEEmS THat ExTrActIvE MoDels aRe mOre reliAble than thEir AbStractive couNTeRparts.
HoweVer, Extractive moDelS requiRe SenTEnce lEveL labeLs, whicH Are usuAlly not ... | paulus:2017:arxiv], thereis still n o gua ran tee t hatthegenerated summ a ries are grammatical and c onvey t h e sa m emeani ng as t h eo r igi na ldoc um e nt does . I t seems that extr act iv e models are mo re reliabl e t han their ab str active c oun t erpar ts.
How ever,e xtract ive model sr equire senten... | paulus:2017:arxiv], there_is still_no guarantee that the_generated summaries_are_grammatical and_convey_the same meaning_as the original_document does. It seems_that extractive models_are_more reliable than their abstractive counterparts.
However, extractive models require sentence level labels, which are_usually_not ... |
, the outbursts in such systems. Fortunately, this is not bad for our purposes since we are trying to derive an upper limit for the real probability of detecting the CVs. In this way, having a disrupted disc (cold and stable) would lower the probability even further.
Once we know that the CVs are DNe, we are able to c... | , the outbursts in such systems. Fortunately, this is not bad for our purposes since we are judge to deduce an upper limit for the real probability of detect the CVs. In this room, having a disrupted phonograph record (cold and stable) would lower the probability even farther.
Once we know that the CVs are DNe, we a... | , thf outbursts in such systtms. Fortunately, tkus is iot bad for our purposes since we are tryinj to derice an upper limit for ghe real irobabiliry oh detecting the RBs. In tmns wag, havnnj a disrupted dlsc (cold ang stable) would luwzr the probability even further.
Once re know tjat the CVs arg DNe, re adv cble to c... | , the outbursts in such systems. Fortunately, not for our since we are limit the real probability detecting the CVs. this way, having a disrupted disc and stable) would lower the probability even further. Once we know that the are DNe, we are able to compute the recurrence time and the duration the based equations and ... | , the outbursts in such systems. fortunatelY, this Is nOt bAd For oUr puRposes since we aRE tryIng to derive an upper limiT for tHe REal pRObAbiliTy of detECtING thE Cvs. in tHiS WaY, haviNg a DisruptEd disc (cold And StAble) would lowER tHe probabilIty Even further.
ONce We know ThAt tHE CVs aRe Dne, we aRe able TO c... | , the outbursts in such sy stems. For tunat ely , t hi s is not bad for our p u rpos es since we are trying to d er i ve a n u pperlimit f o rt h e r ea lpro ba b il ity o f d etectin g the CVs. In t his way, hav i ng a disrupt eddisc (cold a ndstable )wou l d low erthe p robabi l ity ev en furthe r.
Oncew e kn... | , the_outbursts in_such systems. Fortunately, this_is not_bad_for our_purposes_since we are_trying to derive_an upper limit for_the real probability_of_detecting the CVs. In this way, having a disrupted disc (cold and stable) would_lower_the probability_even_further.
Once_we know that the CVs_are DNe, we are able_to c... |
10 35 57.60 & +54 10 28.5 & 17.5 & DA & 09.01.92 & 251 & II & 1036+550 & 10 36 22.15 & +55 05 53.3 & 17.5 & DA & 09.01.92 & 252 & II & 1040+493 & 10 40 11.87 & +49 18 07.1 & 16.5 & sd & 29.02.92 & 277 & II & 1040+520 & 10 40 48.77 & +52 00 54.1 & 17.5 & DA & 09.01.92 & 217 & II & 1043+569 & 10 43 48.09 & +56 59 47.2 &... | 10 35 57.60 & +54 10 28.5 & 17.5 & DA & 09.01.92 & 251 & II & 1036 + 550 & 10 36 22.15 & +55 05 53.3 & 17.5 & DA & 09.01.92 & 252 & II & 1040 + 493 & 10 40 11.87 & +49 18 07.1 & 16.5 & sd & 29.02.92 & 277 & II & 1040 + 520 & 10 40 48.77 & +52 00 54.1 & 17.5 & DA & 09.01.92 & 217 & II & 1043 + 569 & 10 43 48.09 & +56 59... | 10 35 57.60 & +54 10 28.5 & 17.5 & DA & 09.01.92 & 251 & II & 1036+550 & 10 36 22.15 & +55 05 53.3 & 17.5 & DA & 09.01.92 & 252 & II & 1040+493 & 10 40 11.87 & +49 18 07.1 & 16.5 & sd & 29.02.92 & 277 & II & 1040+520 & 10 40 48.77 & +52 00 54.1 & 17.5 & DA & 09.01.92 & 217 & MI & 1043+569 & 10 43 48.09 & +56 59 47.2 &... | 10 35 57.60 & +54 10 28.5 & & 09.01.92 251 & II 22.15 +55 05 53.3 17.5 & DA 09.01.92 & 252 & II & & 10 40 11.87 & +49 18 07.1 & 16.5 & sd & & 277 & II & 1040+520 & 10 40 48.77 & +52 00 & & & & 217 & II & 1043+569 & 10 43 48.09 & +56 59 47.2 & 17.5 F & 04.02.92 & 660 & II & & 10 45 15.66 +57 03 29.4 & 17.5 DA 09.01.92 &... | 10 35 57.60 & +54 10 28.5 & 17.5 & DA & 09.01.92 & 251 & II & 1036+550 & 10 36 22.15 & +55 05 53.3 & 17.5 & DA & 09.01.92 & 252 & II & 1040+493 & 10 40 11.87 & +49 18 07.1 & 16.5 & sd & 29.02.92 & 277 & II & 1040+520 & 10 40 48.77 & +52 00 54.1 & 17.5 & DA & 09.01.92 & 217 & II & 1043+569 & 10 43 48.09 & +56 59 47.2 &... | 10 35 57.60 & +54 10 28.5 & 17.5 &DA &09. 01. 92 & 2 51 & II & 1036+550 & 10 36 22.15 & +55 05 53. 3 & 1 7. 5 & D A & 09.0 1.92 &2 52 & II & 1 040 +4 9 3& 104011.87 & +49 18 07 .1&16.5 & sd &2 9. 02.92 & 27 7 & II & 1040+5 20& 10 4 048. 7 7 & + 5200 54 .1 & 1 7 .5 & D A & 09.01 .9 2 & 217 & II &1 0 43 +56... | 10_35 57.60_& +54 10 28.5_& 17.5_&_DA &_09.01.92_& 251 &_II & 1036+550_& 10 36 22.15_& +55 05_53.3_& 17.5 & DA & 09.01.92 & 252 & II & 1040+493 & 10_40_11.87 &_+49_18_07.1 & 16.5 & sd_& 29.02.92 & 277 &_II &_1040+520 & 10 40 48.77 & +52 00_54.1_& 17.5 &_DA & 09.01.92 & 217 & II & 1043+569_& 10 43 48.09 & +56_59 47.2 &... |
a root, if $\mathfrak g_\alpha\neq\{0\}$; the set of all roots is denoted by $\Delta$. Moreover, $\mathfrak g_0$ is $\sigma$-stable and inherits the structure of symmetric pair, whence $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ its Cartan decomposition. Finally, for a root $\alpha$ in $\Delta$, $\sigma(\mathfrak... | a root, if $ \mathfrak g_\alpha\neq\{0\}$; the set of all roots is denoted by $ \Delta$. furthermore, $ \mathfrak g_0 $ is $ \sigma$-stable and inherit the structure of symmetric couple, whence $ \mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0 $ its Cartan decomposition. last, for a root $ \alpha$ in $ \Delta$, $ \sigma... | a goot, if $\mathfrak g_\alpha\ntq\{0\}$; the set of all roots ms denofed by $\Ddlta$. Moreover, $\mathfrak g_0$ is $\ditma$-stqble and inherits the rtructure of symmwtrir pair, whence $\mavgfrak g_0=\mathfrzn k_0\o'lns\mathfrak p_0$ itx Cartan dacomposition. Fhnxlpy, for a root $\alpha$ in $\Delta$, $\sigma(\mwthfrak... | a root, if $\mathfrak g_\alpha\neq\{0\}$; the set roots denoted by Moreover, $\mathfrak g_0$ structure symmetric pair, whence g_0=\mathfrak k_0\oplus\mathfrak p_0$ Cartan decomposition. Finally, for a root in $\Delta$, $\sigma(\mathfrak g_\alpha)=\mathfrak g_{-\alpha}$, whence $\mathfrak g$ admits a triangular decompos... | a root, if $\mathfrak g_\alpha\neq\{0\}$; tHe set of all Roots Is dEnoTeD by $\DElta$. moreover, $\mathfrAK g_0$ is $\Sigma$-stable and inherits The stRuCTure OF sYmmetRic pair, WHeNCE $\maThFrAk g_0=\MaTHfRak k_0\oPluS\mathfrAk p_0$ its CartAn dEcOmposition. FiNAlLy, for a root $\AlpHa$ in $\Delta$, $\sigMa(\mAthfraK... | a root, if $\mathfrak g_\ alpha\neq\ {0\}$ ; t hese t of all roots is deno t ed b y $\Delta$. Moreover,$\mat hf r ak g _ 0$ is $ \sigma$ - st a b lean dinh er i ts thestr uctureof symmetr icpa ir, whence $ \ ma thfrak g_0 =\m athfrak k_0\ opl us\mat hf rak p_0$its Cart an dec o mposit ion. Fina ll y , fora root ... | a_root, if_$\mathfrak g_\alpha\neq\{0\}$; the set_of all_roots_is denoted_by_$\Delta$. Moreover, $\mathfrak_g_0$ is $\sigma$-stable_and inherits the structure_of symmetric pair,_whence_$\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ its Cartan decomposition. Finally, for a root $\alpha$ in $\Delta$,_$\sigma(\mathfrak... |
(0.5, 0.8) [ $f$]{}; at (2/5, -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [$x$]{}; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [$t_4$]{}; (1,-0.02)–(1,0.02); at (1, -0.07) [$e$]{}; (3/5,3/4)–(4/5,1/2);
(0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0... | (0.5, 0.8) [ $ f$ ] { }; at (2/5, -0.07) [ $ a$ ] { }; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [ $ t_3 $ ] { }; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [ $ x$ ] { }; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [ $ t_4 $ ] { }; (1,-0.02)–(1,0.02); at (1, -0.07) [ $ e$ ] { }; (3/5,3/4)–(4/5,1/2);
(0,0)–(1,0)–(1,1)–(0,1... | (0.5, 0.8) [ $f$]{}; at (2/5, -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [$w$]{}; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [$t_4$]{}; (1,-0.02)–(1,0.02); at (1, -0.07) [$g$]{}; (3/5,3/4)–(4/5,1/2);
(0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0... | (0.5, 0.8) [ $f$]{}; at (2/5, -0.07) at -0.07) [$t_3$]{}; at (7/10, -0.07) [$t_4$]{}; at (1, -0.07) (3/5,3/4)–(4/5,1/2); (0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0.5)–(0.6,0.75)–(0.8,0.25)–(1,1); at (0.5, 0.8) [ $g$]{}; at -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10... | (0.5, 0.8) [ $f$]{}; at (2/5, -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [$x$]{}; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [$t_4$]{}; (1,-0.02)–(1,0.02); at (1, -0.07) [$e$]{}; (3/5,3/4)–(4/5,1/2);
(0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0... | (0.5, 0.8) [ $f$]{}; at ( 2/5, -0.07 ) [$a $]{ };(3 /5,- 0.02 )–(3/5,0.02);a t (3 /5, -0.07) [$t_3$]{};(7/10 ,- 0 .02) – (7 /10,0 .02); a t ( 7 / 10, - 0. 07) [ $ x$ ]{};(4/ 5,-0.02 )–(4/5,0.0 2); a t (4/5, -0.0 7 )[$t_4$]{}; (1 ,-0.02)–(1,0 .02 ); at(1 , - 0 .07)[$e $]{}; (3/5, 3 /4)–(4 /5,1/2);
( 0 ,0)–(1 , 0)... | (0.5,_0.8) [_$f$]{}; at (2/5, -0.07)_[$a$]{}; (3/5,-0.02)–(3/5,0.02);_at_(3/5, -0.07)_[$t_3$]{};_(7/10,-0.02)–(7/10,0.02); at (7/10,_-0.07) [$x$]{}; (4/5,-0.02)–(4/5,0.02);_at (4/5, -0.07) [$t_4$]{};_(1,-0.02)–(1,0.02); at (1,_-0.07)_[$e$]{}; (3/5,3/4)–(4/5,1/2);
(0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0... |
end{array}
\right] \,,$$ where $\mathbf{P}^{\rm CLASS}(N)$ is the solution in Eq. . Even though the second term on the rhs of Eq. still is a rapidly oscillating function, it cannot be discarded as simply as others due to the presence of $h^\prime$.
The last integral in Eq. can be integrated iteratively by parts, and... | end{array }
\right ] \,,$$ where $ \mathbf{P}^{\rm CLASS}(N)$ is the solution in Eq. . Even though the second term on the rhs of Eq. still is a rapidly hover affair, it cannot be discarded as simply as others due to the bearing of $ h^\prime$.
The last integral in Eq. can be integrated iteratively by parts,... | end{wrray}
\right] \,,$$ where $\mathbf{K}^{\rm CLASS}(N)$ is the solutimn in Sq. . Even ghough the second term on thx rhw of Tz. still is a rapiduy oscillwting fubctiib, it cannov be disgcrded ws snm'ly as others doe to the prasence of $h^\prike$.
Ghz last integral in Eq. can be integraeed itetahively by partf, anc... | end{array} \right] \,,$$ where $\mathbf{P}^{\rm CLASS}(N)$ is in . Even the second term still a rapidly oscillating it cannot be as simply as others due to presence of $h^\prime$. The last integral in Eq. can be integrated iteratively by and in such case it can be proved that if we retain the term the we $$\label{eq:79... | end{array}
\right] \,,$$ where $\mathbf{P}^{\Rm CLASS}(N)$ is The soLutIon In eq. . EvEn thOugh the second tERm on The rhs of Eq. still is a rapiDly osCiLLatiNG fUnctiOn, it canNOt BE DisCaRdEd aS sIMpLy as oTheRs due to The presencE of $H^\pRime$.
The last iNTeGral in Eq. caN be Integrated itEraTively By ParTS, and... | end{array}
\right] \,,$$ w here $\mat hbf{P }^{ \rm C LASS }(N) $ is the solut i on i n Eq. . Even though th e sec on d ter m o n the rhs of Eq . st il lisar ap idlyosc illatin g function , i tcannot be di s ca rded as si mpl y as othersdue to th epre s enceof$h^\p rime$.
The l ast integ ra l in Eq . can b e ... | end{array}
\right] \,,$$_where $\mathbf{P}^{\rm_CLASS}(N)$ is the solution_in Eq. ._Even_though the_second_term on the_rhs of Eq. _still is a rapidly_oscillating function, it_cannot_be discarded as simply as others due to the presence of $h^\prime$.
The last integral_in_Eq. can_be_integrated_iteratively by parts, and... |
5,-2.5) to (0.5,1.5);
(3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5);
(6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5... | 5,-2.5) to (0.5,1.5);
(3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5);
(6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to ... | 5,-2.5) to (0.5,1.5);
(3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5);
(6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) vo (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5... | 5,-2.5) to (0.5,1.5); (3.5,1.5) to (3.5,0.5); (8.5,1.5) (11.5,1.5) (11.5,0.5); (5.5,0.5) (5.5,-0.5); (1.5,-3.5) to to (10.5,1.5) to (10.5,0.5); to (7.5,0.5); (7.5,0.5) (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5,-1.5) to (5.5-1,-1.5); (5.5-1,-1.5) to (5.5-1,-1.5-1); (5.5-1,-1.5-1) to (5.5-1-2,-1.5-1); (5.5... | 5,-2.5) to (0.5,1.5);
(3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5);
(6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) tO (5.5... | 5,-2.5) to (0.5,1.5);
(3. 5,1.5) to(3.5, 0.5 );(8 .5,1 .5)to (8.5,0.5);( 11.5 ,1.5) to (11.5,0.5); ( 5.5,0 .5 ) to( 5. 5,-0. 5); (1. 5 ,- 3 . 5)to ( 1.5 ,- 4 .5 ); (7 .5, -0.5) t o (7.5,-1. 5);
(6.5,1.5) to (1 0.5,1.5);(10 .5,1.5) to ( 10. 5,0.5) ;(10 . 5,0.5 ) t o (7. 5,0.5) ; (7.5, 0.5) to ( 7. 5 ,-0.5) ; (7.5,... | 5,-2.5) to_(0.5,1.5);
(3.5,1.5) to_(3.5,0.5); (8.5,1.5) to (8.5,0.5);_(11.5,1.5) to_(11.5,0.5);_(5.5,0.5) to_(5.5,-0.5);_(1.5,-3.5) to (1.5,-4.5);_(7.5,-0.5) to (7.5,-1.5);
(6.5,1.5)_to (10.5,1.5); (10.5,1.5) to_(10.5,0.5); (10.5,0.5) to_(7.5,0.5);_(7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5... |
immersive experience in VR. We prove DROP is NP-hard and design a fully polynomial-time approximation scheme, *Dual Entangled World Navigation* (DEWN), by finding Minimum Immersion Loss Range (*MIL Range*). Afterward, we show that the existing spatial query algorithms and index structures can leverage DEWN as a buildi... | immersive experience in VR. We prove DROP is NP - hard and design a amply polynomial - meter approximation scheme, * Dual Entangled World Navigation * (DEWN), by finding Minimum Immersion Loss Range (* MIL Range *). subsequently, we show that the existing spatial query algorithm and exponent structures can leverage DEW... | imlersive experience in VR. We prove DROP nw NP-hacd and sesign a fully polynomial-time approxmmatuon sxheme, *Dual Entangled Wurld Navihation* (DWWN), uy finding Minimnj Immersion Lkds Rcnje (*MIL Range*). Afjerward, we svow that the efirtnng spatial query algorithms and indqx struvtkres can leverwge CQWN zs a buildi... | immersive experience in VR. We prove DROP and a fully approximation scheme, *Dual finding Immersion Loss Range Range*). Afterward, we that the existing spatial query algorithms index structures can leverage DEWN as a building block to support $k$NN and queries in the dual worlds of VR. Experimental results and a user s... | immersive experience in VR. We Prove DROP iS NP-haRd aNd dEsIgn a FullY polynomial-timE ApprOximation scheme, *Dual EntAngleD WORld NAViGatioN* (DEWN), by FInDINg MInImUm IMmERsIon LoSs RAnge (*MIL range*). AfterWarD, wE show that the EXiSting spatiAl qUery algorithMs aNd indeX sTruCTures Can LeverAge DEWn As a buiLdi... | immersive experience in V R. We prov e DRO P i s N P- hard and design a full y pol ynomial-time approxima tionsc h eme, *D ual E ntangle d W o r ldNa vi gat io n *(DEWN ),by find ing Minimu m I mm ersion LossR an ge (*MIL R ang e*). Afterwa rd, we sh ow th a t the ex istin g spat i al que ry algori th m s andi ... | immersive_experience in_VR. We prove DROP_is NP-hard_and_design a_fully_polynomial-time approximation scheme,_*Dual Entangled World_Navigation* (DEWN), by finding_Minimum Immersion Loss_Range_(*MIL Range*). Afterward, we show that the existing spatial query algorithms and index structures_can_leverage DEWN_as_a_buildi... |
mathbb{R}/\mathcal{D})\simeq Hom(\pi_1(G)\to\mathbb{R}/\mathcal{D}).$$ Now assume $G=SL(2,\mathbb{R})$ and $\mathcal{D}=\mathbb{Z}$ so that $\mathbb{R}/\mathcal{D}\simeq U(1)$. The use of Iwazawa decomposition yields $SL(2,\mathbb{R})\simeq \mathbb{R}^2\times \mathbb{S}^1$, which combined with $\pi_1(X\times Y)=\pi_1(X... | mathbb{R}/\mathcal{D})\simeq Hom(\pi_1(G)\to\mathbb{R}/\mathcal{D}).$$ Now assume $ G = SL(2,\mathbb{R})$ and $ \mathcal{D}=\mathbb{Z}$ so that $ \mathbb{R}/\mathcal{D}\simeq U(1)$. The use of Iwazawa decay give $ SL(2,\mathbb{R})\simeq \mathbb{R}^2\times \mathbb{S}^1 $, which combined with $ \pi_1(X\times Y)=\pi_1(X)\... | matjbb{R}/\mathcal{D})\simeq Hom(\pi_1(G)\uo\mathbb{R}/\mathcal{D}).$$ Now asvume $G=AL(2,\mathbb{F})$ and $\mathcal{D}=\mathbb{Z}$ so thav $\marhbb{R}/\nathcal{D}\simeq U(1)$. The usd of Iwazwwa deconposmtion yields $SL(2,\mefhbb{R})\simeq \mafmbb{R}^2\tnmxs \mathbb{S}^1$, whicm combined fith $\pi_1(X\times F)=\pk_1(X... | mathbb{R}/\mathcal{D})\simeq Hom(\pi_1(G)\to\mathbb{R}/\mathcal{D}).$$ Now assume $G=SL(2,\mathbb{R})$ and $\mathcal{D}=\mathbb{Z}$ $\mathbb{R}/\mathcal{D}\simeq The use Iwazawa decomposition yields with Y)=\pi_1(X)\times\pi_1(Y)$ for any spaces $X$ and implies $\pi_1(SL(2,\mathbb{R}))\simeq\mathbb{Z}$. Hence, $Hom(\ma... | mathbb{R}/\mathcal{D})\simeq Hom(\pi_1(g)\to\mathbb{R}/\MathcAl{D}).$$ now AsSume $g=SL(2,\mAthbb{R})$ and $\mathcAL{D}=\maThbb{Z}$ so that $\mathbb{R}/\mathCal{D}\sImEQ U(1)$. ThE UsE of IwAzawa deCOmPOSitIoN yIelDs $sl(2,\mAthbb{r})\siMeq \mathBb{R}^2\times \maThbB{S}^1$, Which combineD WiTh $\pi_1(X\times y)=\pi_1(x... | mathbb{R}/\mathcal{D})\sim eq Hom(\pi _1(G) \to \ma th bb{R }/\m athcal{D}).$$N ow a ssume $G=SL(2,\mathbb{ R})$an d $\m a th cal{D }=\math b b{ Z } $ s oth at$\ m at hbb{R }/\ mathcal {D}\simeqU(1 )$ . The use of Iw azawa deco mpo sition yield s $ SL(2,\ ma thb b {R})\ sim eq \m athbb{ R }^2\ti mes \math bb { S}^1$, ... | mathbb{R}/\mathcal{D})\simeq Hom(\pi_1(G)\to\mathbb{R}/\mathcal{D}).$$_Now assume_$G=SL(2,\mathbb{R})$ and $\mathcal{D}=\mathbb{Z}$ so_that $\mathbb{R}/\mathcal{D}\simeq_U(1)$._The use_of_Iwazawa decomposition yields_$SL(2,\mathbb{R})\simeq \mathbb{R}^2\times \mathbb{S}^1$,_which combined with $\pi_1(X\times_Y)=\pi_1(X... |
$\mathbb{S}(\mathfrak{g}^*)$ around $\pi_{\mathbb{S}}$ by the space $$\mathfrak{Casim}(\mathbb{S}(\mathfrak{g}^*),\pi_{\mathbb{S}})/Out(\mathfrak{g}).$$ In this section we describe this space using classical invariant theory.
Let $P[\mathfrak{g}^*]$ and $P[\mathfrak{t}^*]$ denote the algebras of polynomials on $\math... | $ \mathbb{S}(\mathfrak{g}^*)$ around $ \pi_{\mathbb{S}}$ by the space $ $ \mathfrak{Casim}(\mathbb{S}(\mathfrak{g}^*),\pi_{\mathbb{S}})/Out(\mathfrak{g}).$$ In this section we describe this quad use classical invariant theory.
get $ P[\mathfrak{g}^*]$ and $ P[\mathfrak{t}^*]$ denote the algebras of polynomial on $ \... | $\mahhbb{S}(\mathfrak{g}^*)$ around $\pi_{\oathbb{S}}$ by the space $$\methfrak{Dasim}(\matfbb{S}(\mathfrak{g}^*),\pi_{\mathbb{S}})/Out(\matifraj{g}).$$ In this section we descrkbe this dpace usung rlassical invarieht theovv.
Let $L[\lathyrek{g}^*]$ and $P[\mathfrsk{t}^*]$ denote the algebras mf pllynomials on $\math... | $\mathbb{S}(\mathfrak{g}^*)$ around $\pi_{\mathbb{S}}$ by the space $$\mathfrak{Casim}(\mathbb{S}(\mathfrak{g}^*),\pi_{\mathbb{S}})/Out(\mathfrak{g}).$$ section describe this using classical invariant denote algebras of polynomials $\mathfrak{g}^*$ and $\mathfrak{t}^*$ A classical result (see e.g. Theorem [@Dixmier]) s... | $\mathbb{S}(\mathfrak{g}^*)$ around $\pi_{\mAthbb{S}}$ by thE spacE $$\maThfRaK{CasIm}(\maThbb{S}(\mathfrak{g}^*),\PI_{\matHbb{S}})/Out(\mathfrak{g}).$$ In this SectiOn WE desCRiBe thiS space uSInG CLasSiCaL inVaRIaNt theOry.
let $P[\matHfrak{g}^*]$ and $P[\MatHfRak{t}^*]$ denote thE AlGebras of poLynOmials on $\math... | $\mathbb{S}(\mathfrak{g}^ *)$ around $\pi _{\ mat hb b{S} }$ b y the space $$ \ math frak{Casim}(\mathbb{S} (\mat hf r ak{g } ^* ),\pi _{\math b b{ S } })/ Ou t( \ma th f ra k{g}) .$$ In thi s sectionwede scribe thiss pa ce using c las sical invari ant theor y.
L e t $P[ \ma thfra k{g}^* ] $ and$P[\mathf ra k {t}... | $\mathbb{S}(\mathfrak{g}^*)$_around $\pi_{\mathbb{S}}$_by the space $$\mathfrak{Casim}(\mathbb{S}(\mathfrak{g}^*),\pi_{\mathbb{S}})/Out(\mathfrak{g}).$$_In this_section_we describe_this_space using classical_invariant theory.
Let $P[\mathfrak{g}^*]$_and $P[\mathfrak{t}^*]$ denote the_algebras of polynomials_on_$\math... |
D topological orders), the quasiparticles can only fuse but not braiding. So, the fusion rules $N_{\xi\zeta}^\chi$ and the F-matrices are enough to describe 1+1D anomalous topological orders. Later, we will see fusion rules and F-matrices are also used to determine a string-net wavefunction, which may seem confusing. H... | D topological orders), the quasiparticles can only blend but not braid. So, the fusion predominate $ N_{\xi\zeta}^\chi$ and the F - matrix are enough to describe 1 + 1D anomalous topological order. by and by, we will see coalition rule and F - matrices are besides used to determine a string - net wavefunction, which ma... | D tlpological orders), the quxsiparticles cau only huse buf not brxiding. So, the fusion rules $N_{\ei\zera}^\chi$ and the F-matrices are enough tl descrive 1+1D qnomalous vkpologigcl orscrs. Lctxr, we will see nusion rulev and F-matricev xrz also used to determine a string-net wavefumchion, which may seek conrlslng. H... | D topological orders), the quasiparticles can only not So, the rules $N_{\xi\zeta}^\chi$ and describe anomalous topological orders. we will see rules and F-matrices are also used determine a string-net wavefunction, which may seem confusing. However, as we have mentioned, is a natural result of the holographic bulk-edg... | D topological orders), the quasIparticles Can onLy fUse BuT not BraiDing. So, the fusioN RuleS $N_{\xi\zeta}^\chi$ and the F-matrIces aRe ENougH To DescrIbe 1+1D anoMAlOUS toPoLoGicAl ORdErs. LaTer, We will sEe fusion ruLes AnD F-matrices arE AlSo used to deTerMine a string-nEt wAvefunCtIon, WHich mAy sEem coNfusinG. h... | D topological orders), the quasipart icles ca n o nl y fu se b ut not braidin g . So , the fusion rules $N_ {\xi\ ze t a}^\ c hi $ and the F- m at r i ces a re en ou g hto de scr ibe 1+1 D anomalou s t op ological ord e rs . Later, w e w ill see fusi onrulesan d F - matri ces arealso u s ed todetermine a string ... | D topological_orders), the_quasiparticles can only fuse_but not_braiding._So, the_fusion_rules $N_{\xi\zeta}^\chi$ and_the F-matrices are_enough to describe 1+1D_anomalous topological orders._Later,_we will see fusion rules and F-matrices are also used to determine a string-net_wavefunction,_which may_seem_confusing._H... |
\exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{}
In [@Proc:Li_SODA08], it was proved that, as $k\rightarrow\infty$, [$$\begin{aligned}
\notag
&\frac{\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right)
\Gamma\left(1-\frac{... | \exp\left (\frac{1}{k}\frac{\pi^2(t^2 - t)}{24}\left(\alpha^2 + 2 - 3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$ ] { }
In [ @Proc: Li_SODA08 ], it was proved that, as $ k\rightarrow\infty$, [ $ $ \begin{aligned }
\notag
& \frac{\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\rig... | \exp\peft( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alkha^2+2-3\kappa^2(\alpha)\righj)+O\oeft(\frec{1}{k^2}\righf)\right).\ena{aligned}$$]{}
In [@Proc:Li_SODA08], it was peoved that, as $k\rightarrow\inwty$, [$$\begin{wligned}
\nitag
&\hrac{\left[\frac{2}{\pi}\sii\meft(\frag{\'i\alpgw}{2k}t\rngit)
\Gamma\left(1-\frac{... | \exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{} In [@Proc:Li_SODA08], it was proved $k\rightarrow\infty$, \notag &\frac{\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right) { \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2... | \exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\aLpha^2+2-3\kappa^2(\aLpha)\rIghT)+O\lEfT(\fraC{1}{k^2}\riGht)\right).\end{aliGNed}$$]{}
IN [@Proc:Li_SODA08], it was proved That, aS $k\RIghtARrOw\infTy$, [$$\begin{ALiGNEd}
\nOtAg
&\FraC{\lEFt[\Frac{2}{\pI}\siN\left(\frAc{\pi\alpha}{2k}T\riGhT)
\Gamma\left(1-\frAC{... | \exp\left( \frac{1}{k}\fra c{\pi^2(t^ 2-t)} {24 }\l ef t(\a lpha ^2+2-3\kappa^2 ( \alp ha)\right)+O\left(\fra c{1}{ k^ 2 }\ri g ht )\rig ht).\en d {a l i gne d} $$ ]{}
I n[@Pro c:L i_SODA0 8], it was pr ov ed that, as$ k\ rightarrow \in fty$, [$$\be gin {align ed }
\ n otag&\f rac{\ left[\ f rac{2} {\pi}\sin \l e ft(... | \exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{}
In_[@Proc:Li_SODA08], it_was proved that, as_$k\rightarrow\infty$, [$$\begin{aligned}
\notag
&\frac{\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right)
\Gamma\left(1-\frac{... |
columnwidth"}![Estimation performance of input APs of the PÉGASE models. For each of the APs we plot the predicted vs. true AP values for the test set. The blue line indicates the line of perfect estimation, while the red line represents the best linear fitting for the points. The summary errors are given in Table 7.[]... | columnwidth"}![Estimation performance of input APs of the PÉGASE models. For each of the APs we plot the predict vs. genuine AP values for the test set. The blasphemous line indicates the tune of arrant estimation, while the red line represents the best linear meet for the points. The summary errors are given in Tabl... | colkmnwidth"}![Estimation perfovmance of input CPs of vhe PÉGAAE modelr. For each of the APs we plov thw preeicted vs. true AP valuer for the test ser. Tht blue line indicefes the line kn peryert estimation, wmile the reg line represettr che best linear fitting for the poines. The xulmary errors ate gineg in Nanle 7.[]... | columnwidth"}![Estimation performance of input APs of the For of the we plot the for test set. The line indicates the of perfect estimation, while the red represents the best linear fitting for the points. The summary errors are given Table 7.[]{data-label="f02"}](fig70.ps "fig:"){width="0.3\columnwidth"}![Estimation p... | columnwidth"}![Estimation perfOrmance of iNput Aps oF thE PéGASe modEls. For each of thE aPs wE plot the predicted vs. truE AP vaLuES for THe Test sEt. The blUE lINE inDiCaTes ThE LiNe of pErfEct estiMation, whilE thE rEd line represENtS the best liNeaR fitting for tHe pOints. THe SumMAry erRorS are gIven in tAble 7.[]... | columnwidth"}![Estimationperformanc e ofinp utAP s of the PÉGASE models . For each of the APs we pl ot th ep redi c te d vs. true A P v a l ues f or th et es t set . T he blue line indi cat es the line of pe rfect esti mat ion, while t hered li ne re p resen tsthe b est li n ear fi tting for t h e poin t s.... | columnwidth"}![Estimation performance_of input_APs of the PÉGASE_models. For_each_of the_APs_we plot the_predicted vs. true AP_values for the test_set. The blue_line_indicates the line of perfect estimation, while the red line represents the best linear_fitting_for the_points._The_summary errors are given in_Table 7.[]... |
+\lambda_j-(\lambda_2+1)<\lambda_j$, by Lemma \[vanishbar\], the term $p_{\lambda_2+1}$ is not a factor of any term of the lowest degree in the power sum expansion of $Q_{(\lambda_i,\lambda_j)}$. Now we are left with the case when $\lambda_i+\lambda_j>\lambda_1-1$ and $\lambda_i\leq \lambda_1-2$. Since $\lambda_i+\lamb... | + \lambda_j-(\lambda_2 + 1)<\lambda_j$, by Lemma \[vanishbar\ ], the term $ p_{\lambda_2 + 1}$ is not a factor of any term of the lowest academic degree in the exponent sum expansion of $ Q_{(\lambda_i,\lambda_j)}$. Now we are leave with the lawsuit when $ \lambda_i+\lambda_j>\lambda_1 - 1 $ and $ \lambda_i\leq \lambda... | +\lamhda_j-(\lambda_2+1)<\lambda_j$, by Lemoa \[vanishbar\], thg rerm $p_{\nambda_2+1}$ is not x factor of any term of the poqest eegree in the power suo expansiln of $Q_{(\lqmbde_i,\lambda_j)}$. Now we are lefb witg the rase when $\lambds_i+\lambda_j>\ldmbda_1-1$ and $\lambga_k\lzq \lambda_1-2$. Since $\lambda_i+\lamb... | +\lambda_j-(\lambda_2+1)<\lambda_j$, by Lemma \[vanishbar\], the term $p_{\lambda_2+1}$ a of any of the lowest expansion $Q_{(\lambda_i,\lambda_j)}$. Now we left with the when $\lambda_i+\lambda_j>\lambda_1-1$ and $\lambda_i\leq \lambda_1-2$. Since by Lemma \[vanishbar\] the term $p_{\lambda_1-1}$ does not appear as a ... | +\lambda_j-(\lambda_2+1)<\lambda_j$, by LemMa \[vanishbaR\], the tErm $P_{\laMbDa_2+1}$ is Not a Factor of any terM Of thE lowest degree in the poweR sum eXpANsioN Of $q_{(\lambDa_i,\lambDA_j)}$. nOW we ArE lEft WiTH tHe casE whEn $\lambdA_i+\lambda_j>\lAmbDa_1-1$ And $\lambda_i\leQ \LaMbda_1-2$. Since $\lAmbDa_i+\lamb... | +\lambda_j-(\lambda_2+1)<\ lambda_j$, by L emm a \ [v anis hbar \], the term $ p _{\l ambda_2+1}$ is not a f actor o f any te rm of the lo w es t deg re einth e p owersum expans ion of $Q_ {(\ la mbda_i,\lamb d a_ j)}$. Nowweare left wit h t he cas ewhe n $\la mbd a_i+\ lambda _ j>\lam bda_1-1$an d $\lam b da_i... | +\lambda_j-(\lambda_2+1)<\lambda_j$, by_Lemma \[vanishbar\],_the term $p_{\lambda_2+1}$ is_not a_factor_of any_term_of the lowest_degree in the_power sum expansion of_$Q_{(\lambda_i,\lambda_j)}$. Now we_are_left with the case when $\lambda_i+\lambda_j>\lambda_1-1$ and $\lambda_i\leq \lambda_1-2$. Since $\lambda_i+\lamb... |
use in CCTV applications[^3], which provides a bias and clock for the sensor. The headboard reads the digital image values and sends them to the FPGA for further processing. It also incorporates a synchronous serial port, which may be used to modify the internal camera registers when needed. The pixel stream and the t... | use in CCTV applications[^3 ], which provides a bias and clock for the sensor. The headboard read the digital prototype values and sends them to the FPGA for further processing. It also incorporate a synchronous serial port, which may be use to change the internal camera registers when needed. The pixel stream and the ... | usf in CCTV applications[^3], wmich provides a yuas anv clock for the sensor. The headboard reads vhe eigitql image values and sevds them no the FPTA fie further 'docessiky. It zpso nnrorporates a sykchronous sarial port, whiwh mcy be used to modify the internal caiera rebidters when neeqed. Uhe pixsl stream and the t... | use in CCTV applications[^3], which provides a clock the sensor. headboard reads the them the FPGA for processing. It also a synchronous serial port, which may used to modify the internal camera registers when needed. The pixel stream and three sync signals for the pixel, line, and frame clocks are generated by CMOS an... | use in CCTV applications[^3], whicH provides a Bias aNd cLocK fOr thE senSor. The headboarD ReadS the digital image values And seNdS Them TO tHe FPGa for furTHeR PRocEsSiNg. IT aLSo IncorPorAtes a syNchronous sEriAl Port, which may BE uSed to modifY thE internal camEra RegistErS whEN needEd. THe pixEl streAM and thE t... | use in CCTV applications[ ^3], which prov ide s a b iasandclock for thes enso r. The headboard reads thedi g ital im age v alues a n ds e nds t he m t ot he FPGA fo r furth er process ing .It also inco r po rates a sy nch ronous seria l p ort, w hi chm ay be us ed to modif y the i nternal c am e ra reg i ster... | use_in CCTV_applications[^3], which provides a_bias and_clock_for the_sensor._The headboard reads_the digital image_values and sends them_to the FPGA_for_further processing. It also incorporates a synchronous serial port, which may be used to_modify_the internal_camera_registers_when needed. The pixel stream_and the t... |
reparametrized by $\theta : [0,1] \rightarrow [0,n]$. The reparametrized curve is denoted by $Q_\theta$.
#### [Fréchet matching]{}s
We are given two polygonal curves $P$ and $Q$ with $m$ and $n$ edges. A (monotonous) *matching* $\mu$ between $P$ and $Q$ is a pair of reparametrizations $(\sigma,\theta)$, such that $... | reparametrized by $ \theta : [ 0,1 ] \rightarrow [ 0,n]$. The reparametrized curve is denoted by $ Q_\theta$.
# # # # [ Fréchet matching]{}s
We are given two polygonal curves $ P$ and $ Q$ with $ m$ and $ n$ edge. A (humdrum) * matching * $ \mu$ between $ P$ and $ Q$ is a pair of reparametrizations $ (\sigma,\th... | reoarametrized by $\theta : [0,1] \rightarrow [0,n]$. Tkw repacametriaed curvd is denoted by $Q_\theta$.
#### [Fréchev marchint]{}s
We are given two polhgonal cugves $P$ ane $Q$ xith $m$ and $n$ edgxa. A (monotonoua) *matehmng* $\mu$ between $K$ and $Q$ is a pair of repardmdtxizations $(\sigma,\theta)$, such that $... | reparametrized by $\theta : [0,1] \rightarrow [0,n]$. curve denoted by #### [Fréchet matching]{}s curves and $Q$ with and $n$ edges. (monotonous) *matching* $\mu$ between $P$ and is a pair of reparametrizations $(\sigma,\theta)$, such that $P_\sigma(t)$ matches to $Q_\theta(t)$. The distance between two matched points ... | reparametrized by $\theta : [0,1] \righTarrow [0,n]$. The ReparAmeTriZeD curVe is Denoted by $Q_\thetA$.
#### [frécHet matching]{}s
We are given Two poLyGOnal CUrVes $P$ aNd $Q$ with $M$ AnD $N$ EdgEs. a (mOnoToNOuS) *matcHinG* $\mu$ betwEen $P$ and $Q$ is A paIr Of reparametrIZaTions $(\sigma,\TheTa)$, such that $... | reparametrized by $\theta : [0,1]\righ tar row [ 0,n] $. T he reparametri z ed c urve is denoted by $Q_ \thet a$ .
## # #[Fréc het mat c hi n g ]{} s
W e a re gi ven t wopolygon al curves$P$ a nd $Q$ with$ m$ and $n$ e dge s. A (monoto nou s) *ma tc hin g * $\m u$betwe en $P$ and $Q $ is a pa ir of rep a rametr... | reparametrized_by $\theta_ : [0,1] \rightarrow_[0,n]$. The_reparametrized_curve is_denoted_by $Q_\theta$.
#### [Fréchet_matching]{}s
We are given_two polygonal curves $P$_and $Q$ with_$m$_and $n$ edges. A (monotonous) *matching* $\mu$ between $P$ and $Q$ is a pair_of_reparametrizations $(\sigma,\theta)$,_such_that_$... |
}+e^{1256}+e^{3456} +e^{1367}+e^{1457}+e^{2357}-e^{2467},
\end{split}$$ where $*_{\varphi}$ is the Hodge operator determined by $g_{\varphi}$ and $\operatorname{vol}_{\varphi}$, and $e^{ijk\cdots}$ is a shorthand for the wedge product of covectors $e^i{\wedge}e^j{\wedge}e^k{\wedge}\cdots$. We shall call both $\mathcal{... | } + e^{1256}+e^{3456 } + e^{1367}+e^{1457}+e^{2357}-e^{2467 },
\end{split}$$ where $ * _ { \varphi}$ is the Hodge operator determined by $ g_{\varphi}$ and $ \operatorname{vol}_{\varphi}$, and $ e^{ijk\cdots}$ is a shorthand for the wedge product of covectors $ e^i{\wedge}e^j{\wedge}e^k{\wedge}\cdots$. We shall visit... | }+e^{1256}+e^{3456} +f^{1367}+e^{1457}+e^{2357}-e^{2467},
\end{split}$$ where $*_{\varphl}$ is the Hodge okeeator vetermihed by $g_{\xarphi}$ and $\operatorname{vol}_{\vacphi}$, and $t^{pjk\cdots}$ is a shorthavd for thv wedge peodurt of covectors $x^j{\wedge}e^m{\cedge}s^n{\wedye}\rdots$. We shall gall both $\mdthcal{... | }+e^{1256}+e^{3456} +e^{1367}+e^{1457}+e^{2357}-e^{2467}, \end{split}$$ where $*_{\varphi}$ is the determined $g_{\varphi}$ and and $e^{ijk\cdots}$ is product covectors $e^i{\wedge}e^j{\wedge}e^k{\wedge}\cdots$. We call both $\mathcal{B}$ $\mathcal{B}^*$ [*adapted*]{} bases to the G$_2$-structure On the other hand, giv... | }+e^{1256}+e^{3456} +e^{1367}+e^{1457}+e^{2357}-e^{2467},
\end{split}$$ where $*_{\varphi}$ Is the Hodge OperaTor DetErMineD by $g_{\Varphi}$ and $\operaTOrnaMe{vol}_{\varphi}$, and $e^{ijk\cdotS}$ is a sHoRThanD FoR the wEdge proDUcT OF coVeCtOrs $E^i{\WEdGe}e^j{\wEdgE}e^k{\wedgE}\cdots$. We shAll CaLl both $\mathcaL{... | }+e^{1256}+e^{3456} +e^{13 67}+e^{145 7}+e^ {23 57} -e ^{24 67},
\end{split}$$ wher e $*_{\varphi}$ is the Hodg eo pera t or dete rminedb y$ g _{\ va rp hi} $a nd $\op era torname {vol}_{\va rph i} $, and $e^{i j k\ cdots}$ is ashorthand fo r t he wed ge pr o ductofcovec tors $ e ^i{\we dge}e^j{\ we d ge}e^k { \w... | }+e^{1256}+e^{3456} +e^{1367}+e^{1457}+e^{2357}-e^{2467},
\end{split}$$_where $*_{\varphi}$_is the Hodge operator_determined by_$g_{\varphi}$_and $\operatorname{vol}_{\varphi}$,_and_$e^{ijk\cdots}$ is a_shorthand for the_wedge product of covectors_$e^i{\wedge}e^j{\wedge}e^k{\wedge}\cdots$. We shall_call_both $\mathcal{... |
in L^p(0,T;X_1)$ and $\frac{d}{dt} f\in L^{p'}(0,T;X_1')$, $1< p <\infty$, cf. Zeidler [@ZeidlerIIa Proposition 23.23]. In particular, (\[eq:HNormDifferential\]) implies $$\label{eq:StrongBUCConv}
\sup_{t\in [0,T]}\|f(t)\|_{H}^2
\leq 2\left(\|\partial_t f\|_{L^2(0,T;X_1')}\|f\|_{L^2(0,T;X_1)} + \|f(0)\|_H^2\right).$... | in L^p(0,T;X_1)$ and $ \frac{d}{dt } f\in L^{p'}(0,T;X_1')$, $ 1 < p < \infty$, cf. Zeidler [ @ZeidlerIIa Proposition 23.23 ]. In particular, (\[eq: HNormDifferential\ ]) implies $ $ \label{eq: StrongBUCConv }
\sup_{t\in [ 0,T]}\|f(t)\|_{H}^2
\leq 2\left(\|\partial_t f\|_{L^2(0,T;X_1')}\|f\|_{L^2(0,T;X_1) } + ... | in P^p(0,T;X_1)$ and $\frac{d}{dt} f\in L^{p'}(0,T;W_1')$, $1< p <\infty$, cf. Zeneler [@ZemdlerIIz Proposktion 23.23]. In particular, (\[eq:HNormViffwrentual\]) implies $$\label{eq:StrungBUCConn}
\sup_{t\in [0,T]}\|f(t)\|_{I}^2
\leq 2\left(\|\partial_t f\|_{L^2(0,T;X_1')}\|f\|_{L^2(0,T;X_1)} + \|f(0)\|_G^2\vight).$... | in L^p(0,T;X_1)$ and $\frac{d}{dt} f\in L^{p'}(0,T;X_1')$, $1< cf. [@ZeidlerIIa Proposition In particular, (\[eq:HNormDifferential\]) 2\left(\|\partial_t + \|f(0)\|_H^2\right).$$ Replacing by $tf(t)$ and one easily derives from the latter $$\label{eq:StrongBUCConv'} \sup_{t\in [0,T]}\|f(t)\|_{H} \leq C_T\|f\|_{H^1(0,T;... | in L^p(0,T;X_1)$ and $\frac{d}{dt} f\in L^{p'}(0,T;X_1')$, $1< p <\iNfty$, cf. ZeidLer [@ZeIdlErIia propOsitIon 23.23]. In particulaR, (\[Eq:HNOrmDifferential\]) implies $$\Label{Eq:sTronGbUcConv}
\Sup_{t\in [0,T]}\|F(T)\|_{H}^2
\LEQ 2\leFt(\|\PaRtiAl_T F\|_{L^2(0,t;X_1')}\|f\|_{L^2(0,T;x_1)} + \|f(0)\|_H^2\Right).$... | in L^p(0,T;X_1)$ and $\fra c{d}{dt} f \in L ^{p '}( 0, T;X_ 1')$ , $1< p <\inft y $, c f. Zeidler [@ZeidlerII a Pro po s itio n 2 3.23] . In pa r ti c u lar ,(\ [eq :H N or mDiff ere ntial\] ) implies$$\ la bel{eq:Stron g BU CConv}
\ sup _{t\in [0,T] }\| f(t)\| _{ H}^ 2
\le q 2 \left (\|\pa r tial_t f\|_{L^2 (0 ,... | in L^p(0,T;X_1)$_and $\frac{d}{dt}_f\in L^{p'}(0,T;X_1')$, $1< p_<\infty$, cf._Zeidler [@ZeidlerIIa_Proposition 23.23]._In_particular, (\[eq:HNormDifferential\]) implies_$$\label{eq:StrongBUCConv}
\sup_{t\in_[0,T]}\|f(t)\|_{H}^2
\leq 2\left(\|\partial_t f\|_{L^2(0,T;X_1')}\|f\|_{L^2(0,T;X_1)}_+ \|f(0)\|_H^2\right).$... |
This can also be done by considering ${{{\mathrm{Dih}}}_2\triangleleft{{\mathrm{Dih}}}_4}$ such that the graded structure is $$\begin{matrix}
{\mathbb{R}}[{{\mathrm{Dih}}}_2]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{R}}^{{\oplus 2}}&&\\
{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{... | This can also be done by considering $ { { { \mathrm{Dih}}}_2\triangleleft{{\mathrm{Dih}}}_4}$ such that the graded social organization is $ $ \begin{matrix }
{ \mathbb{R}}[{{\mathrm{Dih}}}_2]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{R}}^{{\oplus 2}}&&\\
{ \mathbin{\text{\rotatebox[origin = c]{-90}{$\hookrightarro... | Thls can also be done by cunsidering ${{{\mathtm{Eih}}}_2\trienglelert{{\mathrm{Aih}}}_4}$ such that the graded strnctuee is $$\begin{matrix}
{\mathbb{R}}[{{\matfrm{Dih}}}_2]&=&{\matjbb{R}}^{{\opluw 2}}\opoys{\mathbb{R}}^{{\o'mus 2}}&&\\
{\matmyin{\tesb{\rotaceuox[origin=c]{-90}{$\hookrlghtarrow$}}}}&&{\madhbin{\text{... | This can also be done by considering that graded structure $$\begin{matrix} {\mathbb{R}}[{{\mathrm{Dih}}}_2]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{R}}^{{\oplus M_2({\mathbb{R}})&=&{{\mathrm{Cl}}}_{0,1}^{{\oplus \end{matrix}$$ $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]&={{\mathr... | This can also be done by considEring ${{{\mathrM{Dih}}}_2\tRiaNglElEft{{\mAthrM{Dih}}}_4}$ such that thE GradEd structure is $$\begin{matrIx}
{\matHbB{r}}[{{\matHRm{dih}}}_2]&=&{\maThbb{R}}^{{\opLUs 2}}\OPLus{\MaThBb{R}}^{{\OpLUs 2}}&&\\
{\MathbIn{\tExt{\rotaTebox[origiN=c]{-90}{$\hOoKrightarrow$}}}}&&{\mAThBin{\text{... | This can also be done byconsiderin g ${{ {\m ath rm {Dih }}}_ 2\triangleleft { {\ma thrm{Dih}}}_4}$ such t hat t he grad e dstruc ture is $$ \ b egi n{ ma tri x} {\ mathb b{R }}[{{\m athrm{Dih} }}_ 2] &=&{\mathbb{ R }} ^{{\oplus2}} \oplus{\math bb{ R}}^{{ \o plu s 2}}& &\\
{\ma thbin{ \ text{\ rotatebox [o r igin=c ... | This_can also_be done by considering_${{{\mathrm{Dih}}}_2\triangleleft{{\mathrm{Dih}}}_4}$ such_that_the graded_structure_is $$\begin{matrix}
{\mathbb{R}}[{{\mathrm{Dih}}}_2]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{R}}^{{\oplus_2}}&&\\
{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{... |
left\langle{ -i}\right\rangle}_N}.$$ The multiplication is $$e_{ri}e_{sj}=\delta_{rs}e_{r{\left\langle{ i+j}\right\rangle}_N}.$$
This is the direct sum of $N$ group algebras of ${\mathbb{Z}}_N$. There are $N^2$ 1-dimensional simple modules. We use ${\lfloor{\cdots}\rceil}$ to denote a composite label. The simple modul... | left\langle { -i}\right\rangle}_N}.$$ The multiplication is $ $ e_{ri}e_{sj}=\delta_{rs}e_{r{\left\langle { i+j}\right\rangle}_N}.$$
This is the direct sum of $ N$ group algebras of $ { \mathbb{Z}}_N$. There be $ N^2 $ 1 - dimensional childlike modules. We use $ { \lfloor{\cdots}\rceil}$ to denote a composite label.... | lefh\langle{ -i}\right\rangle}_N}.$$ Tht multiplication nw $$e_{ri}e_{vj}=\deltz_{rs}e_{r{\lefg\langle{ i+j}\right\rangle}_N}.$$
This id rhe durect sum of $N$ group augebras ov ${\mathbb{Z}}_N$. Tiere are $N^2$ 1-dimensional sliple lodunxs. We use ${\lfloot{\cdots}\rceil}$ do denote a cokpusnte label. The simple modul... | left\langle{ -i}\right\rangle}_N}.$$ The multiplication is $$e_{ri}e_{sj}=\delta_{rs}e_{r{\left\langle{ i+j}\right\rangle}_N}.$$ the sum of group algebras of simple We use ${\lfloor{\cdots}\rceil}$ denote a composite The simple modules can be labeled two numbers ${\lfloor{ ri}\rceil}$. The basis is $$M_{{\lfloor{ ri}\r... | left\langle{ -i}\right\rangle}_N}.$$ ThE multiplicAtion Is $$e_{Ri}e_{Sj}=\DeltA_{rs}e_{R{\left\langle{ i+j}\rIGht\rAngle}_N}.$$
This is the direct sUm of $N$ GrOUp alGEbRas of ${\Mathbb{Z}}_n$. thERE arE $N^2$ 1-DiMenSiONaL simpLe mOdules. WE use ${\lfloor{\CdoTs}\Rceil}$ to denotE A cOmposite laBel. the simple modUl... | left\langle{ -i}\right\ran gle}_N}.$$ Themul tip li cati on i s $$e_{ri}e_{s j }=\d elta_{rs}e_{r{\left\la ngle{ i + j}\r i gh t\ran gle}_N} . $$ Thi sis th ed ir ect s umof $N$group alge bra sof ${\mathbb { Z} }_N$. Ther e a re $N^2$ 1-d ime nsiona lsim p le mo dul es. W e use$ {\lflo or{\cdots }\ r ceil}$ to deno ... | left\langle{ -i}\right\rangle}_N}.$$_The multiplication_is $$e_{ri}e_{sj}=\delta_{rs}e_{r{\left\langle{ i+j}\right\rangle}_N}.$$
This is_the direct_sum_of $N$_group_algebras of ${\mathbb{Z}}_N$._There are $N^2$_1-dimensional simple modules. We_use ${\lfloor{\cdots}\rceil}$ to_denote_a composite label. The simple modul... |
}}$ are noted by ${\mathbb{E}}_\delta$ and ${\mathbb{E}}$, respectively.
The Markov chain $(Y_n: n\ge 0)$ can be also constructed from a probability space $({\widetilde{\Omega}}, {\widetilde{\cal F}},{\bf P})$ containing an independent family of random variables $\left(\delta^K_n: K\in {{\cal Y}}({{\cal J}}_\rho), n\g... | } } $ are noted by $ { \mathbb{E}}_\delta$ and $ { \mathbb{E}}$, respectively.
The Markov chain $ (Y_n: n\ge 0)$ can be also constructed from a probability quad $ ({ \widetilde{\Omega } }, { \widetilde{\cal F}},{\bf P})$ control an independent family of random variables $ \left(\delta^K_n: K\in { { \cal Y}}({{\cal J... | }}$ arf noted by ${\mathbb{E}}_\delta$ xnd ${\mathbb{E}}$, reskextivelb.
The Madkov chakn $(Y_n: n\ge 0)$ can be also constcuctwd frim a probability space $({\widetildv{\Omega}}, {\wieetioee{\cal F}},{\bf '})$ contaiknng ah indzpxndent family on random vasiables $\left(\dentx^K_u: K\in {{\cal Y}}({{\cal J}}_\rho), n\g... | }}$ are noted by ${\mathbb{E}}_\delta$ and ${\mathbb{E}}$, Markov $(Y_n: n\ge can be also $({\widetilde{\Omega}}, F}},{\bf P})$ containing independent family of variables $\left(\delta^K_n: K\in {{\cal Y}}({{\cal J}}_\rho), 1 \right)$, where $\delta^K_n$ takes values in ${\mathbb{D}}_{1,2}(K)$ and $${\bf P}(\delta^K_n=... | }}$ are noted by ${\mathbb{E}}_\delta$ and ${\Mathbb{E}}$, resPectiVelY.
ThE MArkoV chaIn $(Y_n: n\ge 0)$ can be alSO conStructed from a probabiliTy spaCe $({\WIdetILdE{\OmegA}}, {\widetiLDe{\CAL F}},{\bF P})$ CoNtaInINg An indEpeNdent faMily of randOm vArIables $\left(\deLTa^k_n: K\in {{\cal Y}}({{\cAl J}}_\Rho), n\g... | }}$ are noted by ${\mathbb {E}}_\delt a$ an d $ {\m at hbb{ E}}$ , respectively .
Th e Markov chain $(Y_n:n\ge0) $ can be also constr u ct e d fr om a pr ob a bi lityspa ce $({\ widetilde{ \Om eg a}}, {\widet i ld e{\cal F}} ,{\ bf P})$ cont ain ing an i nde p enden t f amily of ra n dom va riables $ \l e ft(... | }}$ are_noted by_${\mathbb{E}}_\delta$ and ${\mathbb{E}}$, respectively.
The_Markov chain_$(Y_n:_n\ge 0)$_can_be also constructed_from a probability_space $({\widetilde{\Omega}}, {\widetilde{\cal F}},{\bf_P})$ containing an_independent_family of random variables $\left(\delta^K_n: K\in {{\cal Y}}({{\cal J}}_\rho), n\g... |
= 2\sqrt{\rho_R}$, $\rho$ can vanish (there is a so-called vacuum point) and a modification of the solution is required. To find a simple wave solution for large piston velocities, we must derive new conditions for the parameters $r_i$. We modify the DSW solution by introducing a locally periodic TW between the piston... | = 2\sqrt{\rho_R}$, $ \rho$ can vanish (there is a so - called void item) and a modification of the solution is required. To witness a bare wave solution for large piston velocities, we must deduce new conditions for the parameters $ r_i$. We change the DSW solution by inaugurate a locally periodic TW between the piston... | = 2\sert{\rho_R}$, $\rho$ can vanish (tmere is a so-callgd vacuuk poinf) and a oodification of the solution iw reqyired. To find a simple wave solltion for larje piston velocivjes, we must dsvive uex conditions fot the parameders $r_i$. We modhfh che DSW solution by introducing a losally prrlodic TW betwegn tht pystoh... | = 2\sqrt{\rho_R}$, $\rho$ can vanish (there is vacuum and a of the solution simple solution for large velocities, we must new conditions for the parameters $r_i$. modify the DSW solution by introducing a locally periodic TW between the piston the trailing edge of the DSW. When $v_p = 2\sqrt{\rho_R}$, the DSW forms vacu... | = 2\sqrt{\rho_R}$, $\rho$ can vanish (there Is a so-calleD vacuUm pOinT) aNd a mOdifIcation of the soLUtioN is required. To find a simpLe wavE sOLutiON fOr larGe pistoN VeLOCitIeS, wE muSt DErIve neW coNditionS for the parAmeTeRs $r_i$. We modify THe dSW solutioN by Introducing a LocAlly peRiOdiC tW betWeeN the pIston... | = 2\sqrt{\rho_R}$, $\rho$ can vanis h (th ere is a so- call ed vacuum poin t ) an d a modification of th e sol ut i on i s r equir ed. Tof in d a s im pl e w av e s oluti onfor lar ge pistonvel oc ities, we mu s tderive new co nditions for th e para me ter s $r_i $.We mo dify t h e DSWsolutionby introd u cing... | =_2\sqrt{\rho_R}$, $\rho$_can vanish (there is_a so-called_vacuum_point) and_a_modification of the_solution is required._To find a simple_wave solution for_large_piston velocities, we must derive new conditions for the parameters $r_i$. We modify the_DSW_solution by_introducing_a_locally periodic TW between the_piston... |
model.
Then, we obtain an equation for $x$ and $y$: $$F(x)=\int_0^x\frac{f(y)dy}{\sqrt{x-y}},$$ where $$F(x)=T\frac{\Delta\omega}{\omega}\frac{1}{\sqrt{x}},$$ $$f(y)=\frac{\Delta c}{c}\frac{1}{2y^{3/2}\left(\dfrac{d\log c}{d\log r}+1\right)}.$$
To solve for $f(y)$ we multiply both sides of Eq.(12) by ${dx}/{\sqrt{z-... | model.
Then, we obtain an equation for $ x$ and $ y$: $ $ F(x)=\int_0^x\frac{f(y)dy}{\sqrt{x - y}},$$ where $ $ F(x)=T\frac{\Delta\omega}{\omega}\frac{1}{\sqrt{x}},$$ $ $ f(y)=\frac{\Delta c}{c}\frac{1}{2y^{3/2}\left(\dfrac{d\log c}{d\log r}+1\right)}.$$
To solve for $ f(y)$ we breed both slope of Eq.(12) by $ { ... | mofel.
Then, we obtain an equxtion for $x$ and $y$: $$F(x)=\inv_0^x\frac{f(g)dy}{\sqrt{x-h}},$$ where $$F(x)=T\frac{\Delta\omega}{\omeja}\frqc{1}{\sqru{q}},$$ $$f(y)=\frac{\Delta c}{c}\frac{1}{2y^{3/2}\ueft(\dfrac{f\log c}{d\lig r}+1\cight)}.$$
To solve foc $f(y)$ we multipmn botk wides of Eq.(12) by ${dx}/{\sqrt{z-... | model. Then, we obtain an equation for $y$: where $$F(x)=T\frac{\Delta\omega}{\omega}\frac{1}{\sqrt{x}},$$ c}{c}\frac{1}{2y^{3/2}\left(\dfrac{d\log c}{d\log r}+1\right)}.$$ multiply sides of Eq.(12) ${dx}/{\sqrt{z-x}}$ and integrate respect to $x$ from 0 to $$\int_0^z\frac{F(x)dx}{\sqrt{z-x}}=\int_0^z\frac{dx}{\sqrt{z-... | model.
Then, we obtain an equatiOn for $x$ and $y$: $$f(x)=\int_0^X\frAc{f(Y)dY}{\sqrT{x-y}},$$ wHere $$F(x)=T\frac{\DelTA\omeGa}{\omega}\frac{1}{\sqrt{x}},$$ $$f(y)=\frac{\delta C}{c}\FRac{1}{2y^{3/2}\LEfT(\dfraC{d\log c}{d\LOg R}+1\RIghT)}.$$
TO sOlvE fOR $f(Y)$ we muLtiPly both Sides of Eq.(12) bY ${dx}/{\SqRt{z-... | model.
Then, we obtain a n equation for$x$ an d$y$: $$F (x)=\int_0^x\f r ac{f (y)dy}{\sqrt{x-y}},$$where $ $ F(x) = T\ frac{ \Delta\ o me g a }{\ om eg a}\ fr a c{ 1}{\s qrt {x}},$$ $$f(y)=\f rac {\ Delta c}{c}\ f ra c{1}{2y^{3 /2} \left(\dfrac {d\ log c} {d \lo g r}+1 \ri ght)} .$$
T o solve for $f(y )$ we ... | model.
Then,_we obtain_an equation for $x$_and $y$:_$$F(x)=\int_0^x\frac{f(y)dy}{\sqrt{x-y}},$$_where $$F(x)=T\frac{\Delta\omega}{\omega}\frac{1}{\sqrt{x}},$$_$$f(y)=\frac{\Delta_c}{c}\frac{1}{2y^{3/2}\left(\dfrac{d\log c}{d\log r}+1\right)}.$$
To_solve for $f(y)$_we multiply both sides_of Eq.(12) by_${dx}/{\sqrt{z-... |
nabla \phi}{{\left \lvert \nabla \phi \right \rvert}} \right) \cdot \nabla \phi = 0.$ Note that $\nabla \phi/ {\left \lvert \nabla \phi \right \rvert}$ is a unit vector which is normal to the level contour $\Gamma(t)$. Thus, locally, this equation models advection in the direction normal to $\Gamma(t)$ with velocity $v... | nabla \phi}{{\left \lvert \nabla \phi \right \rvert } } \right) \cdot \nabla \phi = 0.$ Note that $ \nabla \phi/ { \left \lvert \nabla \phi \right \rvert}$ is a unit vector which is normal to the level contour $ \Gamma(t)$. therefore, locally, this equality model advection in the direction normal to $ \Gamma(t)$ with v... | nabpa \phi}{{\left \lvert \nabla \pmi \right \rvert}} \rntht) \cdmt \nabma \phi = 0.$ Note that $\nabla \phi/ {\left \lvxrt \babla \phi \right \rvert}$ is a jnit vectlr which is iormal to the letsl contour $\Gajla(t)$. Chns, locally, this equation kodels advectimn iu the direction normal to $\Gamma(t)$ witr velocotj $v... | nabla \phi}{{\left \lvert \nabla \phi \right \rvert}} \nabla = 0.$ that $\nabla \phi/ \rvert}$ a unit vector is normal to level contour $\Gamma(t)$. Thus, locally, this models advection in the direction normal to $\Gamma(t)$ with velocity $v(x)$. This causes to deform with normal velocity $v(x)$. With this in mind, one... | nabla \phi}{{\left \lvert \nabla \phi \Right \rvert}} \Right) \CdoT \naBlA \phi = 0.$ note That $\nabla \phi/ {\leFT \lveRt \nabla \phi \right \rvert}$ is A unit VeCTor wHIcH is noRmal to tHE lEVEl cOnToUr $\GAmMA(t)$. thus, lOcaLly, this Equation moDelS aDvection in thE DiRection norMal To $\Gamma(t)$ with VelOcity $v... | nabla \phi}{{\left \lvert\nabla \ph i \ri ght \r ve rt}} \ri ght) \cdot \na b la \ phi = 0.$ Note that $\ nabla \ p hi/{ \l eft \ lvert \ n ab l a \p hi \ rig ht \r vert} $ i s a uni t vector w hic his normal to th e level co nto ur $\Gamma(t )$. Thus, l oca l ly, t his equa tion m o dels a dvectionin the di r ... | nabla \phi}{{\left_\lvert \nabla_\phi \right \rvert}} \right)_\cdot \nabla_\phi_= 0.$_Note_that $\nabla \phi/_{\left \lvert \nabla_\phi \right \rvert}$ is_a unit vector_which_is normal to the level contour $\Gamma(t)$. Thus, locally, this equation models advection in_the_direction normal_to_$\Gamma(t)$_with velocity $v... |
0$ a.e. in $(0,\infty) \times {{\mathbb{R}^n}}$ provided $f \geq 0$ a.e. in ${{\mathbb{R}^n}}$.
Proposition \[prop:existence-energy-solutions\] is folklore but it cannot be read easily from the current literature. The nature of the time derivative is one of the main concerns here. Typically, the existence class is to... | 0 $ a.e. in $ (0,\infty) \times { { \mathbb{R}^n}}$ provided $ f \geq 0 $ a.e. in $ { { \mathbb{R}^n}}$.
Proposition \[prop: existence - energy - solutions\ ] is folklore but it cannot be read well from the current literature. The nature of the meter derivative is one of the main concern here. Typically, the u... | 0$ a.f. in $(0,\infty) \times {{\mathbb{R}^n}}$ provided $f \geq 0$ a.e. in ${{\kathbb{D}^n}}$.
Proposktion \[prop:existence-energy-soluvionw\] is dolklore but it cannot be read vasily frim tie current literefure. Thc nathve of vhe time derivajive is one mf the main cotcdrus here. Typically, the existence clasf is to... | 0$ a.e. in $(0,\infty) \times {{\mathbb{R}^n}}$ provided 0$ in ${{\mathbb{R}^n}}$. \[prop:existence-energy-solutions\] is folklore easily the current literature. nature of the derivative is one of the main here. Typically, the existence class is too large because of data unnecessarily general our purposes, or the spati... | 0$ a.e. in $(0,\infty) \times {{\mathbb{R}^n}}$ proVided $f \geq 0$ a.E. in ${{\maThbB{R}^n}}$.
prOposItioN \[prop:existence-ENergY-solutions\] is folklore buT it caNnOT be rEAd EasilY from thE CuRREnt LiTeRatUrE. thE natuRe oF the timE derivativE is OnE of the main coNCeRns here. TypIcaLly, the existeNce Class iS tO... | 0$ a.e. in $(0,\infty) \t imes {{\ma thbb{ R}^ n}} $prov ided $f \geq 0$ a. e . in ${{\mathbb{R}^n}}$.
Propo si t ion\ [p rop:e xistenc e -e n e rgy -s ol uti on s \] is f olk lore bu t it canno t b eread easilyf ro m the curr ent literature. Th e natu re of the t ime deri vative is one of the m ai n conce r ... | 0$_a.e. in $(0,\infty)_\times {{\mathbb{R}^n}}$ provided $f_\geq 0$_a.e. in_${{\mathbb{R}^n}}$.
Proposition \[prop:existence-energy-solutions\] is_folklore_but it cannot_be read easily_from the current literature._The nature of_the_time derivative is one of the main concerns here. Typically, the existence class is_to... |
} \in \Aut^{2,q}(P)$ to the gauge-fixing equation, $$d_{A_0}^*(u_{t_0+s}(A_{t_0+s}) - A_0) = 0.$$ In particular, we shall seek a solution in the form $$u_{t_0+s} = e^{\chi_s}u_{t_0},
\quad\text{for } \chi_s \in W_{A_1}^{2,q}(X;\ad P),$$ so the gauge-fixing equation becomes $$\label{eq:Gauge_fixing}
d_{A_0}^*(e^{\chi_s}... | } \in \Aut^{2,q}(P)$ to the gauge - fixing equation, $ $ d_{A_0}^*(u_{t_0+s}(A_{t_0+s }) - A_0) = 0.$$ In particular, we shall search a solution in the human body $ $ u_{t_0+s } = e^{\chi_s}u_{t_0 },
\quad\text{for } \chi_s \in W_{A_1}^{2,q}(X;\ad P),$$ so the gauge - fixing equality becomes $ $ \label{eq: Gauge_fixi... | } \in \Aut^{2,q}(P)$ to the gauge-fixinn equation, $$d_{A_0}^*(u_{t_0+s}(C_{r_0+s}) - A_0) = 0.$$ In pzrticulaf, we shall seek a solution ii thw forn $$u_{t_0+s} = e^{\chi_s}u_{t_0},
\quad\text{wor } \chi_s \in W_{A_1}^{2,q}(X;\qd P),$$ wo the gaujs-fixing equatjln bzcimes $$\label{eq:Gaoge_fixing}
d_{A_0}^*(e^{\whi_s}... | } \in \Aut^{2,q}(P)$ to the gauge-fixing equation, A_0) 0.$$ In we shall seek $$u_{t_0+s} e^{\chi_s}u_{t_0}, \quad\text{for } \in W_{A_1}^{2,q}(X;\ad P),$$ the gauge-fixing equation becomes $$\label{eq:Gauge_fixing} d_{A_0}^*(e^{\chi_s}u_{t_0}(A_{t_0+s}) A_0) = 0.$$ For $s \in \RR$, it will be convenient to define \in ... | } \in \Aut^{2,q}(P)$ to the gauge-fixing eqUation, $$d_{A_0}^*(u_{t_0+S}(A_{t_0+s}) - A_0) = 0.$$ in pArtIcUlar, We shAll seek a solutiON in tHe form $$u_{t_0+s} = e^{\chi_s}u_{t_0},
\quad\teXt{for } \ChI_S \in W_{a_1}^{2,Q}(X;\Ad P),$$ so The gaugE-FiXINg eQuAtIon BeCOmEs $$\labEl{eQ:Gauge_fIxing}
d_{A_0}^*(e^{\chI_s}... | } \in \Aut^{2,q}(P)$ to th e gauge-fi xingequ ati on , $$ d_{A _0}^*(u_{t_0+s } (A_{ t_0+s}) - A_0) = 0.$$In pa rt i cula r ,we sh all see k a s olu ti on in t h eform$$u _{t_0+s } = e^{\ch i_s }u _{t_0},
\qua d \t ext{for }\ch i_s \in W_{A _1} ^{2,q} (X ;\a d P),$ $ s o the gauge - fixing equation b e comes$ $\... | } \in_\Aut^{2,q}(P)$ to_the gauge-fixing equation, $$d_{A_0}^*(u_{t_0+s}(A_{t_0+s})_- A_0)_=_0.$$ In_particular,_we shall seek_a solution in_the form $$u_{t_0+s} =_e^{\chi_s}u_{t_0},
\quad\text{for } \chi_s_\in_W_{A_1}^{2,q}(X;\ad P),$$ so the gauge-fixing equation becomes $$\label{eq:Gauge_fixing}
d_{A_0}^*(e^{\chi_s}... |
(in powers of $v/\Lambda$) giving rise to ALP decays into pairs of SM gauge bosons and fermions, while the additional interactions in (\[LeffD>5\]) are needed to parametrize the exotic decays of Higgs bosons into final states involving an ALP. In computing the various decay rates, we include the tree-level and one-... | (in powers of $ v/\Lambda$) giving rise to ALP decays into pair of samarium gauge bosons and fermions, while the extra interactions in (\[LeffD>5\ ]) are needed to parametrize the exotic decay of Higgs bosons into final state involving an ALP. In computing the versatile decay rates, we include the tree - degree and ... | (in powers of $v/\Lambda$) givinn rise to ALP deeqys invo paira of SM eauge bosons and fermions, whmle rhe aeditional interactions in (\[LeffD&ht;5\]) are nwedev to parametrize the exobnc dedwys mh Higgs bosons lnto final vtates involvitg au ALP. In computing the various decay rates, ee include the ttee-lenej ans one-... | (in powers of $v/\Lambda$) giving rise to into of SM bosons and fermions, (\[LeffD>5\]) needed to parametrize exotic decays of bosons into final states involving an In computing the various decay rates, we include the tree-level and one-loop contributions the relevant operators. We find that fermion-loop corrections... | (in powers of $v/\Lambda$) giving riSe to ALP decAys inTo pAirS oF SM gAuge Bosons and fermiONs, whIle the additional interaCtionS iN (\[leffd≫5\]) Are neEded to pARaMETriZe ThE exOtIC dEcays Of HIggs bosOns into finAl sTaTes involving AN AlP. In computIng The various deCay Rates, wE iNclUDe the TreE-leveL and onE-... | (in powers of $v/\Lambda$ ) giving r ise t o A LPde cays int o pairs of SMg auge bosons and fermions,while t h e ad d it ional intera c ti o n s i n(\ [Le ff D &g t;5\] ) a re need ed to para met ri ze the exoti c d ecays of H igg s bosons int o f inal s ta tes invol vin g anALP. I n compu ting theva r ious d e ... | (in_powers of_$v/\Lambda$) giving rise to_ALP decays_into_pairs of_SM_gauge bosons and_fermions, while the_additional interactions in (\[LeffD>5\])_are needed to_parametrize_the exotic decays of Higgs bosons into final states involving an ALP. In computing_the_various decay_rates,_we_include the tree-level and one-... |
OPERA collaboration, which claimed to have discovered Lorentz violation in the neutrino sector [@OPERA:2011zb]. A large number of theoretical models emerged trying to explain the observed anomaly, for example by Fermi point splitting [@Klinkhamer:2011mf], spontaneous symmetry breaking caused by the existence of a ferm... | OPERA collaboration, which claimed to have discovered Lorentz violation in the neutrino sector [ @OPERA:2011zb ]. A large numeral of theoretical mannequin emerged judge to excuse the observed anomaly, for example by Fermi point splitting [ @Klinkhamer:2011mf ], ad-lib isotropy breaking caused by the existence of a ferm... | OPFRA collaboration, which glaimed to have buscoveced Lorsntz viouation in the neutrino sectoc [@OPWRA:2011zb]. A large number of theuretical lodels energtd trying to explejn the observsf anmnaly, for exampke by Fermh point splitthne [@Nlinkhamer:2011mf], spontaneous symmetry bweaking cwused by the evisttncq of a ferm... | OPERA collaboration, which claimed to have discovered in neutrino sector A large number to the observed anomaly, example by Fermi splitting [@Klinkhamer:2011mf], spontaneous symmetry breaking caused the existence of a fermionic condensate [@Klinkhamer:2011iz], or a multiple Lorentz group structure However, the physics ... | OPERA collaboration, which clAimed to havE discOveRed loRentZ vioLation in the neuTRino Sector [@OPERA:2011zb]. A large numBer of ThEOretICaL modeLs emergED tRYIng To ExPlaIn THe ObserVed Anomaly, For example By FErMi point splitTInG [@KlinkhameR:2011mf], Spontaneous sYmmEtry brEaKinG CauseD by The exIstencE Of a ferM... | OPERA collaboration, whic h claimedto ha vedis co vere d Lo rentz violatio n inthe neutrino sector [@ OPERA :2 0 11zb ] .A lar ge numb e ro f th eo re tic al mo delseme rged tr ying to ex pla in the observe d a nomaly, fo r e xample by Fe rmi point s pli t ting[@K linkh amer:2 0 11mf], spontane ou s symme t ry... | OPERA_collaboration, which_claimed to have discovered_Lorentz violation_in_the neutrino_sector_[@OPERA:2011zb]. A large_number of theoretical_models emerged trying to_explain the observed_anomaly,_for example by Fermi point splitting [@Klinkhamer:2011mf], spontaneous symmetry breaking caused by the existence_of_a ferm... |
can be found. The fact that the construction of the candidate polyhedron fails can mean two things. Either the group is not discrete, or the choice of the basepoint is bad: it gives a Dirichlet domain with a very large number of faces (possibly even infinite see for instance [@GP2]).
Since Mostow’s work, different te... | can be found. The fact that the construction of the candidate polyhedron fails can think of two thing. Either the group is not discrete, or the choice of the basepoint is bad: it gives a Dirichlet world with a very large number of boldness (possibly even countless see for instance [ @GP2 ]).
Since Mostow ’s work, di... | caj be found. The fact that the construction of tie candjdate pouyhedron fails can mean two vhints. Eiuker the group is not discrete, or the xhoire of the basepomht is bad: it flves c Virichlet domaik with a vesy large numbes uf faces (possibly even infinite see fjr instsnfe [@GP2]).
Since Mosjow’s ejrk, spfnerent te... | can be found. The fact that the the polyhedron fails mean two things. discrete, the choice of basepoint is bad: gives a Dirichlet domain with a large number of faces (possibly even infinite see for instance [@GP2]). Since Mostow’s different techniques have been developed to produce fundamental domains. In particular di... | can be found. The fact that the cOnstructioN of thE caNdiDaTe poLyheDron fails can meAN two Things. Either the group is Not diScREte, oR ThE choiCe of the BAsEPOinT iS bAd: iT gIVeS a DirIchLet domaIn with a verY laRgE number of facES (pOssibly eveN inFinite see for InsTance [@Gp2]).
SIncE mostoW’s wOrk, diFferenT Te... | can be found. The fact th at the con struc tio n o fthecand idate polyhedr o n fa ils can mean two thing s. Ei th e r th e g roupis notd is c r ete ,or th ec ho ice o f t he base point is b ad: i t gives a Di r ic hlet domai n w ith a very l arg e numb er of faces (p ossib ly eve n infin ite see f or instan c e ... | can_be found._The fact that the_construction of_the_candidate polyhedron_fails_can mean two_things. Either the_group is not discrete,_or the choice_of_the basepoint is bad: it gives a Dirichlet domain with a very large number_of_faces (possibly_even_infinite_see for instance [@GP2]).
Since Mostow’s_work, different te... |
system weakly bound (1.475 MeV) to breaking up as $\alpha$+d, it is not obvious that a two-body Lee-Suzuki transformation would treat the relevant Jacobi coordinate responsible for the dominant energy dependence. This should be explored.\
$\bullet$ I believe the conclusions about the CG expansion will apply to other e... | system weakly bound (1.475 MeV) to breaking up as $ \alpha$+d, it is not obvious that a two - body Lee - Suzuki transformation would regale the relevant Jacobi coordinate creditworthy for the dominant energy addiction. This should be explored.\
$ \bullet$ I believe the conclusion about the CG expansion will apply to ... | sydtem weakly bound (1.475 MeV) tu breaking up as $\alpha$+v, it is not obvkous that a two-body Lee-Suzukm trqnsfoemation would treat thd relevann Jacobi xoorvinate responsible for tmz domjkant znxrgy dependence. This shound be explored.\
$\tuulzt$ I believe the conclusions about tre CG ecpwnsion will apkly tp othsg t... | system weakly bound (1.475 MeV) to breaking $\alpha$+d, is not that a two-body relevant coordinate responsible for dominant energy dependence. should be explored.\ $\bullet$ I believe conclusions about the CG expansion will apply to other effective interactions. For example, [@achim], a soft potential obtained by integ... | system weakly bound (1.475 MeV) to breAking up as $\aLpha$+d, It iS noT oBvioUs thAt a two-body Lee-SUZuki Transformation would treAt the ReLEvanT jaCobi cOordinaTE rESPonSiBlE foR tHE dOminaNt eNergy dePendence. ThIs sHoUld be exploreD.\
$\BuLlet$ I belieVe tHe conclusionS abOut the cG ExpANsion WilL applY to othER e... | system weakly bound (1.47 5 MeV) tobreak ing up a s $\ alph a$+d, it is no t obv ious that a two-body L ee-Su zu k i tr a ns forma tion wo u ld t rea tth e r el e va nt Ja cob i coord inate resp ons ib le for the d o mi nant energ y d ependence. T his shoul dbee xplor ed. \
$\b ullet$ I beli eve the c on c lusion ... | system_weakly bound_(1.475 MeV) to breaking_up as_$\alpha$+d,_it is_not_obvious that a_two-body Lee-Suzuki transformation_would treat the relevant_Jacobi coordinate responsible_for_the dominant energy dependence. This should be explored.\
$\bullet$ I believe the conclusions about the_CG_expansion will_apply_to_other e... |
^m}\rightarrow 0$ as $m\rightarrow\infty$ since $w(\varepsilon)>0$, so since $A$ is complete, the sum $\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}$ converges in $A$, and hence $a^{p^m}\rightarrow a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}\in A$.\
So let $b:=a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}$, then $b^p=a... | ^m}\rightarrow 0 $ as $ m\rightarrow\infty$ since $ w(\varepsilon)>0 $, so since $ A$ is complete, the sum $ \underset{m\geq 0}{\sum}{\varepsilon^{p^m}}$ converges in $ A$, and hence $ a^{p^m}\rightarrow a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}\in A$ .\
So let $ b:=a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}$,... | ^m}\rihhtarrow 0$ as $m\rightarrow\lnfty$ since $w(\vargpwilon)>0$, vo sinde $A$ is zomplete, the sum $\underset{m\gee 0}{\wum}{\vaeepsilon^{p^m}}$ converges iv $A$, and hvnce $a^{p^m}\rughterrow a+\underset{m\jsq 0}{\sum}{\varepsimln^{p^m}}\nn A$.\
So let $b:=a+\undgrset{m\geq 0}{\suk}{\varepsilon^{p^m}}$, dhdn $b^p=a... | ^m}\rightarrow 0$ as $m\rightarrow\infty$ since $w(\varepsilon)>0$, so is the sum 0}{\sum}{\varepsilon^{p^m}}$ converges in 0}{\sum}{\varepsilon^{p^m}}\in So let $b:=a+\underset{m\geq then $b^p=a^p+(\underset{m\geq 0}{\sum}{\varepsilon^{p^m}})^p=a+\varepsilon+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}=b$ as required. ... | ^m}\rightarrow 0$ as $m\rightarrow\iNfty$ since $w(\VarepSilOn)>0$, sO sInce $a$ is cOmplete, the sum $\uNDersEt{m\geq 0}{\sum}{\varepsilon^{p^m}}$ cOnverGeS In $A$, aND hEnce $a^{P^m}\rightARrOW A+\unDeRsEt{m\GeQ 0}{\SuM}{\varePsiLon^{p^m}}\in a$.\
So let $b:=a+\unDerSeT{m\geq 0}{\sum}{\varePSiLon^{p^m}}$, then $b^P=a... | ^m}\rightarrow 0$ as $m\ri ghtarrow\i nfty$ si nce $ w(\v arep silon)>0$, sos ince $A$ is complete, thesum $ \u n ders e t{ m\geq 0}{\su m }{ \ v are ps il on^ {p ^ m} }$ co nve rges in $A$, andhen ce $a^{p^m}\ri g ht arrow a+\u nde rset{m\geq 0 }{\ sum}{\ va rep s ilon^ {p^ m}}\i n A$.\ So let $b:=a+\u nd e rse... | ^m}\rightarrow 0$_as $m\rightarrow\infty$_since $w(\varepsilon)>0$, so since_$A$ is_complete,_the sum_$\underset{m\geq_0}{\sum}{\varepsilon^{p^m}}$ converges in_$A$, and hence_$a^{p^m}\rightarrow a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}\in A$.\
So_let $b:=a+\underset{m\geq 0}{\sum}{\varepsilon^{p^m}}$,_then_$b^p=a... |
Communication
=============
According the above analysis, self-consistent is very important for category representation. However, even when $\mathfrak{A}^{\alpha}$ is self-consistent, $\mathfrak{A}^{\alpha}$ may be something wrong. For example, [@Putnam1963Brains] has stated that madmen sometimes have consistent delu... | Communication
= = = = = = = = = = = = =
According the above analysis, self - consistent is very important for category theatrical performance. However, even when $ \mathfrak{A}^{\alpha}$ is self - reproducible, $ \mathfrak{A}^{\alpha}$ may be something wrong. For example, [ @Putnam1963Brains ] has express that lun... |
Comlunication
=============
According the xbove analysis, self-convistenf is verh important for category repcesebtatiin. However, even when $\mxthfrak{A}^{\appha}$ is welf-ronsistent, $\mathfczk{A}^{\alpha}$ may gc somztiing wrong. For gxample, [@Putndm1963Brains] has sdageb that madmen sometimes have consistqnt deli... | Communication ============= According the above analysis, self-consistent important category representation. even when $\mathfrak{A}^{\alpha}$ something For example, [@Putnam1963Brains] stated that madmen have consistent delusional systems. Why? As final goal of a category is to help communication, different personal c... |
Communication
=============
According the Above analySis, seLf-cOnsIsTent Is veRy important for CAtegOry representation. HowevEr, eveN wHEn $\maTHfRak{A}^{\aLpha}$ is sELf-CONsiStEnT, $\maThFRaK{A}^{\alpHa}$ mAy be somEthing wronG. FoR eXample, [@Putnam1963bRaIns] has statEd tHat madmen somEtiMes havE cOnsIStent DelU... |
Communication
=========== ==
Accord ing t heabo ve ana lysi s, self-consis t entis very important forcateg or y rep r es entat ion. Ho w ev e r , e ve nwhe n$ \m athfr ak{ A}^{\al pha}$ is s elf -c onsistent, $ \ ma thfrak{A}^ {\a lpha}$ may b e s omethi ng wr o ng. F orexamp le, [@ P utnam1 963Brains ]h as sta t ... |
Communication
=============
According the_above analysis,_self-consistent is very important_for category_representation._However, even_when_$\mathfrak{A}^{\alpha}$ is self-consistent,_$\mathfrak{A}^{\alpha}$ may be_something wrong. For example,_[@Putnam1963Brains] has stated_that_madmen sometimes have consistent delu... |
_{{\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i}} \overline{J_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i},{\boldsymbol{w}}_{-{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i})]_{i=1}^Q$ arranged as a row vec... | _ { { \boldsymbol{w}}_{{{\ifmmode { \mathcal{C}}\else $ { \mathcal{C}}$\fi}}_i } } \overline{J_{{{\ifmmode { \mathcal{C}}\else $ { \mathcal{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmode { \mathcal{C}}\else $ { \mathcal{C}}$\fi}}_i},{\boldsymbol{w}}_{-{{\ifmmode { \mathcal{C}}\else $ { \mathcal{C}}$\fi}}_i})]_{i=1}^Q$ arra... | _{{\bolfsymbol{w}}_{{{\ifmmode {\mathcal{C}}\tlse ${\mathcal{C}}$\fi}}_i}} \overlinx{J_{{{\ifmmose {\mathcxl{C}}\else ${\mathcal{C}}$\fi}}_i}}({\boldsymbop{w}}_{{{\ufmmoee {\mathcal{C}}\else ${\mathcau{C}}$\fi}}_i},{\bolddymbol{w}}_{-{{\idmmove {\mathcal{C}}\else ${\mathcal{C}}$\nn}}_i})]_{i=1}^Q$ zvrangzd as a row vec... | _{{\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i}} \overline{J_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i})]_{i=1}^Q$ as a vector with $\zeta_i>0$. ${\mathcal{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmode ${\mathcal... | _{{\boldsymbol{w}}_{{{\ifmmode {\mathcal{c}}\else ${\mathcAl{C}}$\fi}}_I}} \ovErlInE{J_{{{\ifMmodE {\mathcal{C}}\else ${\mAThcaL{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmodE {\mathCaL{c}}\elsE ${\MaThcal{c}}$\fi}}_i},{\bolDSyMBOl{w}}_{-{{\IfMmOde {\MaTHcAl{C}}\elSe ${\mAthcal{C}}$\Fi}}_i})]_{i=1}^Q$ arranGed As A row vec... | _{{\boldsymbol{w}}_{{{\ifm mode {\mat hcal{ C}} \el se ${\ math cal{C}}$\fi}}_ i }} \ overline{J_{{{\ifmmode {\ma th c al{C } }\ else${\math c al { C }}$ \f i} }_i }} ( {\ bolds ymb ol{w}}_ {{{\ifmmod e { \m athcal{C}}\e l se ${\mathca l{C }}$\fi}}_i}, {\b oldsym bo l{w } }_{-{ {\i fmmod e {\ma t hcal{C }}\else $ {\... | _{{\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else_${\mathcal{C}}$\fi}}_i}} \overline{J_{{{\ifmmode_{\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i}}({\boldsymbol{w}}_{{{\ifmmode {\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i},{\boldsymbol{w}}_{-{{\ifmmode_{\mathcal{C}}\else ${\mathcal{C}}$\fi}}_i})]_{i=1}^Q$_arranged_as a_row_vec... |
) leads to a simple mathematical representation of the final basic equation (if a complex-valued state function is introduced). One may also say that the linearity of the equations is a consequence of this principle of maximal disorder. This is the deeper reason why it has been possible, see [@klein:schroedingers], to ... | ) leads to a simple mathematical representation of the concluding basic equality (if a building complex - valued state function is introduce). One may also say that the one-dimensionality of the equations is a consequence of this principle of maximal disorder. This is the deep reason why it has been possible, experienc... | ) lewds to a simple mathematlcal representatnin of vhe finzl basic equation (if a complex-valued srate dunction is introduced). One may wlso say thau the linearity oh the eqmctiona is c ronsequence of jhis principne of maximal giroxder. This is the deeper reason why ie has brej possible, see [@hleim:fchrkvdlngers], to ... | ) leads to a simple mathematical representation final equation (if complex-valued state function say the linearity of equations is a of this principle of maximal disorder. is the deeper reason why it has been possible, see [@klein:schroedingers], to derive equation from a set of assumptions including linearity. Besides... | ) leads to a simple mathematicaL representAtion Of tHe fInAl baSic eQuation (if a compLEx-vaLued state function is intRoducEd). oNe maY AlSo say That the LInEARitY oF tHe eQuATiOns is A coNsequenCe of this prIncIpLe of maximal dISoRder. This is The Deeper reason Why It has bEeN poSSible, See [@Klein:SchroeDIngers], To ... | ) leads to a simple mathem atical rep resen tat ion o f th e fi nal basic equa t ion(if a complex-valued s tatefu n ctio n i s int roduced ) .O n e m ay a lso s a ythatthe linear ity of the eq ua tions is a c o ns equence of th is principle of maxim al di s order . T his i s thed eeperreason wh yi t hasb een ... | ) leads_to a_simple mathematical representation of_the final_basic_equation (if_a_complex-valued state function_is introduced). One_may also say that_the linearity of_the_equations is a consequence of this principle of maximal disorder. This is the deeper_reason_why it_has_been_possible, see [@klein:schroedingers], to ... |
genceless condition implies that only two components are independent.
A similar procedure applies to the symmetric tensor $h_{ij}$, which can be decomposed as h\_[ij]{}=2C\_[ij]{}+2 \_i\_j E+2\_[(i]{}E\_[j)]{} +\_[ij]{}, \[hij\] with $\overline{E}^{ij}$ transverse and traceless (TT), i.e. $\nabla_i\overline{E}^{ij}=0$... | genceless condition implies that only two components are autonomous.
A exchangeable procedure applies to the symmetrical tensor $ h_{ij}$, which can be decomposed as h\_[ij]{}=2C\_[ij]{}+2 \_i\_j E+2\_[(i]{}E\_[j) ] { } + \_[ij ] { }, \[hij\ ] with $ \overline{E}^{ij}$ cross and traceless (TT), i.e. $ \nabla_i\overl... | genfeless condition implies that only two eimponeits are indepenaent.
A similar procedure applmes ro tht symmetric tensor $f_{ij}$, which can be eeconposed as h\_[mn]{}=2C\_[ij]{}+2 \_i\_j E+2\_[(i]{}E\_[j)]{} +\_[jm]{}, \[hij\] xith $\overline{E}^{im}$ transversa and tracelesv (GT), i.e. $\nabla_i\overline{E}^{ij}=0$... | genceless condition implies that only two components A procedure applies the symmetric tensor as \_i\_j E+2\_[(i]{}E\_[j)]{} +\_[ij]{}, with $\overline{E}^{ij}$ transverse traceless (TT), i.e. $\nabla_i\overline{E}^{ij}=0$ (transverse) and (traceless), and $E_i$ transverse. The parentheses around the indices denote sym... | genceless condition implies That only twO compOneNts ArE indEpenDent.
A similar prOCeduRe applies to the symmetriC tensOr $H_{Ij}$, whICh Can be DecompoSEd AS H\_[ij]{}=2c\_[iJ]{}+2 \_i\_J E+2\_[(i]{}e\_[j)]{} +\_[IJ]{}, \[hIj\] witH $\ovErline{E}^{Ij}$ transverSe aNd Traceless (TT), i.E. $\NaBla_i\overliNe{E}^{Ij}=0$... | genceless condition implie s that onl y two co mpo ne ntsareindependent.
A sim ilar procedure applies to t he symm e tr ic te nsor $h _ {i j } $,wh ic h c an be deco mpo sed ash\_[ij]{}= 2C\ _[ ij]{}+2 \_i\ _ jE+2\_[(i]{ }E\ _[j)]{} +\_[ ij] {}, \[ hi j\] with$\o verli ne{E}^ { ij}$ t ransverse a n d trac e less (... | genceless condition_implies that_only two components are_independent.
A similar_procedure_applies to_the_symmetric tensor $h_{ij}$,_which can be_decomposed as h\_[ij]{}=2C\_[ij]{}+2 \_i\_j_E+2\_[(i]{}E\_[j)]{} +\_[ij]{}, \[hij\]_with_$\overline{E}^{ij}$ transverse and traceless (TT), i.e. $\nabla_i\overline{E}^{ij}=0$... |
multiplying the second row by the scalar $x^{N-1}+x^{N-2}+\cdots+x+1\in R$ yields the zero element $(0,0) \in R^2$.
Acknowledgment {#acknowledgment.unnumbered}
==============
The authors gratefully acknowledge contributions to the proofs by Pascal Vontobel and Lance Small. The authors would also like to thank Darius... | multiplying the second row by the scalar $ x^{N-1}+x^{N-2}+\cdots+x+1\in R$ yields the zero element $ (0,0) \in R^2$.
Acknowledgment { # acknowledgment.unnumbered }
= = = = = = = = = = = = = =
The writer appreciatively acknowledge contributions to the proofs by Pascal Vontobel and Lance Small. The generator wou... | muptiplying the second row by the scalar $r^{B-1}+x^{N-2}+\cdovs+x+1\in R$ yields ghe zero element $(0,0) \in R^2$.
Acknowpeegmenu {#acknowledgment.unnjmbered}
==============
Thv authors grauefully acknowledjs contrlyutiohd to vhe proofs by Psscal Vontmbel and Lance Soapl. The authors would also like to trank Datiks... | multiplying the second row by the scalar yields zero element \in R^2$. Acknowledgment acknowledge to the proofs Pascal Vontobel and Small. The authors would also like thank Dariush Divsalar for useful discussions. [^1]: This work was presented in part the IEEE International Symposium on Information Theory, Austin, Texa... | multiplying the second row by The scalar $x^{n-1}+x^{N-2}+\cdOts+X+1\in r$ yIeldS the Zero element $(0,0) \in R^2$.
aCknoWledgment {#acknowledgmenT.unnuMbERed}
==============
THE aUthorS gratefULlY ACknOwLeDge CoNTrIbutiOns To the prOofs by PascAl VOnTobel and LancE smAll. The authOrs Would also likE to Thank DArIus... | multiplying the second ro w by the s calar $x ^{N -1 }+x^ {N-2 }+\cdots+x+1\i n R$yields the zero elemen t $(0 ,0 ) \in R^ 2$.
Acknowl e dg m e nt{# ac kno wl e dg ment. unn umbered }
======== === == =
The autho r sgratefully ac knowledge co ntr ibutio ns to the p roo fs by Pasca l Vonto bel and L an c e Smal ... | multiplying_the second_row by the scalar_$x^{N-1}+x^{N-2}+\cdots+x+1\in R$_yields_the zero_element_$(0,0) \in R^2$.
Acknowledgment_{#acknowledgment.unnumbered}
==============
The authors gratefully_acknowledge contributions to the_proofs by Pascal_Vontobel_and Lance Small. The authors would also like to thank Darius... |
likely to have come from EAS and to reduce to about 20 Hz the rate of events to be sent to the central station. All ToT triggers are directly promoted T2 whereas T1 threshold triggers are requested to pass a higher threshold of 3.2 $I_{VEM}^{est}$ in coincidence for 3 PMTs. Only T2 triggers are used for the definition... | likely to have come from EAS and to reduce to about 20 Hz the pace of event to be sent to the central place. All ToT trigger are directly promoted T2 whereas T1 doorway triggers are requested to evanesce a higher threshold of 3.2 $ I_{VEM}^{est}$ in concurrence for 3 PMTs. Only T2 triggers are used for the definition o... | linely to have come from EXS and to reducg ro abont 20 Hz fhe rate of events to be sent to the cwntrao station. All ToT trigeers are firectly pronited T2 whecsas T1 tmxeshomf trngjers are requesjed to pass d higher thresvoud of 3.2 $I_{VEM}^{est}$ in coincidence for 3 PMEs. Only T2 triggers are osed gjr tgv befinition... | likely to have come from EAS and to 20 Hz rate of events central All ToT triggers directly promoted T2 T1 threshold triggers are requested to a higher threshold of 3.2 $I_{VEM}^{est}$ in coincidence for 3 PMTs. Only T2 are used for the definition of a T3.\ The probability for a station pass trigger strongly on the inte... | likely to have come from EAS anD to reduce tO abouT 20 Hz The RaTe of EvenTs to be sent to thE CentRal station. All ToT triggeRs are DiREctlY PrOmoteD T2 whereAS T1 THResHoLd TriGgERs Are reQueSted to pAss a higher ThrEsHold of 3.2 $I_{VEM}^{esT}$ In CoincidencE foR 3 PMTs. Only T2 trIggErs are UsEd fOR the dEfiNitioN... | likely to have come fromEAS and to redu cetoab out20 H z the rate ofe vent s to be sent to the ce ntral s t atio n .All T oT trig g er s are d ir ect ly pr omote d T 2 where as T1 thre sho ld triggers ar e r equested t o p ass a higher th reshol dof3 .2 $I _{V EM}^{ est}$i n coin cidence f or 3 PMTs . Only T 2 ... | likely_to have_come from EAS and_to reduce_to_about 20_Hz_the rate of_events to be_sent to the central_station. All ToT_triggers_are directly promoted T2 whereas T1 threshold triggers are requested to pass a higher_threshold_of 3.2_$I_{VEM}^{est}$_in_coincidence for 3 PMTs. Only_T2 triggers are used for_the definition... |
in chemical engineering in 2013 from the University of New Hampshire, Durham, NH, USA. She received the M.S. degree in electrical and computer engineering in 2017 from Oregon State University, Corvallis, OR, USA, where she is currently working toward the Ph.D. degree in electrical and computer engineering.
She is cur... | in chemical engineering in 2013 from the University of New Hampshire, Durham, NH, USA. She received the M.S. degree in electrical and calculator technology in 2017 from Oregon State University, Corvallis, OR, USA, where she is presently working toward the Ph.D. degree in electric and computer engineering.
She is cur... | in chemical engineering in 2013 from the Univgrwity oh New Hzmpshire, Durham, NH, USA. She received vhe N.S. detree in electrical and computer engineeeing un 2017 from Ocsgon State Unjyersicy, Corvallis, OR, OSA, where sha is currently wurning toward the Ph.D. degree in electwical amd computer engigeerpnd.
She ps cur... | in chemical engineering in 2013 from the New Durham, NH, She received the computer in 2017 from State University, Corvallis, USA, where she is currently working the Ph.D. degree in electrical and computer engineering. She is currently a Graduate at Idaho National Laboratory, Idaho Falls, ID, USA. Her research interests... | in chemical engineering in 2013 frOm the UniveRsity Of NEw HAmPshiRe, DuRham, NH, USA. She reCEiveD the M.S. degree in electricAl and CoMPuteR EnGineeRing in 2017 fROm oREgoN STaTe UNiVErSity, COrvAllis, OR, uSA, where shE is CuRrently workiNG tOward the Ph.d. deGree in electrIcaL and coMpUteR EnginEerIng.
ShE is cur... | in chemical engineering i n 2013 fro m the Un ive rs ityof N ew Hampshire,D urha m, NH, USA. She receiv ed th eM .S.d eg ree i n elect r ic a l an dco mpu te r e ngine eri ng in 2 017 from O reg on State Unive r si ty, Corval lis , OR, USA, w her e sheis cu r rentl y w orkin g towa r d thePh.D. deg re e in el e ... | in_chemical engineering_in 2013 from the_University of_New_Hampshire, Durham,_NH,_USA. She received_the M.S. degree_in electrical and computer_engineering in 2017_from_Oregon State University, Corvallis, OR, USA, where she is currently working toward the Ph.D._degree_in electrical_and_computer_engineering.
She is cur... |
= 2^{-18} \lambda^{-2}$.
\[lem:stability\] Let $\lambda {\geqslant}3$ and $n,k \in {\mathbb{N}}$, with $k {\geqslant}(\log n)^4$ and $k {\geqslant}2^{480} \lambda^{20}$. We have $$|\Lambda \setminus \Lambda^*| {\leqslant}\frac{n^2}{k} \cdot |{\mathcal{I}}| + \exp\bigg( - \frac{\delta k}{2^{10} \lambda} \bigg) {\lambd... | = 2^{-18 } \lambda^{-2}$.
\[lem: stability\ ] Let $ \lambda { \geqslant}3 $ and $ n, k \in { \mathbb{N}}$, with $ k { \geqslant}(\log n)^4 $ and $ k { \geqslant}2^{480 } \lambda^{20}$. We have $ $ |\Lambda \setminus \Lambda^*| { \leqslant}\frac{n^2}{k } \cdot |{\mathcal{I}}| + \exp\bigg (- \frac{\delta k}{2^{10 } \l... | = 2^{-18} \pambda^{-2}$.
\[lem:stability\] Let $\lxmbda {\geqslant}3$ cbd $n,k \mn {\mathgb{N}}$, with $k {\geqslant}(\log n)^4$ and $k {\geqslent}2^{480} \oambdq^{20}$. We have $$|\Lambda \setmivus \Lambdw^*| {\leqslabt}\frec{n^2}{k} \cdot |{\mathcal{I}}| + \exp\blyg( - \fdwc{\denva k}{2^{10} \lambda} \bign) {\lambd... | = 2^{-18} \lambda^{-2}$. \[lem:stability\] Let $\lambda {\geqslant}3$ \in with $k n)^4$ and $k \setminus {\leqslant}\frac{n^2}{k} \cdot |{\mathcal{I}}| \exp\bigg( - \frac{\delta \lambda} \bigg) {\lambda k / 2 k}.$$ To prove Lemma \[lem:stability\], we will successively refine $\Lambda \setminus \Lambda^*$, at step show... | = 2^{-18} \lambda^{-2}$.
\[lem:stability\] Let $\lambDa {\geqslant}3$ And $n,k \In {\mAthBb{n}}$, witH $k {\geQslant}(\log n)^4$ and $k {\GEqslAnt}2^{480} \lambda^{20}$. We have $$|\Lambda \sEtminUs \lAmbdA^*| {\LeQslanT}\frac{n^2}{k} \CDoT |{\MAthCaL{I}}| + \Exp\BiGG( - \fRac{\deLta K}{2^{10} \lambda} \Bigg) {\lambd... | = 2^{-18} \lambda^{-2}$.
\[lem:sta bilit y\] Le t$\la mbda {\geqslant}3$ and$n,k \in {\mathbb{N}}$ , wit h$ k {\ g eq slant }(\logn )^ 4 $ an d$k {\ ge q sl ant}2 ^{4 80} \la mbda^{20}$ . W ehave $$|\Lam b da \setminus \L ambda^*| {\l eqs lant}\ fr ac{ n ^2}{k } \ cdot|{\mat h cal{I} }| + \exp \b i gg( -\ frac{\... | =_2^{-18} \lambda^{-2}$.
\[lem:stability\]_Let $\lambda {\geqslant}3$ and_$n,k \in_{\mathbb{N}}$,_with $k_{\geqslant}(\log_n)^4$ and $k_{\geqslant}2^{480} \lambda^{20}$. We_have $$|\Lambda \setminus \Lambda^*|_{\leqslant}\frac{n^2}{k} \cdot |{\mathcal{I}}|_+_\exp\bigg( - \frac{\delta k}{2^{10} \lambda} \bigg) {\lambd... |
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rho_{a_1}(W)}$ such that ${\alpha}_1(u_1) = {\alpha}_1(u_2)$ and if $P_{\rho_{a_1}(W)}(\operatorname{\textsf{fact}}, u_1, u_2) = s_1 s_2 s_3$, then $$\label{in7}
\min(|s_2|, |s_1|+|s_3|) \ge \tfrac 13 | {\partial}\rho_{a_1}({\Delta})| = \tfrac 13 |W|_{\bar a_1} \.$$ It follows from definitions and Lemma \[b1\] that... | rho_{a_1}(W)}$ such that $ { \alpha}_1(u_1) = { \alpha}_1(u_2)$ and if $ P_{\rho_{a_1}(W)}(\operatorname{\textsf{fact } }, u_1, u_2) = s_1 s_2 s_3 $, then $ $ \label{in7 }
\min(|s_2|, |s_1|+|s_3|) \ge \tfrac 13 | { \partial}\rho_{a_1}({\Delta})| = \tfrac 13 |W|_{\bar a_1 } \.$$ It follows from definitions and... | rho_{w_1}(W)}$ such that ${\alpha}_1(u_1) = {\alpma}_1(u_2)$ and if $P_{\rho_{a_1}(C)}(\iperatmrname{\fextsf{fazt}}, u_1, u_2) = s_1 s_2 s_3$, then $$\label{in7}
\mmn(|s_2|, |s_1|+|s_3|) \gt \tfrac 13 | {\partial}\rhu_{a_1}({\Delta})| = \tfrac 13 |W|_{\bac a_1} \.$$ It follows hdom defluitiohd anb Oemma \[b1\] that... | rho_{a_1}(W)}$ such that ${\alpha}_1(u_1) = {\alpha}_1(u_2)$ and u_1, = s_1 s_3$, then $$\label{in7} | = \tfrac 13 a_1} \.$$ It from definitions and Lemma \[b1\] that are some vertices $v_1, v_2 \in P_W$ such that $\beta(v_j) = u_j$, $j=1,2$, vertices ${\alpha}(v_1), {\alpha}(v_2)$ belong to ${\partial}\Gamma$, where $... | rho_{a_1}(W)}$ such that ${\alpha}_1(u_1) = {\alpha}_1(u_2)$ And if $P_{\rho_{a_1}(w)}(\operAtoRnaMe{\TextSf{faCt}}, u_1, u_2) = s_1 s_2 s_3$, then $$\labEL{in7}
\mIn(|s_2|, |s_1|+|s_3|) \ge \tfrac 13 | {\partial}\rho_{A_1}({\DeltA})| = \tFRac 13 |W|_{\BAr A_1} \.$$ It foLlows frOM dEFIniTiOnS anD LEMmA \[b1\] thaT... | rho_{a_1}(W)}$ such that $ {\alpha}_1 (u_1) ={\a lp ha}_ 1(u_ 2)$ and if $P_ { \rho _{a_1}(W)}(\operatorna me{\t ex t sf{f a ct }}, u _1, u_2 ) = s_1 s _2 s_ 3$ , t hen $ $\l abel{in 7}
\min(|s _2| , |s_1|+|s_3| ) \ ge \tfrac13| {\partial} \rh o_{a_1 }( {\D e lta}) | = \tf rac 13 |W|_{ \bar a_1} \ . $$ Itf ol... | rho_{a_1}(W)}$ such_that ${\alpha}_1(u_1)_= {\alpha}_1(u_2)$ and if_$P_{\rho_{a_1}(W)}(\operatorname{\textsf{fact}}, u_1,_u_2)_ =_s_1_s_2 s_3$, then_$$\label{in7}
\min(|s_2|, |s_1|+|s_3|)_\ge \tfrac 13 |_{\partial}\rho_{a_1}({\Delta})| = _\tfrac_13 |W|_{\bar a_1} \.$$ It follows from definitions and Lemma \[b1\] that... |
exp(a|\xi|_p^{2}).$$
See [@kuo], Theorem 8.6.
Limit theorems for Volterra processes {#volterra}
=====================================
One-dimensional case
--------------------
Consider a Volterra process $B=( B_t )_{t\geq 0}$ of the form $$\label{star}
B_t=\int_0^t K(t,s)\,\mathrm{d}W_s,$$ where $K(t,s)$ satisfies ... | exp(a|\xi|_p^{2}).$$
See [ @kuo ], Theorem 8.6.
Limit theorems for Volterra processes { # volterra }
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
One - dimensional case
--------------------
Consider a Volterra procedure $ B= (B_t) _ { t\geq 0}$ of the human body $ $ \label{s... | exp(w|\xi|_p^{2}).$$
See [@kuo], Theorem 8.6.
Limiu theorems for Volterra 'rocessss {#voltefra}
=====================================
One-dimensional case
--------------------
Considxr a Volttgra process $B=( B_t )_{t\geq 0}$ of the vorm $$\labwl{ster}
B_t=\int_0^t K(t,s)\,\mathcj{d}W_s,$$ whcxe $K(t,a)$ satnshies ... | exp(a|\xi|_p^{2}).$$ See [@kuo], Theorem 8.6. Limit theorems processes ===================================== One-dimensional -------------------- Consider a 0}$ the form $$\label{star} K(t,s)\,\mathrm{d}W_s,$$ where $K(t,s)$ $\int_0^tK(t,s)^2 \,\mathrm{d}s<\infty$ for all $t>0$ and is the Brownian motion defined on the... | exp(a|\xi|_p^{2}).$$
See [@kuo], Theorem 8.6.
Limit Theorems foR VoltErrA prOcEsseS {#volTerra}
=====================================
One-dimensIOnal Case
--------------------
Consider a Volterra pRocesS $B=( b_T )_{t\geQ 0}$ Of The foRm $$\label{STaR}
b_T=\inT_0^t k(t,S)\,\maThRM{d}w_s,$$ wheRe $K(T,s)$ satisFies ... | exp(a|\xi|_p^{2}).$$
See[@kuo], Th eorem 8. 6.
L imit the orems for Volt e rraprocesses {#volterra}===== == = ==== = == ===== ======= = == = = === ==
One -d i me nsion alcase
-- ---------- --- -- ---
Conside r a Volterrapro cess $B=( B_ t ) _{t\ge q0}$ of th e f orm $ $\labe l {star}
B_t=\int _0 ^ t K(t, s )\,\... | exp(a|\xi|_p^{2}).$$
See [@kuo],_Theorem 8.6.
Limit_theorems for Volterra processes_{#volterra}
=====================================
One-dimensional case
--------------------
Consider_a_Volterra process_$B=(_B_t )_{t\geq 0}$_of the form_$$\label{star}
B_t=\int_0^t K(t,s)\,\mathrm{d}W_s,$$ where $K(t,s)$_satisfies ... |
within the supergravity inspired constrained minimal supersymmetric model. We perform a global statistical $\chi^2$ analysis of these data and show that the allowed region of parameter space is bounded from below by the Higgs limit, which depends on the trilinear coupling and from above by the anomalous magnetic momen... | within the supergravity inspired constrained minimal supersymmetric model. We perform a ball-shaped statistical $ \chi^2 $ psychoanalysis of these datum and show that the allowed region of argument space is bounded from below by the Higgs terminus ad quem, which count on the trilinear coupling and from above by the ano... | wihhin the supergravity inrpired constraiuwd minmmal sulersymmegric model. We perform a globel sratisupcal $\chi^2$ analysis of ghese datw and shiw tiat the allowed csgion on parzletex wpace is boundgd from belof by the Higgs lkmnt, which depends on the trilinear cotpling snf from above bi the wnomzlous magnetic momen... | within the supergravity inspired constrained minimal supersymmetric perform global statistical analysis of these allowed of parameter space bounded from below the Higgs limit, which depends on trilinear coupling and from above by the anomalous magnetic moment $a_\mu$. The newest \to X_s \gamma~ $]{}data deviate 1.7 $\s... | within the supergravity inspIred constrAined MinImaL sUperSymmEtric model. We peRForm A global statistical $\chi^2$ aNalysIs OF theSE dAta anD show thAT tHE AllOwEd RegIoN Of ParamEteR space iS bounded frOm bElOw by the Higgs LImIt, which depEndS on the trilinEar CoupliNg And FRom abOve By the AnomalOUs magnEtic momen... | within the supergravity i nspired co nstra ine d m in imal sup ersymmetric mo d el.We perform a global st atist ic a l $\ c hi ^2$ a nalysis of t hes eda taan d s how t hat the al lowed regi onof parameter s p ac e is bound edfrom below b y t he Hig gs li m it, w hic h dep ends o n the t rilinearco u plinga nd f... | within_the supergravity_inspired constrained minimal supersymmetric_model. We_perform_a global_statistical_$\chi^2$ analysis of_these data and_show that the allowed_region of parameter_space_is bounded from below by the Higgs limit, which depends on the trilinear coupling_and_from above_by_the_anomalous magnetic momen... |
=1,2$. Assume that the numbers $ b_1(3), b_2(3)$ satisfy the condition $b_1(3) b_2(3) \ge 0$. We attach ${\Delta}_{b_1}$ to ${\Delta}_{b_2}$ by identification of the vertices $(s_1)_+$ and $(s_2)_-$ and obtain thereby a disk diagram ${\Delta}'_b$ such that $ {\partial}{\Delta}'_b = s_1 s_2 (r_1 r_2)^{-1}$ and $|{\Delt... | = 1,2$. Assume that the numbers $ b_1(3), b_2(3)$ satisfy the condition $ b_1(3) b_2(3) \ge 0$. We bind $ { \Delta}_{b_1}$ to $ { \Delta}_{b_2}$ by designation of the vertices $ (s_1)_+$ and $ (s_2)_-$ and obtain thereby a disk diagram $ { \Delta}'_b$ such that $ { \partial}{\Delta}'_b = s_1 s_2 (r_1 r_2)^{-1}$ and $... | =1,2$. Asdume that the numbers $ b_1(3), b_2(3)$ satisfy the einditimn $b_1(3) b_2(3) \ge 0$. We xttach ${\Delta}_{b_1}$ to ${\Delta}_{b_2}$ by iventuficaupon of the vertices $(s_1)_+$ and $(s_2)_-$ anf obtain theceby a disk diagczm ${\Delta}'_b$ sucg thac $ {\partial}{\Delta}'_b = s_1 s_2 (r_1 r_2)^{-1}$ and $|{\Delt... | =1,2$. Assume that the numbers $ b_1(3), the $b_1(3) b_2(3) 0$. We attach of vertices $(s_1)_+$ and and obtain thereby disk diagram ${\Delta}'_b$ such that $ = s_1 s_2 (r_1 r_2)^{-1}$ and $|{\Delta}'_b(2) | = |{\Delta}_{b_1}(2) | + |{\Delta}_{b_2}(2) see Fig. 6.10(a). Note that ${\varphi}( a(c) ) \equiv {\varphi}( s_1 ... | =1,2$. Assume that the numbers $ b_1(3), b_2(3)$ satIsfy the conDitioN $b_1(3) b_2(3) \Ge 0$. WE aTtacH ${\DelTa}_{b_1}$ to ${\Delta}_{b_2}$ by iDEntiFication of the vertices $(s_1)_+$ And $(s_2)_-$ aNd OBtaiN ThEreby A disk diAGrAM ${\delTa}'_B$ sUch ThAT $ {\pArtiaL}{\DeLta}'_b = s_1 s_2 (r_1 R_2)^{-1}$ and $|{\Delt... | =1,2$. Assume that the num bers $ b_1 (3),b_2 (3) $sati sfythe condition$ b_1( 3) b_2(3) \ge 0$. We a ttach $ { \Del t a} _{b_1 }$ to $ { \D e l ta} _{ b_ 2}$ b y i denti fic ation o f the vert ice s$(s_1)_+$ an d $ (s_2)_-$ a ndobtain there bya disk d iag r am ${ \De lta}' _b$ su c h that $ {\part ia l }{\Del t ... | =1,2$. Assume_that the_numbers $ b_1(3), b_2(3)$_satisfy the_condition_$b_1(3) b_2(3)_\ge_0$. We attach_${\Delta}_{b_1}$ to ${\Delta}_{b_2}$_by identification of the_vertices $(s_1)_+$ and_$(s_2)_-$_and obtain thereby a disk diagram ${\Delta}'_b$ such that $ {\partial}{\Delta}'_b = s_1 s_2__(r_1 r_2)^{-1}$_and_$|{\Delt... |
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S. Kurokawa and E. Kikutani_[*et al.*]{}, Nucl._Instr. and Meth. A_[**499**]{}, 1 (2003).
Belle_Collaboration,_A. Abashian [*et al.*]{}, Nucl. Instr. and Me... |
$ vertices of the first type of $\Gamma$ and hitting the origin: by the previous arguments, we know that $p\leq (n-1)+1=n$, because the vertex with an edge to $\infty$ has an additional edge, which may or may not hit the origin. The polynomial degree of the differential operator associated to $\Gamma$ equals $n-(2n-1-p... | $ vertices of the first type of $ \Gamma$ and hitting the beginning: by the former arguments, we know that $ p\leq (n-1)+1 = n$, because the vertex with an boundary to $ \infty$ has an extra edge, which may or may not hit the lineage. The polynomial academic degree of the differential operator associated to $ \Gamma$ e... | $ vegtices of the first type of $\Gamma$ and hnrting vhe orifin: by tfe previous arguments, we knox thqt $p\ltz (n-1)+1=n$, because the vdrtex witj an edgw to $\unfty$ has eh additljnal cdge, chmch may or may kot hit the origin. The ponyvolial degree of the differential opewator axslciated to $\Gamia$ eatals $n-(2n-1-p... | $ vertices of the first type of hitting origin: by previous arguments, we the with an edge $\infty$ has an edge, which may or may not the origin. The polynomial degree of the differential operator associated to $\Gamma$ equals which must be greater or equal than $0$: this forces immediately $p\geq n-1$. fact, with prev... | $ vertices of the first type of $\GAmma$ and hitTing tHe oRigIn: By thE preVious arguments, WE knoW that $p\leq (n-1)+1=n$, because the vErtex WiTH an eDGe To $\infTy$ has an ADdITIonAl EdGe, wHiCH mAy or mAy nOt hit thE origin. The PolYnOmial degree oF ThE differentIal Operator assoCiaTed to $\GAmMa$ eQUals $n-(2N-1-p... | $ vertices of the first ty pe of $\Ga mma$and hi tt ingtheorigin: by the prev ious arguments, we kno w tha t$ p\le q ( n-1)+ 1=n$, b e ca u s e t he v ert ex wi th an ed ge to $ \infty$ ha s a nadditional e d ge , which ma y o r may not hi t t he ori gi n.T he po lyn omial degre e of th e differe nt i al ope r at... | $ vertices_of the_first type of $\Gamma$_and hitting_the_origin: by_the_previous arguments, we_know that $p\leq_(n-1)+1=n$, because the vertex_with an edge_to_$\infty$ has an additional edge, which may or may not hit the origin. The_polynomial_degree of_the_differential_operator associated to $\Gamma$ equals_$n-(2n-1-p... |
H)$ and assume for some $f\in \PF'(H)$, $\RF(f)$ has only one positive entry in each row. Then we have:
1. After suitable permutation of indices, we can assume $$\RF(f) = \left( \begin{array}{cccc} -1 & \alpha_2-1 & 0 & 0\\
0 & -1 & \al_3-1 & 0 \\
0 & 0 & -1 & \al_4-1\\
\al_1-1 & 0 & 0 & -1 \end{array}\ri... | H)$ and assume for some $ f\in \PF'(H)$, $ \RF(f)$ has only one positive entrance in each rowing. Then we have:
1. After suitable permutation of index, we can assume $ $ \RF(f) = \left (\begin{array}{cccc } -1 & \alpha_2 - 1 & 0 & 0\\
0 & -1 & \al_3 - 1 & 0 \\
0 & 0 & -1 & \al_4 - 1\\
\al_1 - 1 &... | H)$ ajd assume for some $f\in \PN'(H)$, $\RF(f)$ has only one posmtive ehtry in dach row. Then we have:
1. After syitaboe permutation of indizes, we caj assume $$\RF(f) = \left( \begii{zrray}{ccge} -1 & \amiha_2-1 & 0 & 0\\
0 & -1 & \al_3-1 & 0 \\
0 & 0 & -1 & \dl_4-1\\
\al_1-1 & 0 & 0 & -1 \evd{crray}\ri... | H)$ and assume for some $f\in \PF'(H)$, only positive entry each row. Then permutation indices, we can $$\RF(f) = \left( -1 & \alpha_2-1 & 0 & 0 & -1 & \al_3-1 & 0 \\ 0 & 0 & -1 \al_4-1\\ \al_1-1 & 0 & 0 & -1 \end{array}\right).$$ 2. In this case, we $f' then = \left( \begin{array}{cccc} -1 & \alpha_2-2 & \al_3-1 & 0\\... | H)$ and assume for some $f\in \PF'(H)$, $\RF(F)$ has only onE posiTivE enTrY in eAch rOw. Then we have:
1. AfTEr suItable permutation of indIces, wE cAN assUMe $$\rF(f) = \leFt( \begin{ARrAY}{CccC} -1 & \aLpHa_2-1 & 0 & 0\\
0 & -1 & \aL_3-1 & 0 \\
0 & 0 & -1 & \aL_4-1\\
\Al_1-1 & 0 & 0 & -1 \End{arRay}\Ri... | H)$ and assume for some $f \in \PF'(H )$, $ \RF (f) $hasonly one positivee ntry in each row. Then wehave:
1 . A f te r sui table p e rm u t ati on o f i nd i ce s, we ca n assum e $$\RF(f) =\l eft( \begin{ a rr ay}{cccc}-1& \alpha_2-1 &0 & 0\ \ 0 & - 1 & \al_ 3-1 &0 \\
0 & 0 & - 1 & \al _ 4-1\\
\ al_1 ... | H)$ and_assume for_some $f\in \PF'(H)$, $\RF(f)$_has only_one_positive entry_in_each row. Then_we have:
1. _After suitable permutation of_indices, we can_assume_$$\RF(f) = \left( \begin{array}{cccc} -1 & \alpha_2-1 & 0 & 0\\
_0_& -1_&_\al_3-1_& 0 \\
_ 0 & 0 &_-1 &_\al_4-1\\
\al_1-1 & 0 &_0_& -1 \end{array}\ri... |
m)^*S^0$ for some $x\in S^0$;\
$(2)$ $xax=x$, $^\circ{x}=\!\!~^\circ(a^m)$, $^\circ(x^*)\subseteq\!\!~^\circ(a^m)$ and $^\circ(a^m)^* \subseteq\!\!~^\circ (x^m)$ for some $x\in S^0$.
Suppose $a$ is \*-DMP with index $m$. Then $a^{\scriptsize\textcircled{\tiny D}_m},~(a^m)^{\dag}$ exist with $(a^{\scriptsize\textcircl... | m)^*S^0 $ for some $ x\in S^0$;\
$ (2)$ $ xax = x$, $ ^\circ{x}=\!\!~^\circ(a^m)$, $ ^\circ(x^*)\subseteq\!\!~^\circ(a^m)$ and $ ^\circ(a^m)^ * \subseteq\!\!~^\circ (x^m)$ for some $ x\in S^0$.
Suppose $ a$ is \*-DMP with index $ m$. Then $ a^{\scriptsize\textcircled{\tiny D}_m},~(a^m)^{\dag}$ exist with $ (a^{... | m)^*S^0$ vor some $x\in S^0$;\
$(2)$ $xax=x$, $^\circ{x}=\!\!~^\girc(a^m)$, $^\circ(x^*)\subsgtwq\!\!~^\circ(e^m)$ and $^\dirc(a^m)^* \sjbseteq\!\!~^\circ (x^m)$ for some $x\in D^0$.
Syppost $a$ is \*-DMP with inddx $m$. Then $a^{\scriptwize\uextcircled{\tiny D}_m},~(a^m)^{\dag}$ ewnst wjbh $(a^{\sermptsize\textcirck... | m)^*S^0$ for some $x\in S^0$;\ $(2)$ $xax=x$, and \subseteq\!\!~^\circ (x^m)$ some $x\in S^0$. index Then $a^{\scriptsize\textcircled{\tiny D}_m},~(a^m)^{\dag}$ with $(a^{\scriptsize\textcircled{\tiny D}_m})^m=(a^m)^{\dag}$ Lemma \[1\]. Take $x=a^{\scriptsize\textcircled{\tiny D}_m}$, then by Lemma \[7\]. Further, from... | m)^*S^0$ for some $x\in S^0$;\
$(2)$ $xax=x$, $^\circ{x}=\!\!~^\cirC(a^m)$, $^\circ(x^*)\suBseteQ\!\!~^\ciRc(a^M)$ aNd $^\ciRc(a^m)^* \Subseteq\!\!~^\circ (x^m)$ FOr soMe $x\in S^0$.
Suppose $a$ is \*-DMP witH indeX $m$. tHen $a^{\SCrIptsiZe\textcIRcLED{\tiNy d}_m},~(A^m)^{\dAg}$ EXiSt witH $(a^{\sCriptsiZe\textcircL... | m)^*S^0$ for some $x\in S^ 0$;\
$(2)$ $xax =x$ , $ ^\ circ {x}= \!\!~^\circ(a^ m )$,$^\circ(x^*)\subseteq\ !\!~^ \c i rc(a ^ m) $ and $^\cir c (a ^ m )^* \ su bse te q \! \!~^\ cir c (x^m) $ for some $x \i n S^0$.
Su p po se $a$ is\*- DMP with ind ex$m$. T he n $ a ^{\sc rip tsize \textc i rcled{ \tiny D}_ m} , ~... | m)^*S^0$ for_some $x\in_S^0$;\
$(2)$ $xax=x$, $^\circ{x}=\!\!~^\circ(a^m)$, $^\circ(x^*)\subseteq\!\!~^\circ(a^m)$ and_$^\circ(a^m)^* \subseteq\!\!~^\circ_(x^m)$_for some_$x\in_S^0$.
Suppose $a$ is_\*-DMP with index_$m$. Then $a^{\scriptsize\textcircled{\tiny D}_m},~(a^m)^{\dag}$_exist with $(a^{\scriptsize\textcircl... |
:= \{ {\mathbf y}\in (L')^{n-1} \ | \langle {\mathbf y}, {\mathbf y}\rangle = T_{2} \text{ and } {\mathbb D}^+_{y} \cap \overline U = \emptyset \}.$$
Using the estimates and, and standard arguments for convergence of theta series, it follows that the sum $$\label{eqn:xi_0 sum}
\sum_{T = {\left( \begin{small... | : = \ { { \mathbf y}\in (L')^{n-1 } \ | \langle { \mathbf y }, { \mathbf y}\rangle = T_{2 } \text { and } { \mathbb D}^+_{y } \cap \overline U = \emptyset \}.$$
Using the estimates and, and standard arguments for overlap of theta serial, it stick to that the sum $ $ \label{eqn: xi_0 kernel }
\sum_{T = ... | := \{ {\mathbf y}\in (L')^{n-1} \ | \langle {\oathbf y}, {\mathbf y}\ranglx = T_{2} \test{ and } {\oathbb D}^+_{y} \cap \overline U = \em'tyswt \}.$$
Uwing the estimates and, and stanfard argymenus for convergencx of thebc serjcs, it hollows that thg sum $$\label{exn:xi_0 sum}
\rul_{T = {\left( \begin{small... | := \{ {\mathbf y}\in (L')^{n-1} \ | y}, y}\rangle = \text{ and } = \}.$$ Using the and, and standard for convergence of theta series, it that the sum $$\label{eqn:xi_0 sum} \sum_{T = {\left( \begin{smallmatrix}* & * \\ * T_2 \end{smallmatrix} \right)}} \sum_{\substack{({\mathbf x}_1, {\mathbf y}) \in \Omega(T) \\ {\mat... | := \{ {\mathbf y}\in (L')^{n-1} \ | \langle {\mathbf y}, {\mAthbf y}\rangLe = T_{2} \teXt{ aNd } {\mAtHbb D}^+_{Y} \cap \Overline U = \emptySEt \}.$$
UsIng the estimates and, and sTandaRd ARgumENtS for cOnvergeNCe OF TheTa SeRieS, iT FoLlows ThaT the sum $$\Label{eqn:xi_0 Sum}
\SuM_{T = {\left( \begin{sMAlL... | := \{ {\mathbf y}\in (L' )^{n-1} \| \la ngl e { \m athb f y} , {\mathbf y}\ r angl e = T_{2} \text{ and } {\ma th b b D} ^ +_ {y} \ cap \ov e rl i n e U = \ emp ty s et \}. $$
Usingthe estima tes a nd, and stan d ar d argument s f or convergen ceof the ta se r ies,itfollo ws tha t the s um $$\lab el { eqn:xi ... | :=_ \{_{\mathbf y}\in (L')^{n-1} \_| \langle_{\mathbf_y}, {\mathbf_y}\rangle_= T_{2} \text{_and } {\mathbb_D}^+_{y} \cap \overline U_= \emptyset _\}.$$
Using_the estimates and, and standard arguments for convergence of theta series, it follows that_the_sum $$\label{eqn:xi_0_sum}
__ _ \sum_{T = {\left( \begin{small... |
_i(s)^2 ds \right) \right| \geq\epsilon_n\right] \nonumber\\
\label{eq.estgitntnpl1}
&= 4\,\mathbb{P}\left[ B_{i,n}^*\left( \int_{t_n}^{t_{n+1}}g_i(s)^2 ds \right) \geq\epsilon_n\right]
= 4\,\left\{ 1-\Phi\left(\frac{\epsilon_n}{\sqrt{\int_{t_n}^{t_{n+1}}g_i(s)^2 ds}} \right) \right\},\end{aligned}$$ where ... | _ i(s)^2 ds \right) \right| \geq\epsilon_n\right ] \nonumber\\
\label{eq.estgitntnpl1 }
& = 4\,\mathbb{P}\left [ B_{i, n}^*\left (\int_{t_n}^{t_{n+1}}g_i(s)^2 ds \right) \geq\epsilon_n\right ]
= 4\,\left\ { 1-\Phi\left(\frac{\epsilon_n}{\sqrt{\int_{t_n}^{t_{n+1}}g_i(s)^2 ds } } \right) \right\},\end{a... | _i(s)^2 fs \right) \right| \geq\epsilok_n\right] \nonumber\\
\labxl{eq.estfitntnpl1}
&= 4\,\mathbb{P}\left[ B_{i,n}^*\left( \int_{v_n}^{t_{n+1}}t_i(s)^2 dw \right) \geq\epsilon_n\rigft]
= 4\,\levt\{ 1-\Phi\ledt(\frec{\epsilon_n}{\sqrt{\inv_{f_n}^{t_{n+1}}g_i(s)^2 ds}} \rifmt) \riyhv\},\end{aligned}$$ whete ... | _i(s)^2 ds \right) \right| \geq\epsilon_n\right] \nonumber\\ \label{eq.estgitntnpl1} B_{i,n}^*\left( ds \right) = 4\,\left\{ 1-\Phi\left(\frac{\epsilon_n}{\sqrt{\int_{t_n}^{t_{n+1}}g_i(s)^2 have the fact that s\leq t} W(s)$ the same distribution as $|W(t)|$ when is a standard Brownian motion, the symmetry of the distri... | _i(s)^2 ds \right) \right| \geq\epsilon_n\Right] \nonumBer\\
\laBel{Eq.eStGitnTnpl1}
&= 4\,\Mathbb{P}\left[ B_{i,n}^*\LEft( \iNt_{t_n}^{t_{n+1}}g_i(s)^2 ds \right) \geq\epsIlon_n\RiGHt]
= 4\,\leFT\{ 1-\PHi\lefT(\frac{\epSIlON_N}{\sqRt{\InT_{t_n}^{T_{n+1}}G_I(s)^2 Ds}} \rigHt) \rIght\},\end{Aligned}$$ wheRe ... | _i(s)^2 ds \right) \right| \geq\epsi lon_n \ri ght ]\non umbe r\\
\label { eq.e stgitntnpl1}
&= 4\ ,\mat hb b {P}\ l ef t[ B_ {i,n}^* \ le f t ( \ in t_ {t_ n} ^ {t _{n+1 }}g _i(s)^2 ds \right ) \ ge q\epsilon_n\ r ig ht]
=4\, \left\{ 1-\P hi\ left(\ fr ac{ \ epsil on_ n}{\s qrt{\i n t_{t_n }^{t_{n+1 }} g... | _i(s)^2 ds_\right) \right|_\geq\epsilon_n\right] \nonumber\\
_ \label{eq.estgitntnpl1}
__ _&=_4\,\mathbb{P}\left[ B_{i,n}^*\left( \int_{t_n}^{t_{n+1}}g_i(s)^2_ds \right) \geq\epsilon_n\right]
_ =_4\,\left\{ 1-\Phi\left(\frac{\epsilon_n}{\sqrt{\int_{t_n}^{t_{n+1}}g_i(s)^2 ds}}_\right)_\right\},\end{aligned}$$ where ... |
bottleneck is the sample efficiency in the real environment and computation in the approximate model is relatively cheap.
Despite the computational intensity, in the next subsection we provide some proof-of-concept experimental results showing the variance reduction benefits of the new estimator compared to prior bas... | bottleneck is the sample efficiency in the real environment and computation in the approximate exemplar is relatively brassy.
Despite the computational intensity, in the next subsection we provide some proof - of - concept experimental solution showing the variance decrease benefits of the new estimator compare to p... | bohtleneck is the sample enficiency in the real eivironmsnt and zomputation in the approximave midel us relatively cheap.
Desoite the bomputatiinal untensity, mh the ncrt sugdectnoi we provide soke proof-of-woncept experikevtcl results showing the variance redustion brnffits of the ngw esuimwtor bonpared to prior bas... | bottleneck is the sample efficiency in the and in the model is relatively in next subsection we some proof-of-concept experimental showing the variance reduction benefits of new estimator compared to prior baselines. Acknowledgement {#acknowledgement.unnumbered} =============== We gratefully thank Ching-An and his coau... | bottleneck is the sample effiCiency in thE real EnvIroNmEnt aNd coMputation in the APproXimate model is relativelY cheaP.
DESpitE ThE compUtationAL iNTEnsItY, iN thE nEXt SubseCtiOn we proVide some prOof-Of-Concept experIMeNtal resultS shOwing the variAncE reducTiOn bENefitS of The neW estimATor comPared to prIoR Bas... | bottleneck is the sampleefficiency in t herea lenvi ronm ent and comput a tion in the approximate mo del i sr elat i ve ly ch eap.
D e sp i t e t he c omp ut a ti onalint ensity, in the ne xtsu bsection wep ro vide somepro of-of-concep t e xperim en tal resul tsshowi ng the varian ce reduct io n benef i ts of t ... | bottleneck_is the_sample efficiency in the_real environment_and_computation in_the_approximate model is_relatively cheap.
Despite the_computational intensity, in the_next subsection we_provide_some proof-of-concept experimental results showing the variance reduction benefits of the new estimator compared_to_prior bas... |
{mk}\end{aligned}$$ In fact, substituting (\[mk\]) into (\[eq\]) and integrating through test functions, we find that the solution assumed in the form (\[mk\]) is reduced to zero or single-kink solution (\[kink1\]). In paper [@QXL], we proposed the peakon-kink interactional solutions in the form of $$\begin{aligned}
u=... | { mk}\end{aligned}$$ In fact, substituting (\[mk\ ]) into (\[eq\ ]) and integrating through test functions, we witness that the solution assume in the form (\[mk\ ]) is reduce to zero or unmarried - kink solution (\[kink1\ ]). In paper [ @QXL ], we nominate the peakon - crick interactional solutions in the form of $ $ ... | {mk}\ejd{aligned}$$ In fact, substiuuting (\[mk\]) into (\[eq\]) and invegratihg throueh test functions, we find thet tye sooution assumed in the worm (\[mk\]) id reducee to zero or siifle-kink solutjln (\[knnj1\]). In paper [@QXL], we proposad the peakon-khny nnteractional solutions in the form jf $$\begim{apigned}
u=... | {mk}\end{aligned}$$ In fact, substituting (\[mk\]) into (\[eq\]) through functions, we that the solution is to zero or solution (\[kink1\]). In [@QXL], we proposed the peakon-kink interactional in the form of $$\begin{aligned} u=p_0(t)sgn(x-q_0(t))\left(e^{-\mid x-q_0(t)\mid}-1\right)+\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(... | {mk}\end{aligned}$$ In fact, substitUting (\[mk\]) intO (\[eq\]) anD inTegRaTing ThroUgh test functioNS, we fInd that the solution assuMed in ThE Form (\[MK\]) iS reduCed to zeRO oR SIngLe-KiNk sOlUTiOn (\[kinK1\]). In Paper [@QXl], we proposeD thE pEakon-kink intERaCtional solUtiOns in the form Of $$\bEgin{alIgNed}
U=... | {mk}\end{aligned}$$ In fac t, substit uting (\ [mk \] ) in to ( \[eq\]) and in t egra ting through test func tions ,w e fi n dthatthe sol u ti o n as su me d i nt he form (\ [mk\])is reduced to z ero or singl e -k ink soluti on(\[kink1\]). In paper [ @QX L ], we pr opose d thep eakon- kink inte ra c tional so... | {mk}\end{aligned}$$ In_fact, substituting_(\[mk\]) into (\[eq\]) and_integrating through_test_functions, we_find_that the solution_assumed in the_form (\[mk\]) is reduced_to zero or_single-kink_solution (\[kink1\]). In paper [@QXL], we proposed the peakon-kink interactional solutions in the form_of_$$\begin{aligned}
u=... |
antisymmetry are successfully mapped into static Hamiltonians with an order-two crystalline symmatry/antisymmetry, whose classfication has already been worked out in Ref. [@Shiozaki2014]. Thus, the latter result can be directly applied to the classification of unitary loops.
We first summarize the $K$-theory-based met... | antisymmetry are successfully mapped into static Hamiltonians with an order - two crystalline symmatry / antisymmetry, whose classfication has already been worked out in Ref. [ @Shiozaki2014 ]. therefore, the latter resultant role can be directly applied to the categorization of unitary loop.
We first summarize th... | antlsymmetry are successfuluy mapped into static Iamiltohians wigh an order-two crystalline sbmmarry/anupsymmetry, whose classwication jas alreqdy ueen worked out mh Ref. [@Shljzakj2014]. Thuv, the latter rexult can ba directly appnidd to the classification of unitary ljops.
We gigst summarize jhe $K$-uhejry-bzsed met... | antisymmetry are successfully mapped into static Hamiltonians order-two symmatry/antisymmetry, whose has already been Thus, latter result can directly applied to classification of unitary loops. We first the $K$-theory-based method used for classifying static Hamiltonians, and then finish the classification unitary loo... | antisymmetry are successfulLy mapped inTo staTic hamIlToniAns wIth an order-two cRYstaLline symmatry/antisymmeTry, whOsE ClasSFiCatioN has alrEAdY BEen WoRkEd oUt IN REf. [@ShiOzaKi2014]. Thus, tHe latter reSulT cAn be directly APpLied to the cLasSification of UniTary loOpS.
We FIrst sUmmArize The $K$-thEOry-basEd met... | antisymmetry are successfu lly mapped into st ati cHami lton ians with an o r der- two crystalline symmat ry/an ti s ymme t ry , who se clas s fi c a tio nha s a lr e ad y bee n w orked o ut in Ref. [@ Sh iozaki2014]. Th us, the la tte r result can be direc tl y a p plied to theclassi f icatio n of unit ar y lo... | antisymmetry are_successfully mapped_into static Hamiltonians with_an order-two_crystalline_symmatry/antisymmetry, whose_classfication_has already been_worked out in_Ref. [@Shiozaki2014]. Thus, the latter_result can be_directly_applied to the classification of unitary loops.
We first summarize the $K$-theory-based met... |
is the second set of rotation periods published for this cluster. @Messina.08 have recently published periods for 122 cluster members. Using this data we have investigated the Rossby number-amplitude and period-color distributions.
We find that for stars with $(B-V)_{0} < 1.36$ the amplitude and Rossby number are ant... | is the second set of rotation periods published for this cluster. @Messina.08 have recently publish period for 122 cluster members. Using this datum we have investigated the Rossby number - amplitude and period - semblance distributions.
We find that for star with $ (B - V)_{0 } < 1.36 $ the amplitude and Rossby num... | is the second set of rotatlon periods publnwhed fmr thia clustef. @Messina.08 have recently publmshee peruods for 122 cluster membdrs. Using this dara wt have investigatxs the Rossby hmmber-cm'litude and perlod-color divtributions.
We xivd that for stars with $(B-V)_{0} < 1.36$ the amplieude anc Gossby number wre sgt... | is the second set of rotation periods this @Messina.08 have published periods for data have investigated the number-amplitude and period-color We find that for stars with < 1.36$ the amplitude and Rossby number are anti-correlated, and are related via \[eqn:arrofit\]. Extrapolating this relation to higher Rossby number... | is the second set of rotation pEriods publIshed For ThiS cLustEr. @MeSsina.08 have recenTLy puBlished periods for 122 clustEr memBeRS. UsiNG tHis daTa we havE InVEStiGaTeD thE ROSsBy numBer-AmplituDe and perioD-coLoR distributioNS.
WE find that fOr sTars with $(B-V)_{0} < 1.36$ thE amPlitudE aNd ROSsby nUmbEr are Ant... | is the second set of rota tion perio ds pu bli she dforthis cluster. @Mes s ina. 08 have recently publi shedpe r iods fo r 122 cluste r m e m ber s. U sin gt hi s dat a w e haveinvestigat edth e Rossby num b er -amplitude an d period-col ordistri bu tio n s.
W e f ind t hat fo r stars with $(B -V ) _{0} < 1.36... | is_the second_set of rotation periods_published for_this_cluster. @Messina.08_have_recently published periods_for 122 cluster_members. Using this data_we have investigated_the_Rossby number-amplitude and period-color distributions.
We find that for stars with $(B-V)_{0} < 1.36$ the_amplitude_and Rossby_number_are_ant... |
well as enjoy a conserved energy, that is, \[funcional\] f&&\~\_[a=0]{} 0(a\^[-u]{})\
f&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v with $0<u<1$ and $v>\frac{1}{2}$.
If we want moreover the now well defined energy to be finite it is clearly necessary (owing to the $a^{\prime}$ factor in the denominator) that when $r\rightarrow \i... | well as enjoy a conserved energy, that is, \[funcional\ ] f&&\~\_[a=0 ] { } 0(a\^[-u]{})\
f&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v with $ 0 < u<1 $ and $ v>\frac{1}{2}$.
If we want furthermore the nowadays well defined energy to be finite it is intelligibly necessary (owe to the $ a^{\prime}$ factor in the denominator) th... | wepl as enjoy a conserved tnergy, that is, \[fuuxional\] f&&\~\_[a=0]{} 0(a\^[-u]{})\
r&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v wkth $0<u<1$ and $v>\frac{1}{2}$.
If we want moceovwr tht now well defined dnergy to be finire iu is clearly necessary (owlug to bhe $a^{\'rmme}$ factor in tme denominador) that when $s\rkgktarrow \i... | well as enjoy a conserved energy, that f&&\~\_[a=0]{} f&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v with and $v>\frac{1}{2}$. If well energy to be it is clearly (owing to the $a^{\prime}$ factor in denominator) that when $r\rightarrow \infty$ \[mu\] T\~e\^[- \^2 r]{} with \[finita\] \^2 > is much faster that required by (\[funci... | well as enjoy a conserved enerGy, that is, \[fuNcionAl\] f&&\~\_[A=0]{} 0(a\^[-u]{})\
F&&\~\_[a=|[A]{}]{}(|[a]{}-a)\^v With $0<U<1$ and $v>\frac{1}{2}$.
If we wANt moReover the now well defineD enerGy TO be fINiTe it iS clearlY NeCESsaRy (OwIng To THe $A^{\primE}$ faCtor in tHe denominaTor) ThAt when $r\rightARrOw \i... | well as enjoy a conserved energy, t hat i s,\[f un cion al\] f&&\~\_[a=0]{ } 0(a \^[-u]{})\
f&&\~\_[a=| [a]{} ]{ } (|[a ] {} -a)\^ v with$ 0< u < 1$an d$v> \f r ac {1}{2 }$.
If we want more ove rthe now well de fined ener gyto be finite it is cl ea rly neces sar y (ow ing to the $a ^{\prime} $f actori n the d ... | well_as enjoy_a conserved energy, that_is, \[funcional\]_f&&\~\_[a=0]{}_0(a\^[-u]{})\
f&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v with_$0<u<1$_and $v>\frac{1}{2}$.
If we_want moreover the_now well defined energy_to be finite_it_is clearly necessary (owing to the $a^{\prime}$ factor in the denominator) that when $r\rightarrow_\i... |
{circ}})^{2/3},b,\tau_i\}$, we also allow each transit epoch to have a unique out-of-transit normalization factor, $OOT_m$ where $m$ denotes the epoch number. The zeroth epoch is defined to be that which has the lowest mutual correlation to the orbital period. Finally, the orbital period is a free parameter too.
Direc... | { circ}})^{2/3},b,\tau_i\}$, we also allow each transit epoch to have a alone out - of - passage normalization factor, $ OOT_m$ where $ m$ denotes the epoch number. The zeroth epoch is define to be that which has the lowest reciprocal correlation coefficient to the orbital period. Finally, the orbital period is a free ... | {cirf}})^{2/3},b,\tau_i\}$, we also allow eacm transit epoch jo have e uniqus out-of-tfansit normalization factor, $LOR_m$ whtge $m$ denotes the epocf number. Nhe zeroty epixh is defiisd to bc thaf whieh has the lowesj mutual corselation to tha uryital period. Finally, the orbital peryod is s vree parameter too.
Cyrec... | {circ}})^{2/3},b,\tau_i\}$, we also allow each transit epoch a out-of-transit normalization $OOT_m$ where $m$ zeroth is defined to that which has lowest mutual correlation to the orbital Finally, the orbital period is a free parameter too. Direct -MAP Priors {#sub:directpriors} In most cases (system with less than five... | {circ}})^{2/3},b,\tau_i\}$, we also allow each tRansit epocH to haVe a UniQuE out-Of-trAnsit normalizaTIon fActor, $OOT_m$ where $m$ denotes The epOcH NumbER. THe zerOth epocH Is DEFinEd To Be tHaT WhIch haS thE lowest Mutual corrElaTiOn to the orbitAL pEriod. FinalLy, tHe orbital perIod Is a freE pAraMEter tOo.
DIrec... | {circ}})^{2/3},b,\tau_i\}$ , we alsoallow ea chtr ansi t ep och to have au niqu e out-of-transit norma lizat io n fac t or , $OO T_m$ wh e re $ m$de no tes t h eepoch nu mber. T he zerothepo ch is definedt obe that wh ich has the low est mutua lcor r elati onto th e orbi t al per iod. Fina ll y , theo rbitalp e ... | {circ}})^{2/3},b,\tau_i\}$, we_also allow_each transit epoch to_have a_unique_out-of-transit normalization_factor,_$OOT_m$ where $m$_denotes the epoch_number. The zeroth epoch_is defined to_be_that which has the lowest mutual correlation to the orbital period. Finally, the orbital_period_is a_free_parameter_too.
Direc... |
In particular, the $D=5$ potential admits a second, non-supersymmetric $AdS_5$ critical locus (first found by Romans [@Romans:1984an] in $D=5$ $N=8$ supergravity), at \[RomansCP\] e\^[4U]{}=e\^[-4V]{}=, =e\^e\^[i\^]{}, -a=ie\^[-]{}, where $\theta^\prime$ is an arbitrary phase. All other scalars of the $D=5$ theory are... | In particular, the $ D=5 $ potential admits a second, non - supersymmetric $ AdS_5 $ critical locus (foremost discover by Romans [ @Romans:1984an ] in $ D=5 $ $ N=8 $ supergravity), at \[RomansCP\ ] e\^[4U]{}=e\^[-4V]{}=, = e\^e\^[i\^ ] { }, -a = ie\^[- ] { }, where $ \theta^\prime$ is an arbitrary phase. All other sca... | In particular, the $D=5$ potentlal admits a second, non-vupersgmmetric $AdS_5$ critical locus (first fonnd vy Ronans [@Romans:1984an] in $D=5$ $N=8$ sjpergraviny), at \[RomqnsCK\] e\^[4U]{}=e\^[-4V]{}=, =e\^e\^[i\^]{}, -a=ie\^[-]{}, whxde $\theta^\prime$ ls an erbitrary phase. All other scalars of tha $A=5$ cheory are... | In particular, the $D=5$ potential admits a $AdS_5$ locus (first by Romans [@Romans:1984an] \[RomansCP\] =e\^e\^[i\^]{}, -a=ie\^[-]{}, where is an arbitrary All other scalars of the $D=5$ are set to trivial, and the $AdS_5$ radius at these points is $2{\sqrt Feeding (\[RomansCP\]) into (\[upliftmet\]), (\[upliftflux\])... | In particular, the $D=5$ potential Admits a secOnd, noN-suPerSyMmetRic $ADS_5$ critical locuS (FirsT found by Romans [@Romans:1984an] In $D=5$ $N=8$ sUpERgraVItY), at \[RoMansCP\] e\^[4u]{}=E\^[-4V]{}=, =E\^E\^[I\^]{}, -a=iE\^[-]{}, wHeRe $\tHeTA^\pRime$ iS an ArbitraRy phase. All OthEr Scalars of the $d=5$ ThEory are... | In particular, the $D=5$potentialadmit s a se co nd,non- supersymmetric $AdS _5$ critical locus (fi rst f ou n d by Ro mans[@Roman s :1 9 8 4an ]in $D =5 $ $ N=8$sup ergravi ty), at \[ Rom an sCP\] e\^[4U ] {} =e\^[-4V]{ }=, =e\^e\^[i\^ ]{} , -a=i e\ ^[- ] {}, w her e $\t heta^\ p rime$is an arb it r ary ph a se. ... | In_particular, the_$D=5$ potential admits a_second, non-supersymmetric_$AdS_5$_critical locus_(first_found by Romans_[@Romans:1984an] in $D=5$_$N=8$ supergravity), at \[RomansCP\]_e\^[4U]{}=e\^[-4V]{}=, =e\^e\^[i\^]{}, -a=ie\^[-]{},_where_$\theta^\prime$ is an arbitrary phase. All other scalars of the $D=5$ theory are... |
--------------
: Table for different values of $a$ and $Q$ for ABG black hole. Parameter $\delta^{a}$ is the region between static limit surface and event horizon ($\delta^{a}=r_{+}^{sls}-r^{EH}_{+}$).
---------------------------------------------------------------------------------------------------------------... | --------------
: Table for different values of $ a$ and $ Q$ for ABG black fix. Parameter $ \delta^{a}$ is the area between static limit open and consequence horizon ($ \delta^{a}=r_{+}^{sls}-r^{EH}_{+}$).
---------------------------------------------------------------------------------------------------------... | --------------
: Table for different valmes of $a$ and $Q$ for ABG ulack hkle. Paraoeter $\delta^{a}$ is the region bxtwewn stqtic limit surface and event hogizon ($\delra^{a}=r_{+}^{wos}-r^{EH}_{+}$).
---------------------------------------------------------------------------------------------------------------... | -------------- : Table for different values of $Q$ ABG black Parameter $\delta^{a}$ is surface event horizon ($\delta^{a}=r_{+}^{sls}-r^{EH}_{+}$). -- -- -- {width="0.230\linewidth"}{width="0.245\linewidth"}{width="0.245\linewidth"}{width="0.245\li... | --------------
: Table for different values of $A$ and $Q$ for ABg blacK hoLe. PArAmetEr $\deLta^{a}$ is the regioN BetwEen static limit surface aNd eveNt HOrizON ($\dElta^{a}=R_{+}^{sls}-r^{EH}_{+}$).
---------------------------------------------------------------------------------------------------------------... | --------------
: Table for diffe rentval ues o f $a $ an d $Q$ for ABGb lack hole. Parameter $\del ta^{a }$ is t h eregio n betwe e ns t ati cli mit s u rf ace a ndevent h orizon ($\ del ta ^{a}=r_{+}^{ s ls }-r^{EH}_{ +}$ ).
------ --- ------ -- --- - ----- --- ----- ------ - ------ --------- -- - ------ - ... | --------------
_ :_Table for different values_of $a$_and_$Q$ for_ABG_black hole. Parameter_$\delta^{a}$ is the_region between static limit_surface and event_horizon_($\delta^{a}=r_{+}^{sls}-r^{EH}_{+}$).
---------------------------------------------------------------------------------------------------------------... |
G$.
Suppose, for contradiction, that $$c: = \max_{x \in G} ( u(0,x) - h(x) ) > 0.$$ Note that the maximum $c$ is finite and achieved in the interior since $h(x) \ge u^{*}(x) \geq u(0,x) $ for $x \in \partial G$ and $u$ and $h$ are continuous. Let $x_0 \in G$ be such that $$c = u(0,x_0) - h(x_0).$$ Consider now ... | G$.
Suppose, for contradiction, that $ $ c: = \max_{x \in G } ( u(0,x) - h(x) ) > 0.$$ Note that the maximum $ c$ is finite and achieve in the department of the interior since $ h(x) \ge u^{*}(x) \geq u(0,x) $ for $ x \in \partial G$ and $ u$ and $ h$ are continuous. Let $ x_0 \in G$ be such that $ $ c = u(0,... | G$.
Skppose, for contradiction, that $$c: = \max_{x \iu G} ( n(0,x) - h(x) ) > 0.$$ Note that the maximum $c$ is finitx ane achueved in the interior rince $h(x) \he u^{*}(x) \gew u(0,x) $ for $x \in \'zrtial N$ and $m$ and $i$ are continuoux. Let $x_0 \in G$ be such thad $$z = u(0,x_0) - h(x_0).$$ Consider now ... | G$. Suppose, for contradiction, that $$c: = G} u(0,x) - ) > 0.$$ is and achieved in interior since $h(x) u^{*}(x) \geq u(0,x) $ for $x \partial G$ and $u$ and $h$ are continuous. Let $x_0 \in G$ be that $$c = u(0,x_0) - h(x_0).$$ Consider now the function $h_c = h c$. choice $c$, have that $u(0,x) \le h_c(x)$ for $x \i... | G$.
Suppose, for contradiction, tHat $$c: = \max_{x \in g} ( u(0,x) - h(x) ) > 0.$$ notE thAt The mAximUm $c$ is finite and AChieVed in the interior since $h(X) \ge u^{*}(x) \GeQ U(0,x) $ foR $X \iN \partIal G$ and $U$ AnD $H$ Are CoNtInuOuS. leT $x_0 \in G$ Be sUch that $$C = u(0,x_0) - h(x_0).$$ ConsiDer NoW ... | G$.
Suppose, for contrad iction, th at $$ c:= \ ma x_{x \in G} ( u(0,x) - h (x) ) > 0.$$ Note that thema x imum $c $ isfinitea nd a chi ev ed in t h einter ior since$h(x) \geu^{ *} (x) \geq u(0 , x) $ for $x\in \partial G$ an d $u$an d $ h $ are co ntinu ous. L e t $x_0 \in G$ b es uch th a t $$c = u (... | G$.
Suppose,_for contradiction,_that $$c: = \max_{x_\in G}_(_ _u(0,x)_ - h(x)_) > 0.$$_Note that the maximum_$c$ is finite_and_achieved in the interior since $h(x) \ge u^{*}(x) \geq u(0,x) $ for $x \in_\partial_G$ and_$u$_and_$h$ are continuous. Let $x_0_\in G$ be such that_$$c =_ u(0,x_0) - h(x_0).$$ Consider now_... |
!\tau_1\mu_k
\frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg)
\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)
\bigg(1+\frac{M_{r}}{q}\bigg),$$ where $\mu_k$ and $\tau_3$ are as in Lemma \[Costo\] and $M_{r}$ is defined as in Theorem \[average de DDF\].
According to, we estimate the pro... | ! \tau_1\mu_k
\frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg)
\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)
\bigg(1+\frac{M_{r}}{q}\bigg),$$ where $ \mu_k$ and $ \tau_3 $ are as in Lemma \[Costo\ ] and $ M_{r}$ is defined as in Theorem \[average de DDF\ ].
According to, we e... | !\tau_1\lu_k
\frac{M(r)}{k\,r}+ \tau_3\frac{\mathcxl{U}(r)}{k\,r}\bigg)
\bigg(1+\ftax{15\delta_{{\uoldsymgol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bige)
\bigg(1+\frac{M_{r}}{q}\bigg),$$ where $\mu_k$ aid $\tqu_3$ art as in Lemma \[Costo\] and $M_{r}$ id definee as un Theorem \[average de DDR\].
Wccoxdmng to, we estimste the prm... | !\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg) \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \bigg(1+\frac{M_{r}}{q}\bigg),$$ where $\mu_k$ are in Lemma and $M_{r}$ is de According to, we the probability $P[\mathcal k}^ {sq} ]$ that a random f \in \mathcal { A}$ is squa... | !\tau_1\mu_k
\frac{M(r)}{k\,r}+ \tau_3\frac{\mathCal{U}(r)}{k\,r}\bigG)
\bigg(1+\FraC{15\deLtA_{{\bolDsymBol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)
\bigg(1+\fRAc{M_{r}}{Q}\bigg),$$ where $\mu_k$ and $\tau_3$ are As in LEmMA \[CosTO\] aNd $M_{r}$ iS defineD As IN theOrEm \[AveRaGE dE DDF\].
ACcoRding to, We estimate The PrO... | !\tau_1\mu_k
\frac{M(r)}{k \,r}+ \tau _3\fr ac{ \ma th cal{ U}(r )}{k\,r}\bigg) \big g(1+\frac{15\delta_{{\ bolds ym b ol}G } ^{ {13}/ {6}}}{q ^ {{ 1 } /{2 }} }\ big g) \b igg(1 +\f rac{M_{ r}}{q}\big g), $$ where $\mu_ k $and $\tau_ 3$are as in Le mma \[Cos to \]a nd $M _{r }$ is defin e d as i n Theorem \ [ avera... | !\tau_1\mu_k
\frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg)
\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)
\bigg(1+\frac{M_{r}}{q}\bigg),$$_where $\mu_k$_and $\tau_3$ are as_in Lemma_\[Costo\]_and $M_{r}$_is_defined as in_Theorem \[average de_DDF\].
According to, we estimate_the pro... |
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