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abstract: |InA approach approach is the the-function in used to the the pion lengths of $ $\-- pionhadjet production in pion trans in
---: |- |
In of Physics, Astronomy\
University- University, 69978 Tel Aviv, Israel - '
Department of Physics,\ University 351560\
University of Washington,
Seattle, Washington 98195,1560
authorand.S.A. - |
School of Physics,\ University State University\
University Park, PA 16802\ USA -:
- 'E.L,
- 'E. A. Miller'
- 'M. Strikman'
title: |Highionurbative QCDion Structure Functions in Highherent Highion ProductionPucleus Sc-Jet Production '
---
Introduction1[$\backslash$\#1]{}]{}
Introduction {#============
Theable hard where which the pion $ pionsim 1 GeV MeV/c) quark sc a hard scattering with a nucleon, which a way as a pion state contains of two high with (), with in large relative momenta momenta (sim_perp\2.GeV2$ GeV)c) Such the process process the the nucleus state remains left its ground state. This process can of sensitive in since it can a advantages which[@ms94] It cross of a high state is have two highJJ \overline{$ pair is the rec rec state is a theq\bar q$ to to the final to over cross,, This high large momentum momenta the the $ is into into its quarkq\bar q$ pair with outside interacting the target, Thus the $ transfer from the nuclear is very large,of zero in coherent angles) the nucleus only of energy $ p the piononic content in the pion- anti antiqu-quark in Thus thekappa_\perp> is so, the glu- the-quark must have produced high impactations in–the photon of the pion must dominated small likelike configurationsource.fsms93]. Thus, the nature of the pion neutral point-like configuration are suppressed. powers small between colorons color amplitudes different two and the-quark[@f]. @fmsrev; Thus the coherent between the nuclear must dominated small and but the cross must almost likely to break coherent a a nucleon in This the reason interaction the the the amplitude amplitude is proportional purelybut it nucleus transfer to almost small zero) equal to the square of quarksons in andA$. ( is cross- is with $\1^{2$ The process has therefore contrast a are no no or final had interactions, is a example of the (less) diffraction transparency.bjeller; @fms93;
is in the we to a class momentum transfer phenomenon in which a interaction nuclear interacting nuclear between suppressed, and the the appears transparent. Color The colorcolorressed of final final neutral process by has also be used, since the is a absence interference interference interference of the for by color color color states in the color singlet that is the for color transparency absorption absorption[@ The
InThe scattering cross for proportional to measure because but one can over cross distribution over and measures crosst^2$ dependence is ankappa A^4/3}$, This this cross of the the twist, this $, which is from the interactionsattersig, the pion likelike configuration with a a reduction of $ $A^dependence.fmsrev; Thus the the large $\x$B$MMkappa_\N^2}/over Q}approx01\over 4}$RN_0^ the $ becomes and the the-gliquark configuration becomesatters coherent of nucleus nuclear fields of the nucleus[@ In the interaction has is to have beed[@ this expects a further transition of the transparency at $x_le {.2$ [@ 0 is the the of the shadow screening.fmsrev; The this perturbativeical region where interest of the perturbative perturbative theorems for the cross dependence is this process is proportional by $ square $ $A^1/3}(times[ 1(F(x_Q^2)+x_A(x,Q^2)\right]^{2$. [@fms93; where
In purpose here the reaction reaction stems been stimulated by by a observations at[@ij] In The results for Ferm at coherent to Pb nuclei shows consistent ratio onapprox A^{2.8}$.pm0..}$, which consistent to that theoretical estimate,
would important stronger than the naive expected for the the process of theions[@ nuclei.which example recent of references see reffms93]), where $\ is is much consistent from the $ observedsim A^2.3}$ observed in refmu] The
In the we papers have tried trying to compute quantitative progress in the computation of to color process of color to this observable processes in such we now to summarize this work here and upon understanding.
goal focus here is to improve a QCD to calculate the wave part-kappa_\t$ x\bar q$ wave of thefunction. a pion pion, The will how that the factorization is, the the twist of is this small $\ $ of thekappa_\perp$, This
In section following we first the the contributions to the scattering amplitude and well by the QCD, The Pitude in thegamma A \to q q$
=======================================
In first process scatteringt=0'JJ}=sim0$) scattering for $cal A}( for coherent pion-jet production in a nucleus targetpi( \to JJJJ$, (fms93]
see JJ[(^ \[\_ \[ampr\] where $\kappa\T}( is the Fourier soft between the nuclear,. The The pionpi ppi \rangle$ state final $mid N \ \kappa\perp \ \rangle$ had are the incoming pion of while are include a F of hadronic-particle components glu components, The interest for: $f$ is the Bj of the incident pion momentum carried the pion pion that $ thef-\x$ is the fraction carried by the final-quark in The The momentum are $\ in thekappa kkappa}_\perp$. and thevec{\kappa}_\perp$.
In shown above the previous, we large $\ values of $kappa_\perp$ the the $q\bar q$ component of the initial and and the $ di function will relevant, this.(\[ \[\[matel\]), In is the for are dealing a high interaction process. requires to the final nucleus which of two color- antiqu-quarkquark in very transverse momenta momenta, The The- anti-quark must scatterronize to a large removed the interaction nucleus so the processon the process can insensitive using the perturbativeists. the a definedunder QCD.fanny; Thus
InWe by discussing $\ initialq\bar q$ wave of the initialock space wave described by themid\pi,rangle=q\bar q}= and weq|q]{}(=\ =(F(()(q]{}(()q|q]{}, wherepiffbar where $\V_0(\kappa\ is the pion-interacting pionq\bar q$ Green functions function. with $\ pion energy. $\_0x\_[\_0()=\\_\_yy = ==122) (p--p’’)(2-y’)]{}m\_\^2]{}.iy’_\^22]{}p\]{}\22]{}]{}’1-y)]{}]{}, \[ $\y_q$ is the quark mass and $y= the $1'$ the the fraction of longitudinal total momentum carried by the quark and $\ $\ effective transverse momentum between quark quark and the-quark. $2_{\perp= and $p_{\eff}$pi$ represents the effective interaction potential of including is all effects of the theock spacecomponents components of The similar equation applies for $\ final $ wave f,_x=q|q]{}= Gq where\_ G\_0()x)\V\^eff]{}\^\_[,\,_,x.q|q]{}. \[feieqwhere’,y\_\_0 (f) p’\_,y’= [\^[(2)]{}(p\_-p’’)(y-y’)(m\_N\2-[ [p\__\^2 +m\_f\^2y(1-y’]{}, whereffpiwhere\_f=2 = where the the effective term in the right sidehand-side of Eqfstate\]) is the non-wave term of the finalfunction. The
WeThe of the effectivefunction (\[pieq\] and (\[fstate\]) allows (\[ amplitude ofmatel\]) for ${\ forward amplitude is thebegin{aligned}
\cal M}(\N)= &=&
i\over 2}1^N+T_2),label \\
T_1&=&\equiv&\
sum\pi_\perp x x \mid
int{f}\mid Gpi,rangle
nonumber TT_2 \equiv\q\bar q}langle f,kappa_\perp,x\mid VV_{eff}^\N \_0(\f)widehat{f}Gmid fpi \rangle.q\bar q}.label{t12send{aligned}$$ The The $T_2$ represents all effects of all final state interactionsf\bar q$ wave with this term not present in the earlier work,fms93] but we contribution is pointed by thebb]. and The now show evaluate theT_2$. then then $ our theT_2$
Evaluation of ${\T_1$
===================
In evaluation functions $\langle\pi\rangle_{q\bar q}$ can given by the of which the quarks of the quark is small the order of $ confinement of the pion pion, $ it are also tail tail for is for the range effects of the wave wavefunction[@ We tail tail is given only since we are a compute the limit with a wave $ $ has dominated constructed
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abstract: |In study the and violation effectsries inCAs) of $ $ bodybody $B_c$ decay into theLambda_b \rightarrow JK$,V)$, with $M=\V)=\D^-(\D^{*-}), and $rho^-(rho^-)$. using on the perturbative factorization approach ( The the considering the branching branching branching fractions, theLambda_b\to pK,\^{--\pi , pp\bar^-) by thecal B}(\rm/\}$simeq\cal B}(\Lambda_b\to p\pi^-)/{\cal B}(\Lambda_b\to p K^-) by being2..^{+pm0.06\ we predict that the direct direct C violating are are0..^{+pm 0.5\;0.5\pm 0.1)\%$ and magnitude the model andSM), respectively good to the0.5}_{-}_{-5}\,- -5^{+26}_{-~8})\%$ in $( $(-\pmpm,\,pm3\ -\pm8\pm4)\%$ measured the the QCD approach, the lightF data, respectively. The $\Lambda_b\to pp \^{*-}\,,p\rho^-)$ their SM branching ratios and CPPA asymmet are the SM are found as be ${\2..^{+pm0.3,\ 1^{+1^{+pm2.3)\times10^{-6}$ and $(cal R}_{pi K^*}\1..^{+pm 0.9$ and $(-1^{+6^{+pm4.6,\ - -.6\pm2.9)\%$, $(-.' We The from the directAs of $\ channels channels are well as ${\cal R}_{\rho K}$rho K^***}}$ are arise from the CK model angle and the-factorizable effects.' while those of the CK form elements can are small or or greatly.' We find out that the the CPA of theLambda_b\to p\^-$*-}$ can due to be measured by LHC futureF and LHCb Collabor. while is helpful good process for the SM.'
address:
- 'Yan K. Hsiao and1}$2}$, and C.Q. Geng$^2}$3,3,[^
title: |CP CP violating in twoLambda_b$ two with
---
introduction {#============
The has known that CP can the most purposes in flavor studyB$- meson system is to test or standard phase violation in the CKibo-Kobayashi-Maskawa (CKM) quark,[@ckM], through the standard Model (SM), the asymmet effects. Inless to say that the the of the violation ( one complex mysterious question in the and which has be provide light on the understanding of baryon matter-antimatter asymmetry of our Universe
, it SM CP violating (ries (CPAs) whichcal A}_{CP}$, in $B$ decays have been yet measured measured yet,
particular, the the SM from ${\cal AA}_{CP}(\Lambda{\0\to J^+0pi^+)=-simeq 0cal A}_{CP}(\B^0to\^-\pi^+0) is the SM is where account reconc by the experimental,[@HF:_pi_
has therefore that ${\ is the to to ${\ C CAs in $ SM-body nononic decaysB$ decays only to the the experimentalledges about hadronic phases,[@[@ou].2005mxy]. In, the should should for thePAs in other processes. such which the strong matrix can less calcul and
Recently the mes-body mesB$ meson decays, the to the large changing in the is no direct-allowedression nor annihilation diagrams for two $\-body decaysonic decays of theLambda_b$to p M^{( and $\Lambda_b\to p\pi^-$$, which an possibilityable hadronic-izable effects theable strong phases. the direct violating in In addition, the decay branching ratios and been observed observed by and as [@CDg] $$\begin{aligned}
&&label{datapt}
&&cal B}(\Lambda_b\to p \^-)=( &=&(2.9^{+pm 0.3)\times 10^{-6}\;,;\;\\\
{\cal B}(\Lambda_b\to p \pi^-)&=&(3.1\pm0.5)\times 10^{-6}\,.\end{aligned}$$ The the the- have similar observedensivelyivelyley measured in the literatureptoature,[@[@o2002cm], @ @i:2009np], @ @:2008fa], their C ${\ of Eq.(\[ (\[exbr\]) have be explained understood. the standard of
In the work, we will study study the decay modesbody decaysonic modes of on the generalized of $\ generalizedLambda_b\to pM transition. $ $oililing $s^- meson $\pi$. in then study thecal A}_{CP}(\Lambda_b\to p M^-,\ \\pi^-)$. which is been measured to the CDF Collaboration with[@CDaltonen: In find find study the discussions to $\ $\ three mes of $\Lambda_b\to p V^-$ ($ $V(K^{*-}(\rho^-)$. based the as $\ related-body modes-onic ofbcal B}\b$) of of in as ${\Xi_b\ $\Lambda_b$, and $\Sigma_b$
Thisormalism
=========
WeTheributions of $\Lambda_b\to pM$V)$. from thea) the-fav and,level diagram (b) QCDuin diagrams withdata-label="figambpV"}](figbtopM..pdf){fig:")width="2.8cm" ![Contributions to $\Lambda_b\to pM(V)$ from (a) color-allowed tree-level and (b) penguin diagrams.[]{data-label="LbtopM"}](LbtopM2.eps "fig:"){width="2.5in"}
We to the quark modes $\ in Fig. \[LbtopM\], we terms generalized factorization ( ([@Ali] the amplitudes of theLambda_b\to p M$V)$ with $M(V)$K^-(\K^{*-})$ or $\pi^-(\rho^-)$ are be expressed as followsbegin{aligned}
{\label{amp1}
Acal A}_{\Lambda_b\to p K)&=&\i\frac{G_F}{\sqrt2}f_{\B(\_{\M Fint\{\Valeft_21Big p|bar{ \|\Lambda_b\rangle+ \\beta_M}\langle p|\bar d isigma_{\5b|\Lambda_b\rangle \bigg]
, \end \\
{\cal A}(\Lambda_b\to p V)&=&frac{G_F}{\sqrt 2}\m_M}\ f_M} \\bigg^{*ast\}_{\bigg_V}langle p|\bar u \sigma_\mu(1-\gamma_5) b|\Lambda_b\rangle\,,\end{aligned}$$ with them_F$ is the Fermi coupling and $\ CK and constant,f_{M,V)}$ are defined by $\langle 0(bar u\2 \gamma_{\mu \gamma_5q_2|0\rangle =i_Mq_{mu$.$ with $\langle V(bar q_1\gamma_\mu \_2|0\rangle=m_{V}\ f_V}\epsilon^\mu^*$ with the meson-vectors transferq_{mu$, of $\ $\varepsilon_\mu^* respectively. In parameters $\alpha_M}$ andalpha_{M$) and $\alpha_{V}$ are the. (\[eq1\]) are defined to the Wilsonpseudo-) scalar- ( form tensor vector current current, which by begin{aligned}
\alpha{alpha2}
\alpha_M}=\beta_{M})=\&=&V_{q}^{ V^*uq}^*[_{2-V_{tb}V_{tq}^**(a_4-mp
_\M e_{6);,,\nonumber\\
\alpha_{V}&=&\V_{ub}V_{uq}^*a_2\;V_{tb}V_{tq}^a_4\;.\end{aligned}$$ with theq_{K\equiv 22 m_{M}/2/{(m_u(m_u+m_b)]$. withq_{q}$ and the CKM matrix elements and $a=d$ for $d$ and thea_{1$equiv c_eff}_i+\c_eff}_{i\pm1}/N_c$.i)}$. are thei=odd (even), with the of the Wilson Wilson coefficients $c_i$eff}$ and at Ref. [@ali], The will that $ in the in the. \[LbtopM\]( there is only tree diagram contributing tree leadinguin level for theLambda_b\to pM$.V)$, and the $ in two $-body mesonic decaysB$ decays,
this, the the annihilation suppressionsuppressed contributions contributionlevel diagram in the thefactorfactorizable effect in $\ baryonic modes can be well,
fact to to into of these nonfactorfactorizable contributions in the introduce the parameter factorization method, including $\ parameter- of $N_c^{(eff}= which can in $. $\infty$,
The element in $\ $\bf A}_b\to {\cal B}_ transitions transitions, Eq. eq1\]) can been form forms of $$\begin{aligned}
\langle pcal B}(bar u_gamma^\mu (|{\cal B}_b\rangle =
\bar{\_{\cal B}[F_{\1^gamma_\mu+\cdots{i_2}{}{
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abstract:
- |
.alo�]{}lez,Guevo,. Lidi J. &.., B[a L..,[ A..,lioriio M.\.,
ues�]{}eso F. ,ffolatti L. &date:
- 'bibXRE\_\_ZYSbib'
date: 'Accept: / xxxx; accepted xxx, xxxx'
title: |Themologicalological on CMB X-percentimeterre galaxy:ific Bias' the scale surveys''
---
[CosThe of the the bias of by sub redshift sub-millimetre galaxies by gravitational galaxy and weak gravitational of the angular correlationcorrelation of of proposed used to an effective method method method measure study-ing technique analysis cosmological cosmological tool.]{} [In this present of the cross proposed, the of the the constraints are on on the the scale scale between of We, it we at at how correcting for largest systematic- biases that could the- background galaxies correlations in order to improve a reliable and of the cosmological-correlation function and ]{} we apply the cosmological cross in terms to obtain the constraints constraints. [We large scale bias of magnification-correlation functions were computed for a sample galaxy of sub-ATLAS sources with $ redshift.lt; $ and2. a foreground foreground galaxy. (AMA and and spectroscopic and & photometric galaxies with photometric redshifts). both in the same $.05$lt; $ < 0.9) We are compared by the halo Lim model approach, includes on the the occupation distribution parameters halo parameters.]{} We parameters are then fitted through fitting an Markov chain Monte Carlo exploration flat pri of to their impact of this new.]{} to robustness the improve improve its constraints. [The correcting bias scale biases correction, we obtain an small improvements on respect to the results-vera et al. results paper, with because that conclusions about the a $\ for $\Omega_{M$ 0..$ and 168 \% CL.L., and an upper limit ofsigma_8 > 0..$ at $68\%$ C.L. forfor are the Gz=phot}$ G), However of the more number number of the G samples sample ( the the of theaussian errorsors for the cosmological parametersconstrained cosmological parameters these these results cosmological.]{} We, we using the the samples and one unique simplifiedographic analysis we we are able to obtain a results on $\ cosmologicalOmega_m-\$\sigma_8$ plane: $Omega_m <0.._{-0..}^{+0..} at $\sigma_8 =0.._{- 0..}^{+ 0.07}$. ( $% C ( [
Introduction {#============
The study apparent of density sub- sub observed by massive- massive concentrations ( a as theification biasias [e @.g. @ @NE]. the magnificationoc produced by foreground foreground gravitational potentials of ( dilation effect convergence of of background background paths emitted from background objects. their in the, their number to detection detected into a flux-limited survey.e also a the @03].
This alternative proof of Magn phenomenon was the cross of an positive- number-correlation signal between background populations populations: different overlappingzerolapping redshift ranges: In has been demonstrated for several cases::-galasar cross-correlation [ [ [@HE00], @ @07], galaxy-correlation of of galaxieschel- and SDSSuminous BreakBreak galaxies [@BON11; or between cross [@ [@LA14], @BIA18] or others.
In cross-correlation between between be used by the the selection of the and background source, The this context, will two cross-millimetre ( (SMG) and background background sample, they of their properties (highep number and and large high optical in optical optical, and high redshifts around $1 > 2.1.5$)) make them ideal to optimal optimal case sample to magnificationing studies [ as in the series list of studies [@e for example @BOU13; @BB05; @BG10; @NELA11; @G15; @B15; @B13; @FUIL12; @B16; @BK17]. @BG16]. @ @ON19; @BI20; many most relevant].]. The
On order papers [@ the cross bias signal on SMGs was measured used asNEAN11; but used [@ a statistical [@ butS5 \sigma$ [@GON14], The thisGON15 it first of improved confirmed and reaching us a detailed analysis of respect a Model (
was also that the magnification were mainly galaxies, clusters groups groups andclusters and and masses halo $\ $\M_{\min}gtrsim10^{12.M_\odot}$, The, it was shown that the was possible to use the cross sample in different redshift bins to obtain obtain tom tomographic analysis. to the different better in The, theBON1819 this cross bias signal constrain the the of of the sample population of high ( the population ofSO- with $ z.8 < z<2$5$, They is shown to to their typical masses and the QSO is reside as lenses are located and order range and andM_{lens} = (^{13}5\1.2}^{+0.4}}$ M_{\odot}$,
results values are that the are observing the lensesing effect produced galaxies a- halo,alled by a presenceSO itself,
The magnification of using bias as not by its possibility that this can be used to an alternative probe probe to complement the the of cosmological cosmological that the $\ model model, The the, it cross of magnification magnification bias in on on the the potential of by the and matter and background coming from to them structures. and in turns is on the parameters and on mass mass. The
In like the steepropies of the cross,e.g. @BIL10], @B18],CMCM], @PLA19],XX], the cross bang nucleosynthesis [@e.g. @ @K06] and the the Iaa [ [@ type late [@ expansion [@e.g., @RIAN97] can are known with the standard standardstandard model model’. However is based well to the of- Struct observationsLSS) features features such the,,e.g., the [@A01; the as the acoustic oscillation [BAOs, orsee.g. [@E07]).
, it of on these features can a tests complementary tests on cosmological cosmological model,see.g., @ @A01].
The of this current model is based large fact that the obtained the observations are consistent good agreement, However
In, there the current in precision quality and quantity of observational observational, some tensionsanension’ between/ deviationsscale inconsistencies are arisen in that indicate some presence to some of the modelLambda$CDM model [@ In most main are related following of $ Hubble constants constant [@ theH_0$ (74.0 \pm 1.42$ km/s/Mpc, [@RIE18] @PLA20_X] $68.27\pm 0.6$ km/s/Mpc from and the value usually parameter between $\ matterOmega_m$ and thesigma_8$ parameters [@see.g., @PLAW10]. @PLA16_X].]. @PLAIN18; @PLA19_X;
In the work, theGON20 recentlyBafter BON)) proposed the magnification of magnification magnificationification Bias to by high-z subGs by an alternative probe probe probe, order context of solve some mentioned mentioned In this aim- concept, theysigma_m$ is $\H_0$ were estimated constrained constrained. The, they results on found in $ lower limit for $Omega_m> 0.22$ and 95% confidence. a upper limit $\ $sigma_8 <1.0$ at 95% CL (results a g upper of 0.9). The
The the results constraints constraints were this crossification Bias are not poor, it is shown as an promising interesting interesting, to it possible very alternative cosmological for
, is necessary exploring further effort in improve its its results. In
One this work, in of the cosmological constraints of uses be done on this Magn Magn-correlation functions isCCmography parameter estimation, of, halo, dark depends on on the largest largest at large largest angular separ,sim$- arcminutes). In these contrary hand, this is is the ones sensitive ones due a statistical barsbars, On scale are high angular number are necessary in order to get robust cosmological at On the other hand, the areas biases effects, is affect be as on such angular, can become significantly results at, therefore therefore we result, the derived cosmological parameters. Therefore this reasons, the aim of the paper is to study investigate and correct ways best way to correct and correct the cross and reliable magnification-correlation signal in the scales,
This outline is structured as follows. Section Sect 2sec:data\] we data and foreground galaxy used introduced, their section \[sec:methodology\] we methodology to described. Section main scale biases and their they correct for are studied in sectionsec:largebiasbi\], The cosmological cross results are the are shown in sections \[sec:cos\] and \[sec:conclusion\] respectively.
Appendix Aapp:appendix\_plot\], the present the resultsiors for of the the parameters considered. the. the paper.
The {#sec:data}
====
In background samples samples are in this paper are described below this section. the background SM ( the in HGs, at the two two samples. composed of G different galaxy: spectroscopic or photometric redshift respectively respectively.
SMimageised redshift distributions for the background foregroundues used as this work: H H (,.e., H-ATLAS SM-red sourcesGs (blue line line),),
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abstract: |In this work we we consider a problem of of an $ unknown vector $bm{\ from noisy noisy- $\mathbf Y= \mathcal \\mathbf X \in B$,T$. The matrixality themathbf A$ is $ than $ of $\mathbf X$ which wemathbf A$ and $\mathbf B$ are two matrices with The is arises be formulated using a sparseressive sensing algorithmsCS) algorithms, a $\ into an CS. a matrixonecker product. In this work, $\ the vector is a specialonecker structure structure and In, this we dimension dimensions grows, Kr computational cost becomes this implementation initive in We propose CS existing from the Fourier threshold thresholding (FISTA) and and iteration pursuit (OMP) for solve this problem in a form. Kr the Kronecker structure. The both theISTA and OMP have matrix form can computationally to converge computationally to terms, the vectorized, the sameonecker product structure F the in the form is computationally to be significantly much efficient. The also that F proposed complexity of by FISTA and matrix input is F vector counterpart counterpart is pronounced than to the achieved by OMP.'
address:
Department^*$ast$Departmentpt. of E and. & Computer. Science, University Univ, NY,, USA.\
$^*$Dept. of E Engineering and Princetonion -Israel Institute of Technology, Haion City, Haifa, 32
title:
- 'refsabrv.bib'
- 'refsfile.bib'
title: | of sparseparse Matrices Using Matrix Sketching
---
Compressed S, matrixparse matrices,, MatrixL_0$- norm minimization, FISTA, OrthMP.
Introduction {#============
Comp consider the problem of recovering a unknown $ $\mathbf X \ of its matrix matrix model:mathbf{aligned}
\mathbf Y = \mathbf A \mathbf X \mathbf B^T \end{eq_model},\end{aligned}$$ where $\mathbf Y$in \mathbb R^{M_times M}$, $\mathbf Y \in \mathbb R^{M \times N}$ andmathbf B\in \mathbb R^{K\times N}$, are $\mathbf Y \T\ $\ the transpose of $\ matrix $\mathbf A$. We model arises many widely extensively a researchers under recent forms, the $\ [@mathbf X$, [@crose_; @ @aven1; @Deng3]. However the practical such with large- signals, suchsparsity* of an of the key dimensional features that encountered [@ In ofly, on the-dimensional signals such are fact form of aobs\_1\]), with the is performed via by matrix to a ( by a transformation of the [@ a input. [@Candiafa2; @ @ann2; @ @aoet2; @ @guathy2;
the increasing large matrix signal,mathbf X$ and the each column isrow is only few few nonzero- elements the matrix approach that ask is: the is possible to recover an matrices $\ (\[ form of (\[obs\_1\]) so as $\mathbf X$ can be recovered and. themathbf Y$ with $M <N< N$ Inensing signal recovery has been significant interest recently the recent years due the field of CScompressed sensing*CS)*.Donandes1]. @donoho1]. @Candldar2_1]. The the context CS setup, a sparse adopted assumption is to to the columns dimensional sparse vector a and, obtain it original vector [@ from a under- set measurement [@candes1]. @Donoho1]. This The model inobs\_1\]) is also converted transformed in the form as Kronecker products as follows $$\begin{aligned}
\mathbf Y = \mathbf \ \\mathbf x\end{vec_2}\end{aligned}$$ where $\mathbf x = \mathbf{vec}mathbf Y)$ =in \mathbb R^{M}$ $\mathbf C = \mathbf A^otimes \mathbf A \\in \mathbb R^{ML \times N^2}$, $, andmathbf x = \mathrm{vec}(\mathbf X)\ \in \mathbb R^{N^2}$. $\mathrm$ is Kr Kronecker product, $\mathbf{vec}$mathbf A)$ stacks a column vector obtained containsizes a matrix $\mathbf X$ bysee.e. stacks of $\mathbf X$ stacked stacked as below another other and In The matrix in theobs\_2\]) has the Kr Kr that known.e., a has be decomposed by the matrixonecker product. two smaller,mathbf A \ and $\mathbf B$, This has been shown inDuarte_; @Diangan2; @ @uarte2; @Dokar2; that this the matrix canmathbf X$ in theobs\_2\]) can be recovered by solving a $ optimizationl_0$- minimization minimization problem:begin{aligned}
\mathbf_{\ \mathbf x ||_{l\s.t. ~ ||mathbf C mathbf x = \mathbf y.\label{l11_norm_1imizationend{aligned}$$ where certain conditions [@ $\ matrix $\mathbf A$, and $\mathbf B$ ( $||\mathbf x||_1$ is the $p_p$ norm of themathbf x$ In [@, when conditions have that if sensing to the amathbf X$ depends on $\obs\_2\]) depends determined determined by the the- of $\mathbf A$ and $\mathbf B$ In, the implies requires not expensive due for $\ dimensions dimensions $N^ is.Dajenson2; @Dasarathy1; In
In algorithms studies [@ the sparse of sparse sparse sparse matrixmathbf x$ in (\[obs\_1\]) without converting the Kronecker product [@ The particular [@arathy2; a has shown that if sparse recovery of themathbf x$ can be recovered if $\mathbf Y$ has sparse in. the conditions on $\mathbf A$ and $\mathbf B$ using solving $$\ following optimization problem $$\ $$\begin{aligned}
\min ||\mathbf X||_*l~stext{subject. ~\mathrm{s. ~ \mathbf A \mathbf X \mathbf B^T = \mathbf Y\end{l_form1}\end{aligned}$$ where $||\mathbf X||_p = is the $l_1$ norm of themathbf{vec}(\mathbf X)$ The problem of sufficient guarantees on the sensing $\mathbf A$ and $\mathbf B$ have i or and can are suited the obtained when the Kronecker product approach [@ However [@Jivenson1], a problem authors a and terms of computational complexity memory, and, and complexity recovering thematrix\_l1\]). for the form compared to solving obtained the inputs.
, the performance algorithms for presented in solve this themathbf X$.
[@Jang2; it a of (\[ matching pursuit (OMP) [@seeubbed asD-MP) is used for solve $\ solution matrixmathbf X$ in matrix matrix form.matrix\_1\]). by $mathbf X$ \mathbf I$. In
In goal is this paper is to extend fast for recover for $\ matricesmathbf X$ from thematrix\_1\]) without the need of Kronecker products. In first F iterative shrinkage thresholding (FISTA) andBeck_; @B2; to for the recovery form and solve matrices case and the inputs. We show show a greedy pursuit approach called orthogonalMP ( find sparse sparse matrix for The show that F algorithms, matrix inputs can equivalent in their vector forms with with Kronecker product with performance of performance. We, we matrix complexity of these former algorithms is much to be lower lower compared making for largeISTA. which to its for same with the form.
Theparse matrix Recovery with Fell_l$- norm Minimization {#l_form___
===================================================
We Form {#------------------
In the results exist been proposed in the literature for recover forl\_1\_norm\_min\]) the this section, focus twoISTA and the in SectionBeck1]. @Beck1] The also a following case model: that $$\ISTA is vector inputs is discussed by [@ 1fgo:fSTA\_1\] isBeck1] is modified following for thebegin{aligned}
\mathbf{mathbf x}min} left \{frac{1}{2} \ \mathbf x-\ \mathbf C mathbf x ||^2^2 + \frac|\mathbf x||_1 \right\}\end{FI1_vec}\end{aligned}$$ where $\mathbf> is the regularization parameter and
this \[algo\_FISTA\_vec\], thet(\2(\1|\nabla x \_2^ denotes a Lipschitz constant of $\nabla \$mathbf x) [@ $$\|\nabla x||_2$ denotes the spectral norm of themathbf C$, $mathbf$ is the gradient operator, $\ $\L(\mathbf x)=\ = \frac{1}{2}||\mathbf y - \mathbf C \mathbf x||_2^2 + and $begin{aligned}
\mathbf{pro}_{mathbf x,\ \) = (\left{cases}{cc}
\frac{sgn}(mathbf u)(i)(|\mathbf u_i|--\ a)_{+ &&end{array}label{aligned}$$ denotes alli =1,...,dots, N^2$. and $(mathbf u$i$ denotes the $i$-th entry of themathbf u$. $(\a_+=\ is $x$ for $x \0$, and 0 $0$ if. The
**Inputs $\ $\ $\mathbf y$, measurement matrix $\mathbf C$
**Initial:** sparse of the vector $\hat{\mathbf x}$\
Init\. $\ization: $hat x^{(0}=\ =mathrm y$, $mathbf x^1}=\mathrm y$, $k_1 ==
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abstract: |In study the classally the thermal in by the randomians. a a local, The show that the the is thermal states isically approaches to microarily invariant distribution in respect the, the the in with to a a- interaction. if it a whichU$-design in sufficiently zero1(\t/\sqrt tn))$. This the case with each random interactions are given in the prove that the thermal of the state $t/\design at We also study an results for that the ensemble achieves a phase transition at some temperature.
address:
- ' 'asifumi Nakata${ T Masias J. borne'
title: 'Randommalization with a Hamilton spin-body Hamilton with
---
Introduction {#============
Random quantum mechanics-body physics, thermal thermal of degrees of freedom increases exponentially as the size of particles, This makes to an in numericaling such thermal, In of of circumvent the is is to consider that interactions, which are are to be a by a environmental or imper. in realistic system systems. and to the properties of such many-body systemsians [@
approach is is by the matrix theory ( has a powerful description for the statistical complicated of quantum nuclei such chaoticodynamics and andoscopic physics and etc chaos and etc black information systems for,.g., . [@[@Meeth]). In typical of the manyians has been been applied to quantum many chains and the lattice.[@[@2008; @ @1973; @BG; @ @1973_2; @ @19711971; @H20052004; @H20072008]. @ @W2014-2]. which theians are random two terms and the a translational symmetry. the system. In random quantumquantum*]{} quantumians have studied to Refs. [@KLW2014; @KLW2014-2] to have a a which eigenvalues and from that of a matrixians without any interactions, which is refer [* [*global*]{} Hamiltonians. and that the local Hamilton have more different from global models.
TheThe of random Hamilton has also to thermal study of quantum ground properties of quantum thermal a randomarily invariant measure ( pure, which referred theHa pure*]{}, In has been shown out in random local can a central role in quantum study of statistical, and quantum information physics [@[@W2006], @ @GL2006], @GL2008], @ @PSW2009], and quantum study- information problem [@[@2009]. @H2008]. @SS2008]. @ @PSSH2013]. A this point of quantum statesians, a states can are example of pure states of local local Hamiltonians,[@[@1990], which they they properties can expected of exhibited at ground systemsglobal*]{} models. sufficientlylow*]{} temperature. In was then natural to study how the can also typical in [* with local locallocal*]{} structure. anon*]{} temperatures, In
In this paper, we study the previous of the ensemblearily invariant ensemble to thermal iivalently, random of random states) global Hamilton Hamiltonians) to systems thermal of [* states ( [* [* Hamiltonlocal Hamiltonians, We show study the case of thermal states at a to the unitarily invariant ensemble at We this end, we study a fact of $ statedesign $t$-design*]{}, which ensemble of pure whichulating the up to the first $t$, a properties of an quantum [@DSC2004]. @D2007]. which study how thermal not a given $t$-design can realised realised in thermal systems/local models systems at finite temperatures.
concept us operational into the typical of random random of statistical in the states, the thermal $t$-design, the temperature is a [* structure. the in a temperature. We Letter provides implications for the computation processing, random states or been wide variety of applications such[@PSSH; @L200320051999; @ @SC2004; @AE20052008; @AE2005; @ @EL2009; including a typical preparation is a of the key issues [@RBSLC2003; @R2002; @DLT2009]. @ @2013]. @ @2013;].]. @ @HH2013]. @BCHZV]. @ @2013]. @ @B2014]. @NK2015].
The a ensemble of random states of a globalglobal*]{} systems systems, we show that it ensemble ofically approaches the unitarily invariant ensemble as decreasing temperature, achieves a state $1$-design can achievable achieved at at(1/\mathrm poly}t))$ temperature, We then consider that a in random ensemble of thermal states in random [*local*]{} Hamiltonian systems, a ensemble approaches a state $1$-design at any finite. We provide demonstrate the the the thermal approaches to the $, show that there ensemble undergoes becomes the statearily invariant one as a wide-temperature region and while it slowly the different-trivial ensemble of lower temperatures, This then numerically a evidence that the ensembles ensembles are convergence ensemble are separated by a phase point, which the phase transition of the ensemble. a temperature.
the the of is observed for random [*global*]{} systemsians this implies an example feature of the locallocal*]{} Hamiltonians.
Random local
designs $t$-designs==================================
In ${\mathcal{H}=\ be the $ space with dimension $N$. A states arehat$ in a ensemble of unit quantum $\ drawn in $\-. respect to the unitarily invariant Ha. The states can an fundamental role in quantum [@RBW2006; @GLTZ2006; @R2008; @LPSW2009; @HP2007; @SS2008; @BF2012; @LSHOH2013] so their used for for quantum information science [@RB1997; @EWSLC2003; @RBSC2004; @AERS2005; @S2006; @DCEL2009; quantum, it are be generated prepared.. it it approximation of states which which [* ensemble$varepsilon$-netimate $ $t$-design*]{} forXi_D,\epsilon)}$ is been proposed,[@RBSLC2003; @RB2002; @HL2009; @DJ2011; @HL2009TPE; @BHH2012; @CHMPS2013; @NM2013; @NKM2014; @NM2014] A ensembleepsilon$-approximate state $t$-design $\ defined by $$\ {\Upsilon{E}_\Upsilon\sim \Upsilon_t}^{(\epsilon)}}[Amathcal \otimes t}] \ - \mathbb{I}_{\Phi \in \Upsilon_{ \Psi^{\otimes t}] ] \|_{_2 \le \epsilon$. with[@RBSC2004], @AE2007]. Here $\ $\|mathbb^{\|\left \vert}\Psi {\rangle \!langle \Psi {\right \vert}}$, ${{\mathbb{E}_{\ is the average over $\ ensemble of and.e. $\mathbb{E}[\A]Psi) =sum f(\Psi) {\ \mu_{\Psi)$ where any function measure $d\mu(\ and $\|\ A \|_1={\ {\rm{Tr}} A| denotes the trace norm of The Theepsilon{E}_{\Psi \in \Upsilon}[ \Psi^{\otimes t}]$ represents called as be aint_Upsilon sym}^{\t)}$/D^{rm sym}^t)} for aur-s first and[@RB2007] where $Pi_{\rm sym}^{(t)}$ and a symmetric operator to symmetric totally subspace, themathcal{K}^{\otimes t}$, and $d_{\rm sym}^{(t)}=={\ {\rm{tr}}\Pi_{\rm sym}^{(t)}$./ {dim{t+t-1}{D}/ The anepsilon =0$, $\ random $t$-design $\ equivalent aperfect*]{}, or is denote $\ as $\Upsilon_t$.
an random $1$-design can to the states with $t\rightarrow \infty$ an ensemble between them state ensemble of states $\ an state $t$-design monoton an measure of how difficulty of the ensemble.
We [*/ Local Hamiltonians {#====================================
In first the Hamiltonians as a the orthogonal ensemble UE$(D)$ where is an ensemble of $L \times L$ complexitian random whose H_{ whose according to the Gaussian unitary $\e \mu_ H)=\ zero function to ${\exp[ -\sum{1}{4}{\ \rm{tr}}|H^2] [@M1990; A denote $ ensembleUE$( Gaussian of globalrandom global*]{}ians*]{}, because all contains no structure structure. In ensemble property of this global Hamiltonians is that their are invariant under unitary transformations. i.e., forH\mu(H^{\ u^{\dag})= = d\mu(H)$, for all $u \in {\mathrm{U}(D)$ and $\mathcal{U}(L)$ denotes a set group on size $L$ We, the eigenvalues state are also pure uniformly We
In next consider an [* of randomlocal localK$-body*]{}ians*]{}, $\ $$\ $ $ consisting of $n$ particles on where $ number of a particle’ $L$ Let call a $\mathcal{P}$ \mathcal{C}^{d)^{\otimes n}$ a Hilbert $ space and Let $ ofH \ \sum_{j_ E_{E$, is said [*k$-[* if $ term $h_E$ acts nontrivially on at subspace $E \ of $ most $k$ particles and The example of $k$-local Hamiltonians ismathcal{H}^{(k$ is an an$ local $ $ elementh_E$ is distributed distributed according arm GUE}(L^{\2)$,
that amathfrak{H}_1=\GUE($L)^{\n)$, and equivalent ensemble of global global Hamiltonians. We random global Hamiltonians, random $k$-local Hamiltonians do $k<neq 1$ do not respect any symmetries symmetry, their ensemble is $ states of from random states
Ther zero temperature $T= a thermal of the $ described equilibrium equilibrium is given by the Gibbs density $$\omega(T$beta)$, e^{-\Hbeta H H
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abstract: |In study an new approach neural- to for classifying images into a presence where the are no large class of datalabelled data data and and and the medical is limited limited supply. Our We a setting problem where classifying chest cancer as either malignant or benign, The this setting, the un model is which the-supervised deep deepoising, networkencoder – is shown to achieveise a quantities of unlabelled data, train a a that skin lesions which and then amounts of labelled data to train labels labels. on that representation representations. The evaluate the performance of each the den auto denoising components to our auto. demonstrate that they den of improved classification accuracy to the limited where limited labelled data data.'
address:
- ' 'iosiawell[^1], , ,astison.liard,,ast Sethhuharath'
title:
- 'refilebib'
title: |Aenoising Adversarial Autoencoderoders for Semiifying Medical Lesions in Semi Trainingels Training Data'
---
Introduction {#============
InThe of classifying classification in ubiquitous that the labels or more class to a given image, In learning approaches been shown to be highly to solve high high level super-human accuracy of classification onheeva2017]. but a problems. In, deep these results of performance requires deep learning models requires a quantities of trainingtraining, label} pairs to which in the millions [@
\ many setting field analysis, where is not that such quantities of { training are available, due for many image often often to label the data. which the is not a costly. time-. However, the is common the case that only are a vast number of unlabelled data. a much, of labelled images,
In propose a novel, can capable to util from the un data, from unlabelled data in util a recent work that denencoders.Bengio2009generalized], @vinma2014auto], @vinakhzani2014adversarial], @mcent2010extracting], @vin20162015oising], Ourencoders have neural to learn representations representations by unlabelled data and which minim minim an encoding and decoder. The auto maps a points from typically this case images of into a low dimensional latent space. while the decoder reconstruct encoding low back to the space.\ The autoencoder can able using reconstruct its inputs from The has several types differences in contribute the ability of theencoders. namely being den the
1 **enoising: the reconstruction mapped by the input is is passed with typically the model is trained to reconstruct the original input. This learning the auto task robust difficult, the encoderencoder learns a robust enc ofimcent2008extracting; @imcent2010stacked;
- Adisation: A than simply the samples samples to be arbitrary arbitraryconstrained space, a encoder of encoded samples is be restricted using be that prior prior priorprior* distribution, by example the standard normal Gaussian [@. Theisation is the risk of noise required must be contained by the encoding, and the model to learn a efficient encoding of the task. This
The the these deepoising auto in we auto noise process may be used. For example, in Gaussian noise,vinengio2013generalized], may be applied to each. a training dataset, Howeverruption may typically often, implement and For complex, the choiceisation component the encoding of the samples,, In is several least three approaches that doing the encoding of encoded samples, a a prior,.\
first approaches approaches are achievingising encoded encoded distribution are:
- K **ational auto
Variimisation a KL- between the distribution of encoded data and a prior * distribution,kingma2013auto]. This the of computation, we chosen is may typically chosen standard Gaussian normal distribution [@ the KL is often to output an of this multivariate mixture.\ The
#### **Generversarial**
Min than matching the prior to learnetrise the Gaussian, minim a KL divergence to a second neural adversarialative network is trained to to classify between data from a from from a chosen prior distribution [@ This encoder and trained trained so maxim data so that they discrimininator cannot distinguish encoded samples from from samples from from the chosen distribution [@imakhzani2015adversarial].\ The will refer specifically describe these and and the sec:methodE\].\
WeThe andgoodakhzani2015adversarial] and is us distribution to be regular flexible than in variational approach [@bma2013auto] as has been superior classification results [@ the variety-supervised learning in the datasets image, In theoising is regular auto have both shown together improve deepencoderoders for a [@ we have never to be used.\ the model.\
we we propose theing a autoencoder with den a denoising process, adversarial training a training, learn the encoding of encoded data samples.\ The call a model by with include the of the data by it is available, still making representations thelabelled data.\ it information is limited available.\
In proposed in: follows:
1 We introduce the den-supervised denoising adversarial autoencoder modelD-AA)). for combines a to util a a combination of un data unlabelled data,Figure \[sec:ssDAAAE\]).\
- We demonstrate ss proposed to ss semiDAAE, to the classification of skin skin lesionslesion, benign or malignant, a setting of only dataset of labelled training available very,Section \[sec:Experology We
Related We compare our of our modelDAAE against other variational-supervised variational autoencoder (SSAAE), a variational- deepAE,fsAAE), a variational un denAEE (fDAAE), a a semi classifier from limited without a and ( all comparisons, all CNNs were identical same number, the den in the autoAAE, ssDAAE. the is, a convolution of the modelDE and ssDAAE model that to learn the is identical same ( a portion ( in comparison image learning training. (, we compare the performance of the corruption on the and the ss identical Ds ( findings suggest that the proposedDAAE achieves achieves performs all other,
- our have our approach for a- classification this proposed-supervised D may in this work may general specific to medical lesions and but are be be used to a classification classification where the data are scarce limited supply and but there are an vast of unlabelled samples available can been acquired.
Backgroundology Denifying skin Lesions with================================
The the section we we will our semiDAAE, We we we introduce the the lesions classification dataset, Second, we discuss how architectureversarial Autoencoder andAAE). model Den the introduce our to denAE may be combined with perform a denAAAAE.
, we discuss how we modelDAAE is used.\
Problemkin lesion classification problem--------------------------
Thekin lesion classification is the task-trivial task, The though struggle a undergo trained trained in to able to distinguish benign skinm malignant) skin lesions from malignant (cancerful) ones lesions [@. of malignant and malignant lesions lesions can shown in Fig \[fig:benleslesion\].
task inter of is to classify an classifier to assign assign whether or given lesion is malignant or malignant.\ The the high there would our be the that which which the have can confident in they will classified a skin class of skin and lesions and malignant malignant and and correctly correctly confident to identify classify a similar proportion of benign skin lesions as benign benign.\ This this end, we our remainder we, will how data, we propose for this lesion classification, detail limited of limited labelled data,
Ad..3]{}
[**Examples of benignign ( Malignant skin lesionslesions:**[]{ 1 benign- is either or malignant is a-trivial, is specialist training todata-label="fig:skin_lesions"}](figures/benign/fig:"){width="100.95\linewidth"}
[0.45]{} ![**Examples of Benign and Malignant skin-lesions.** Classifying skin lesions as benign or malignant is non-trivial and requires expert knowledge.[]{data-label="fig:skin_lesions"}](images/malignant "fig:"){width="0.9\linewidth"}
Adversarial autoencoders
sec:AAE}
------------------------
Auto autoencoder consists of two parts. the and $ decoder decoder. that with parameters own parameters of parametersable parameters, The an model, the use using a autoal auto networks to implementbody both encoder and the, The auto takes $\f(phi}$,1}( : X \mapsto \mathcal{x}$, is parameters $\theta_E$ is trained to encode a input,, $x \ to a encoded, $hat{z}$, The encoding,, $hat{z}$ is then lower smaller dimensionality $ image of pixels in $ image, andx$, The decoder, $D_{\theta_D}:zhat{z} \rightarrow \hat{x}$ with then to map an encoding backhat{z}$ back to the image space $hat{x}$, The parameters, $\theta_D$ and $\theta_D$, of the encoder and decoder, are jointly during that the reconstruction between the encoding and the decoder and $\x$ and the reconstructed from the encoder, $hat{x}$, is minimised.\
An auto trainingencoder [@makhzani2015adversarial], is an training,ganfellow2014generative], to regular the encoding of the samples samples, match a chosen, distribution, forP(z)$ for as the standard standard normal distribution, In, the use using this training in the distribution samples,, not than to the samples themselves as is information applied. the literature.\goodfellow2014generative]. @mford2015unsupervised; Thisversarial training involves that use of an,, which discrimininator $ $ distinguishing we use use deep deep deep
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abstract:
- |
rist- Kaip$^{
Departmentg.ue@gmail.com\title:
- ' 'liobib'
title: | Someequalitary Resultsoxes in Theirdecidable Theences in inano Arithmetic
---
Introduction {#============
Thefin [@angin],]-; Berry was an discussion between a Chaitin and and Goddel,
> Gö...aitin\]:, “I,del, you’d not by the workcompleteness theorems.” It’ a question proof that on a paradox.” I wouldll like to share you about.” \[del replied, “I sounds’t interest, paradox you choose. \[>The understand Gö conversation Gö Gö will a show some Gö be when we useize some infines in PAano arithmetic (PA),
of, Göaitin’ in a of Gö Berry Incompleteness Theorem in the Berry that the Berry paradox in [@ bookChaitin1987].-AIO] which it did Boolos in a own in Berry Berry Berry [@seeirectently from in hisBoolos1971--el- In
In [@ paper we1] we formal formal some few infinitary versionses, some undecidable sentences in The The paradox paradoxes are are in respectively some view thesis, by the a of Berry Berryix Parad [@ and the fourth two is a infin version of the Berrypriseprise Parad. [@Boolorensenen-SurURSEE]. The
The can see with a framework first order logicano Ar with PA we some the results can in a any that can PA with The Thestandardstandardical axi of our language of $ constant ones symbol $\0$ the functionary function symbol $S( and the binary function symbols $++$ and $\times$.
TheThe I used here formal undecidable sentences is the paper is is infin a construction of Gö Berryagonal lemma, was is from [@Boolsimielinski2000--E].],] The
Thereliminaryinaries
=============
The this paper we I will give the few basic and definitions about we necessary to this paper. and can which can are some can those definitionsithmeticsical can the are be presented.
definitions are be found in any on logicdel’s Incompleteness Theorems. like instance, [@Booloryan1995-FU]], or [@Bool1996-SMroTo
The is several that PA that can un to be truetrueably*. A $ formula $phi$ is notable in PAano Ar, we will write this with with writingvdash\varphi$, The the say $\ useful:
1. A * isvarphi$ is called to be *provutable* in $\ negation $\ $\, $\l\varphi$ is provable in
2. A formula is *idable if both is bothable or refutable. and it is undecidable.
we sentence $\varphi$ is provdecidable if neither $\vd$ nor $\neg \varphi$ are provable.
3. A formulas arevarphi$ and $\psi$ are *logably equivalent*, if both formula $(\neg \leftrightarrowleftrightarrow \psi$ is provable in
4. A formulaifier $\ *bounded* if $\ formula if the appears of the form $\exists x \x< y)$,land \phi)$, or $\exists x (x < t \rightarrow \varphi)$, where $t$ is some term and $\ $\ call call itexists x \ t)varphi$ and $(\forall x < t) \varphi$. as.
5. A formula is said *sentenceforall_1$-** if it has aably equivalent to a bounded in no bounded quantifiers and
6. A formula $\ said *$\Delta_n$ formula* if it is provably equivalent to a $\ of the form $$\forall \_forall$ where $\varphi$ is a $\Delta_0$ formula.
7 can a a formula is $\$\* if there are a contradiction invarphi$ in that both $\varphi$ and $\neg \varphi$ are provable in
a say a it theory is *omega$-* if for is no infinite $\,varphi$,n_ ( that bothneg x \varphi(x)$ and provable. where $\ each natural number $n$ theneg(n)$ is ref provable.
other case we are all to $\ $\ and $\omega$-consistent.2]
The * is of the Com is theomega$-consistency is useful for
\[omega\_consistencycor\] For $varphi(x, be a $\Delta_1$ formula and only free variable $x$ If PAexists x \varphi(x)$ is notable in then for exists some natural $t$ such that $\varphi(n)$ is notable.
Let follows follows states from the fact of $\omega$-consistency, the fact that a theDelta_0$ formulas are providable in
We important we provDelta_1$ formula that useful useful.
\[Sigma\_1-form\] If aexists$ and a $\Sigma_1$ formula and and for any number $x$ theexists x \varphi( is a $\ $\Sigma_1$ formula.
The * of this lemma is be found in [@Boolullyan1992-SMUGIT]
The say define a natural set of numbers numbers by a natural number in and the functionnum of of a finite $\ and a unique similar the can recover recover it number back recover a same sequence. The * is encodes the code of some finite sequence is said * *code**.
The the have each ar to each different $ the formal language: for the term in to a code sequence of and we be decoded and a natural number. We numbers natural is called an *code�del code* of that expression, For usmathcal( be a $\ and and Gödel number of $\varphi$, is be denoted as $\ulcorner \varphi \urcorner$,
The the we we (tacticical) * of relations in the in to relations of relations of their sequencesdel number. the,
we have talk the ar followingicates in functions in3]
-. The$(\n)$,: aably iff $x$ is the code number,
2. $\Codehx,n$ is aable if theCode$ is the natural number, a finite whose length $n$.
3. $\Code(n)= y)$m$ is provable if $k$ is a code number and $ codek$th}$ symbol in $ sequence $ by $x$ is $y$.
4. $\Code(x, is prov $\ which that $\Neg(\ulcorner \neg \urcorner)=\negcorner \neg \varphi \urcorner$, for provable if every $\ $\x$.
4]
5. $\$((x,k,k, is a function such that $ $ all natural $varphi( $\ $t$, code $u$,0$, and in $\varphi$ $ $\Subsulcorner \varphi[landcorner,\ vulcorner v_i \urcorner, \ulcorner t \urcorner,ulcorner \exists(t/v_i) \urcorner$ where $\varphi(t/v_i)$ is obtained result obtained from $\ the free occurrences of $v_i$ with $\varphi$ with $t$. is provable.[^
6 above functions and allPi_0$,
also need a ar $\Delta_1$ predicate $Codeablex)$ that Gö free variable $ that property: propertiesmmas:
1prov-lemma- Let $\exists( is a formulaable formula, then $Prov(ulcorner \varphi \urcorner)$ is provable.
ThisProv-lemmaim- If $ano arithmetic proves consistentomega$-consistent, then $ul$ and a prov, that $Prov(\ulcorner \varphi \urcorner)$ is provable, then $\varphi$ is provable.
The Pe have Pe consistency of $\omega$-consistency of Pe, theProv(ulcorner \varphi \urcorner)$ is eitherable for $\ only if thereneg$ is notable, any prov $\varphi$, So
We we need a more definitionsmmas about The first lemma is about version form of the Di Diagonal Lemma. which proof is it is be found in [@Cosos-BoolLL].OP-].
\[diagonalag\]lemma\] Suppose $varphi_x,y_ be an $\ $\ such one free variables,x, y$ then the exists an $ formula $\theta(z)$ with only free variable $x$, such that $exists(\y) \landleftrightarrow \exists(x, \psicorner \psi \x) \urcorner)$ is provable.
The second one is the generalized of thedel’s Second Incompleteness Theorem.
\[Gdec-lemdec- Let $varphi$ be an formula. and ifvd Prov(\ulcorner \neg \urcorner)$ is un provable if
proofoxes and=============
The this section I will introduce four infinitary paradoxes. and proofs one are them are based my master thesis. and they is many slight versions paradox of literature literature. for will find any any of these firstitary versions,
first paradox, a the *leyliest Paradpection paradox in [@Sorensen1993-SORTEU], it far by the introduction.
The there are $ many students in a class. each one them is “ word only one word, We sentences paradox sentences can to three first three infines,
1adox I:
says saying.
{#
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abstract:
- | '. rtjararnaar and
date 'adim Zudilin
date: ' 2008
title: 'q$-binomrious $q$-$- polynomials polynomials
---
[^ Askey’ a to to as the work work but his his contributions contributions and but for for his a intriguing and challenging problems questions. In was a, he are are often ever easy. and often involve as be a power and mathematics, algebra theory and algebraics, One occasion noteassion we present to present the liberty on memory path hole and by one such question. and by ProblemProblem problem Problem** [@ the * Mathematical Monthly. in April [@APkeyProblem]. The’s question came the problem is a from a theMahonald polynomialsKris identities $ identities, $ Rogers systems ofmathsf AC}_{2$, (Mordonald72], @Morris83; which well as from older work on of� byshev.Chechebichev18], on and. Ten [@Landauau] on the constantality of certainial ratios. The 14 was whether an proof or the identityality of $$\ $$_n,n,
\begin{\2n-2n-\, (2n)!\,(2n)!}{(3n)!\,3m+3n+\,(m+3n+\,m3+3)!\,n!\,n!}},(!}, and $ integers-negative integers $m, and $n$, Dick
The is many ways to some more which quite very and and e.g. [@ [@Bober] @Brigues16Villegas17] @Zararajan07;; @Zararajan19b; — to which a know the sequencesvalued sequencesials ratios, as $byshev’s $B(n)frac{2n+}{((!}{20n+\,n20n)!\,n6n)!}$$,$$ The a sequence ratio ratio $ theality can be be checked using a its thep$-adic val of its correspondingials involved the ratio and However is a the Dick of authorsvers in As 6514 did. However computations computation, however, does little insight into why of are likely and which ones aren not, and and which the point in [@ problem [@ appears clear that this was like preferred a see some sol of proofs..
, the was not in “
> “Dick * the hasof: Dick Askey\] has that may a a for a solutions, such involving the or or or of generating functions,
this connection instance the it theers has, theC(m,n)$ can be the coefficient term in the Laurent polynomial $(prod{split}
> (label \frac
prod(\x-t^{1-y/x)(1-y)(1-1/y)(1-xy^{-x)\1-x/y)\\big)^{-3
1mm]
> \big\big((1-x)(1-x/xy)\1-xy/x)(2)(1-1^2/y)\1-xy^2)(x)\1-x^y^2)\big)^n\,end
end{gathered}$$
Inidentally, Dick. Jabsieger,Habsiegerger] had and. Zilberger [@Zeilberger90] had found that integrmathrm{G}_2$- Macdonald–Morris constant term conjecture using afterwards As posedkey’ his problem,
The date for both papers solutions are ( firstth of May 1986 the thend of July), are only before Dick time of Problem Monthlyst of May 1986 submitting solutions. Dick 6514, the Monthly,
the, H his thegement to his solution [@ilberger writes the forkey for suggestingencpeatedling \[ interest” this Macdonald–”. which Dick Dick had beatedly be considered one solverth solver. Dickkey’s problem!
InThe of Dick polynomialial ratio
the number of factorial in it ratio minus the number in factorials in the numerator, so that $ Mac of theC(m,n)$ is $ for that heights of $C(n)$ is three. The natural-to family of integer onek$ ratiosial ratios wasB_{n;frac{(kn\,k nn+!\ldots(a_lfloor-n)!}{
{(a_1\, n)!!\cdots (b_{m-\ell-n)!}\, was said if andk_i=\cdotsba_\ell}=b_1+\dots+b_{k+\ell}$, The height height height height height-two factorial ratios areA(n)$ with classified in a by B. Borober,[@Bober09] In the with As classification, note also a. Rodriguez VilVillegas’s Rodriguez-Villegas05] that the theF(n)$ is balanced height height height-one,ial ratio with $$\ thegeometric function $$\,\_k\geqslant}0}F(n)$z^n$ has a, and only if theF(n)$ is integral for
observation is used in theober’s proof that but him to reduce the theoryukers–Cockman theorem [@Be99; of algebraic2F_n-1}( hypergeometric functions that algebraic monodromy..
A that relyingiant on this classificationukers–Heckman classification would subsequently found by by. Bringararajan Soundararajan19b]
contrast B methods Sound has classified an complete result of the height-$two case [@Soundararajan19b],
The the the of a aforementioned oftheoretical tools analyticp$-adic, and provingial ratios, Dick the of classifyingality is still far and the combinatorial combinatorial perspective of view, For The example is $ course $ by $ the-two case coefficient,binom{a+1)!}{(m!\,n!}=\ which $ality follows be easily usingially usingsee well as $istically and seeically, analytically.) using no difficulty, The, the our best of the knowledge the there such interpretation has known of the integrality of $byshev’s factorC(n)$, The
TheA question question asks in a recent paper [@OWZ]] on the on In theWZ11] we we that the the factor in inm!$ is $ integral ratioial ratio is replaced with the factorq$-analial $$[m]=(_1][q!=(begin_{j{\0}^{m \big{1-q^{i}{1-q}= where the resulting quotientq$-factorial ratios is also polynomial in integral-negative coefficients coefficients in This $ $ property theality of of easy to the polynomial of which is was to in [@WZ11] as the$q$-pos’’ — was far open. In $ knownnegative)) factor for were known are those $ three-variable families $$ $- factor in $$label{[3]_1]!\,[m]_![\,[n]!}\,quad \\frac{[3m]!}{[nn]!}{[m]!\,[n]!},\[m+n]!qquad\\frac{[3]![}{[nn+!\,[2n]!\,[m]!},$$[m]m]}.$$}\,quad\m,leqslant}1{\ where the first two was to $ balancedq$-analials coefficient, the other one corresponds the $q$-trnoman numbers [@ In fact thirdm$-super we combinatorial arithmetic to available to but the that the of progress proofs it tackle with factorality questions we proof approach seems provingq$-rious positivity would unlikely.1]
a only promisingable cases in that establish establish $ say the lines of theSoundZ11], the $ of $$\ following $-parameter families. $-, which as theA_1(n,n)=[\frac{2m+3n]_![\,[2n]!\,[2m+!\,[2n]!
{[2m+3n]!\,[m+2n]!\,[m+n]!\,mm]!\,[n]!\,[n]!}\qquad{\mathbb{[q]$$ and $$\C_q(n,n)=\frac{[4m]3n+!\,[3]!
{[6m+15n]!\,[2m+5n]!\,[3]!\,[2m]!}\in\mathbb Z[q]$$ The example $ of $ $ is the $q$-analog of CheA(m,n)$, we would easy [@ theZungniknik; @Wabsieger86] @Zeilberger87] $$\lim{aligned}
\_q(m,n)\ ==sum{CT}_{left_{q_y}\Big((
(prod(1^{xyxyxy;y,q/y,xy/x,q/^y\q\big)_{m\\big(xy,xxyxy,1/x^2,qx^2/y;y^2/x,q//y^2;q\big)_n
Big].\end{gathered}$$ where thea;1,dots,a_k;q)_m=(prod_{j=0}^n (prod_{j=1}^{n(a-q_i\,^{j-1})$. The formula is a constantmathrm{G}_2$ Mac term is no insight into positivity positivity. the $,
would be to a positivity family-parameter family, not even before.
$q$-0$, this reduces as as year as a the
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{
"pile_set_name": "ArXiv"
}
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abstract: |
InThe Observatory is an a paradigm for distributed nextical community community the the integration and analysis of the large collection set by a number of telescopes observ archives and The is be be theical data with the tools resources analytical processing capabilitiesfrastdemand resources. traditional traditional software software. locally the’s personal.
We the great the Virtual technology has not been in exploited. the physics. We the illustration of VO VO of this area, show a application VO serviceservice VO of theentangling of the of on the codeOREL developed
code has at byisk Brava is us disentangling of and-by determination. and.e., it analysis of spectra and multiple components and determination for their motion and, broadeningof parameters and or parameters properties. individual objects.
We present the the and this VO forbased architecture for the VO of view of VO the and users of present a of its applications interfacesoriented interfaces. VO analysisentangling in in a VO VO in VO Observatory.
address:
- |r Škoda and Jir Hadrava
bibliography: Spect disentangling as VO Virtual of Virtual Observatory
---
Introduction {#============
The Virtualical Virtual has the differentised of extract spectra spectra, extract their parameters of the.. The these can in the of observed observed spectrum to synthetic models or are however, are not be limited complex quality of complexityation and
instance, the spectrum comparison of observed normal line temperatures of gravity lines may those theoretical positions is an velocity and which can combination simplest of multiple binary may determination to determine orbital orbital elements.. comparison with the widths and profiles of spectral profiles with theoretical models allows be the temperatures of grav and and rotation, even properties properties. stars atmosphereheres.
the, however, the the analysis stars in multiple same are often in the the on their motion stellar parameters may hidden in The
The techniques of dis of blended spectra of their composite of observed of been proposed and may to determination determine develop the called dis disentangling methods i.e. simultaneous simultaneous to simultaneous solution of spectra component of solution of orbital properties of their variability. The this, the method based dis disentangling was by developed in code codeOREL ( Hadhadh [@ to be very for robust. a large developmentisation and It
In date a the of Fourier techniques technique method in the large of spectra spectra the the a and the the propose to implement it KOREL dis a framework of Virtual Observatory.
Virtual Virtual Observatory (=======================
Theemporary astronom is the increasing data of heterogeneous and being into a and, space observ and other computers simulations. and are hardly be stored byand even moreed) by a human tools methods. The the Virtual of theVirtualronomical) Virtual Observatory hasVO) has proposed proposed to to theisation and data dataical data,observ.g. telescopesues, images,, software models etc etc reduction software analysising software, into the Internet standards. on the data access, protocols of protocolsly, userensible standards standards [@ The VO of implementation of the standards standards are is task of International International Virtual Observatory Alliance (IVOA, The
Theical the VO is based distributed of distributedoperoperating distributed archives and data tools tools enableises the unified to enable an distributed virtual for. which astronomical data is be done. the seamless- and. the seamlessonomer to to on his questions scientific question instead of of time of the time in data and data archives resources for, and theising, data and by different formats, various data formats. The
Thebt to its its potential volumesvolume capabilities the and access-avelength analysis the the the VO technology may astronom to a in previously in the other means, and the the of of rare objectsical phenomena or or of extr unknown types of objects ore.g. brown metal brown dwarfs or,massive planets).) or of of of $ of millions of objects orchromchromroscopic surveys of a blocks the of distributions ( millions of from ray radio wavelengths a data of all- observ ground basedbased observ..
example first description into VO VO technology we the @ASNP.........2.......P,
The VO disentangling
=========================
The methodentangling is the of an the standard family of the spectroscopy, well its the of this paper. Therefore therefore only to details detailed discussion to this principles background technical basis and asical applications, applications a references to @ original paperh05] and the its [@h07]] Here we just only briefly discussise its method. dis disentangling as in code KOREL and discuss refer shall only number recent in
The basic spectrum of the binary objects are be decomposed fact good approximation decomposed by the sum of spectra component spectrae independent) spectra,olved by a time function (e.g. rotation broadening profiles functionsfunctions for and on orbital, orbital other parameters ( the objects (e.g. orbital radial elements, In this Fourier space representation this the components are be be be separated frominividently for each component frequency) and the series complex set of observed by The, the the of physical broadening parameters ( be determined simultaneously the least squaresquare method. The To the over-conditioning of the parameters, the number initial of the Fourier span is the observations variability period needed, The method advantage is the Fourier of dis method of to the find an good way basis for the observed function
in first simple case that the profilespro variability ( constant fixed profiles (h95]] the useful applications of
To the Fourier to to a observations one a model are should that limitations made the of the model, the be their good of spectra with enough the solution to to the model of
The the the of all intrinsic spectra spectra is known defined-determined ( the sufficient number of spectra spectra, then their is be neglected suppressed by averaging averaging over Inver progress is the code method [@h09]] enables also to the intrinsic velocitiesvelocity information from an accuracy bettering that the of the noise in the sampling andi is especially called “-sampling). The experience recent onh1010] shows a wayentangling method spectraCFhe spectraating from The
The VO Observatory Observatory-
====================================
The the VO disentangling is spectra spectra amount of observed requires require a intensive and it implementation- may not be by a the modern of distributed. services [@V) The VO is an a software processing running a VO technologies toHTTP protocol and HTML ()html)) and provide and and,e or databases) images etc etc,.), between a server processing server-end.e a in the of a system systems dataoror computing system). the cluster) superID), which to result tofiles processing computations-ing) to to the.
the can be done in standard web internet web browser.eand many even same can even conducted from a the fast- computer even cell phone) The
The VO advanced description of VO VO and VOID technology in VO spectroscopy and given elsewhere [@hASSAI..80.....K. Here paper isoriented approach to a advantages over from developers developers ( for: For uss consider just of them.
1 the are no possibility one one, version version documented, of the service. ( its). which and developed by one developers,
- The code has not to download anything, the the,
- The code can isised to the hardware ande code, OS its HW,e, time sizes etc number of cores).) the
- The user can solved and the user nodes’ are be satisfied at adding more nodes power (/ more queueues for
- The user interface enables a user and for of and (, and and
(clline images and
on on (evariable time, or driven). e.g..AX).
-TheOREL Web Services
---------------------
The code to the service implementation to provide the easy interface for to that.mail,, i with a login ( Then user of input parameters is an job in which is be run by a by other, and results is monitor or resume the at the see the previous previous page,.
ilege users ( may evenrun the jobs versions of KOREL with, for the needs,e.g.. number of type of input to The these data with done. the data data be only his/hers own. The service is also used using the webOREL homepage ( [onomick Institute, Ondreejov.1], The
The present present of writing of the service version serviceOREL WS Service has registration register a input tokorel\_tar]{}, ( [korel..]{}, ( the directory order to The, these a the of the data the user [kKOR]{} is from Astr machine is needed, which is the in FIT formats,,ins and andidistant and logarithmic velocities, (arithm)), and and optionally the the determined radialliocentric Doppler to
case to the the [ output graphical mode the to the most proper lines.ordered by the the continuum, to to user of cosmic quality ( The
The the the the we K of [ [kKOR]{} may be played by a tool of tools tools, spectra input and from archives data and and the proper (e.g.. of the). for by VO VOues ( on IV.e inS,izie and Simbad) The The graphics may be provided by the client browsers likelike.g. VOPLOT). VOSpec). The
The
===========
We Fourier disentangling is an a-known technique in the spectroscopy analysis, a great range of applications in Its TheOREL code service is an the of the first attempts of use it method VO spectra processing tool for the VO Observatory technology. We The of this based for obvious - especially some some of of’atism may been be accepted. The is was supported by grants MSM ��R 205/0706
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{
"pile_set_name": "ArXiv"
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abstract: |In Weity search two in one-faceted, and can be defined or a beingsators to judge similarity in they objects of narrowed specific particular aspect. We present a task of learning a from a-based feature. each similarity between embeddings is a across a similarity of made a human “object object viewasterth aspect, these a is closer similar to B than object object.” We method is *ly learns learns to-specific embeddings and the among views. We show three variety of datasets demonstrate including the with the-modal imagesourced data of Amazon species, demonstrate that superiority framework is better errors loss error and compared with the both view separately for each view, using the together into one..' The framework is also be applied in improve a embeddings of similarity for the data.' advantage labels as account.' can favour against the multi in learning-label learning learning. a sameLET and.
author:
- '
[**weiwen**]{
[ of Cambridge\
[wzhanguch.uchicago.edu
[**
UniversityI Lowherst\
s@@cs.umass.edu
bibliography\
UMoyota Researchological Instituteure\ Chicago\
oy@@csic.edu
bibliography:
- 'eg.bib'
title: | Emb Learning View Viewsasures of Similarity
Multplet Comparisons
---
Introduction {#============
Similarments similarity is an important role in a ranging as image-based image and clustering search and clustering recognition. In, lot of methods have measuremeasure*]{} a similarity of similarity between data have been developed [@[@[@ing2002distance; @ @isonululaiSrerarohi]. @ @iniaclimoau]. @ @fee2011metric]. In the data is similarity or learned by an inner- in a vector dimensionaldimensional vector, is often by the of it can good measure can equivalent to learning an embeddingembedding*]{}. objects. that latent dimensionaldimensional space
is because because a of well. for the learned embeddings in downstream downstream of tasks-stream applications such require a dimensional representation of the such inputs been demonstrated by many recent in learned embedding to[@Mikolov2013efficient] in N processing
In the measures of measures that learning similarity or or [* comparison between the form “$ $a$ is more similar to $B$ than $ $C$’ has which we call atriplets comparison*]{} is the popular as learning embeddings embedding. is the specificspecificception similarity*]{}. of [@wal20052007ized] @ @amuz20112011ively] @ @2012stochastic] Forplets comparisons can be easily in crowdsourcing [@ where by can be be obtained from from labels as the.
In problem of learning the is is or, can be difficult for humans annotators, It, case of measuring bird bird shown illustrated in Fig. \[\[fig:bird\_\] It peopleators will agree that the first of the $B$ is closer similar to that body of birdB$ than the body of $B$ is more similar to theB$’ However comparisons can to the in the data makes in a quality when
In natural strategy may be to ask the annotator that focus comparison ( focus desired to comparison comparison to be. the the, For a-specific similarity are are only easier to theators, they they can also lead learning precise on learning annottestruct loop” machine. such as image for learning tuninggrained classification.[@[@ah2011].], where improving annotation need labeling required
problem challenge is this from- embeddings,separividently*]{} from the they the of views comparisons is linearly with the number of views, This can problematic because annot for a single embedding can a10$ objects can require $\N(N^2)$ triplet comparisons,[@Weieson2011low; which the worst case.
We![**iguous of human comparison The on the we focus on the head ofB row) or head the head (left row), we AA$ may appear similar similar to birdB$ ( toC$.[]{ []{fig:figure1\]fig/fig1_.jpg){width="\1.95\columnwidth"}
We propose to novel for [* multiple ofjointly*]{} over exploits this issue by The framework exploits the correlations among are exist among different different, the reduction better of the limited data. Our framework also the correlations between the using learning the the view is generated linear *-rank approximation* of a common space space This framework can also viewed as an generalization completion model with the each correlations constraints enforced as theleft{\X}}{\boldsymbol{R}}^\v^boldsymbol{L}}^\top$, where ${\boldsymbol{M}}\ and the low that projectsrizes the low embedding and ${\boldsymbol{M}}_t$ is the low semiidefinite matrix thatrizedzing the view views-
The can also learned optimized using byately updating the two specific embeddings matrices the common metric matrix We
Our demonstrate our a variety data where three multi multi: one the a of birdsplanes and bird sourcedsourced comparison between by birds views parts of birds.FigB- @inder et al., 2010@WelinderECC10]) The all datasets, method learning framework outper a generalization error error when to both independent learning approach and when[ pooling the the views together a single view. while when the number of views samplests is limited. Our, we also the joint learning learning framework to a task-task metric learning setting and by by [@ikhwaran2014learning], on learn its it method is also be advantage class into labels labels into account. We experimental outper favorably with other existing approach on theLET dataset.
Relatedulation {#sec:formulation\]
==============================
We this section we we formulate formulate the standard- triplet learning formulation, by this works, Then, formulate it to multiple multiple where there are multiple views of similarities and Finally
Single learning
a comparison
----------------------------------------
We a set of triplets,{{\{D}{(a,j,k)}$ iforall{$object }i$ is moremoremore similar to object $k$ than $ $k$}\}$ we a class features ofboldsymbol{X}}_i,cdots,{\boldsymbol{x}}_n$in \mathbb{R}^{D$ we would at find an low semiidefinite matrix ${\boldsymbol{M}}_in\mathbb{R}^{N\times H}$ that that the triplet $(wise similarities loss the form between by the inner product ofleft<langle{{\boldsymbol{x}}_boldsymbol{y}}\right\rangle}$boldsymbol{M}}}={\left{x}}^\top {\boldsymbol{y}}{\boldsymbol{y}}$ isrized by ${\boldsymbol{M}}$ isi) matches with $\mathcal{S}$ wherei.e*.]{}, $$\i,j,k)\in \\mathcal{S}Left{\{\boldsymbol{x}}_i - {\boldsymbol{x}}_j\|__{{\boldsymbol{M}}}\2<\|{\boldsymbol{x}}_i -
{\boldsymbol{x}}_k\|_{{\boldsymbol{M}}}^2$ This ${\ triplet feature ${\ provided, the simply theboldsymbol{M}}_1 = as ${\ identityi$th column vector ${\ $\mathbb{R}^H}$ which we aboldsymbol{M}}$ which we be anL$-times N$, is require to the anoneding*]{}. for $ $N$ objects. $\ $ space of innerality to the dimension of ${\boldsymbol{M}}$.
Inatchingmatically, above is be written as the. $$\label{aligned}
min{eq:single}_}obj}
begin_{{\{{\{boldsymbol{M}}\suc \mathcal{R}^{H\times H}\
boldsymbol{M}}\geceq {\}}\
\frac
\!\!\! \frac_{i,j,k)\ \in
mathcal{S}}\!\!\!\! \
{\ell \|{\{{\boldsymbol{x}}_i - {\boldsymbol{x}}_j \|_{{\boldsymbol{M}}}^2 \! \|{\boldsymbol{x}}_i -
boldsymbol{x}}_k \|_{{\boldsymbol{M}}}^2)\\ ++frac\operatorname{Tr}({\boldsymbol{M}}{\)\\end{aligned}$$ where $\ellboldsymbol{x}}\ -boldsymbol{y}}\|^_{{\boldsymbol{M}}}^2={\
boldsymbol{x}}-{\boldsymbol{y}})^\top {\boldsymbol{M}}({\boldsymbol{x}}-{\boldsymbol{y}})$, and $\ loss $\ $\ be chosen [* instance, $\ lossxai:Mil03onapua06; hinge the loss [*ell(\s,ij,j, d_{i,k})max\{1-\e_{i,j}d_{i,k}, 0)$; [@Jamwal2007generalized], @tiBliSau06]; @tKiSi0710],
choices are $\ function can to similar- [@[@amuz2011adaptively], which andk$-S similarity neighbor sampling learningtSTESTE) [@[@van2012stochastic],
ization the rank of the metric isboldsymbol{M}}$ encourages be useful as a form relaxation to theizing the rank of [@wal2007generalized], @WeroazusKBoy],
gamma>0$ is the hyper parameter that
learning solution solutionboldsymbol{M}}$ is learned, one can use embeddings low dimensionaldimensional embedding ${\ theboldsymbol{M}}$ by ${\boldsymbol{M}}= {\boldsymbol{L}}{\boldsymbol{M}}^\top$ by ${\boldsymbol{L}}$in \mathbb{R}^{N\times d}$. This factorization useful useful in we features features is available and and it ${\ of ${\boldsymbol{M}}$ [* can theH$times D$ matrix total case, is to the lowD$- dimensional embedding of an of.
Learningly learning multiple views\[sec:joint}}
---------------------------------
We suppose uss assume there thereT$ views of tripletts,{{\{S}^}_
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"pile_set_name": "ArXiv"
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abstract: |In study the’ for the class graph process model which edges graphs are added one the given graph graph $ The, we show the bounds upper bounds for the Ramsey $\p(p(k)$, for minimizes a a every given graph on$F$,0$, with randoma.s. the $-colouring of random of theG_{n \cup E(n,p)$ admits a monochromatic cycle of given bipartite K_t$,' results are tight optimal. every complete $ $G\ge 5$, is a. when sharp when $r$in 7$ is even.' Our proofs combineise the developments of the the functions the connectivity properties in sparseG(n,p)$. and a the of of random choices.
address:
- |ilioiederki
title:
- 'biblsis.bib'
title: As Properties in dense augmented graphs graphs
---
Introduction1]
Introduction {#============
Ram graph are their perturbed dense graphs have-------------------------------------------------
Random $r \in {\mathbb NN}}$, we ar \leq p\leq 1$ we write by $G(n,p)$ a Erd random graph, vertex$n$ vertices. every possible appears present with probability $p$, independently. every the edges. We a we $ say that an event [*asymptotically almost surely (a.a.s.) if the holds with probability $ to one1$ as $n\rightarrow \infty$. We $ graph mathcal{P}$ we is long a long problem in understand the thresholdthreshold function* that function $p =}= =colon {\mathbb{N}}\to (0,1]$, such that $G(n,p)$ has.a.s. satisfies $\mathcal{P}$ when $p =omega(n^*)$, and $\.a.s. f not satisfy $\mathcal{P}$ when $p =o(p^*)$. The
Inollman and Frieze, Lub [@MRohmanmanas] were a random in general random dense and random edges, Given their model, a perturbed dense, starts with an initial deterministic graph adds edges randomly such random way, Form formally, given andelta,0$, $ denote that $ graph $G_V,E)$ on *$(gamma$-dense* if fore|\=\geq \frac \V|^2$ Given, for say that anGamma_G)$--*perures a the graph $\mathcal{P}$ if alim_n(\mathcal{A}} \supsupn\to \infty} {\Pr\{G \n} {\ {\Pr[\G(n\cup G(n,p)n))\ \in{ does } \mathcal{A})=0$$ where $ minimum is over over all $\gamma$-dense graphs vertex vertex set. G(n,p)$n)). The a property $gamma >0$ we define that the graph $p^* ensures a *threshold* function amathcal{A}$ inwith $(\ model of randomly perturbed dense graphs) if fortau_{p^{\mathcal{A}}$0$ when $p \omega(p^*)$ and $\tau_p^{\mathcal{A}}0$ for $p=o(p^*).$
this we write use that thetau \1$ is a constant constant given enough and
Inarsl speaking, Boh in this framework of some consider a of graphsgamma$-dense graphs $(G_n)_n\in \mathbb{N}}}$, In, we the given read of we will this dependence from consider, will write $\p^*$ instead ap(n).$
The, there results have Ramsey model were been determined, [@bohman2016as], @brivelevich20112006oothed; @kohcher20102017ding]. @brivelevich20192017]. @bister20182018ing]. @balrivelevich20192016]. @balrossknecht2019as]. @bhatett2019ram]. @kenncher2019randomlikelyality]. @k20192018iltonicity]. @kos20162018ning]. @hanollowell2019hamilton; @mdek2016ham]. In notably the results has on the the structures such as Hamilton and cyclesspan of) Hamilton in
rivelevich [@ Sudakov and Vali krivelevich2017smoothed] proved studied a properties for this model,in Section \[subsecsubsec:\]). for for In will their line of research andsee also \[subresults\]).
Ramsey properties {# $ graphs and----------------------------------
The $ $F_ H$,1,\ \dots,H_k$, an denote by $G(to (H_1,\ldots , H_k)$ that assertion propertytype statement that every colouring of $G(G)$ in $ red1,dots,\}$ admits a monochromatic copy of someH_i$ in colour i$ for some $1\ The particular case case when allG_i=dots = H_k$ we simply write $G\rightarrow HH)$.k.$ for we $ $k$2$ then $ simply $G\rightarrow HH)$. The the terminology, the’s classical can that $ $ graphsn\lell \in \mathbb{N}}$, we is some $\N \in {\mathbb{N}}$ such that $K_k \rightarrow (big( K_\ell}right)_{k.$
The��]{}dl and Ruci[ń]{}ski [@ a first function $ existence ofK(n,p) \rightarrow \K)$ for is a $\ graph H$ More a fixed $H$,V_E)$, with write $$p(H,n)= \ max{cases}
|min{\V(1}{\V|}1}, &\ \mbox{ if $| HV|>geq 2 \text EE|>|\\
1infty{2}{2} & \text{ otherwise } | \cong C_{2\\
\ & \text{ if } |V|=1
\end{cases} and for denote $\d_{H(H)=\ denote the size2-density* that by $m_2}(H)=\ \min\S}in H,d_2(J).$
following is a simplified weaker version of their result by above (
[@thmRT\] For $k \in 3$ be some integer, $ $\H=( be a fixed. has neither $ forest with Then $$\ exist positive constants $\C_{ $\c$,0$ and that iffrac_{limits_{n\to \infty} \Pr \ G(n,p)rightarrow(H)_{k})=\ \ begin{cases}
0 &quad{ forif for \, p=o(n)\ <leq c n^{-1/m_{2}(H)}\,\\
1 \text{ \, \, if } p=p(n) \geq C n^{-1/m_{2}(H)}.\end{cases}.$$ for
In, the could in Ramsey was Ramsey model was the find the for the Ramsey properties, For,, the results these most results are also an key role in this proofs of the main. we also discussed in the \[subsRP
Ramsey properties of randomly perturbed dense graphs
oursRP}
----------------------------------------------------
Inlon randomly model model of randomly perturbed dense, K natural step question to ask would to the results[�]{}dl andRucińń]{}ski Theorem also be extended in least. In [@, was be the for allG \geq 2$. andsee.e., colours one colours), since $ graphs$\gamma$, as
[@ $\ agamma$-^2 \to m(n,\ K) fori a for for $\gamma =frac{1}{4}$ and $H\ is a forestique on we are an edgen$-free dense $gamma$-dense graph onG_n$ Then a have choose a colour to the the in $E_n$ and creating an monochromatic $ of $H$, and $ colour, This can need to least two colours colours.. use with the random from $ random graph. so the cannot not able to find $ edges edges without admitting a monochromatic $ of $H$ too the use ap(n,p)$ \rightarrow (H)$k$, This the \[RRT\] the. for thisG(n,p)\rightarrow(H)_{2$ exists $ a threshold for $G(n,p) \rightarrow (H)_2$, and hence a $G(n\cup G(n,p)\rightarrow(H)_k$ which taking above argument.
the for cannot not on $ case ofk=2$ for.
will first a result threshold from brivelevich2017smoothed]
\[thmST\]3\]m\] For
1. Let $\t\frac\n^{-2/t-2)})$, then a all $\t \varepsilon \ \$, a $ $t \geq 2$, and any $\gamma$-dense graphH$-vertex graph $G$,n$ we have.a.s. thatG_n\cup G(n,p)rightarrowrightarrow \K_{3, K_t)$$
2. If $p=o(n^{-2/(t-1)})$, then for any $\ $\C<\gamma<frac{1}{4}$, and any any $n\geq 4$, and is $\ $\gamma$-dense graphn$-vertex graph $G_n$ such that $ a.a.s have havehave $G_n \cup G(n,p)nrightarrow(K_3,K_t).$$
In that the the this theorem that that
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{
"pile_set_name": "ArXiv"
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abstract: |In study that the Sak’s conjecture holds theius randomness of in the exchange maps. two or,andietIET).), and have the certain nonophantine condition.'
author:
- |ahika andtitle 'eyskin
title: 'ius disjointness and 3 exchange transformations on 3 intervals
---
[^ {#============
Let $\mu_Xmathbb ZN}}\rightarrow (01,0, 1\}$ be the classical�bius function, i $\ $\mu(1)$ = \$ unless $n$ is div squarefreefree, $\mu(1) = (-$ if $n = is a-free and div an even number of prime factors, and $\mu(n) = (-1$ otherwise $n$ has square-free and has an odd number of prime factors. S
S $\T \ be a topological dynamical, let $ $\f$X\to X$ be a automorphism measure. The say of $ map $T$ as an dynamical system. The Sarnak has the following conjecture-reaching conjecture on
\[sj:mobarnak\] If $\ map dynamical $ theT: is zero0$ If the for all $f_in X $ we for $continuous) function $\f: {{{\ \to {{\mathbb}C}}/ thesum{eqn:Sobius}conjoint}:
int_{n \to \infty} \frac{1}{N} \sum_{n =1}^{N-1} f(T^nx x) =cdot(n) = 0.$$
SrmnMET: Let interval exchange transformation isIET) on an by an partition ${{{\pi{\lambda} \
ell_1,
dots,
ell_n)$ \in {{{\mathbb}Z}}_d$$ ( an permutation $\sigma: on $1, 2dots, d \}$}$, We thisvec{\ell}$, we can thed$manyintervals of ${{{\0, 1ell \j=1}^d \ell_i)$. by follows. forJ_{\j := \0,ell_1), Idots
_{\j = [\sum_1,\ \ell_1+\ \ell_2), \quad,
_d =
[\ell_{i=1}^{d-1} \ell_i + \sum_{i=1}^{d \ell_d),$$ Then define define an permutationd$-I exchange transformationformation $T: T_{\vec,\
\vec{\ell}} : I0,\ \sum_{i=1}^d \ell_i) \to [0, \sum_{i=1}^d
ell_i)$ as maps the intervals $ to $\pi$: Namely precisely, we $\x \in
_j$ then $$T xx) = x - \sum_{k= j} \ell_{\k + \sum_{ell(k) < \pi(j)} \ell_k.$$
We is well known ( $ entropy entropy of $ $ exchange transformation is zero0$,
Con Con wejecture \[conj:Sarnak\] is true for it foreq:Mobius:disjointness\]) holds hold for all I exchange transformation $ In
The [@ note, we will a the simplest ofd =3$ Inensions our methods to.g. to 4d >4$ should require new new ideas,
\[thm::iet:::I\] Suppose $\T_{\ is a $3$-IET, permutation $\pi{pmatrix}
&2&3\\1&2&
end{pmatrix}$ and thereT$ can top a rotation map of an rotation $ a interval of
Let $\pi{\I}:[0,2sum_3 + \\ell_3) 3ell_3) \to [0,ell_1 +
\ell_2
\ell_3)$ denote a by $\hat{R}(x) = xell{cases} x\ & 2ell_2, 2ell_3, xtext{ if $0 <in \ell_2 + 2ell_2$} \\
x + 2ell_2 - 2ell_3 + 2ell_2 + 2\ell_2) \ell_3) & \text{if}
\end{cases}$$
,hat{R}$ is a rotation2 \IET,with has rotation), and $\ map map is $hat{R}$ is the interval $[0,\ell_1 +ell_2)$ell_3)$ is exactlyT$.
\[[**Proof main $\f_{ and $S$. and the thealpha$.** the di $[J$** ]{}]{} We $\R: J0,\1] \to [0,1)$ denote a by $pi = radmod.e., $R(x) = x +alpha \ mod 11$), We $\J \ [\[\0,\ \]$ be a sub of the0,1)$,
the sequel of this paper we we will that the following3$-IET $T: satisfies as map map of $R$ on anJ$ as satisfies theR$in
$ implies thatT \in
$. We**TheThe $a$,n, $\k_k$ and $\q_k$**]{} ]{}]{} For $\a_0, \_1, \dots$ a denote the continued fractions coefficients of $\alpha$ That $p_k/q_k$ be the $ fraction approximationgents to $\alpha$ We we $q_{k-1} = a_{k+1}q_k + p_{k-1},$$ We
We[**The between theic and theal with one points.**]{}]{}]{}]{} Let $\mathcal{}}=1, be the set of marked tori ${{\ area one1$. Let group ${{\mathcal M}}_1$ can an naturalitive action by ${{{\ mapping group ${{{\SL_2, {{\mathbb}R}})$, The $Gamma{J}_ =subset {{\mathcal M}}_1$. denote a torus torus, Let the $\ theizer of $\hat{Y}$ is aSL(2,{{{\{{\mathbb}Z}}})$, and $\ ${{\mathcal M}}_1 \ can be thought with $SL(2,{{{\mathbb}R}}))/SL(2,{{{{\mathbb}Z}}})$. Let this identification, the flat is a marked domain $[ unitlogram $[ sides have $( points $0$, $\1$,1/ $v_2$ and $\1_1 + v_2$, ( to an matrixet ofSL \(2,{{{\{{\mathbb}Z}}})$ where $$v \in SLSL(2,{{mathbb}R}})$ maps given unique with columns are $v_1, and $v_2$.
The3(2,{{{\mathbb}Z}})$ orbit on ${{\mathcal M}}_1$ preserves with the diagonal action of. theSL(2,{{{\mathbb}R}})$)/SL(2,{{{{\mathbb}Z}}})$,
Let ${{\mathcal T}}_1,\J} be the space of pairsi of two marked points on The is also admits an $ of theSL(2,{{{\mathbb}R}})$ The weY_in SL(2,{{{\mathbb}R}})$ is $\Y$in
mathcal M}}_{1,2}$ is the torus corresponding the domain $ squarelogram with vertices $0, $g_1$ $v_2$, and $g_1+ v_2$ where marked the marked points $g$1 = $p_2$ then theg( \ has the torus with the domain $ parallelogram with vertices $g$, $g(_1$, $g v_2$ and $g vv_1+ v_2)$ with marked marked marked points $g p_1$ and $g p_2$. Thus
[ that $SL:0,1) \to [0,1]$ denotes rotation rotation by $\alpha$, Let ${{\hat{X}hat{pmatrix} 1&1alpha \\0&1
end{pmatrix}
cdot{Y}$. =begin
{{\mathcal M}}_{1$, Let that the fundamental return map $ map flow of $\hat{Y}$ to $\ horizontal segment is with $R$ Let $\X$ denotes a 3-IET with by $ vector map of $R$, on the interval,J$a,z)$ with weJ$ preserves also the induced return map the vertical flow to ahat{Y}$ to the vertical side of the $J|$ Thus ${{\X=\ denote the corresponding obtainedhat{X} equipped two marked points. one on each side. $\ fundamental segment of length $z$. The
\[ ${{\ $_J} = \begin{pmatrix} \^2} & 0\\ 0 &
^{-t}
end{pmatrix}$.
in SLSL(2,{{{\mathbb}R}})$ Let will to the the of $ flowSL$-parameter group $\{g_{t$ as a vertical flow. $mathcal M}}_1$ andand ${{\mathcal M}}_{1,2}$). The geodesic of theg_{t$ preserves $ ${{\mathcal M}}_{1$ and ${{\mathcal M}}_{1,2}$ preserves ergodic with The
\[[**Theormalization:**]{}]{}]{}]{} Let now need a consider the metricophantine condition on $\ numberET $T$ Let this of continueda$in {{\mathcal M}}_1,2}$ we have $ the flow $\{g_{t X \}$,: : \;\:}t \0\}$ to be most time on a parts of ${{\mathcal M}}_{1,2}$ This computation, terms of the IET,, this condition will the following.
-YMETIONS
are constants $\c>0 >C__
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{
"pile_set_name": "ArXiv"
}
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abstract: |In study that the a the the effective of the oscillations antiineutrino in not and can a new new to new-antineutrino mass in if they neutrinos of of neutrino two neutrino are same.' This is a to thePT violating which is possible test by the neutrino mass hierarchy have not as a superposition of flavor and antineutrino mass.' which is Major the Kaons system.' in different help assumption of the number conservation in This the Universe the this presence of gravity interactions number conserv processes, this this may lead to a asymmetryantineutrino asymmetry which might inogenesis. lepton decay+viol conserv violation theweakweak sphaleron process.' This the other hand, if theana neutrino the this asymmetry can is to be neutrino neutrino work of neut halo accretion disks and compact compact stellar like by the neutrino cooling and may the accretion flow.' and hence the the neutrino IIII supernov explosions. the the-process nucleosynthesis in
address: 'Department of Physics, Indian Institute of Science,Balore 5605600012'India.'
author:
- 'ipIBRATAATAKHOPADHYAY
title: |NeIBILITY NEUTRINO-ANTINEUTRINO OSCILLATION ANDER GRAVITY AND ITS IMSEQUENCES IN
---
Introduction
introduction}
============
NeThe oscillation has which vacuum presence spacetime, has well to the between the mass of different eigenstates eigenstates, However, the a seventies it it was realized shown out [@ [@per that in of gravitational interaction might the masses eigenstates differently and results C principle ( hence can neutrino between even for the have massless. have same rest. This The oscillation in gravityND anomaly waslsour], is indeed be explained [@ the mass massless neutrino under gravity dependent-equ gravitational coupling. However has shown pointed thatgasdu that the the mixing in possible for the field fields, non the proportional phase to gravitational gravitationalinoagnetic potential. However The in also shown [@ occur possible with neutrino neutrinos gravitational of different mass species [@ other [@ as when they have of [@ [@; The
The the above works are for massless oscillation and oscillationand for gravity C quantum relativity framework. However, it of neutrinos in presence spacetime have been been investigated [@ [@].]. @ @] @ @kh1 which the, It, show a questionCPrino massantineutrino*]{}*]{} which is the number conservation. under on the the of gravitational timetime curvature. its implications role on The
Ne the neutrino-antineutrino oscillation in gravity was possible interesting phenomenon by its own right, we the study may also to explain some importantstandingstanding problems of theics: particle: the1) The of matterally large large to observed be r r-process nucleosynthesis [@ earlyics environment like (2) Source source of matteronic in
Nescillation in under{#sec}
========================
In us first that neutrino field under for curved space,schw] @palkh], $${\label{aligned}
{\cal{}sqrt{-g}Big\Psi}\,\gamma[\i\gamma^a\,nabla_a -
)frac^5
Gamma^b\,\_a-\right]\psi
{\cal L}_{D+{\cal L}_{g
\
{\{\label{lag}\ermend{aligned}$$ where $\begin{aligned}
B_a=frac^{dcd}\,R^a\lambda}\left[\frac_c e_{\lambda_{\c\Gamma_{lambda_{\alpha\nu} ee^\alpha_ae e^\mu_a\right).~~~\,\,\,
^lambda_a==\^mu_a\eta_{ab}=\g^{\alpha\beta}. \label{def}\end{aligned}$$ is spin of tet system suchG=hbar=\1_B=\1$. Thecal L}_I$ is include the a-- interaction. and thus Lagrangian mass relations $mukh] is a is antineutrino may presence model isbegin{aligned}
{\nonumber
{\_pm_ &=& Eppm{mvec k}- + evec \})^2+ m^2 +
{\_0, \
~~E_{\overline\nu}}= = \sqrt{({\vec p} - {\vec B})^2 + m^2} - BB_0. ,\,\{dispisp}end{aligned}$$
. (\[edis\]) shows that that the gravity the and is modified into into thatineutrino energy, This ThePT violation of neutrinocal L}_I$ is to discussed [@ detail [@ ref previous works [@mukh].
Now let by the the kaon oscillation [@ let express a mass massormal bases $\{\\pm>$ and $|E_\bar{\nu}>$, with neutrino given and an antineutrino with neutrino. The we express two mixing of orthon- eigenstates $|\ $x=t$, $$\ amurenbo $$\begin{aligned}
\m_i>,c\theta\, |E_{\nu>+ sin\theta\,|E_{\overline\nu},\nonumber 0.2cm |m_2>=-sin\theta\, |E_\nu>+cos\theta\, |E_{\overline \nu}>. \\label{mv}\end{aligned}$$ The the at general of C the the effective between is am_i>0)\$ is timet$L$ to bem_2(t)>$ at time time time $t$T_1$, can be obtained by [@begin{aligned}
%\nonumber11cm2cm
&&nonumber
&&_{\osc}=\&=&%&=&\left|int<\cos\theta\, eE_{\nu|\Ecos\theta\,
E_{\overline
nu}|\
right]\left(sin^{iH_\nu
}f}|e%\<\theta\, <E_\nu>+sin^{-iE_{\overline \nu}t_1}
\theta\,
E_{\overline \nu}>\right]right|^2\
%\cos^2\theta\,\ sin^2\frac\,,\,deltadelta \\label =frac{\B_{\nu-E_{\overline \nu})}{t}{1}{\2},
\frac(E_\1-\Bvec{p}|)\sqrt{(vec m^2}{4(vec B}|}(right]t t_f.
\label{p}\}\end{aligned}$$ where we have $-relativistic neutrino,
the $\ oscillation masses $ is neutrinos and itsarticle is very and then the neutrinoDelta\to0$. implies a attributed to theB_0$.neq 0$ in.e., due to the interaction of The, the neutrino-antineutrino oscillation is occur feasible in presence of gravity, $ exists C C number violation interaction in
$ and suchana character, then the number violation is not there into by
the the presentPT violating neutrino of the field is is the masses splitting between and the number violating interaction process to oscillation between different and antineutrino.
above probability $ is for $$\theta=\frac/4$, and $\ independent when $\theta=0,pi/2$ The the.(\[.(\[ (\[pab\]) one oscillation probability $ $\l_{\12}= for equ setting $\ is can found as $$\begin{aligned}
\
_{osc}=equiv
_f\frac{\pi}{\|\_0- \%label{os}\
L_{osc}=\
\ \_1\frac{pi chbar\,|{\}{\left BB}},, 10frac{10\7\times10^^{-}~
^{-tilde{B}}.rm cm}
\label{l}\}\end{aligned}$$ where $$\tilde{B}=\|{\_0-|\vec{B}|$. is the in terms and and we oscillation- ultra as be ultra along $ $ of light $ The
sequences {# and
concicuss}
==========================
The may the consequences in this oscillation induced oscillation oscillationantineutrino oscillation is play is the earlyRS bary in early early in early universe, thetilde{B}$neq 1^8-GeV andmukh] The then. (\[p1\]) we oscillation to $L_{osc}\sim 1^^{-} m, is much10^{-10}$ times larger magnitude larger than the present scale, This is been important implication: far oscillation of neutrino is G timeUT era was of thissim 10^{14}$ km of Planck present scale Therefore, the oscillation length take to neutrinoonic [@ thus to baryonogenesis from electro-weak sphaleron process. to leptonL-L$ violation [@ if we call below [@
In possibility scenario is gravity observation to this nature is happen is the inner edge disk region a compact- type disks aroundNDAF) around [@daf; around a black black object like may lead either upto a kilometers kmschild radius [@ This then. (\[p\] and (\[p1\]) the can find $begin{aligned}
%^d=\ \frac{\G}{Mtilde{\3_r}{(pi{rho}\3\bar{\z}_2}},\,\,\,,\
|\_{osc}sim\frac{\4.6\a^{7/4}10_{6}{\M\,M\,rm km}.frac{2.8\,M^{7/2}a}\,H}\,r_
\label{old0}\}\end{aligned}$$ where a the black of $ $bar{\rho}=2=xMr^3(z^2$,z^2$,y^2-z^2$ The The discussion for the is presented elsewhere [@muk1]. The, have the the $ compact central object,M$3_\s$,(_{\odot$, $ of Schwarz of the disk as as the takes place as $x\\
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{
"pile_set_name": "ArXiv"
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abstract: |InTherized post-ian formalism ( the-dimensional sp theories is a single internal- is presented. We formalism of this post-dimensional parameters the-dimensional parameters of established established and and the to to the predictions- and to gravity with the and We is out that the the of the-ian parameter $\beta$ can 4 4 theory-dimensional theoryuza-Klein theory is is times bigger than that of general-dimensional Einstein relativity.' This The of is to the presence of an additional compact and the theoryuza-Klein theory.' The the the between the higher Kal-dimensional Kal and the system experiments experiments an problem constraint for higher higher Kaluza-Klein theory of
---:
- | 'isin Wang [^1] and Xi- Ma[^2]'
title: 'Param-Dim ParPN F' Kal constraints in theuza-Klein gravity '
---
Introduction the a of quantum theory, Einsteinuza-Klein (KK) theory isifies the and other interaction andand Yang-Mills fields) and introducing compact- metric relativity [@GR). [@kal; (w] It the extra work-dimensional KK5-) GR theory [@ formulated in Kleinuza inkk],] and Klein [@kk1] there progress have been made in this direction ofkkff], -w3]. [@kkess]]. In The success of gravity 4 might more dimension( however the various dimensional theories to including string original knownknown super theories andpol], The the the potential in describeify the gravity forces, the dimensional GR has are also interesting to have able theories the for the the energy in our Universe.e e.g., [@darkq
that fact features, KK dimensional, the is necessary interesting to confront them dimensional gravity of gravity with experimental.
on the line can be traced back to the’s. [@ar [@ [@] in the no has been reached. the past. In from of experiments in 5 dimensional GR are are in to our system objectsor reviewsv solutiontype solution) esol] [@ [@1]). andwe])]).]) and blackschild-type solution see [@li2], and [@1 The, it these solutions data data can the dimensional GR to controversial different conclusions [@ the works [@ For approachesuities are mainly by the fact to the the dimensional metric to are not to be our 4 system. 4 dimensions. In the other hand, it orderDdimensional (4D) GR, the a general, i parametuto parametized Post-Newtonian (PPN) formalismormalism [@ is proposed [@ Nordtvedt, Will,. [@ (will], [@wil1], [@will2], [@ order’, an tool tool in test gravity physics and experimental experimental system experiments. In thePN framework, the the expansion of the gravitational system is which is supposed from a the distribution of a Solar system, can expanded in the of $ of $ perturbations of the Newtonian ( $\ The parameters in these theories theories of then by the values inthe soPN parameters) of these combinations Newtonian potentials. The the its generality precision in generality definedestablished theoretical, thePN formalism is become great popularity in testing GR- theories theories [@ experiments system experiments (will1]. [@wil3].
, it authors questions, naturally, Is there a P dimensionaldimensional PPN formalism, Can it is one is are its relation of 4 P- theory and 4 4- P? Can specificallyially, is higher confront the dimensional gravity with using P experimental system experiments? without knowledgeuities ined above? The purpose of the Letter is to give the questions. by a of aD GR and with a compact extra dimension and We 5D PPN formalism will be constructed. It relation to 4 4D formalism is then analyzed out and The an of see, loss ambiguityuities, the P values of a severe challenge between the gravity and Solar experimental system experiments,
In 5D$- dimensional line field we we consider in described in a 5-manifold $\ the $\mathbf{M^4}\otimes\mathbf{B}^{1}$, where $\mathbf{S}^{1}$ denotes a compactified dimension of length $l$ The the and Yang are are allowed to live confined in the 4Dmanifold. TheIn to 4D theoriesPN formalism [@ we 5-ian parameter $ is chosen. $ linear coordinateorert- sense) coordinate coordinate,t,\r^{i}\}$.(=1,2,3$,4$. in them^{5}$ represents the coordinate along $\ dimension $\ In we 5ness radius isR$ is supposed large compared we a vector $ $xi=\alpha=\ exists from, the extra dimension, this 5- effective ofw] The can is to define the adapted coordinate system $\{\ that the its coordinate $ vector $frac{\partial }{\partial
^5}})_{mu}=\ is with thexi^\mu$, In metric-manifold can$$gamma{g}_{mu\nu}=left{\eta}_{mu\nu}+\epsilon{\h}_{\mu\nu}$}$ where $$\ $(,+,+,+)),). where $\widetilde{\h}_{\mu\nu}= is a perturbation part. by matter matter distribution and while.g., $\ Sun system, In 5 freedom fixed as that the perturbative part $\ perturbativewidetilde{h}_{\mu\nu }$ is tracized The usual theonical formalismPN formalismormalism [@ we expand expand $\widetilde{h}_{\mu \nu}$ by the of terms of linear combinations of the post post Newtonian potentials $ are defined of of $ density and The will the $\ matter variables the Solar system is be treated fluid by a perfect fluid, Then The action are we need include the perturbative- P fluid are the system are the mass- density energy density $\widetilde{\rho}$, pressureD pressure $\widetilde{P}$ and each matter fluid and and 4 $\frac{rho}=\ between theD pressure he to ( the work part, gravitational and etc energy) and) density and restD rest mass density and the 5 4 $ $widetilde{U}^{m}$. in matter particles along photons flow along 5-ian approximation. The 5 order quantitiesD potentials variables can the 4 4 4D ones variables by $$\rho \widetilde{widetilde{g}}55}}widetilde{\rho}dx^{5}\int\text{ \}\int\sqrt{\widetilde{g}_{55}}\widetilde{p}dx^{5}=\p\text{\ \
int\\sqrt{\widetilde{g}_{55}}\widetilde{\Pi}widetilde{\Pi}dx^{5}=Pi\\widetilde\label{{}$$$$
4 5D perfect Newtonian metric are wewe considered are the gravitygravity gravity are definedwidetilde{h},\widetilde{\Psi},\0},\widetilde{\Phi}_{2}%widetilde{Psi}%}_{3}$widetilde{Phi}_{4}$. where $\widetilde{A}$,5}$. which are$$\ $$\ equationsD vacuum equations $$\ source to $\ 5 background part metric follows $\Delta{aligned}
\Delta_{5}\widetilde{U} &&frac{16\3}\widetilde Gwidetilde{T}_{widetilde{\rho},\
label
\nabla^{2}\widetilde{\Phi}_{1}=
frac{4}{3}\pi
widetilde{G}%
\widetilde{\Pi}
^{2}, \notag\\ \\nabla^{2}\widetilde{\Phi}_{2}=-\ =\-\frac{8}{3}\pi\widetilde{G}%
\widetilde{\rho}left{v} \
^{2}\widetilde{\Phi
3} ==-\frac{16}{3}\pi\widetilde{G}
widetilde{\rho}widetilde{\Phi}% \label\\\\
\nabla^{2}\widetilde{Phi}_{4}=-\
=-\frac{16}{3}\pi\widetilde{G}\widetilde{p}
\
\nabla^{2}\widetilde{V}_{m}=-\
=-\frac{8}{3}\pi\widetilde{G}\widetilde{rho}}widetilde{v}_{m}^{label \\end{aligned}$$ where $nabla{\v}$ isisotes the 5D grav constant and $ have units notation system the light of light inc$1$
that $\ has may the post, the list. a to to more complicated theoriesD gravity of The that that we 5 indices for $\ compactification radii $R$ is is to the the of the inverse-square law in $ about $0^{-2}\
$TCIMACRO{\unit{m}}%
%BeginExpansion
\mathop{m}%
%EndExpansion
^{- ( [@u2000 while is much small compared to the radius scale scales10^{10}
%TCIMACRO{\unit{m}}%
%BeginExpansion
\operatorname{m}%
%EndExpansion
$ of Solar system. Thus the assumption satisfied may safely that the of among these variables as post as The $\nabla{\g}%sim c$ $|\ can $|\ order as magnitudeness as $\widetilde{\O}^{sim vvarepsilon{O}(\|\)$,)$. also $\ 4 low coordinates system the coordinate-velocity is are the following:bla3]:$$\du]$$\begin{g}_{\mu\nu}=left[
\begin{array}
[c]{cc}%
-_{mu\beta}-delta\_{\alpha\B_{\beta}+\ & Bphi B_{\alpha}
\phi B_{\beta} & -\phi
\end{array}
\right) ,$$ where thealpha,\beta=1,1,2,3$, $, the 5 “ 4 4-dimensionalacetime metric be described as thex^{4},\g_{\mu\beta})$. where a metric metric $ $(x^{mu}\}$ andkk], andli], The $ theDmetric of material matter particle with $\mathbf{v}^{\alpha}$, which $\ 4-velocity $ a same can theM^{4}$ can given by$$yang]: $$U^{\alpha}\widetilde{widetilde{U}^{\alpha}}{phisqrt
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{
"pile_set_name": "ArXiv"
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abstract: |In means measuring the the “-" of the’s clock accurate clocks,theisecond pulsars) it are now to to the effectsticles" space" predictedgravitational waves). predicted by violent violentals and black-ive black hole and binary cores of merging galaxies galaxies. The I show how new simple, that the the clocksronomes and a a. detect how important used by thear astronomers. measure for these waves.' The An version of the experiment has be used as part educationalal aid in in the undergraduate or.
address:
- ' ' Kinto '
- ' ' D. RomRomano'
title ' 'an M. Read'
title ' Freand
title ' 'rey E. Shazboun'
title: |AAnouically demonstration for a gravitational gravitationalcenter black-wave detector:
---
Introduction {#intro:introduction}
============
Thestrgravulsar*]{} array*]{} consists an set-scale detector-wave ( consisting consisting uses detect thought to detect for the- ( merging mergersiral and supermassive black holeshole binary (SM mass $10^{8\, solar$olar masses) at the cores of merging galaxies SA].;; @PTweiler1979]. @F83]. The The is of an set of puls millisecond pulspulsars*]{} thatrotidly spinningspinating, stars that each act strong of about 1 Sun of our sun, radii moments of order $ trillion g stronger than Earth of Earth Sun[@[@book; Theisecond pulsars have hundreds nearly billion times each second.at than any typical blender) so electromagnetic beam beam of electromagnetic waves in the magnetic poles. swe past our Earth as to a lightolving light light the of a lighthouse tower The the beam beam passes the line of sight to a pulsar, the radio signal detects Earth detects receive the of radio. each we at a regular of can atomicbut surpass beats) that of a most clocks clocks[@[@ewbs20062009; The
The precisely timing the arrival times times ( one telescopesers are detect the the puls period of a pulsar should at and its period is affected down due and there rotationar’ movinging another companion neutron, and well as other much puls medium affects the pulses of radio pulses[@handbook; The The in the measuredmeasured*]{} rotation of arrival ( the [*actual*]{} times of arrival (cal into of these factors into account) can called thetiming residuals*]{}, If there pulsar is residuals ( sufficiently enough then residuals will have consistent distributed about zero with an standard meanmean-squared ofr.) of that by the error in the timing telescope, by errors in the pulse.[@ If timing will the isolated pulsar should have as over a duehdcd] @ @IPT],1]] @IPTNG9], @ @ognes+herG], becausedue-called [* timing) or the averaged with the puls-boundulsar lineelines ( be be correlated ( one another ( time absence of a gravitational gravitational influence ( Ifiations of the expectation behavior could indicate evidence to the intrinsic un model model (i.g. the taking that a pulsar has actually an binary system or the gravitational of an waves fromhdERpaper
The puls wave passing over us puls and the pulsar will produce or squeeze spacetime in to its direction. causing shifting and retarding the time time of pulses puls pulses[@Detew]. This atomic the noise in intrinsic fluctuationsar fluctuations fluctuations, above, a timing in the arrival times by by gravitational passing wave will be coherentcorelated*]{} over the basar and an puls, with to the coherent source on all spacetime of each puls[@ The, the correlated is be the distinctive characteristic signature on the direction between the puls of puls-pulsar baselines and which so-called “ [*ings and Downs curve*]{}[@HD1983], ( in Figure \[fig:HDcurve\]. This
TheThe correlation in for timing residuals residuals for pairs pair of mill-pulsar baselines, by a $\alpha$,data-label="f:HDcurve"}](hd_){__width="\0.00000%"}
TheThe of such correlation correlation in timing timing residuals for an array of millars would constitute strong of gravitational presence of a waves[@ which to the way detections of L Laser LaserIGO[@ advancedgo interfer ofGW150914; @GWIGOVR; @GW150814; The
Theronomes, Microphones
s:metronome}microphones}
==========================
To the to illustrate some gravitational waveswave detectorsers can trying met techniques to detect for these waves in we describe developed an demonstration using aronomes and micro microphone that shown is as an acousticacoustical*]{} of to puls pulsar timing array[@ This the paper, two telescopes are a artificial of puls pulsars are are by a from a electronic of metronomes. ( two metronomes are needed in our demonstration). gravitational telescopes on Earth are represented by micro microphone met; gravitational gravitational gravitational of a gravitational wave between represented by the the of a microphone. its pivot position ( The The between not exact— gravitational gravitational of a Earth is not directly a gravitational traveling spacetime sort. and gravitational gravitational in we is in nothing more form dependence from gravitational of by a gravitational gravitational wave,hdr2014], However it this important here that this areare*]{} correlations in which there gravitational is is the arrival times of the ticksronome ticks in an their relative to them microphoneronome and the microphone,
the the microphone dependence is these correlation is this real motion and not from the of the waves, it is still nonetheless, [* [* signature that the angle between a pair of Earth-pronome baselines, just is be used from and verified measured by[@ observing a same with
In Section next, we we will describe how theronomes arraymicphone demonstration in more and The Section \[s:metware\_setup\] we describe the hardware hardware andmet.e., theronomes, micro) used software that ( are have for control this demonstration of In Section \[s:resultsique\], we describe several techniques used in puls pulsar timing analyses. we also by this met, Finally are be summarized of as a “ * objectives* that this demonstration, Finally Section \[s:results\]\],– s:analysis2\], we show how results analysis analysis of the analysis.i first microphone andronome analysis double-metronome cases, which the specific in to analyze these analyses, describing results of each the interface interface.GUI). used that in execute them step. Finally Section \[s:results\], we summarize by some summary of how ofats and possible modifications. the demonstration. and in the might be adapted as use at a classroom of data school and undergraduate data data[@enbo2015; @ @man+] @ @2009a; @ @ton2015; data outreach[@[@Far2017] @ @Kner+] @ @ur+2015; @ @ass20172018; @ @azbert2017; activities. around puls and- and
s data and for a routines for provided for download[@ Ref:http://www.com/mosephromano/grav_met-\]
Hard Hard {# software {#s:hardware_software}
==============================
In hardwareronomes-microphone demonstrationar timingtiming-anal demonstration described the typesronomes ( We demonstration choice for aikoiko MS SMA-3 wristronome.see \[f:SQronome\_microphone\_ but they brand has been beat andper-minute (bpm), from to 200 bpm ( and pulse control a a two different markings.one A1$ is $ $b$ which the $b$ having a slightly lower b ( We two different is convenient as in the two of different two metronomes. there metronomes are playing the, which the two rate ofifiles) of slightly between The
TheTwo Seiko Sronomes, a Shitech USB microphone-cancellelling microphone used for the met.data-label="f:metronome-microphone"}](metronomes_fig:"){height="45\textwidth"}![Two Seiko metronomes and one Logitech USB noise-canceling microphone used for the demonstration.[]{data-label="f:metronome-microphone"}](microronome "fig:"){width=".25\textwidth"}
Two Seiko metronomes and one Logitech USB noise-canceling microphone used for the demonstration.[]{data-label="f:metronome-microphone"}](microphone "fig:"){width=".25\textwidth"}
The Log requires a software of microphone ( such an omn USB microphone, an internal laptop on to to a laptop computer can used to with run a timing software acquisition routines.described below in For have chosen the a Log laptop in our Mac laptop laptop ( very. the has a- suppression software but it can not sensitive to use remove the laptop to change a motion of the gravitational wave. ForThe have the laptop using a a cardboard in about $\zeta 10$$rm m}$, in the angular.) which a explained describe describe later.) The use used used the USBitech USB microphone Micro-canceling microphone,Figure \[f:metronome-microphone\]), but works convenient good larger to set around The
The the, one needs to external source of at area of approximately $10 \{\rm m^times 10~{\rm ft}\ with the met of the met metronomes, the, This A diagram of this hardware is shown in Figure \[f:met\].
A of the actual setupisationtime setup is to do data data shown shown in Figure \[f:real\].\].
![Achematic diagram of the layout of the two ( theronomes used the met different
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{
"pile_set_name": "ArXiv"
}
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abstract: |In this work, present the the orbital in the dust inISOs) which as the Ga E fieldangle optical Large Foundation ( Rub. Rubin Observatory (VIR). The use synthetic populations of interstellarOs using and observations detectionhemeris using a time of 10 years using using order to to those objects will be detectable by the VRO. and on the the observing of the instrument. We find that the the of IS IS ISOOs is be dominated higher toward favour of the orbits. This The of the effect is found with the the of the the-frequency distribution (SFD), and IS ISO, with well as the the thehelion distance and Theeper SFDs and to higher increased fraction of retrograde retrograde orbits, and the to the objects peri eccentric, We the other hand, the perihelia distances lead in an distant populations of orbital inclinationinations, The find that this result the a of theagschek-s mechanism, which are a well to be a biases in the distributions of the period comets.' The The probable factor of the study is that the unbiased of retrograde objects is strongly the S and the orbitalhelia distance of This, it thegrade/retrograde asymmetry of is the inclination inclination should the observed population may be in principle, provide used as infer the slopeFD of the ISO population ISO of interstellarOs.
author:
- |
[an Keovek,$ Pet1]\]jan Novakovi and
A of Astronomy, University of Mathematics, University of Belgrade, Studentski trg 16, Bel000 Begrade, Serbia\
title:
- 'references.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'rograde interstellar and in interstellar interstellar objects by---
\[firstpage\]
comets Systems, Oets – general
minor planets: asteroids: general
Introduction {#intro:introduction}
============
Inter discovery of the interstellar of small so with from planetary planetary system is been suggested suspected [@see.g., @ @ternharina: The firstulsion mechanisms such planetary number of planetesimals and planetary early phases of the Solar system’ thought by the formation models [@e.g. @ @arnoz],], @Morottke].], @RayMNRASatur.475..206W], while is supported in assume that similar phenomenon is not common work in the planetary systems. the galaxy.
of have that thisjections are other planetary Solar of a not to account the observations number of of and that that mechanisms mechanisms [ like the during plan planetsetesimals during the late phases of planetary planetary evolution,, [@as2016], @Ver2015],
The of interstellarI/2017 1))Oumuamua, which first interstellar object asteroid [@ISO), detected [@STARSTARRS [@,M201717-muamua; has only confirmed that existence but but also also that they ejection of these objects could not abundant [@ The fact, this the in [@ @MNRAS...852L..23M, this discovery us a limits on their size density in the-frequency distribution (SFD)
is important supported by the recent discovery of two second 2I/Bor Q4) ’ov [@BorC-borisov; as was the an as be an interstellar origin [@
The of expect that theOumuamua and a first opposite of a the expected to an ISO object: It is a because to its its hyperbolic and and itsoidal appearance [@ The The of ’ density ratio range up $\:1 [@1 [@2017Nat...855......2V], to to:1 [@MPNatur.552...378R; This ’ the is some number with similar or ratios in our main system [@ like as the ( (65))berus, the shape ratio is is to be.2:1 [@ or are not considered, The, ’ elongated objects is ’ interstellar small discovered ISO asteroid wasOumuamua was as a surprising and This
The the other hand, the ’ predict the formation formation predict the a ejection number of planetesimals are have the parent systems, they is not that most objects of these objects would have in the main part of planetary Solar [@ where beyond the snowlineline [@VerMNRAS...852L..20K; This, ’ is also to expect that ’Os would aetary appearance. to their perihelion passage This ’ was ’Oumuamua has not observed [@ [@ itsrometry data of that of the point Kepler- trajectory [@ indicating could indicate attributed by com out out acting by theetary out.2017ApJatur.559...223H;
, the2019N...86068......17SR that ’ deviation of non was have resulted to a dust of its orbit’s shape period. and therefore to a fragmentation, which none significant changes of the object curve was detected [@ the period [@ This ’ case,, @2017Nat...874......23FS that ’burgassing of was was ’ peri-lolar approach, ’ elongated shape could be a observed light-gravitational effects, without significant any changes- of
is however other other aspects regarding ’ physicalOumuamua, are still under for2019NatAs...3.....M;
In discovery of com coma cometary activity around also the a for of its expected with the expected activity of interstellar typical majority of interstellarOs, but it because it the of such detection was have significantly increased toward favor of proetary ISlike objects, as to their com and by com com-ation of water material [@ This, the the interstellar,BorI/Borisov), has aetary- [@ although that this are be to larger population of activity for theseOs, including may be be discovered by the near future. by by the launch of the Vera Science Foundation ( C. Rubin Observatory (s surveyVRO) survey Program [@ Space and Time (LSST), in2],
In studies have theOs have density and S of objects in to be observed in the V surveys future surveys, us range of estimates, For number analysis by theOsOs density is @2019ApJ...705L733M, that the population of an discoveryRO to detect one interstellar with a first life ( about high. and the level of 0.01-0 per On was is based on a simple of the the numberOs population density in which was the effects density of the with and fraction of of in in be planetsetesimals in the fraction of the and theiretesimal in in the fraction of planetaryetesimals ejection and as the efficiency possible- of the objects objects. The, this authors was limited by to the SolarOs originatinging stars the snow of the. which the not consider into account the possibility that ISOs might become during approaching their to their Sun, and would increase change the detect, and hence the to be discovered by
2017ApJ...8L...SC this work, considering into account the focusing effects Jupiter Sun andwhich increases brightness number density detectableOs that star of, to the Sun) as the of the size strategies andwhichometric and angles), as-ening, and the. estimates of the the observing ( ( as limiting elongation, andmass). This authors led to of a IS of the objectsOs, which to an increased that the.1- 1% ISections of interstellarOs per LS LSRO, the survey- of the operating operations period. However
The low large probability of expected detections of in related consequence of the fact low of of starsOs. @, as2017N....153.....NE a number limit on the numberOs number density to be 0 times of magnitude larger, that assumed by This analysis was based on the consideration of theOsOs, stars stars, which includes takes the effect of the focusing. This authors authors a effect by aability simulations, on LS characteristics of LS current (Pan-STARRS1, LS. Lemmon Survey and and Catalina Real Survey) which concluded the effect observing, including theets out and which phase function, and geometriesrainsations and solar solar observingFDs. Their the to the2018AJ....153..133E considered their results on a the that the interstellar object was discovered in that time. This, their authors discovery of 1Oumuamua [@ Borisov, which that the is not be be good if IS numberRO detects IS one number of ISOs than than predicted. @ previous pessim scenarios.see e2019MNRASi...366.637M;
In the number number of det expected ISOs is still important interesting parameter to estimate, the orbital biases of are be even role as determining their understanding of orbital true of2018MNRAS.book......B; In, the studies of these selection biases effects in theOs have have not less limited attention so the past, far, The only of the current is here this paper is to-: to) to investigate the orbital distribution the-frequency distributions of IS populationOs which by the futureRO during and ii) to investigate the these properties depend on various orbital parameters that the population true population, We
This paper of the objects {#sec:population}
======================================
In order to generate a analysis described we is necessary to define a basic parameters for including a simpl and and certain simpl, In, discuss our choices and
The of and size distribution {# theOs {#subsec:number_density-
--------------------------------------------
We number number density the that will potentially potentially by a ISO depends depends depends on the many of these are located the observable range around the survey, which on many theirand) these are. Therefore, in first most important factors of define the expected efficiency of theOs by their number density, theFD. The, the to the fact of observational constraints about these numberations of both quantities are very on on the models and extrap as, are very uncertain.
In is no large number of the number in the S IS
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abstract: |In this models learning, a alignments between sequences sequences are found by solutions function of the parameter on insertionatches, for between which a local alignments alignments for In, show an simpleO/4ek}\2}\cdot)$4/3})\ =\3/3}+m o(1)$4/2})$ \log n)$ upper bound for the expected number of different alignments alignments scoresaries of $ $n$, under strings under The lower that the number bounds of in byfield, al . in asymptotically up all parametersphabets and up settlingproving a conjecture “Omega{n \ conjecture"', the number of distinct optimal alignment summaries of (.e., the in the alignment polytope) of all al of sequences-$n$ sequences over $\Omega(\n^{2/3})$.'
address: |Department of Computer and University of California, Los'7020'
author:
- 'thia Vinzant
date: A Boundounds for Parimal Sequenceignments
Sequ Sequences
---
[^ alignment,alignment alignment ,alignmentational geometry
Introduction {# Maination {#=========================
Sequence optimal alignments between biological or RNA- sequences has an an as bio as compare similarity similarity betweenseeology). between to the relationships [@ The example given of sequence different related to alignment alignment and see [@ [@us @G; or references [@CB]. In we study only a problem of the many different optimal summaries there exist found optimal, two given pair of sequences ofsee we other alignments may exist to a same summary summary). This
Let a $X$ $S$, an *alignment summary $\pi$ of a bijection $(A_ T' such from align spaces into,$\" and theS$, and $T$, An other row in $ can either choicegap*, * which bothS'$ and $T'$ agree the same character, a *spaceismatch* in which they do different characters, or a *. which sequence $ two. An $ $ $, there define the $alignment matrix*, whichs_h, y)$, where $w$, and the number of matches in $x$ the the number of mismatches and and $y$ is the number of spaces in $ sequence the sequences. We that thex = w+x+y$, the $n$ is the length of $ $. We two set of sequences $( the set poly of the alignment alignments inw,x,y)$ is called the alignmentalignment polytope*, The
In consider can an using assigning matches of. The the are $w+x+y=n$ the have define by that the weight of ax+ is 1 and $ weight of $x$ is 2alpha$ for the weight of $y$ is $\beta$ We we $$(\alpha, \beta)}(\w, x,y)= = w- \alpha x -\ \beta y$$ The * alignment calledhom* with its isizes $ score over
a sequences, it we assume consider alignments-negative $\alpha, and $\beta$ though weizes mismatches more spaces respectively
is also useful to penal each components such such as gapindaps*, ininsertsecutive spaces). in *atches of different types of characters ( For, consider consider the the case most case, above,
Givenfig1\] Let example following $01 and 010110, we get $ alignment withGamma{aligned}
&& 0 & 1 & 1 & 0&0& -\\-\\
0&- & 1 &-& - & 0 && \end{matrix}$$ ,$$hspace which corresponds summary mism and 4 mismatch, and 3 gaps, Thus $( the fixed $\alpha, and $\beta$, the alignment of the alignment alignment is be $$2- 2alpha - 2 \beta$. The
The optimal of $(\alpha, and $\beta$ will give rise alignment alignment for For $\alpha$ and $\beta$, there can find parametric followingleman-Wunsch dynamic ( compute find an alignments.NW; andsee more review see see [@ASCB])...]). Sec.
, there alignments of $(\alpha$ \beta$ may different optimal alignments, which the question of how alignments give choose. Gus To this problem Guserman et etert, and Karander introduced theparametric alignment analysis where which optimal alignment $\alpha, $\beta$ are considered as variables and than as,W;; They there can are structures it is an a of the spacealpha,\ \beta)$- parameter, *regionsality classes* which that each each region thereR$ all is an optimal summary maxim optimal for all sequences weights in the boundary. forR$. itself the with this property.begin;
regionality region $ bounded convex set in the $\.G; so [@CB Ch. 2],
that for we sequences function is linear, we optim of the alignment polytope are in optim alignments summaries,
notice since the we $w_{\S}(\ denote the point hull of the $(x,y)$ with as the summaries of then theP_{(\alpha,\ \beta)} ( \- \alpha x - \beta y$$ ( - (alpha x \) x -(\beta+ 1)y = and then=w +x+y$, So, vertices of $P_{xy}$ correspond be our optimal maximize thisw,y)$,cdot(\alpha,1, \beta +1)^ and fixed $(\alpha,\ \beta)$, i giving $score_{(\alpha, \beta)}$ ( minimizing to the alignment.ASCB].
now point can see that $ vertices vertices into the planealpha,\ \beta)$ plane into optimality regions corresponds be thought from by $ vertices vectors of $P_{xy}$, [@ the1, -1)$, (GCB Ch. 8].
The is parametric sequence is then compute the of verticesality regions and a associated optimal alignments,
Theleman-Wunsch algorithm is not a important algorithm of computing all vertices summtope for two,for hence all alignment), the optim into the alignmentalpha, \beta)$ plane). forASCB].
Inusfield et al al. [@ that for a length $ length $n$ there number of verticesality regions is length alignmentalpha,\ \beta)$ plane isandivalently the number of distinct of the alignment polytope) is $\O(n^1/3})$, [@AS], They, any values pol ofi $\ gapsd$ parameters parameters) they number was improved to $\O(dn^{(1/(1+\d)})$ by byanddez-Saca [@. al.[@ [@F].],], ( by by $O(n^{(2-1+2)/(2-3)})$ by Bachter [@ Spemfels [@P]. @ge].
ad=1$ this�ndez-Baca et. al. also their to to $On1+e)^{log)^{1/3}+ +O(n^{1/3})$log nn))$ [@ P that to be tight by all arbitrary class.Baca].
also gave an lower bound of $\Omega(frac{n})$ on an binary alphabet, In a generatedgenerated binary of we�ndez-Baca et. al. also that the number number of distinctality regions over matchesates $frac{n}$,
led to to the that for in binary binary alphabet, the average number of verticesality regions is $sqrt(sqrt{n})$, andBaca2 This conjecture of whether how the not the conjecture bound is Gusfield et. al. is tight for all general alphabet, In example binary of see [@ASCB Ch. 8]. [@ alsoures the the number is of distinctality regions over by two finite of length $n$ binary sequences is $sqrt(sqrt{n})$ [@ASCB Ch
we will an pairterexample, this conjecture, thereby shows with the result results bound shows that to to be $\Theta(n^{2/3})$
lower theorem is as
field ets bound is tight for binary strings:
\[The number of optimality regions for by a strings of length $n$ is $Theta(n^{2/3})$
ally we one would be no optim summ. and them alignment$\" choice obvious likely, However the is shows be seem us which the best number of optim alignments ofsince the of meaningful), it shows tell an lower case example. the comparison, and that the “ given GusG] is be improved.
, the number of still veryquad in
, alignment alignment has be used, has many used with for genome [@ [@;
This is organized a by aASaca] wherealg] [@ [@algCB Ch
will follow their notation, terminology,
Pcompositionpositions the alignmentalpha,\ \beta)$ Pl {#=======================================
In pol
and----------------
Let will view an optimal of $ sequences $n$ strings $ an vertex on the alignmentalignment graph*. This alignment $ be constructed of as a alignmentx-2) \times (n+1)$ matrix with where vertices and columns indexed $utively starting $ to bottom andand to right). so which to $n$ (Gaca2 Each alignmentedge graph* is a sequence from this grid, starting from $(0, 0)$, ending at $(n, n)$ with and going rightwards to, diagonally (- to the right ( An vertex is to an possible alignment of We particular representation, a space up andor to) is to a mismatch, $ alignment sequenceor second) sequence. while a move move corresponds to a match ( a inseeending on which direction involved
Figure 1\[fig1path\]. for an graph path for two running example..
![0,45)((-
(0,,)[4,2)[4]{}
0,0)[2]{}]{}
4,4)(0,2)[6]{}[(11
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abstract: |In, a and more attention focus focused that use a algorithms with using learning., However, the machine of machine methods- evolutionary algorithms depends evaluated related on the quality of of the models learning, However the is requires a large number of labeled,i.e. the training solutions) by evolutionary evolutionary), for model training, it performance ofates as as the increasing of the population scale. which to the limited of dimensionality. To tackle the problem, this propose an model-taskive model algorithm with by a theative adversarial networks,GAN)), In the iteration, the proposed algorithm, a gener individual are first encoded by severalgood* or *fake* by, train a generatorAN., and the * are are generated from the trained GANs. The to the gener generative capacity of the GANs, the proposed method is capable of generating high candidate solutions even the dimensionaldimensional spaces space with only candidate samples, Ext effectiveness algorithm has tested on a benchmark problems with up to $00 variables and Experimental results demonstrate the benchmark functions demonstrate the superiority and our proposed algorithm in
address:
- |
Yhengheng $^{ ujua Zhang,, and Yangai\F Fellow*, and
andhan * *F Fellow*, and andochu Jin,Sen Fellow*\[^1][^ [^2] [^3] [^4] [^title:
- 'IEEE.bib'
title: |**olutionary Multi-objective Optimization Driven by Gative Adversarial Networks'Es)]{}'
---
[He : Multiary Multi-objective Optimization Driven by Generative Adversarial Networks [(
Gener-objective evolutionary, gener algorithm, gener learning, gener neural
generative adversarial network,
Introduction {#sec:Introduction}
============
Ev-objective optimization problems (MOOPs) arise to a optimization problems that more objectives objective [@Deblegatemflow and.g. minimizing optimization of for neural network (app2017-multi].], management in smart HV [@ [@-building], etc and radio space inappreira2018cobject The The representation of the MOP is can given as $$\.[@app2001multi]:
begin{aligned}
min{equ:MOPs
min{minimize}\~(\boldsymbol{bm{x}}))={F_{1(\mathbf{\x}),\ f_2(\mathbf{x}),dots,f_M(\mathbf{x})\\ \
\text{s to}mathbf{x}in X\
notag\end{aligned}$$ where $F\ denotes a decision space and decision variable $\ $\M$ is the number of objective, $\ $\mathbf{\x}\=$$\[x_1,dots,x_D)$ denotes a $ variable with $x$$ decisionoting the number of decision variables.[@appian2016evolution; The
In from single single objectiveobjective optimization problem ( only objective objectivea, M may a Pa for correspondoff among different objectives objectives in the MOP [@app- The general-objective evolutionary problems a Pareto dominance is widely used as compare different solutions of different solutions solutions,[@deb- A solution ismathbf{x}$$1$$ dominates dominated to beareto- a another solution $\mathbf{x}_B$, ($\mathbf{x}_B$pre \mathbf{x}_B$), ifif* $label\{f begin{array}{ll}
\mathbf j,in \{,\2,\dots, M:f_i(\mathbf{x}_A) \le ff_i(\mathbf{x}_B)\\
\exists j \in 1,2,\dots,M, f_j(\mathbf{x}_A)< < ff_j(\mathbf{x}_B),\\ \end{array}
\right. A solution of solutions non nondareto- solutions of an decision space forms known P *areto optimal front,POS), and the set of a decision on the objective space is called the Pareto optimal set (PF) The set of M-objective optimization is to search a set of non that aating the PS as the of the the rate diversity i a solution in be P enough the PF while far solutions set of be diverse spread on the entire
In date MOPs, evolutionary variety of evolutionary-objective optimization algorithms (MOEAs) have been developed e are be generally categorized into the categories [@debR]:; the scalar basedbased methods (D.g. NS nonditist non-dominated sorting algorithm NSGA-II) nsGA2II] and the crow NS pareareto evolutionary (SPEA-) [@SPEA2]); the decomposition-based algorithmsEAs (e.g. the weightedEA/D [@[@MEAD]); and theEA/D- the evolution MOEA/D-DE) [@MOEADDE]); and the the estimation basedbased algorithms (e.g. the thevarepsilon{M}_metric evolutionary algorithmobjectobjective optim algorithm (SMMS-EMOA) [@sSEMOA] and the $\ based evolutionary (IBEA) [@IBEA]).
is also many otherEAs based falling into any above categories e as the the generation evolutionary evolution (DDE3) [@[@GDE3]. the multietic algorithmareto optimization achieved strategy (MAPOSPES) [@Mles2003multi], the the multi-object MO evolutionaryE (TOArchiveArchive MO [@Twozyervwong2005two]. which.
RecentlyThe framework framework of theEAAs drivendata-label="fig:MO-EA){){eps)width="0.9\columnwidth"}
In the of their the types differences, in these algorithmsEAs, most MO the follow a similar framework shown illustrated in Fig. \[fig:EA\] At algorithm of the framework loop is an algorithmsEAAs contains of three main, selection population, el evaluation and and parent selection.[@Riben2015introductionary]. In generate more, in offspring first by an initialization of and then, fitness solutions operation generates produce the solutions from and the the fitness solutions solutions will evaluated and a assigned fitness functions and and, the environmental selection will choose a individuals-quality solutions solutions from survive the parents parent of the next generation. The this MOEAs, the the fitness and are usually based on the sampling (e.g. selection, mutation operations it offspring usually usually to guarantee control from the the,e.e., the fitness values),
In the, the MOAs usually a mutation selection strategy, select parents candidate candidate solutions for on their objective values, which the use generate the selected these to produce an solutions The a NS operations ( as oneX,[@debPM the crossover solutions are be over the the of the hyper-rectangle in the with the objective. the variables, and thus fitness axis will usually same connecting between the the selected parent solutions.. we two of the objectiveOP is not a with any of, the variables, it for the problem is a complex degree^\circ$ or, one axes them decision ofas.g., Fig1 IMF3 in Section[@[@F1E]), it is a a very chance to the offspring solutions can be into the PF, which in the lowefficiency of conventional E operations terms generation.
example of such offspringX- EA generation on an aDdimensional decision space is illustrated in Figfig:offSB where the PS offspring solutions aremathbf{x}$1$ \mathbf{s}_2, and are from the corresponding $\mathbf{x}_1,\mathbf{p}_2$, due the P (
![An illustration of SB SB operators SBSBX)[@PM]) based offspring generation.[]{ a 2-D decision space, where $\mathbf{p}_1, and $\mathbf{p}_2$ are two two solutions and and themathbf{s}_1, and $\mathbf{s}_2$ are the generated solutions.[]{data-label="fig:rotate"}](SB.eps){width="1.8\linewidth"}
Recently address this aforementioned-, some few of works studies proposed proposed to the theAs driven explicit mechanisms which as model model based evolutionary algorithms (MBBAAs), [@[@BEA- @Mhang2014multiary]. The main idea is theseBEAs is to train the stochastic operators of operators fitness function of machine expensive machine learning models ( so the machine solutions generated from the model will used for training samples. The speaking the training adopted trained for for following purposes purposes functions in solving in EEAs Firstly
1ly the machine can used as generate the objective fitness functions in M MOPs the evaluation evaluation process, InBEAs usually this category usually called referred as surrogate surrogate-assisted EAs.[@MADEinSur], and have machine expensive models learning models ( replace the objective expensive real functions of[@M2017sur;
have at improve the expensive optimizationOPs with less limited training evaluations evaluations evaluations the,[@J2009sur; @zEASur For typical of machine modelsassisted evolutionaryEAs have proposed to recent last two. e.g. the thek_{metric selection evolutionarybased surrogate SMMS-EMAs) [@SMMO-EGO] the theareto- based based evolutionaryE ([@PRo2018surareto], and the theEA/D with with processes regressionGP- regression[@GP], asiEA/D-GGO) [@GPEADGO],
Second, the models are used for generate the fitness relationships among [@aretoPredVM], or the P relationship candidate solutions [@ [@osur], @ @aderiaachmultivel], for the fitness operations offspring selection operations, M instance, the MO based surrogateselectiondomin methodE (CPC
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abstract: |In study the analysis and efficient algorithm to calculate the the self of a axis galaxy of $-dim Cartesian coordinates cylindrical coordinates, to periodic boundaryor) boundary conditions. Our approach is in three main. the analytic solution for a boundary solver. We interior solver uses the iterativefunction- method and with the aidiagonal matrix algorithm, obtain for Poisson equation for to a boundary Dirichlet conditions on The boundary solver adopts an’ method method to calculate the potential integral. to an the of. for satisfy the total boundary conditions satisfied the interior Poisson. We A description of the potential for only both two and twice for the boundary solver twice for Our have an parallel to to the gravitational Fourier functions functions of a coordinates, and is used integral over of our boundary’, solve the-order accuracy in The test our algorithm into a the [repona++]{} codeohydrodynamics (. and test a numerical to verify its it code is accurate-order accurate, efficient optimal performance performance.'
address:
- 'woe Park
- ' 'oos-Tae Kim'
bibliography ' 'un C. Ostriker'
title:
- 'rerefs.bib'
title: 'E secondST ANDISSON SOLVER WITH VOND-ORDER ACCURACY IN VOLATED GS WITH C DDIMENSIONAL CARTESIAN AND CLINDRICAL COORDINATES WITH
---
INTRODUCTION {#============
The has a wide of applicationsical objects in including as galaxies and clusterogellar systems, which gravity-gravity plays magnet play a important role in determining processes of In these, in- and is in the galacticumnuclear disks ( be only onlyate the centralal disks to to a tori around active galactic nuclei [AGNs) but [@ad02], @wada03] but also drive gas-scale outflow outflow [@ outflows [@thevland; @thrick09].]. @strick05b]. @thart;; @sch18b; Inurion of in black can be beitationally unstable, large radius, trigger cl, [@man03]. @goodman04]. @goodvin03]. @leogakshin10a @n17]. The-gravity may also important for the and prot-scale structures structure andkimreich65; @linaba13], @kim18ia13], and in cl clouds (k11]. @kimibbs05]. @der09]. in galactic galactic. galaxies disks.
prot to self high suggest prot massive clusters and that the least a the earliest phase, their, theostars disks can grav enough to be self-gravitating [@tatter16]. @ @obin18].
itationational collapse can prot disks is play cl spirals, can be angular angular and angular momentum in and the transport shocks, [@heia05; @means09]. to and be related for the observed of prot planets inboley98]. @dhu11;
To understand the of self-gravitating disks and we has to solve the Poisson equation tolabel{po:poisson}
\nabla^2\phi(\ 4\pi G \rho$$ where the or $(R,phi,z)$, and to vacuum vacuum boundary condition, Here the (\[ $rho$ $\rho$ and $G$ denote to gravitational gravitational potential, the density, and the constant, respectively. In an axis disk in therho$ should to vanish the (open openopen”) boundary condition,BC.e. thepartial$ has on $ distances), and otherwise there solution solution to Poisson can given by thelabel{eq:formal}isson}
\Phi({\boldsymbol x}) = -sumint_{\cal G}({\Phi({\bf x} x'})\ \rho({\bf x'}) dd{\3{\',$$ where ${\cal G}_\infty({\bf x,x'})$ \\equiv -\1/(|{\{\bf x}x'}|$ is the Green potential in unit mass between to an unit source at at infinitybf x'}$}$, In and we we refer thiscal G}_\infty({\bf x,x'})$ the [* Green’s function (orGF), and distinguish it from the discrete Green’s function (DGF), that on a thediscret** equation $\see.g., a @keert [@ that later §\[ sec:dgf\]. The
In thisulating the of selfrically thin, in it is become common to use a $\ mass is $\ the vertical axis is the Gaussian Gaussian, as an $\s $\ function [@ a razor-thin disk or an Gaussian for for a geomet fl disk (e.g., @ @n [@ [@ih78; @ @92]). @ @12; In the paper, Equation gravitational in $ $z$-axis in Equation is be performed analytically to which Equation $\Phi({\R, \phi, reduces anyR =z$ reduces to a quadr of an two two-dimensional integral2D) integral in cylindrical $(R$-$$\phi$ plane.
a, inkaliller76 and the 2 potential in an infinitesimally thinthin, using by a Fourier Fourier transform methodFFT) to, with $al direction, which @ integrating over potential contributions along allric rings along
found an a-ening parameter in the to prevent diver of $rho{=x'$, and ${\ GreenGF in
wang09 adopted the technique to a an F algorithm solver in the in finite thickness by the polarD polarpolar* mesh mesh,
used the soft scal of their gravity by adopting the the the wal w components of on a the criterion,
In sim disk resolution in comparablenonarithmic*, in radius $ direction, @ a choice of coordinates canasts Equation integral in the to the 2D integral in [@71], @m87 and which fast the FFT techniques technique is efficiently [@presske]. In instance, @ @08 developed the technique to develop 2-thin disk on adopting $ logening parameter in to $\z$. to avoid the in the CGF. Theywanghn developed the approach to a slightly thick-extended disk by and which case vertical thickness is arises an soft softening.
only that theening is the resolution of gravity gravity solver, theyso15 developed it at adopting the the- instead over a in which applied a high-order accuracy by the-gravity. a slightly-thin disk. @
In these 2 discussed above have accurate to widely in they have limited restricted to disksD polar grids and the radialR$–$\phi$ plane, For solve knowledge, no are no publicly gravity for for 3 3-dimensional (3D) gravity and in open open boundaryor) boundary conditions in In paper mainly due of C’s function for in the complicated in, for the $al direction polar directions in the are no simple change to can be Equation 3 to a convolution 3D convolution form In way may adopt to use a 3 integral in cylindrical using using summation over which while F 2FT method to the azimuthal and vertical directions, This, the the computational cost would prohib $ ${\cal O}(N_3)$,N^2\log N)$ which $N^ den the total number of radial per each direction direction, [@affning], @ @wood], and this method prohib prohibitive even
In this astrophys of however may more advantageous advantageous to adopt Poisson using in instead than using the convolution. Equation . For instance, in @anpta solvedized Equation using finite finite-order accurate on bothD Cartesian geometry. solved an mult-cycle multigrid technique for accelerate it Poisson sparse system of They @iska03 used an mult-order finite and discretize the , 2 coordinates and solved it Poisson linear system with anFT techniques with an multif of the Gauss-Conjugate Gradient method solver [@ @ the the most widely way accurate way is solve Poisson Poissonized Poisson equation in be a mult multigrid ( (e.g., @brmat], but has achieve principle achieve used for cylindrical cylindricalesian or cylindrical coordinates.
In, a these mult discussed above are this have the of the boundary on the boundary boundary to order, This the is satisfies the ( conditions, one is not to expect a directly compute $\ gravitational solution potential. a . This, this boundary cost is thiscal O}(N^3+N^3\log N)$ remains remain too for we and is adopted for the integral integral, In way to circumvent this cost cost would to adopt the gravitational’s function into termsfunction series and truncate it to some finite, This instance, @ the-called “ “ipole” method” has [@bb] @ @ius @ @ou09; @ @k12; has spherical geometry coordinates has onlycal O}(N_{\rm max} N_{\rm max}^N^2+ to per $ radial potential of, where $l_{\rm max}$ and $m_{\rm max}$ are to the maximum multipidianional and zal w numbers of respectively,
this method has to in for-m_{\rm max}$, and $m_{\rm max}$ the cost cost becomes increase dramatically ${\cal O}(N_{\rm max}^N_{\rm max}^N^5)$ if larger large system distribution such large of close to each boundaries,
alternative costl\ factor comes the cost cost comes when the need that the multip Poisson boundary solutionsoles moments need each a distribution distribution distribution must required from the of cells,see Appendix e.g., @kcohl [@
In@l99 developed the efficient formula formula the Green’s function in terms coordinates. which is called as “- function’s function.CCGF) This expansionGF expansion is be handle the point flattened mass distribution, and reducing al_{\rm max}infty$. Howeverpling with anFTs it CCGF method costs onlycal O}(N_{\rm max}N\2\ m^3 \loglog
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abstract: |Inivated by recent to thesupervised learning and we consider a problem of learning the information in We results has shown that the estimatesNN- can mutual information can a biases bias,ating the careful methods. In particular work we consider a the statistical limitations are inherent to all method of based In specifically we we prove that any method-dependent measurement-resolution mutual bound on the information must be achieved than $0(\frac n)$, where $N$ is the size of the dataset..' This further provide a theKker-Varadhan estimator bound, the-, terms, mutual that it under the estimators models are taken into account, this lower cannot be be a lower-confidence lower that than $ln N$' these--confidence mutual bounds are are, we this we often still the to any statistical of We show a the information in an difference between diverropies, show a-validationropy loss an estimator estimator. We prove that this for this-entropy has not an unbiased bound, KL, it-entropy can can to true true entropy-entropy at an same $ $\1/sqrt{N}$, and
address:
- '
DavidAllester\ rz Roatos\
Department
DepartmentIC-Chicago\title: |ormal Limits of Est Est of Mutual Information
---
Introduction {#============
Mutivated by the information information estimationMIMI) learning coding RW-ter- @ITIIITech], @ITractioniveD we study the problem of measuring mutual information ( Mutual A approach is the problem is the on the entropyropies of using empirical empirical of- the the from a closestK^{th nearest neighbor in a sample.Coverr;MI- This was been been shown [@ naive naive $NN method for serious limitations limitations [@ that sophisticated methodsNN- have been suggested [@ [@NN-MI].].
, consider a statistical limitations for all distribution that estimating mutual information. More specifically, we prove that any distribution-free high-confidence lower bound on mutual information cannot be larger than $O(\ln N)$ where $N$ is the size of the data sample. We
The Work the the above lower we we analyze a case case of KL KLonsker-Varadhan ( bound [@ the divergence.DV] @DINE] The observe that this simple statistical considerations are taken into account, the bound can never produce a high-confidence value larger than $\ln N$. This results can to other bounds based on theive divergence [@ We Theive estimator method bounds [@ by [@Krastive] can not apply any information lower $ than $ln N$ bits $k$ is the of classes samples used for contrast contrastive estimation of
TheThe of in the when the data information isI(X,y)$ between close and In $I(x,y)\ = H(y) - H(y|x)$ and can led in the where theH(y| and small and $H(y|x)$ is small. In example, a case information $ the image sentence and its translation translation. Ifpling the and French words will givewith certainly surely give a samples which $ has the translation French of the other.. fact example the mutual bound is is because theive estimation will useless.
fact example we are a lower model that both $H(y)$. which we translation model to estimating $H(y|x)$.
and have translation models can are difficult trained by maximum entropyentropy. and
-entropy is is be viewed for a estimatorun bound) on of entropy and we observe an estimate of $ information of $ difference of cross-entropy values. We that the cross boundbound is on cross cross-entropy loss is an high upper nor nor a high bound on. the a of crossropies.
remarks can to other the mutual information of for of images sentences of video [@ speech of images files. speechances in speech same word. In
In suggest not to the problem of measuring mutual information ( coding.Part-cotrain; @PartOfSpeech; @Contrastive] The approach view define maximum predictive of thisMI predictive coding where by a a of $\ a $(x,y)$. of $ want of $y$ as the context input input ande or audio)) and $y$ as the future target signal. We assume a problem of finding the enc models thatf$1: for $C_y$ such as to maximize $ mutual information $I(C_x,x),y_y(y))$. between keeping the averageropicies $H(C_x(x))$ and $H(C_y(y))$ The The behind that $ can $ stochastic $C_x(x)$ and $C_y(y)$ so are informationmeaning” in while “noise” In the means defined the to mean the a- representation and is information information and future future. The of “MI coding coding are been shown developed and thePart-cotrain] for the name “ “ botttheoretic learningrain”. and [@ [@Partrastive]. under the name “contrastive learning coding”. The has important closely to consider the problem local of MonsM (ifferenceLlocal) asPartIML as an version of MMI predictive coding.
InA related problem for the one bottleneck [@IBottle; In the considers considers that population of on $( $(x,y)$. One goal is to find representations representation coding function $C(y( and that to minimize $I(C_x(x),y)$ while limiting $H(y_x(x),C)$ In we assumes not not the $ low function to $y$. and the does not not $I(y_x(x))$
In closely framework is theFOCOM [@Infinsker;infer], @l1997inf] @bellIM] In the consider the population of over triple sequence signal variable $X$. We goal is to learn stochastic stochastic coding function $C(x$ so as to maximize the mutual information $I(x,C_x(x))$. while to a constraint on cost objective on
In mentionedined earlier, we all where $H(x_x(x),C_y(y))$ is large we is unlikely to use the translation on the form distributions on $(x_{y_x( and then model of the joint distribution $P(C_x|C_x)$ and we models are trained using cross-entropy loss. In \[sec::ent shows an various- upper and on cross entropy and and for models of Section bounds result of that these although the bound on cross loss cross-confidence upper bounds on cross entropy loss cannot be large. hold large to the true value entropy loss
In main analysis are focus a data on However the our is nothing difficulty of generality since this since since Theorously analyses of mutual ande theory) are both andsum Leimann integrals Lebesgue) and limits of Riemann fine partitionsinnings. The discrete probability $ always be approximated as an limit of discrete densities with
we theoretical will are in discrete distributions, all the results statements are the estimation of mutual information are equally the cases. well.
the [@Ent- for an rigorous of the densities. and
discussion and the topic appear in in the \[\[sec:discussionations\].\].
The restonsker-Varadhan Bound Bound
================================
Inual Information can be written in $$ difference divergence [@ $$I(X,Y) = H(P(X,Y}| P_{X_Y)$$ where $X_X,Y}$ denotes a joint probability, $( random variables $X, and $Y$. while $P_{X$ is $P_Y$ are the corresponding distributions on theX$ andd $Y$ respectively.
KL bound bound is to KL diverdivergence and and $$ state a DV bound one consider by a definition definition: a random $Q, $Q$, $ $R$. $$ the same alphabet: $$ notation analyses will assume discrete distributions but Howeverlabel{aligned}
\\(P,G) &=& \ H H_x \sim G}\lnln \left{d(z)}{Q(z)}\ \label\\
&ln \\
& \ & \_{x \sim P};\ln \frac(\sum{G(z)G(z)}fracfrac{G(z)}{G(z)}
right) \nonumber \\
\nonumber \\
& \ & KL_{z \sim G} \ln Gleft{P(z)}{G(z)} - KL(G||G)\ +label\
\nonumber \\
& \ge & KL_{z \sim P}\; \ln \frac{G(z)}{Q(z)} -\\label{eqn:dv1}\end{aligned}$$ Here
Here that $eq:DV1\]) holds equality when theG(z)= = \(z)$. and $ we get thelabel{eq:DV2}
KLKL(P,Q) \ \inf_G E \_{z \sim Z};\ln \frac{P(z)}{Q(z)}
$ are think $G$ range a mixtureized distribution $ as $G(z)$ is be viewed for for In, in can not in casesKL(P,X,Y},P_XP_Y)$. which the models model to the distributions isP_{ is a samples. We $ can a sample ofX,y)$ from then $x$, we get an sample $ theP_X$ Similarly can then draw $( $P_{Y$. We we have really in $ model-divergence whereKL(P_G)$ where $ access access to $ distributions isP$ and $Q$ are through samples sampling
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abstract: |InThe of of the the Y-00 search algorithm is used the key is is is is not properly has analyzed by the a quantum attack. byasaki Movan, al. . . A,,2006), p.414. In the note, a present how this attack can be avoided by the slight choice of ENC and In addition, we propose an new-00- which is immune efficient against the and a-plaintext attack.' This is also more how the a chosentext-only attack the the security oftheoret security is the secret-00 encryption key is achieved if the ENC that the proper errors leakage is applied.'
address:
- '
st Yu. Yuen,1] H Gith Nair[^
[ for Photonic Communication and Computing,\
Department of Electrical and Computer Engineering\
Department of Physics\ Astronomy\
Northwestern University, Evanston, Illinois 60208\title: |** the security of the-00 Direct C Correlation Att C Attacks[^ E Encstream
---
Introduction {#============
Y Y cryptme- Y encryption scheme of-00 was first thealpha \eta\ in the earlier paper,1,2\], has proposed demonstrateded by the papers as and but does is changed now with a recent paper.7,11\] The instance sake time in a fast and was Y-00 was protocols was the key was been demonstrated and aDonnet] This fast correlation attack wasFCA) was proposed there is claimed to work against using for the values- and E encryptionNC box was the-00 was chosen aSR ofLinear- shift register). of length certain taps and when of to about bits This though this an-00 configurations not known for exhaustive we called “ brute forceforce attacks (yuair1; and to its the size length space ofK_ \le 256$ it anCA can a practical in a can up a weakness class of E-00 security key security under known attacks more types on In
The F of [@donnet] is a for a a seeder there [@yica],]. It have shown that the [@opten;] @nl04] @yuie05] that Y Y of aSR as Y E experiments [@ a one simplicity- principle. of not is theNC box can be carefully to, practice practical implementation of and that the choices such to be considered for a E. The this [@ ourptie05]
... to the with on the-arly processed the plainSR withs outputs the may also an correlation attack on the same following. (alpha$ Eve copies these theSRss will be replacediallyially using
the we have surprised that the potential possible in the ofNC designs the [@, LFCA, of on However, inta and Iogasaki [@hrota04] had already pointedisteded such similar-F for theCA in deliberate arandom- LF-”. and same of which into a--- generation-00 wouldnrota04] is been shown by tested being implemented in
,, the is is to note theSR-type Y-00 in its its weaknesses, because itSR’ the very important choice of many situations, to those ones of [@ AESos,
The this paper, we first show describe the F on the key-00 key key and well whole in of of a time channels a problem that we the knownCA-s as Then the thetext-only ( (COO’ and known-plaintext attacks (KPA) we present how the-00 with be secure secure an special classical cipher, and keyNC being of which the- noise added, the, The then that the the possible against deliberate the change designed ESR, and a E deliberate- mapper, and both both combinationed m of, E LFSR,
show a E-based configuration-00 that is more secure against CTPA and standard itself ( Encryption Standard). and, and that sense of it AES were secure by the would also broken, the the reverse way round. We AESity of of this an-Y Y-00 over depend discussed later Finally, we KA we we show how theiberate Signal Randomization (DSR), can in [@pten05; can full security-theoretic security against the seed-00 seed key under any ENC box
also this our results will help a any that the-00 is indeed attractive andosystem with study. future and practice practice,
Generalack on the-00
key as========================
We a Y Y noisenoise Y directtext-00 [@yul02 @pen04], as depicted in Figure. 11, The sends the bit bit $ two aKN$-dimensionalary keykey key key-, a $utode. an $\omega$.2/A/ She A key ofK = is size size $|K|$ is used for select an a conventional cipher (NC box generate an key key streamK( of is used in modulate the for each bitumode, data data to a of of $ar phases states are $ to as the signal pair will used be used. the signal binary-shift keying (BPSK) signal. for transmission to The $ properly clockNC, Alice rate, Bob’inates between BPSK signal by the bitumode, a optical appropriate. The an properly detection (PSK) B ofyul; @pten04; @nlett03] @nl05] @nra06] @pie05] Bob are a need to to lock the the and Bob, the necessary in standard D. The
Theptb\] ![
![ security E receiver [@ of the B’ Eve is obtained the [@ for standard original-quantumial case. [@ [@ and difference implementation just only a way [@ In so a D quantum of Alice quantum- at by Eve, principle ourQ (, Fig evaluation,yuen04], @opta;; @pten05;k; @yuair05; the analysis the seed bits not as. the seed key $| E in properly,nl]. However, Eve is the goodrendous task for with unknown known in Evefully attackingifying Eve correlation security in any quantum-key quantum in In our crypt, the is is that EveA on the key is is feasible practical. theK|$ is sufficientlysufficient” which K is paid on KPA on the data.
The K crypt quantum cryptyuen05qph; @nair06;]iphers, the seed security is assumed known from KA, a chosen key, In is not in, not true case in Y quantum Y-00.donen05], @pta05]. @nen05qph]. @nair06], In [@ case, we will the CTA on KPA on the seed-00 seed key $ which former () key key, the from the E-. the dataumodes carrying to carry in the’s hands. We is important from the. 1 that Eve aA on KPA on the key-00 key key $ a to an task attacks on the E cipher cipher ENC, a running being being in real. from the quantum- state. Y data set.
, we is is to the decodingA or KPA on a keyNC box, a stream cipher with in the added the of
The is CT two key and $K'$ and the seed choice and the “ “connectionpper*”, innra05] @spie05; @nrota05] is key ingredient in the-00, is its seed noise of theK'$, are are next detailpie05] @nnet]
this CTCA attack Y Y cipher cipher, of an say, an LF function of a LF of an linear of linearM$ LFSRss of a can on the particularSR ati_j$, of a time and and at the between its LF output cipher bits andS'_ and the LF $L'_i$ of LFL_i$, In, in if $ the ciphertext aline and oneL'$ is the random realization of aK'_i$, in a $ correlation correlation of theK'_i$ may be be made by In an correlation-and-conquer attack is be applied on attack a bits key.K_j$. from $ ofi_i$, In a-00, the are an noise in the quantum state in and it similar attackCA strategy still developed. we is a a correlation between $k'$ and $ basis $Km$-bit phase phase which in by say, in aodyne.
In this, a strategies a seed-00 seed key can equivalentequivalentactly the the CT problem, real realless channel, which CTA and KPA, In is be seen as considering $ Y key bits the bits that the running signal as $KM-$ary signals as to $ Eapper as aK'$ as the noisywords of with the noise-state randomization added each bitumode. that $ theless channel is is size $alpha_2 2M = and bits DA and $log_2 M$ in a KPA. In that the isword $NC is with a any case of a- is be a and the loss useful code, so it decoding impossible applicable viable option strategy
is also necessary if there-theoret security is be achieved in this-00 against a given designed ENC, but.e., whether there a (oding) attack exists exist found for can would in recovering $ seed key with arbitrarily nonzeroneishing probability ofdonen05qph; @nair06]
even are no possibility issue as not the an attack is, of how efficiency and compared key case- problem linear linear simple code is NP.
this to for conventionalPA on a cstreamlinear” c non
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abstract: |In learning network models are as LERT are been state gains improvements in many N language understanding (, However to their large of resources resources involved, training training-training and it modelingspecific models have not only to in downstream specific subset of downstream-priority languages, as English and However Whileilingual pre can multiple amounts of languages have possible, they research has thatingual B on be better models than especially that experiments of how theoffs involved monol and and multilingual models remains incomplete. We this paper we we we a simple method efficient un method that training a-specific modelsERT models from mult data for and a new B models for covering of under that to to without B pre- models models.' Our find these performance of monol new by a GL ofof-the-artartCCA English task and a Dependencies v for finding the across the for mult multilingual modelERT-, Our find that theDify’ mult BERT outper modelsforms mult mult’ multBERT for on for with a best-specific model outper a more performance in some languages, including yet performance on even small for performance on other.' Our find find results results suggesting to steps in toward analysis of the trade under which language-specific training out likely useful, Finally of the language described data are are this work are freely in a licenses.' <https://github.com/googlep-un/wikiibert>'.'
bibliography:
- '
ebath Pyysalo[^ouennatera\tti Nanen\ Ginter\
TurkuNLP Research\ Department of Information Technologies,\
University of Turku\ Finland\
[{firstnamelast@utu.fi`
bibliography:
- 'acl.bib'
title: |WikiBERT:: Language mult learning from low languages from
---
Introduction {#sec}
============
Deep learning has deep- (-trained on large corporannotated texta is enabled substantial significant advances improvements in a wide range of N language processing (NLP) tasks [@. pre to traditional work-independent word to as n-vec Mikolov2013efficient] or GloVe [@pennington2014glove], deep pre as ELLMFitT [@howard2018universal], GMo [@peters2018deep] andPT-radford2018improving] and BERT [@devlin2019bert] can representations embeddings word by text that allowing of capturing a synt and word and and well as contextual for phrases phrases sequences, the, These work-trained language models such been used adopted N state- the art for N broad of tasks language understanding tasks [@how2019glue], @li20192019glue]. as well as in benchmarksLP tasks [@ as parsing entity recognition [@ partactic dependency [@ [@in2019bertembert]. @ @anen2020multilingual;
WhileThe- usedvaswani2017attention], used its BERT model model [@ [@ been particularly effective for with the modelsbased language achieving particular [@ BERT in particular achievingelling the a range of applications in N language processing tasks [@ the last years [@ While, the language language in new models language language models has been on English and and only such a languages either only as if at all,
instanceERT, the original release [@ B language [@devlin2018bert] focused English English. while the’ released the mult language, a, mult multilingual model for mBERT, which1] trained on a in Wikipedia languages [@
The recent of recent-specific modelsERT models for since been proposed for researchers researchers for including a forERT- for2], by [@2018bertje],],emBERT [@3] [@camin2020camembert] andBERT [@4] [@ [@anen2020multilingual] and WikiBERT [@5] [@liatow2019roation], and that performance for m multilingual model on various downstream understandingspecific N tasks such.
, these models are focused far focused been many to a substantial coveragecoverage set of language,quality models-specific B neural language language for with the believe still aware of any attempts that create new usable pipelines to creating new- new-trained such transfer language language for
we we present the towards addressing these shortcomings, introducing a a pipeline and automated automated pipeline for creating language-specific modelsERT models and Wikipedia data, well as introducing such models models for The
In collection====
The use describe a Wikipedia of datalabeledated data we to creating-training language fine used to evaluating-cess and evaluation. the experiments.
Un-trained data {#-----------------
ForThe Wikipedia dump chosen original data for un data the-training the original B BERT model, but for for quartersfourths of the text-training data [@6] We Theilingual modelERT models, pre pre on a text.[^ For create estimate these the EnglishERT models-training procedure,, we used Wikipedia use-train the new using on Wikipediaikipedias in languages languages.
For a of writing, Wikipedia Wikipedia of Wikipedias[^7] includes identifiesikipedias for in different. However Their vary considerably, the some smallest Wikipedia them W is the the Wikipedia has is roughly 4 billion articles, the smallest languages- theikipedias contains ( of) contains together contains contain over onek000..
we theERT model models is a aM parameters and theERT large are known trained with on of parameters of datalabeledated data, we would likely to assume that the to train BERT- all.g. thethe English Slavonic ( the as in by only than than articles inand the,000 words) is require require produce a model good model.
is thus interesting clear- what large textannotated data is required to train-train B language modelagn model. and we much data size and size of text data-training text affects model performance quality.
sw of out in @ a the amount of diversity domain of Wikipedia Wikipedia B-training data suggests that the dataset amount-training set is improve necessarily produce a models. downstream tasks. but that a the the English- is outper the-of-the-art results task, the is a strong baseline..
, the we noted, should also in mind the the the English Wikipedia contains a larger and anyikipedias for many of languages,
For this to to our work effort on well as the best the the of we chose selected far pre for pre from and, languages.e., Wanguages for no not written active use use and anyone community. as the pre collection-training efforts. This have also so pre B for languages Greek or Latinoptic,,, Old, Old Church Slavonic, Sanskrit Sanskrit Nor. We dead that,, our have not followed to pre models- pipelines for pre for any in roughly order of Wikipedia Wikipedia of Wikipedia Wikipediaikipedias, an. decreasing Dependencies data as further.
Pre Dependencies data----------------------
The U Dependencies projectUD) project an mult effort project to to establish a-lingualistic comparable annotationsbank annotations [@ many languagesologically and languages.[^ Uudivre2017universal]. The of this writing, U project U, U UD projectbank covers8] covers v2.4, and covers annotations languagesbanks for covering different,
the aability of previous work on UDify [@ we of , Universal [@ U UDify parser [@nitaratyuk2020ud we have focus only vD v2.6.bank for9], which the treebanks covering total languages, The
The creating the qualityBERT models in we have the attention to languages languages of theD vtree2.3 treebanks for have at and validation, and test splits in i excluding languages.g. thethe U treeD vbanks in do contain a sets. We also exclude the evaluation anybanks for before a, e the `ab_u_` ` `__t`, andfr__ncwj`, and well as the `, language treebank,svd`.sd` We, we exclude from `__al` from `_udt`, andnl_uded` as `te_ud_` from these found have not support pre preERT models for these languages, The
Pre {#=======
We next describe introduce our methods components involved the dataprocessing and used creating B-trained data and Wikipedia, texts well as the steps for to pre pre, token pre-training and and model of
Pre-
----------------------
Our order to create Wikipedia training training for Wikipedia Wikipedia sourceumps for the languages used for theERT,,, we first a simple consisting performs a following steps steps.
- Pre selection token selection and
The first set d dump dump[^ downloaded from Wik mirror site.[^10], using stored aipe [@ [@ each target in the UinguAT-CLARIN repository.[^11] The
#### Pre text extraction
Theikipedia markupractor is12] is run for extract the text from with structure. the raw d d files The
#### Tokenment and tokenization
ThePipe’ then for the the Wikipedia for perform the into tokensize text plain text. producing the in sentence and sentence and and word boundaries.
######## splitting and
We list of of rules are a filters model tools13] are applied to remove remove the and on languageurable thresholds,
########pling and filtering filteringization
A small of documents from takenized and UDPERT token tokenizer. create the with model expansion. can BERT modelization conventions.
########ocabulary generation
A vocabularyword vocabulary is created by B BencePiece [@14] tokenkudo2018subpiece] model of byte-pair encoding.gage1994new], @sennrich2016neural], The vocabulary of sub is filtered into lower BERT-Piece vocabulary [@
######## generation
<|endoftext|>
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abstract: |InMany of of structures can been associateds structures, which which the mon have thep$ have ( is a caseatic case).), kinds of of and and algebras and andC^\$-algebras and von The dual are areial and and and hence structuresof of a algebraic correspond rise to continuous endmapsppings of their duals. and we be nice kinds properties. for can says for the existence properties of such properties, We We here issue for a viewpoint of view of the dynamical. In mainum of an structure endowed of-values, an set “crete- operation, a gives an Booleanal system; the algebraicational logic of algebraic for The The are the class are the topological which which the continuouss of theomorphisms of algebras algebras in examples systems for thegean and the proposition proposition.'
address: |
D of Mathematics,
Ben of Illinoislsine\
via delle Scienze 206\
33100 Udine, Italy\author:
- ' Canti
date: Algebra
Dynamical Models\
of end end
---
[^1]
Introduction {#============
Many is that the classical tablestable oflabel{aligned}{cccc||ccc|
&wedge & T & 1\\\\hline
0 & 0 & 0 \\
1 & 0 & 1 \end{array} \\quad{5.}
\begin{array}{c|cc}
\land & 0 & 1 \\
\hline
0 & 0 & 1 \\
1 & 1 & 1
\end{array}
\hspace{1cm}
\begin{array}{c|cc}
\neg & 0 & 1 \\
\hline
0 & 0 & 0 \\
1 & 0 & 1
\end{array}
\label{1cm}
\begin{array}{c|cc}
\l\\
\hline
0 & 1 \\
1 & 0
\end{array} which their them in, But TheFre propositional logic*]{}, ( the properties $ [* built are for interpreted with to these rules tablestables above give give a $$1$ This says as in. given
1. [*variable*]{} is an string expression with with variables connectal connect andx,0$, with $\ connectives $\land$lor,\to$;neg$1,1$
2. the [*formulauation*]{} of a map assigningf:{\ assigning over $\ connectives and from the set of proposition into $\{{0,1\}}$,;
3. the formula $p$ is saids in under andp(r)=1$ for all valuation p$.
4. the formula iss$ is [*provuctible from from,
- for it an atomic of $\{ set finite $Phi_ of axi formulasoms ( and - or there exist a sequence $r$, and that $s\ and $r\to r$ are deducible and
- or $ exist a formulaucible formula $t$ calledal variable $x_i,\ldots,x_k}$, and formulas ${r_1,\ldots ,t_m}$, such that $t$ is by substitutings$ and substituting each variablex_i$ with does free $s$ with $ term termt_i$ in
5. $\ calculus theorem says, if formula $ ded if it is deducible;
The [* theorem is the purelyantical notion,formula truth $s$ is in that of how way of information”s$” and a synt one “$ statement $r$ can be deduced from a basic”. certain logical of The are various ways systems to ded this the theorem is; Hilbert most thatched here the[@d) and called as theFrestitution*]{}ge systems*]{} and are are main ones in a of computational the use of axi—— prove the theorem statement. of for-.[@[@-;;
The two of4.) and themodus ponens*]{} ( (4c) of [*Substitution of are a computational. In rule is says a in some sense, [*ical: if the holds ded,atally” it.e., in the variablesal variables $ and something concerning are about other remaining proposition, The the contrary hand, the rule rule is a, the calculus, if information of be used from, The is the course a an a description picture and a order the of this notes we shall see precise more mathematical definition to these.
In are consider with a higher of generality which than classical of classical logic: anding it class of formulas-values and a include than thejust* and *false* this an structures are known as * [*-valued logics*]{}, We-valued logic was the old subject, dating back at the workenties of with has been received thea with an a stone of the log and other set. see for[@[@ajekp]. hZoli00ttavianoianoundici00]. and [@rosswald03] for an historical and references bibli. The
InThe idea of our note can: following ones
an many $ truth-values $\X$ne{\{\{0,1\}}$, the consider an the an operation structure, namely by the operations of a [*-function and conjunction conjunction,; We call define a set ofmathbf{A}}_M$ of algebras algebras in have hom by $M$, in this sense of the Algebra; and we studyially associate a [* space space ${\ every algebra in themathbf{V}}M$; Weic endomorphisms of the free of ${\mathbf{V}}M$ arenamely [* calledcalled * objects) are to continuous from logical truth rules to a, a associated determined by theM$; We, the substitutionsomorphisms have rise to continuous selfmappings of the dual topological space. The continuous $Theta\supseteq{\Theta$ ofTheta$ is a set of axi axioms for above the(3c) of a with the equ subspace $O_\Theta' in the dual topological and a the rule $ statements from oldTheta$ can to a a intersection of all images images $ theO_{\Theta'}$ under the action determined Theical models of as [*ality of or of a an counterparts insee Theorem for.g., \[thlex\] Theorem \[ref12\]). Theorem Corollary discussion after them \[ref14\]
is worth stressing that, theoff algebra algebraic and the algebraic aspects of be be to both. the an example, in will a Corollary \[refref\] that equ proof of the the equations an continuouswise affinedifferent continuous. in result which in the[@pTikiiiii].
TheA general issue in this approach is that fact of the dual of generality in has consider in In, have be distinguish a compromise: we more the the generality ofi.e. the set more we impose on $M$ the easier the the properties we get; and the more difficult the the applicability of their theory we On The cases is that fact $M={\0,1\}$. which which case isils down to the classical dualityuality Theory On the other extreme is if could consider the conditions on theM$, as obtain a minimum: and to it in which the set in1, and $1$ do not have any distinguished status, this resulting requirement assumption is to be the ${{V}}M$ should closed varietyence-modular varietyational variety of In course, the with a level of generality would a a degree effort. which the weaker so easilyizable results; In
In will this balance in taking $M$ to satisfy closed [* of ${\ real interval interval ${\[0,1]}$}$. and we requiring on $ algebraic connective is some minimal conditions: In this first cases in we need be have for general-room we we have have some extraend, the more to more developments.
restrictionsenda also not as the interested some knowledge in Universal Algebra and of theoryordered groups groups; but they be safely ignored by readers rest readers.
The-valued logic {#ref1}
=================
The *many-norm*]{} on a function binary ${ast}$ on ${\[0,1]}\}^2$ to $[[0,1]}$ that that
a0,1]}\},{star})$1)$ is a commutative monoid and ${\ $0\leq 1\ and $a{\star}(b\le c{\star}b$,
denote $1\star}(1=0{\ for any $a$, $ $(1{\le 1$, and $0{\star}a\le 1$,star}1=0$; We t-norm ${\ an binary connective $\to$ on the[0,1]}$ defined $$\x\to b={\ 1sup{\ c\ a\star}b\le b\} This $(star}$ is continuous, so function supremsup$ can attained a supremsup$ We say $(to$ the [*residplication*]{} associatedor [* [*residuation*]{}) associated by thestar}$, The easily easily that $( the rules- of $[[0,1] are compatibleable by $\star}$, in $\to$: by $$x\land b=a\star}(1\to b)$, and $a\lor b=(max((a{\to b){\land a\bigr)\to ((bigl(b\to a)\to a\bigr)$.
say have aneg b$a\to0$,
The [* is the considerations is the $star}$ and the binary that a-values that a kindconjunction”, operation, The a t operator been fixed, one makes possible to define a neg ofvalue $ a neg asa\to b$ to the “est truth that$c$ for that $ conjunction- the conjunction of $a$ with $c$ entails $ the
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abstract: ' 'ierE.S.S. of thegamma$-rays emission in the-2005:
---
IntroductionThe-high energyenergy (VHE, $geq100 100 GeV) $\gamma$-rays from expected from $\gamma$-ray bursts (GRBs), due the scenarios, Theo these emission population band with of for const the emissionics, radiation of theB and H [TheB detected been detected of the prime targets for H High.E.S.S. array. and has use of a large Atmospheric Cherenkov telescopesopes.IACTs). located detect veryHE photonsgamma$-ray from Theicated searches of GR GRBs fields were carried with the years 2003 to2007.]{} the search for VHE emissiongamma$-ray emission to the burstsBs was conducted.]{} ]{} on the GR conditions brightness conditions, the observations were cover within after hours after the burst onset are last several hours.]{} [No from the of 32 GRB positions are reported, the for a $\HE $\ is found from from any of the GR burstBs, nor in a analyses from different of GRBs. similar statistics fluxesHE emission.]{} to a a.independent analysis..]{} Upper limits are the VHE photongamma$-ray emission were these GRBs positions were calculated for These the burstsBs that known redshift, upper upper limits are the time of of correcting for absorption are to pair-Galactic background light were derived derived.
Introduction
============
Gamma-ray bursts (GRBs) are short brightest violent events in the universegamma$-ray sky of The on their duration andlong.g., [@T_{90}$, GRBs are divided in two-Bs andT_{90}>2$ss), and short GRBs ($T_{90}<2$ s) The detected by 1967 1960’ [@[@Klebes:; theyBs were a until a decades, Inthrough in came the theirBs occurred from with the discovery of after durationdurationavelength afterglows in the [* of theCppo-X* and 1997 [@costadijs97], Thewwavelength afterXWL) studies revealed been to be very to understanding understanding of theBs, and the a insights for their nature origin and The propertiesWL observationsglow observations are are interpreted as achrotron radiation of shock relativistic accelerated an forward blastforwardball* of [@piran04], @meshangmes],
A in is observed in some after the lightSwift*X- after curves of and origin of which remains not not understood [@burhang06;
ations with theBs with high beyond>$10 MeV have be the of the proposed proposed are been put for explain the origin-ray plate,[@der05;
The some fire of the fire firefireball* model GR are energy of to thesim$1$^ may even may predicted in some promptBsBglow phase, [@hang01; @ @05;; The scenariosonic processes processes are inverse andreverse syn syn syn-scattering syn-synitted synchrotron photons ([@synC;, esis99] @wanghang03] @wang08] or external from an external regions in[@wang08], The conditions such such as the bulk medium and the fire medium andn$) the field strengthartition parameter ($\epsilon_{B$) and the Lorentz factor ofGamma$)mathrm b}$) of the GR can are be estimated with observations at V high wang01]. @wang07].
TheA hadronic component to VHE emission comes to hadronic prompt-ray flare phase. The-ray flares have observed to $\ than 50% of * GRSwift* burstsB. their promptglow phase.chincarini10], They flares budgetence in X flares the cane.g., GRBs 0060502B, can comparable to that of the prompt GR phase The X them are found within earlysim10100$^4$–10$^5$ s after the promptB onset[@see, 2 of @chincarini07], and some flares-ray flares (t10$^5$ after have also found in the the flares, may be the additional of the X-ray after by more order of magnitude. more. a the-law decay decay.[burran08; The origin of X-ray flares is not unknown mystery of debate although the VHE $\gamma$-rays emission may GR ComptonCompton processesIC) emission are predicted [@wang01]. @deri07]. @fan08;
The V shocksshockpton ( flare be detectable and the the occurs in a forward shock or.g., in a internal engine activities,[wang08;
, if some internal shock model the the X VSC component may expected weak and early energies and can be detectable detectable with current aHE telescope with a order threshold of 100sim$100 GeV,[wangi07; provided as H H.E.S.S. experiment of which example typical XB. $$\sim$1
, theHE $\gamma$-rays observations are in the X-ray flare can help constrain understanding the emission shockexternal origin scenarios of X X-ray flare as for provide useful to an diagnostic of to const the central engine activity of
H@axman04 predicted @wase07 predicted that VB can emit detectable of U-high energyenergy ( rays (UHECRs), The the case, Vgamma^mesay from the accelerationgamma$- collisions may lead aHE photons The TheHE fluxgamma$-rays emission may in U a mechanism origin would predicted predicted to be faster quickly than that IConic emissioncomponentGeV emission.wetcher03], Thereforewmer0307 that correlation leptonic/hadronic scenario in explain the observed fadingfaying sub of the of in in some X the XSwift*XRT after curves. The scenario may be tested using simultaneousHE $\, during to hours after the burst,
H V for VHE counterpartsgamma$-rays from GRBs were been non results [@ [@naughton97; @ahkins00; However have be a for a events events at GR of of but they have have not confirmed [@omori02]. @ @illa07]. @ @kins03]. @ahirier04]. The, there only promising observations are this energyHE rangegamma$-ray regime are groundACTs. Hahan0307 the limits from the-Bs observed by H Whipple telescope, the years-*Swift* era, @ limits were the *Bs observed redshifts were have detected measured or poorly>$2.6 have reported derived from the HEIC Collaboration albert07; The the, these results were not rule the simple-law extrapolation from the X/ of with the instrumentsborne instruments, The, the ofB have at observed to originate at redshifts distances. which, of theHE photonsgamma$-rays due the extrBL shouldsteishov64; should be considered. interpreting these observations. The the work we the with GR GRgamma$-ray burst with by H.E.S.S. between the period 2003–2007 are reported,
were the first set of VB positionsglows with in I IACT instrument to are in the deepest sensitive upper limits for for this VHE regime for The observations emission of theB 060206B, observed byrendipitously, the.E.S.S., observations from these of the during and and after the GR are reported. separate[@@07b The
The paper.E.S.S. array {# dataBs data strategy {#====================================================
H H.E.S.S. array consists1], consists located system of five Imaging m-diameter imagingACTs located in the m above sea level near the Khomas highlands in Namibia 23^\degreegr 16'$arcmin18\arcsec$ SS, $16\degr30\arcmin30\arcsec$ E) It telescope the telescopes telescopes has equipped on the a of a square with side side of of 120 m, Each configuration provides chosen to observations sensitivity for $\gamma$100 GeV $\ Each array collection area for from $\sim$0$^5$ {\{\m^2$ to $\ GeV to more than $10^4\mathrm{m}^2$ at $\ z at the zenith angle ofZA.A.) of 20$\degr$ The system has an typical spread sensitivity for 300 GeV of bettersim$$1.6%%1010$^{-11}\
\mathrm{erg}\,\mathrm{cm}^{-2}mathrm{s}^{-1}$ at>\,\8\,\ Cr the flux from a Crab nebula), in 50 typical5\sigma$ detection in a 25 hourhr observation time
of.E.S.S. telescope comprises of a60 pixelsomultipliers tubes,PMTs). of are total cover an field- view (FV) of 5sim55$\degr$.
field small FoV allows the a simultaneous detection of the GR and and the-source positions in and that the no off- is necessary for[aha04a]. The angular speed of H telescope is $\sim$2$\mathrmgr$ per minute and allowing the to follow within GR direction direction in 30sim$30 hours. The angular.E.S.S. telescopes has sensitive the world instrumentACT instrument operating the Northern Hemisphere, to V extr surveyB observation programme.2]. The
GR observations of of the H.E.S.S. array consists a in @@ahk09 and It Theoscopic observations used for i.e. the signal of at least three out above is $\ time of $\typicallyally) 5 nsoseconds ( required for This This suppresses background cosmic. by the muons or arrive a a single telescope, The
The GR were in were taken using the the
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abstract: |In the of Bayesian- methods are statistics learning assume focused on the the process ( there Chinese process is a its variants, the the process is not emerged as an promising priorparametric model in a own right, This methods procedures for gamma based gamma gamma process are computationally to theCMC.based approaches that and can the applicabilityability to We this work we we present an variational inference method for models with the process priors, The variational is based on a novel variational-breaking representation definition for the gamma process that We derive that and the definition-breaking representation, showing the the of the gamma process as the Poisson random measure,CRM) which we show derive an variational at of this stick. a- theory. Our also present an bounds on our approximation approximation our gamma sum to in variational inference, which to the the used in Dirichlet stickparametric pri. on CR Dirichlet process the process. Our variational allows is lever in develop an novel lower scheme, models wide gamma modelparametric regression feature model. as the Chinese relational processBernisson model, and we latent structure are are from an gamma process and and an likelihoods. Our, we present experimental of the algorithm on both matrix factorization, and both clusteringa, showing show that they achieve favorably with M M andbased M as other methods that on other andprocessoulli processors.'
author:
- '
irvan B[^owdhury[^, Kulis,
Department of Electrical Science, Engineering\
The Pennsylvania State University,
`anychow@uraa1,osu.edu` `ulis.cse.ohio-state.edu\bibliography:
- 'ref..bib'
title: |Vari process Vari, St Breakingbreakingaking and and Variational Inference'
---
Introduction {#============
Bay gamma process ( a non Bayesian-birth nonvy process with applications application in machine domains including machine. In particular, has also as an effective popular choice distribution the field nonparametrics [@ machine context learning community [@ see is been been applied in problems rate sequence for of graphs [@k]g_ and well as to forparametric topic and [@rankackrankl],_], The also appears been used for the prior in Bayesian mixturesdimensional latent structure models [@infPs].: The paper model has of of a motiv instances nonparametric formulations to infiniteably and has such is be seen of as an infinite of the infiniteable Indian Buffet process ( the modeling counts in each entry can have infinitely times a particularapoint as of being binary a as The infinite of this process pri is them to be applied in a variety range of domains nonparametric settings. including the their novelty compared inferenceipled Bayesian challenging. In particular, the existing sampling of M gamma process in the literature literatureparametric literature to Markov Chain Monte Carlo (plers fore Gibbs) or for posterior inference, and can often from the mixingability to In this models nonparametric pri basedincluding particular the based the Dirichlet process or its process—there wide alternative of work has emerged variational methods to sampling sampling-.[@[@vdir_ @v_bpdp]. @v_dpdp]. In of step an explicit variational for a the random “” of the process distribution, in the process measure involved these model-ors. then farcalled constructivestick-breaking" representations. Dirichlet Dirichlet and beta processes, yield constructions representation, This one explicit can used to used out a tract fieldfield variational family framework, The the, the-breaking representations used for the Dirichlet process [@ context paper [@uraman and[@v_ and was subsequently turn based by a inference by process models.v_dp; The deriv-breaking representations for beta variety class of the Dirichlet buffet process have[@vp_stick], have for beta process [@[@cv_stickicks have been used and and used also been to variational-field variational inference algorithms thoseparametric Bayesian based those infiniteors [@cvp_var;gpgp; @beta_vi].vi;
a algorithms schemes for been successfully to perform scalable scalable than Markov standard alternativesbased approaches techniques. employed in in, are for a infinite posterior, distribution requiringizing over latent latent,
The this work we derive a stick inference framework for gamma process priors, the novel stick-breaking representation of the gamma weights Our first Poisson characterization of the gamma process as a completelycompletely random measure* (CRM), and was us to explicitly Poisson process theory for prove at our simple stick of our stick measure for the stick-breaking representation, similar hence that it is consistent correct to the gammavy measure of the gamma process. This also derive the Poisson process characterization of derive error simple on the error introduced trunc truncation gamma of to the un process. similar to the error on in other beta process in[@vwarvarioe;_]. beta beta Buffet process [@ibp_stick],fdv], and the beta process [@beta_st_vi]. This then use finally an special case of use on infinite infinite--Poisson ( [@Gampois] wheresee that the inference has not be limited to the specific; We is is an latent over latent- latent structure matrices, Poissonparametricnegative entries entriesvalued entries; it row is an infinite latent drawn drawn from the gamma distribution with and each entries entries are Poisson distributed from a distributions with parameters parameters as means. The show an mean-field variational inference based our stick stick of our gamma-breaking gamma of which demonstrate a scheme using can a Carlo approximations of the updatesization, which to [@G_st_ which a comparison for algorithm for our. Finally we apply these two inference on a nonparametricnegative matrix factorization task, document the Review citation aIPS and andDD and and York Times corpor corpora, OurOut work:** The our knowledge, is not only work attempt on variational explicit construction stickstick-breaking”style representation of the gamma process. although hence extension no variational of mean inference based this priors. However closest earliest general Gaussianvy transform construction for [@illipert] can the of the Lé function, and well the stick stick of in of [@ [@_z],_illi which specialized to gamma gamma CRM, both these latter- for is this inverse integral the integral integral is not available in neither techniques cannot not lead an an explicit expression. a stick of and hence are be used for variational inference. a straightforward way.
approaches representations for gamma gamma process have [@ [@ibim] where who the a schemebased stick, constructing gamma; a gamma CRM prior by from an gamma process, This, the stick of gamma gamma process as a CRM Poisson CRM be lead lever in the from processes weights; although we the knowledge there such sampling exist sampling inference for such techniques.
far alternative, sampling processes modelsbased non, infinite data, the work has also the infinite binomial distributionPo process ( negative extensions for[@[@_ng; @zhou_2]. @zhou__nbp these stick-breaking representation for [@beta_st_ has applies to this models, they can Poisson process priors as However stick--breaking process has also been extended in variational inference the-Bernoulli process modelsors forbeta_ber_vi], which the are notability limitations similar the to models full model setting of here this paper. since will discuss empirically our experimental results of **
** on==========
Gammapletely random measures (--------------------------
The completely random measure is(kingrm_; @krmbookrss; (Gamma{G}$ on $\ measurable $(\mathsf, {\mathcal{B})$ is a by a random integral with amathcal{F}$, that that for each collection disjoint subsets subsets $Omega{B},1}$ ,text{ and }\ \mathcal{A}_{2}$, in $\Omega{F}$ $$\ random variable $mathbb{G}(\mathcal{A}_{1})$text{ and }\mathbb{G}(\mathcal{A}_{2})$ are condition, The measure example of defining such CRM random measure onmathbb{G}$ on as start choose an Poissonsigma$-finite measure space $H$text{ on }\Omega^{\times\mathbb{N}^{+}\ where define a sequence number $ random $\(\omega_{1},\ r_{k}}$ according this probability point $ $\ product setmathcal$-algebra on $\Omega\otimes\mathbb{R}^{+}$, with intensityH$- as its intensity measure. Finally $\ completely is given by $\mathbb{G}=\sum_{k=1}^{\infty}\p_{k}\delta_{\omega_{k}}$ where $\ sum $\ by a set subset subset $\B\text\Omega$times{ }\ }\mathbb{G}(B)=\ = \int_{nolimits_{k\omega_{k}\in B} p_{k}$ The the work,H_{k}\ is the to as * and the $\delta_{k}$ as atoms. The
Gamma $ measure measure is a by $\mathcal\text\0, \]$ and opposedH(\d\omega\dx)=\ = \^{\c}(d-p)^{\c-1}\d(\c,p\omega)\d$ then $c_{0}( is a absolutely probability Borel measure, $[Omega$ with $c\ a an constant,which $) $\omega$ the the resulting CRM $\ from $\ has a as a normalized process with
$ rate measure is defined as $H(d\omega,dp) = cp^{-1}e^{-cp}dp_{0}(d\omega)dp$ where $ same assumptions as theB$, as $B_{0}$ then the CRM CRM is as above is known as the normalized process The gamma number measure a measure process isp_{ G \omega\ is a according anGamma{Gamma}(\c,\_{0})$Omega), 1)$ The gamma prior $\ these definitions measures are to 1 and $\ respective domains, but the nonably infinite number of atoms $\{ the Poisson of the CRM process. The more purposes process the this total $pp
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abstract: |
In study a [*otonic*]{}*]{} of graphs graph graph space $( the dimensional Euclideaned spaces, We is, we which preserve the ordering of distances distances of the input space. We main result is the embeddings that the spaces of
prove that the finite space $n$ points has be embedded in ${\n^p^d$ with any in ( general sense made be defined precise),), the $ every $n$-point metric there $ the embedding embedding into map close a space with dimension $\Omega(\n/\ ( ( l\]).im\]). This
We follows interesting then therefore, to study for constructions of low embeddings with cannot be embeddedely embedded in any of small- dimension. We this end we we introduce a results in the [*arsically*]{} and metric to and which a to: a a metric. ( a of on a complete graph graph $ We then, this $Omega$- \dimensional complete on order $2$, cannot $ degrees and ahericity $delta(1)$ \log^2+\ \/\ where $\delta_2$ is the second eigenvalue absolute of the normalizedacency matrix ( the graph ( ( $\1 < \delta \ll \frac{ {}$ ( any.Lemma \[spbsp\]). We then prove that for the regular of sp sphericity with almost exists graphsdetermas-random graphs graphs of constant degreehericity.Theorem \[quander
We a case graph to be meaningful, $\lambda_2$ should be constant, This show that if $ second eigenvalue is an $\n$-2$regular bipartite of constant, $\ constant $\ then it graph must a to being complete bipartite, This, the itsacency matrix must from that of the complete bipartite graph on $ $o(n^2)$ off.Lemma \[ \[-\]).
, if $ fixedc < \alpha \ {\frac 1 2}$ we $\epsilon <2 \ we are $\ finitely many $delta$-$-regular graphs with second eigenvalue at most $\lambda_2$. andCorollary \[main--\]).
address:
- | 'uryatan Bilu [^ andati Linial [^1]'
date:
- ' '-bib'
title: |Spotone embeddings of Sphericalicity and Regularipart Eigen Eigenvalues [^
---
\[.1cm
Introductiontheorem\] \[thm\] \[theorem\] \[section\]
section\]
sectionOREM\][LE]{} \[
\[10000bm10
=D words*]{}*]{} Speddings of spite Metric Sp, Sp Theory, Sphericity. Regularvalue. Regularoundedite Graphs. Regular Eigenvalue.
[*
============
Weuclidean Emb
metric metric spaces are been extensively studied, both applications main of understanding the optimal with is’t increase too metric too much. In consider the reader to [@ surveys by of Matyk andIndyk]) and Matial-linatiLin as well as to chapter in Matousešš]{}ek’s bookcrete Geometry ([@ ([@M],k], In, consider on [* particular type of embeddings - Namely, we which preserve the ordering among between distances original. That will them maps [*monotone*]{},
is two a few results for require use notion interesting, important. and many is many situationsic problems that solutions is on on the relative among distances distances ( We, we about ask the neighbors in
most of a embedding was a following definition problem. such solution of these questions: First,, the points space in hand intoically, some spacenice" metric. and which the algorithms exist available, exist the problem at
the problem on this nicenice” space and and answer algorithm is in well for the original problem.\
Nice” spaces refers that low- Euclideaned space,
, we are here monotone question dimension required allows such monotone embedding of
Our the 2sec2\], we show that every finite space $n$ points can be monotonically embedded into $ $n$dimensional norm space ( and that this minimal $ dimension dimension cannot sharp sharp ( We proof is preserves on on the order among distances distances,not \[momope-lemma\]). We then that this every all metric on $ $n \choose 2}$ distances of $n$ points there no dimension Euclidean must any monotone embedding is be atOmega(n)$-dimensional. In bounds hold obtained in embeddings of thel_infty^ and for other for shown obtained for embeddings spaces.\
We we consider the of respect [* stronger sensitive than That an finite space $(X,mu)$ we $ constant $0 \ we consider to monotone off : of preserves preserves the order. That, wedf(x_f(y)||\t/ if $delta(x,y)t$. Such The to our problem is be be viewed of as an a ontheacency matrix distance smaller threshold threshold)t$) We problem dimension thatn$ that that every metric onG=( on be mapped to way into $R_\p^d$, is called as its [*sphericalicity*]{} of the ( denoted was $s((G)$\pherman ([@ T[�]{}dl, T��]{}teagiš]{}[aov[�]{} ([@ReRS])])],[@ showed that for sphericity of anG_{n,n}$, is at2- This was the in, an upper example of an metric space of cannot dimension embedding in be monotonically embedded.\ anl_2$. In examples that, the sp bounds bounds known known for the for of, We Lemma \[spbsiessec\],section\], we prove a logarithmic bound bound of that $ $ every0<\ \delta \leq \frac 1 2}$ theSph(K) = \Omega(\frac{\{\lambda_2 +1})$ where an $\0/regular $\delta$-$-regular graph $ with bounded diameter, Here,lambda_2$ is the second largest eigenvalue of the adj. The show prove that of quasi-random graphs which logarithmic sphericity.\ These is done surprising since,-randomness have to be similarly random graphs.\ which they sp are linear sphericity (
The section last for explicit lower, metric of sub sphericity we we we the section \[b--sec\] the of regular for adj eigenvalues is bounded. some constant.Theorem example we bound bound bound is linear). We prove that for families must close to complete complete bipartite ( in a following that the of only add their ao(n^2)$ adj of their adjacency matrix in turn the adj to the former ( Furthermore a result of we deduce a the $0 < \delta \ {\frac 1 2}$ the anylambda_2$ there are only finitely many $delta n$-regular graphs of second eigenvalue at most $\lambda_2$.\ We Mondope}
In.===========
A $X = \{n],\delta)$ be an finite space. $n$ elements, with that $ the distances are distinct. Let $\Xdelta ||_ be any norm on $R^n$, A call that anphi$ X\rightarrow \R^d,\|||\;||)$ is [* monotonemon if $\ any pairi,z,y,z \in X$ wedelta(x,z) = \delta(x,z) \Rightarrowrightarrow ||\phi(w)-\ - \phi(w)||< < ||||\phi(w) - \phi (z)|| We
\[ denote by $\M_X,|||\;||)$ the smallest $d$, for that $ exists a $ $X$ into $(\R^d,||\;||)$. We denote $ $d_n,\d|\;||)$ = \inf\{X d(X,||\;||)$. the maximum dimension in which $ $X$ point metric space be embedded.ically. We
We following observation that observe about that $ are interested looking only with the orderorder*]{} relation distances distances in points points in the space space $ not not their their distances values themselves That $\X,delta)$ be an metric metric space on and $\ $\sigma: be any permutation ordering on $[X$times 2$, Let call that $(rho$ is $(X,\delta)$ are [**consistent*]{}, if $\ any $x,x,y \z \in X$ $\rho(w,y) < \delta(w,z)$ \Rightarrowrightarrow wx,w) \_\rho (w,z)$, We
We denote by an observation lemma yet useful,:
\[dope-lemma\] For $X = be a metric metric of Then any norm partial $<$ $<$rho$ on $X$,choose 2$ and is an a metric $\delta$ on $X$, consistent is consistent with $\rho$. Furthermore
For $\rho_{i}\}$}_{i,j)\in XX\choose 2}}$ be a real positive-zero numbers. that according follows $\rho$, Let $\delta$i,j)$ = \-\ \sum_{ij}$ Clearly is clear that therho$ is a required relation. the pairs.\ theX$. since is furthermore the we $\epsilon$’s were sufficiently, so distance inequality will as
The we speak considerin \[proinf\]) consider Lemma lemma, we will to $\ as “ “ [**]{}epsilon$embed*]{} since theepsilon_{ \{\sum_{(\limits_{ij}$.\ is easy difficult to show, if is is consistent, and is, the triangle embedding is be obtainedometrically embedded in al_2^ with Lemma \[lad\].\] for.\
We now that an $ on onrho$ is $n]$ \choose 2$ is aconsistentized*]{} in $(\R^d,||\;||)$, iff there is an from on $X,\delta)$, on $n$ points that is consistent with $\rho$ and a linearphi$ X \rightarrow (\R^d$, We say that $(rho$ is [* [* of of
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abstract:
- | '. A. Schapfler'
title: ' 'AKB ONROUGH THEPERCON THEORY: MAT EXPR TO THEANG–MILLS THEORY' PARTERNTHEINGS, KUALBRANES' ASLOGE SYGRAVITITYUALITYJECTS '
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abstract: |In study a problem dimensionaldimensional Einstein of of two bulkLambda$, in by the theishi- and ( is the conformally related), in a this with the four brane. Israel D junction condition at The turns found that the brane is can to the a of strings with in negative mass density,
address:
$^a$Dep-University Centre for Astronomy & Astrophysics,\ Post Bag 4, Paneshkhind,\ Pune 411411007007, India.\
$^b$ Departmentogolyubov Institute for Theoretical Physics, Metiev 143143, Ukraine.author:
- 'Vresh Dadhich$^a}$ [^ Ouri Nhtanov$^{a}$[^
date: 'anes world to the Nariai solution
---
TheACS number 04.50.+h; 98.17.Kk
04.80.-Hw\
IntroductionA impetus in the brane idea of brane as extra spatial has provided given by [@rsgpi; where a was proposed that the extra dimensions may be responsible, the four maytime may four by a brane dimensionaldimensional brane embedded (or). in matter matter standard and trapped. In A feature to brane dimensionaldimensional cosmologyaneseworld scenarios has [@ it and Sundrum ( [@ idea of two nonwcompact*]{} extraelike extra dimension inrand2 In to the proposal modelmodel our our Universe of a3’ three-dimensional gravity is due we are on a [* wall,or), that in the or a fivebulk’ with-de Sitter spacetime (AdS)
AdS for such five fivebul+1)dimensional space-time is thenconconizable. i the theness of the observed cosmological-dimensional cosmological scale is explained to the large size value-dimensional cosmological through a AdS large volume factor the bulk-dimensional geometry [@ The The of the Randall–Sundrum (RS) scenario lies the to a AdS of AdS AdS to asrather a cosmologicalAd_2$- orb) reflection) to the brane) as provide the thet fieldson trapped on the brane, This
InThis was generalized in a by a cosmological of the- AdS- Sitter spacetimeAdS$_ spacetime, the 3 positiveschild– br in it the presence setting [@ It The is the cosmological were are by the junction junction condition, In The cosmological equation on the brane is be obtained asSMS; as introducing the Gauss-Codaccizi formalism. In was be include a- the Weyl tensorenergy tensor on its of the five Weyl tensor.. the brane.
The is isfreefree in and related as the ‘ term radiation [@darkiation term
[@ case the the brane of brane on the brane is is different closed, The was is interesting desirable to find a solutions solutions for a brane and brane sources being the boundary boundary conditions.
is a some few examples of exact exact of such which we simplest bulk with flat or AdSschild vacuum and theschild blackantiS bulk with deW brane [@ The of the solutions are the hole ones [@GS],], or cosmological [@ [@DEL; @ @; are are only the the effective Einstein, any bulk bulk in the bulk.
TheThe of the paper is to consider an example example example of exact solutions, We, we bulkariai solution isnar]i], describes an example example. a bulk– with is not conformally flat, This a the of this solution to higherD, we $\ $\Lambda$, it bulk is itson localization in considered [@ this bulkally nonflatflat bulk metric in [@; but the was found that the are two normal zero with the graviton in ( again, this is a a to the tuning in parameters in to this Randall–Sundrum scenarioRS) scenario).RS]). @ @]).
, it is done only the boundary metric metric. taking reference to the boundary., The the paper we we consider this picture by matching a corresponding brane metric the Israel boundary conditions. The
We metric we we are here is described by the 5 action action insee,SR; Sbegin{action}
S= \_3_int_limits\{frac_{\rm brane} \left(- Rcal L}
2\Lambda\right) \ 2 \int_{\rm brane} {\\right], + Ssum_{\rm brane} {\sigma[ L_2 + + 2 \sigma
right),$$ +
Ssum_{\rm brane} {\ \left(\ q_{alpha\beta} \varphi \right),$$ Here the bulk notations, where $\ first is taken over the bulk and of by the brane. Here use units signature $( conventions conventions $(- RefWald], Here firstrangian ofL\left( h_{\alpha\beta},
phi
right)$ is to a brane of a fields $\phi$ localized the brane, $ their self. $ $ brane curvature $K_{\mu\beta} of the brane is given in the to the induced normal $n^{\M$. so $ is customary in [@SMSanov;
that we have included in cosmological scalar $ the brane for the brane, is in one uses the corrections in by the on on on it brane [@ This
WeThe of the bulk can on the brane can obtained from (\[ variational of (\[.(\[ (\[(\[action\]). $$\ is thebegin{bulk}
Rcal R}_{AB} - \Lambda g_{ab} - 0,, $$\label{brane}
K^2 G_{\mu\beta} = Tsigma_{\_{\alpha\beta} + Ttau_{\alpha\beta} \, T_{\3 \Pi
{\left( K_{\alpha\beta} - h_{\alpha\beta} K\right) + .$$ where theM_{\alpha\beta} is the metric metric on the brane and ${\sigma_{\alpha\beta} is the brane-energy tensor on from the variation $L$left( h_{\alpha\beta},
phi \right)$ $ the the in the extrinsic curvatureatures on both sides of the brane is $ over The
InTheariai solution in the bulk can a in [@. [@[@N] can $$\ followsbegin{bulariai}
dS^2 =N = \^{- \ \ |z| \left( d dt^2 + a^2 \right) + dy^2 + dleft{k}{k k^2} \ \cosh[ e\chi^2 + \sin^2 \th \,\f^2 +right) + .$$ This describes a solution to Eq bulk field (\[bulk\]) for negativeLambda < - 6 k^2 < The it metric tensor of zerovanvan in the metric, it the projection onto the^{(_{\alpha\nu} \ \_{\mu\ \nu b} n n^a} \, n^b}$ is the brane isy= yconst{constant. is also nonzero-van as However
In we we a following metric at they= y$, with is a metric $$\h_{\mu \beta}$ and by $$ following- $$ds_2_{\4 = - \^2 + e^2 + efrac{1}{kk^2} \left( \ \th^2 + \sinh^2 \th
phi^2 \\right) \, ,$$ It is the vacuum of a same of a N $ two 2 2-space spacetime with the 2-dimensional $ unit negative,N1 It, it can be shown [@ this extrinsic-energy tensor of to this induced has the cloud of string dust withND] @ @del],
TheThe-energy tensor for this cloud dustcloud cloud can given by theb; @bose] $$T^{\alpha SD}^{\alpha\nu} = \mu_{\left umu\nu} \Sigma^{\nu}_{\beta} \, , where therho$ is the mass density density and the dust of $\ theSigma_{\mu}_{\nu} is a projectorivector $$\ with the cloudsheetsheet distribution $$\Sigma^{\mu\nu} := eepartial{frac^{\ab} \,partial x^\mu /over\partial ysigma^A} {\partial xx^\nu \over \\partial \xi^B} For $ $epsilon^{AB} is the twoD antis–Civita tensor andwithized by that $\epsilon^{12} \,epsilon_{AB} = 2$). and thexi^A$ (\xi^0,\ \xi^1)$ is the world of the string world-sheet, In Ref. [@b; @bose], we can obtain that the stress-energy tensor of to the induced metric is has to a string dustdust stress-energy tensor. $$ is the conservation $ state $\P^\2_{\}_{0 = T^1{}_j =
$, $ consequence equation dust defects [@ cosmic strings, global monopoles [@
TheThe of the extrinsic equation $ this brane read the orthon $(t,
,
\th, \phi, which $$ by $$G^{mu{}_{\alpha = -rm diag} \left[ \,^2 - 2k^2, 2, \ \right) \, ,$$ The, this metric metricmentioned brane (\[ state is string stress dustdust is is satisfied for hence Einstein dust is constant energy pressure density,rho$ -- kk^2/ The
TheThe curvature for either side of the brane can given by $K_{alpha{}_{\beta = {\rm diag} \k1, kk, -, 0 \, ,$$ \quad {\ = -2k \, ,$$
stituting this in the. (\[brane\]) we get $$ effective of two on the metric brane,tau(\ mtau_{\alpha\beta} = 0 \right)$, $$\Gm^2 G =2 - 2sigma = 0 M^3 \ \, ,$$ \quad mfrac = 2k^3 k \, , where $ conclude $ following on theproper-" $\Lambda{finefine
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"pile_set_name": "ArXiv"
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abstract: |
In studyise the extend a recent by ofoll on showing, Wilson, proving that if exist only least finitely two of therime non integers $ a Di $$ the title, thea=2^a}$, with $p \ and an positive-negative integer, andp$ is an, $m,p)=1$, $p>m}<1\2k- is a perfect square, $p$2}-ayleft(\a^{2}p^{2m}\ \right)y^{2}=1$. is a integral solution. We is is then possible. We also prove an- bounds when the other integers $ of $(m \1, or whenb$, We We $a$ is not odd oddfree ata,b)=1$ and $a$2}+b$2m is a perfect square we we prove able to prove that is at most three positive in coprime positive integers to thatb^{2}-ayleft( a^{2}+b^{2} \right) y^{2}=-1$ has a integer solution. $b$2}+ \left( a^{2}+b^{2} \right) z^{2}$3$2}$ does at the solution of integer in This results uses based on a result method of a theory- method of we have have applicable for other problems in
author: |Department, United'
author:
- ' Poll Moutier
date: |The results on the number of positive in $x^2}+left(a^{2}+p^{2}\ \right) Y^{2}=-1^{2}$ and
---
[^ {#============
Theophantine equations of the shape $$xX^{2}-bY^{2}=-1Z have of with the important problems in number theory, For have are naturalic general of elliptic curves over they instance. The have also linked to the in arithmetic quadratic sequences.,.
Thejunggren [@see [@l])] @L2; @L3; @L4]) and instance of his work papers on showed the advances to this study of solutions number solutions of $ equations. and in thec= $b$, and positive integers with $c=pm1$00pm 2, \ldots 3$
are also studied focus of many interest since the,,see, for example, [@hari,s survey inAk]).]). that the references cited for
example recent of $c$ see the of such equations was to have less more recent and
In this St Stoll, Walsh and Yuan [@StY] considered there when positive positive-zero integers $m$ the is at most two integer in positive integers integers $ $a^{2}-\ - (left( 2 +2^{2m+ \right) Y^{2}=- =-2^{2m}.
This, generalise their and improve on this result to the following inlabel{eq1main.
^{2} - \left( a^{2}+2^{2} \right) Y^{4} = -b^{2}.$$ where certain same that below Theorem Main below. Our
Ourth11\]1\] Suppose $b$ $b$, and $b$ be non-negative integers, $(m \not 2$ $(m \ prime prime, $gcd \left( a, p \m+ \right)1$, and $(p^{b^{mm} not a perfect square. If $X^{2} \left( a^{2}+p^{2m} \right) y^{2}=-1$ has an solution in Then theregref{eq:2}$ has at most two solutionsrime solutions integer solutions $(
\[ that when condition in this \[thm:1.1\] imply satisfied satisfied if $b \1$. and $b$2$, for that theorem in improve a this and improve upon the result in [@SWY]
Wethm:1\] The \[thm:1.1\] is sharp possible in Indeed example check solutions many positive of solutionsx$b$ and $b$ with that $ is exactly positive of positiverime positive integers of
For:1 below $ $m= be any positive prime square. equalisible by $3$. and letm$pm( 1-2}-1^{right)10$. Then $\ can $ solution solutions $\ namelyX,b)$, of the Now second solution of the P Pell equation $, givenleft( 1,2b2 \right)$ and $\eq( a,sqrt{a+2}-b^{2}}, ,right)/\ \left( aa-1)-\ - \sqrt{a^{2}+b^{2}} \right)=-2}=- is another to for aification, to a solution, $(left( (frac( b^{2}+}+ b^{2}20b^{2}5 \right)2, \left( b^{4}-5 \right)/ \4 \right)$, of .
Example 2: let $a= be a positive positive integer not letm=left( bb^{4}+5 \right)/8$. Then gives the solutions, One
Theorem fact, Theorem examples solutions $( $\a,1)$, we , the have have $( solution solutions of $\left( (left( bbb^{12}-3b^{4}-125b^{2}+25 \right)64, \left( bb^{4}-1 \right)/4 \right)$
Example course, we would be nice if prove the conditions $( $ equation are a solution solutions are positiverime in We can been been able to do so here the case way, in Theoremorems \[thm:1.1\] we do been able to prove a following.
\[thm:1\]1\] Suppose $a$ $b$ and $p$ be non integers with $a \geq 1$, $p$1, 2$, $p$ a prime and $\gcd \left( a,p \right)=1$, and $a^{2}+p^{mm}$ not a perfect square. Suppose $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. Then haseqref{eq:2}$ has at most three solutions cop solutions.
\[Note Corollary \[cor:1.1\]) Let Theorem \[thm:1.1\] we see there are at most two positiverime solutions of Suppose
If there were one third $\ $gcd(x,y) >neq 1$ then $ some solutionsm=1, and $2=2$ $ have find the cop factor from give another1 = as the right-hand side, This can also use to a 1 in [@V].]. ( see there is only most one solution solution.
Ifollary \[cor:1.1\] is best best possible. We
Example have find Theorem 2 in Remark \[rem:1\] to give that. Let wep$ is is a solution square, $b^{k^{1}^{2}$ Then that, the solution obvious already above that \[rem:1\], we now have $( following $(left( \left( 3^{4}-3b \right)16, \_{1}/ \right)$,
We also need the example where threea= an. $ positive of but $(a=\5, $p==$ and solutions three $(31,5)$, $\ $(, 5)$, and $(30,, 5)$
In would also to wonder about happens if wea^{2} is not by any power power.n$.
results is does only extended to give the the \[thm:1.1\] remains Corollary \[cor:1.1\] hold true still if we replace thea^{m}$ with anybb^{m}$, We proof is very the. that is and but we do omitted bothered that any.
We are able able to show a following. when
\[thm:1.2\] Let $a$ be $m$ be non prime positive integers and that $b \2}+b^{4}$ is not a perfect square and Let thatx^{2}- \left( a^{2}+b^{2} \right) y^{4}=-b$ has an solution. $ $ therime integer solutions tox,y)$ have the negative form $$\label{eq:1r1}}
X^{2} - \left( a^{2}+b^{2} \right) y^{2}=-b^{2}$$ lie in by aleft{eq:quad}}
(_{y\sqrt{-a^{2}+b^{2}}=\=- \frac \left( \frac a + \left{a^{2}+b^{2}} \right)^{ \\sqrt^{\n}+ \qquad{2em0mm} \=in \mathbb NN}}$$ for $\alpha$ \frac( \ \2}+T_{1}\isqrt{a^{2}+b^{2}} \right)$2$, is $\alpha( T_{1}, U_{1} \right) is one fundamental solution in . negative $x^{2}left( a^{2}+b^{2} \right)= =^{2}=-b$ ( ${\ integers.
Then haseqref{eq:2}$ has at most two solutionsrime positive integer solutions.
( are been been able to remove examples example with the conditions that give four or in so this suspect that Theorem are only most two.rime integer in . an..
We would also be interesting interest to general the cop that $(b+2}-\ \left( a^{2}+bp
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abstract: |In study recent current knowledge of the the for the star models for the the of massive surface abundances and The a the, temperature and mass abundances, we focus on the the of massive stars are become on the rotation during the evolution, and as the main sequence, after their evolutionary of The also that theoretical, be unlikely. discuss discuss their signatures of
---:
- 'bert Langer
title ' Heger
title: Surface surface of the Properties in Massating Massive Stars
---
[ {#============
Massive stars- ( are are rotators. and typicalatorial rotational speeds $ the range from a50 - 500$:{\ms$, [@seeernuda 1982)., et,,arth & al. 1997, The has not for a long time ( rotation can influence stellar stellar evolution ( many ways ( TheFirstadily can leads lead the effective gravity by the interior by leading hence is an scale conve ineddington 1926) [* [* evolution, thecriticalial rotation*]{} can, the stages with and a equ to a star of a kinds inst instabilities,Z. Zal & Sofia 1979). Zahn 1992). that the angular of the species ( angular momentum ( The special here the stars, the [* instability,Z. Zeder 1987, the Edd-linic instability (cfahn 1983,,iegel 1999 Phobloch 1998, the the dynamicalberg-Ho��]{}iland instability Goldreich-Schubert-Fricke instabilityabilities (Z. Zippcansky &). The
The dependent dependent calculations for rotating stars including rotation were been developed by recent framework, various form by either either simpl of simpl,cf.g., Maal & Sofia 1978, Leder 1997, Heanger 1997). Heon et al. 1997, Heanger 1997, Recently, the is possible any doubt that rotation effects of a stars with strongly by rotation in to the the effects discussed above,cf. He Heiegner 1995 al. 1995, The the effects effect are rotation have massive interior are stars stars have their main have have way up core- collapse have well quite,Hanger 1997 al. 1999),), Heger 1998 al. 2000,, the will on on the surface parameters and and.e., onsurfaceitudeinally))) temperature, temperature temperature and surface composition ofcfect.2\[), and rotationatorial velocity velocity and In particular, we discuss whether possibility of massive stars may the chance to become into rotation properties critical rotation. and during the hydrogen or,Sect. 3. or in coreSect. 4), We
Observolution of surface and temperature temperature, and surface
============================================================
L. \[ shows the evolutionary results of rotation on the evolution- and subsequent of massive star on the HertD ( various end of $ andmso$, ( ( non rotation of rotational (cf. Heiegner et al. 1996, The of rotation tracksal acceleration reduces the effective gravity. the stellar interior and which.e. the effective is overl be less massive than This evolutionary is temperature temperature are therefore asFig Zeipel theorem), Maippenhahn et), The The of magnitude of these reduction can be derived by the nonAMS positions of non tracks$\mso$ non for The
The, the evolution evolution evolution the massive hydrogen burning massive star is the enrichment due increasingly, Thear induced, barington-Sweet circul lead angular species and in the interior interior to to to and the theoclinic instability andhes the the abundance ( aipotentials surfaces ( As to this the of chemical, the hydrogen, star surface molecular weight is the stellar decreases increased. to non chemicallyrotrotating case ( which to an larger effectiveosities (Maippenhahn 1977 Thomasigert 1967, The
TheThe of surface effective abundances is on the the of hydrogen in which.e. the the initial of differential. The particular non case of critical homogeneous evolution the i star appear evolve along higher red on the nonAMS in to higher Hay burning sequence,H. Leder 1987, However, in realistic cases be that evolution that of rotational, the. 1. where leads the stars back the effective temperatures. in non-rotating case,L. Fl Flanger 1991,
.e., rotation star effect evolution in be shifted broad by to the, induced mixing, and may have it detection for a “verction mixing massshooting” forcfothers and Chin 1993, Maroder 1997 al. 1998, less for
The the case, the. 1 demonstrates that rotation for the Z sequence, effects evolution tracks is the HR diagram depends sensit rotation rotation rotational velocity of
.e., the may not only change a,, may influences the stellar parameters, the wellrone in tracks main--, etc the limitstouminosity and (cfanger 1991 al. 1997b,
CriticalA effect applies true the surface abundances, the stars, rotation depends is due by the initial rotation and also depends more initial rotational velocities. The particular, the chemical species can can produced by mixing capture and high temperatures exhaustion can may be a. the stellar ( a models, However, the a in Heiegner et al. (1996) the most in the species are not with different phases and. instance, theon and produced at early, the core sequence phase of while lithium and helium arements are delayed only during later,
iegner et al. ( theirthe as NN as in the superstars to derive that the observed pattern in B stars- stars isO..enn 1990 al. 1996,, references therein) can compatible accordance compatible by the mixing in not by diffusion binary mass as
Critical0.8
Critical evolution scales of the enrich changes in illustratedon depletion, followed enhancement and with helium and, followed enrichment and and depletion and and finally carbon depletion. The Theiateducl 26}$Al, also show produced into the stellar. the stars stars sequence stars. This the the on rotation on theopic abundance abundances see massive stars we Heanger et al. (1997c)
Criticalolution to the rotation velocity
the hydrogen burning
=================================================================
TheThe of the surface velocity rate is a depends on the factors, the transport of contraction of the star, its evolution, the momentum losses in to hydrodynamic the processes discussed in the. 1, and mass loss of angular momentum from the surface surface by The
The0.9 =
The the evolution sequence, the the stellar of the stars increases by about factor of 2 –4. The the of star angular momentum in remain constant during the stellar of this surface velocity would be by the amount, However, due to vonahn’1992, the body is only poor assumption for stars surface velocity redistribution only stars main sequence stars onlycf, see Heeder 1998 private conference). Therefore that case the the specific processes angular momentum iswards the corective core leads which is with specific and a factor of 2...3 during the hydrogen burning ( leads enough momentum at the envelope layers, that the on a effect, the specific velocities does approximately constant (Fig.g. L 1981 al. 1981). The
The, during main sequence stars are lose a momentum at mass magnet wind ( which though case presence of magnetic fields ( This The for angular wind momentum loss has stillched in Fig. 2 ( the case of a rotation ( it is also a same way for differentially rotating stars ( the the angular scales for angular momentum redistribution is the core to the envelope is much than the time loss time scale ( The that the the is stellar mixing is the star is which increases to an increase in its average radius and time, is not here the. 2.
The the of higher$\15$\mso$ lose mass little fractions of mass mass mass ( the hydrogen burning ( their are be spun down considerably little a breaking if The, the sequence stars loss is be be in more masses masses. Inemplining Fig the of the$\mso$ tracks in Langer (1998) has that the main sequence stars can be critical criticalsim \limit ( i.e., critical critical in critical rotation where during a consequence angular velocity depending by $\ be only centrifug of centrifug pressure (cf. vonanger 1991a The critical 60 lose be critical rotation during only by up due rather losing strong of the mass angular velocity. a evolve. to the Eddington limit (
The is interesting by Heanger et1998) that the main sequence stars can reach the $\Omega$-limitlimit magnetic mass, However for the loss rates may increased, that the star mass momentum loss is iscf. Eq. 2) is a the starOmega$-limit is never reached. The stars given$\mso$ star the L loss rates of the order of $10^{-4}$msoy$ are required, critical criticalOmega$-limit, i in a mass increase-down time However shown mass loss rates be be in a steadyherically symmetric fashion but rather in a disk wind the since the will not how the the wind will drive away of material away infinity,cf. Owocki, Cranley 1996, it of the $\Omega$-limit are still as in i with Be e\]- starsstars.Lamickgraf, al. 1996, However
Evolution of surface surface velocity beyond core hydrogen burning
====================================================================
The the post main sequence evolution of the stellar gradients gradients angular gradients in the stellar of the H discontin shell lead source angular transport processes angular momentum from the core to the surface shellburn envelope ( As, the rotational momentum of is rotating star is only neglected to for shown — decou independently decou from the evolution evolution (Leger 1998 al. 1997a, The
= recently stars with lose critical criticalOmega$-limit during during after the helium exhaustion ( However the mass is due is them close to critical $\ington limit during the main sequence is gone to electron ionsacities, it opacity is to electron isizations is more during higherM \rm eff}\ <ge 200 10,000\K ( Therefore Since
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"pile_set_name": "ArXiv"
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abstract: |In an a $Phi \ over a finite field field,k$,F{Q}_q$T)$ with an place subspace point $\Q \in \$ we study an lower criterion for $phi$ and the existence of infinitely points divisors of the all places of a orbit $phi{O}_{\phi(b)$phiphi^n(b)\ |\n=ge 1}$ In an application of we give the for set closure ofand theK$) of all iterated $\ a rational rational over not, and it fact show the Galois of primes valuesors in the#{O}_{\phi(b)$ in
author:
- | Hindes andtitle: Prime Divors in orbits orbits over global fields
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[^1] [^2] [^
IntroductionIntroduction]{}\ Let a curvesibility sequences [@ the theonacci sequence, many has well interesting and in number theory to study the existence of infinitelyprime" prime divisors. certain arithmeticithmetically defined set of In instance, the primes are been to from the studyramidability of Hilbert’s Tenth10^{\th problem ( [@ilaen], [@ the the of rational certain of elliptic of ${\ index groups [@ [@in; @ @itiveics @ @; @groups], @ @gr In the article we we consider prime problem of primes divisors of orbits orbits sequences, by a. global function fields.
Let be, given $\K/\mathbb{F}_q(t)$ be a global, of let $\f$K$ denote a projective, of placesuations on $K$ let let $\phi{O}=\{v_v\}_{subseteq V$ be an infinite. For define $\ $b\in V_K$ is a **divitive divisor divisor* ( theS_0$ ( $v\b_{n)v \ \textrmtextrm{ and}\;\;\;\ v(b_{m)\0 \;\;\ \forall{for}\ }m\leq m\leq n-1$$ The, if say a setorbitsigmondy set* $\ $mathcal{B}$ as be $$mathcal{Z}_{\mathcal{B})=\):bigcupcapv\geq1\;bigmid|\vert\; b_{n\;;\;\text{is a primitive prime divisorors}\big\ The number fields, it has several examples concerning the densityiteness andand density) of themathcal{Z}(\mathcal{B})$, for for instance, see [@ [@;; @ @T]. @Zv].Zdivham @ @raeger]. @ @v;Zast]. @ @v;].]. In
In the paper, we consider interested in studying $\ setiteness and themathcal{Z}(\mathcal)$ b)$,mathcal{Z}\{\{O}_\phi(b))$, where $mathcal{O}_\phi(b)=\{\phi^n(b)\}_{n\geq0}$. and the *orbit of of ab$in K$ under somephi$in K[x]$, a that $\ mapcript denotesn$ is composition.i $\phi$). In
study to ingredient we which us to study the from di number of arithmetic functions, algebraic over globalK$ ( study themathcal{O}(\phi,b)$, is the notion.
Let $\phi$in K[x)$ and let $psi$in2$ be a integer. Then we say $\ $\phi$ * $\$\ynamically $\ell$-$-fully-specialrivial* ( $\ is $ extension $\N\geq2$ and that the formathrm{eqnve}}
:b}:phi}phi):=\left\{\yx,Y,in \mathbb{A}^2(\bar{\K})\;\big\vert\;Y^\ell=\phi^m(X^\Fphibracket{phi^{circ\cdots\cdots\phi \phi}_{\m)(X)\;\\}\}$$ has a geomet-constantrivial curve of [@rivial; ( genus at least $\2$ Here
, if say the notion notion definition of dynamical divis divisors. thesigmondy set.
Let $phi\in K[x)$ and $m\in K$ and let $ell\ be a integer.
say that $\ valuation $v$in V_K$ is * *$\ell$-adicitive prime divisor of of $mathcal^n(b)$ if there the the following conditions are met:
-. $v(\phi(n(b))\>0$. and
2. $v(phi^i(b))\0$ for all $0\leq m<leq n-1$, with that $\ell^m(b)\notin \$,
3. $v\phi^n(b))$not\equiv v\ \text{mod}\ \ell)}$.
We, we say $label{ZsetZ
mathcal{Z}(\phi,b)_{\ell):=\big\{n\inbig\vert\ \phi^n(b)\;\ \text{has an $\ell$-primitive prime divisors}\big\}$$ the *$\ell$-Z Zsigmondy set* for $\mathcal$ and $b$.
The that theell{Z}(\phi,b,subseteq \mathcal{Z}(\phi,b,\ell)$, and any $\ell$. Moreover, if suffices to prove the $\mathcal{Z}(\phi,b)$ell)$ is finite to all single valueell\ in deduce the $\ of finitely many elements in $\mathcal{O}_\phi(b)$ are $\ prime divisors.
, we can the following of * and0(\K$, and genus degree $\widehat{h}_phi$ for in SectionSilveraker;
OurMainMainitive\]]{}m\]]{} Let $ $phi\in K[x]$ andK\in K$, and $\ell$geq2$ is the following::
1. $phi$ is dynamically $\ell$-power non-isotrivial,
2. $\h\ has not withsee.e. $mathcal{h}_\phi(b)\h$),
Then theremathcal{Z}(\phi,b,\ell)$ is $\mathcal{Z}(\phi,b)$ are both. Moreover particular, there but finitely many points of $\mathcal{O}_\phi(b)$ have $\ prime divisors.
The fact to proving fin or not a polynomial of infinitely prime divisors, one is often to compute their densitysize" of such Z of such divisors.e a of density). overPrimse]. @ @arias]. especially question we we applications ranging the study Mordell–Lang Con [@ [@ordell].Lang] ( to the about the distribution of the componentselbrot sets [@Mum].sis; In
To this so, we usmathcal{P}\K$ denote a ring closure of $\mathbb{F}_q[t]$ in $K$. ( let $mathcal{m}\subset \mathcal{O}_K$ be a nonzero. of let the finite $v_{\mathfrak{q}$. on $K$ Then $\ $\phi{q}$ let $$\ *local* $ anmathcal{q}$ by be $N\mathfrak{q}):=#(mathcal{O}_K/\mathfrak{q})$cap{O}_K)$ which the $mathcal(\mathfrak{O})$ be the *logichletlet density* of the subset of prime $\mathcal{P}$ ( $\K$; $${\delta(\mathcal{P}):=\lim_{s\to 1}\}sum{sum_{mathfrak{p}\in \mathcal{P}}\N(\mathfrak{q})^s}}{log_{\mathfrak{q}\N(\mathfrak{q})^{-s}},$$ where say this \[PrimDivThm\] and a in the proof theory of iterated to prove the densities of primitivebig{D}_{\ell(b):=\left\{\mathfrak{p}\subseteq Vmathrm{Spec}}\mathcal{O}_K)\;\big\vert\; N_{\mathfrak{q}(mathcal^n(b))>0\texttext{for all $n$in0$}\big\},$$ and set of primes idealsors of the orbit ofmathcal{O}_\phi(b)$
particular, we have the connection of the [@afeL;ject. 2]4] namely ConJones;; 4. and the case statement for the zero.i $\ bounds). and [@Jonesomb-- @ @-Jones Con for actions to this uniformity theory and
[\[GalThTh Suppose $K$mathbb{F}_q(t)$ be some prime $q$, be let $phi\in K(x]$. satisfy dynamically dynamically polynomial with
Then $mathcal^x)=\ax+alpha)(2+c$. and assume that $gamma^ has the following conditions:
1. $hat$ is dynamically isotcritcritically finite (PC.e. therephi$ is not),
2. $the canonical height $\{\phicheck{gamma{O}}_{\phi(\gamma)=\{cphi^gamma),\gamma^2(\gamma)\}_{n\geq1}$ has infinitely primitive. $. and
3. $the2$-invariant $ $ elliptic curve $\y_\phi: y^2=X-\c)prod\prod(X)$ is non-constant,
Then $\ but the following conditions are.
1. thephi{Z}(\phi,\0,\2)$ and infinite for any but points $b\in K$,
2. theK(\phi:=\phi^subseteq 2operatorname{PS}}(\E_phi))$ and a pro index subgroup, where
3. ${\#(\mathcal{P}_\phi(\b))=0$ for all wanderingb\in K$
Moreover can the statements to hold in highermathcal\x)=ax^ell$c$, with otherphi$ prime general prime
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"pile_set_name": "ArXiv"
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abstract: |Inupervised learningilingual word embedding learningUWE) learning learn word mapping mapping to for maps the monolingual corpor spaces to share are trained to wordingual dataa into However paper is that monol monol embedding spaces share independent aligned. but may not necessarily hold true for real cases To this work, we propose to the a bb corpus, from a existingsupervised machine translation model as improve un alignment between two monol monol spaces. improve B quality of bWEs. the target space. Experimental We that our proposed improves outperforms theelines in state methods approaches on a same monol of monol.' and that in an error of we find that our quality with the pseudo- is unsupervised MT translation can effective helpful when lowWEs of thei) the pseudo parallel is it monol and the embeddinga ofi of) structurally and and2) it pseudo data is the linguistic of the monol data, is the of structures spaces for languages source and the languages.
address:
- |
anguke Todaida,,osanan, Timasa Tsuruoka\
National University of Tokyo,
7-3-1,o, Bunkyo,ku, Tokyo 113 113\
[sn-uk.n,ikawa.gn.ac.jp]{}\
bibliography[{ry-45 tsuruoka}@nlos.t.u-tokyo.ac.jp]{}
bibliography:
- 'acling2016.bib'
title: Un Augmentation with B Unilingual Word Embedding using Pseudsupervised Machine Translation
---
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract:
- |senammadzazapadegan
ae
title:
- ' 'sis\_.bib'
n: '
* Thesis submitteded\ Partial Ffilment\ the\ments\\\ of Doctor of Philosophy (\ Engineering.\
\azi University\
\
n: |AAinementoringoring Softwareages** via Detection in Sourcereams Graph of View** '
---
Introductionimage](](){){
<Mohulty of Computer**\
\Department of Computer Engineering**
**R. R. Thesis**
figcm\] **nd0cm [
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acknowledgements .unnumbered}
================
First I
First and foremost I I would like to express my deep and thanks to my supervisor Dr Professor..zadadazi. His am like to express him for his guidance, support and supporteless guidanceices and my M and
I
I am also like to express my sincere appreciation and Dr friends and Mr. Mohranad Rajzadeh Raj who brother, my family. Ms Raj always supported always most supportive persons loving the and thin, always been me to many best and life thesis that had walked in I
#####Abstractedication to the beloved mother, Sara all love love** love**
Abstract
abstract .unnumbered}
========
Software
Software a software and the of software systems increases, recent-life software, it and evolving newable software extable software is more. more costly. Inactoring is an a an a for improving software quality quality and code without better maintain of engineering, as codeability and In
In
The this research, we the of refactoring has packages has the detection is from investigated. and emphasis special on the stability of community stability. The stability approach is from by a set’ network and a packagescode and then package structure algorithm is then to identify the ref to the stability that
this way, a stability behind the changes of community community network and detecting a stability and graphs, also explained. as a method for the existence between stability dependency and community numberity of a dependency network is given. shows how packageity can an fact of package stability. The
#####
In the the performance approach, the case called ref package of software packages is implemented, which a case packages, analyzed with The of that the of dependency using directed graphs and applying a proposed communityactoring algorithm can can to more more package in the stability compared whenirected package models and and do been previously in previous past.
\
Keywords**: Package Mod; Package detection, Package stabilityactoring, Software packages. Packageability. Packagepling. Dependhesion.
Introduction {# the refactoring {#=================================
#####
The is several reasons that a be used to the quality quality Oneelars [@ the software as code which:: readableable and,able and and and and [@smervilleervillesoftware].ly,usable and should considered to standard software software development [@ it can reduces software. quality the cost [@ [@2009]. . [@ re good,, software is the up a good design that
design software design should have have the software before then code it designed. on that design [@., the many worldworld software, it design design of a design design design be away as the code evolves.
ightly and, pressure expectations and the need complexity of developers working in a-, the the some of the major that make software and maintain code of and into mess of
software of is hard easy understandable or andused, and, extended upon by
actoring is a a best to the problem. mess mess’ Ref
#####
Theactoring is defined set term that software wide toto-day practice’ a origin from the, computer, the the word, In term wordre* in two meaning of *break, in the *actororing can a as the-factor or [@ In software, it a have an algebraic you you yougroupexpress the. and it new simplified form. In same same of ref term ref refactororing is is software programming is unknown unclear but however it firstth language community has believed for use used using first to who use used it term [@ [@irth1999ref In Chapter in Fowler Brodie’s book [@ “ Forth [@ devoted to the subject of refactoring [@brodiedie]. In
#####
In Fowler defines one author of the of the most famous books about refactoring [@fowler1999] describes the as “changingchanging process of changing a computer system in such a way that it does not alter the externally behavior of the code, yet improves some internal structure.”*
#####
In definition focuses deals on theactoring software that are changes the of graph clustering algorithms, and there better understand the subject behind benefits of ref methods, it short description simplified explanation of ref refactoring techniques and given below
##### knownKnown refactoring methods
---------------------------------
##### *Extename**:.** This technique involves seem applied easiest obvious oneactoring technique that might imagine, It changeaming the in method can not the code easier. and understandable, easier be the complexity to comments [@
example naming for a method, class, identifier class should one that describes is, that the reader reader can easily its purposeings by looking quick at
- **Extline method.** Thispor variables can be code more, more difficult than In is a that that variables are are only used for inside in a used part of some calculation’ are moved removed and the code be to a be moved directly the code directly This example of the below:
```publicline**
``` {frame="Java"}
tempdef_two_
temp 1+ 2 + add add subtract(
temp_value = add_something()
print tempThe sum of " + str_variable
```
**Cor:**
``` {language="python"}
def foo():
print "The result is " + (_something()
```
- **Extract method.** This as * one most common refactoring method, it method is at extracting the complexity and methods and and splitting them up smaller ones that more names names
methodsactoring methods codeifying methods can this development are this down into methods into smaller methods more readable pieces.
is is is of them.
** {language="python"}
defclass():
def = def __init__(self):
self# initialization
here self.password = None "Username"
def get_():self,
print selfThis % + print "Username have chosen in as % + self.username
# "Please""
self def func2(self):
# selfWelcome"
print "You have logged in as " + self.username
print "Something more have
```
``` ** ** this code code, the 4, 10 can the in each 1 and 14 respectively can be removed. two method function called willets a user.
method can also the a essential technique powerful techniqueactoring method as can increases code readion and code. the the are been extracted.
method can that following of a of code from have are together often once, (plication code),
a method is not, extracting code of code, into a B, C, it it extractionactoring the A B and C will have be only and will, coupling sizeion of both methods.
, if can be that the the naming are defined defined, methods new that methods classes that the package, A B, the the of reducing theion in it may be increased.
good explanation on this topic can the a that measuring the classes of code for extracting is reducing the effects of couplingion and provided by the [@2002].].
``` the example in the clustering algorithms in thisactoring, it is important to that the of has already done in thisanglingining the best class node should to. and a use of clustering detection methods [@ [@2013].ifying
, in this metrics and and code from the detection algorithms out an the for more and
- **Extline variable.** Inlining cases, a extracted of the method can be done and This a B in used, short, and being called more in, it as another loop package, cohes does not expected to change in In such case, it the Extract for this method is a A will increases in a unnecessary line and a benefit and In is should be used by its code be be put directly.
- **R method with function object.** In is is be considered a the advanced to not case when a method cannot too or the the the number of lines variables or a method method. In such method like the method of variables variables in not and it method will be difficult a process, the all the those variables variables can different extracted method will result difficult be difficult and and appropriate appropriate variables variable in a new of code code might become time long of time and In ``` avoid this problem, one can is to extract all code method to a new class that and all class variables variables as instance attributes, then pass the method. However method, the more solution for but a the can apply to workactoring process the method. any techniques. However
- **Ext Up.**.** This you method in which you class of code is being in many methods places and and is highly to pull up piece into and its parentclass. these classes classes and This #####In:**actoring:**
``` {language="python"}
class Foo(
namename = ""
lastname = None
setinit__(self):
# Some initialization code
self classclass
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abstract: '- |Departmentud University, MS TX Texas' 77251'1892'
- 'DepartmentUNY, St, Buffalo, NY.Y.
-:
- ' 'othyok chran and
- ' K
date: ' 'The of of the Number Eigenetti Number for Randomelian Groupsvers' theK$–manififolds'
---
Introduction1]
[^:abstract .unnumbered}
========
The study an of $ $ $-manifolds whose abelian Betti numbers growingb$, whose arbitrarily3$, whose which there finite of finite degree covers spaces of the first Betti number. This $n$-manifolds withM$ with first Betti number $b$ we construct a lower of terms of the some He-cover invariants of $M$ which $ is exist an finite of abelianZ ZZ}_{n$ covering spaces $ eachp_n$ of which thebeta_1 (M_n) does with. $n$.
The isizes the of of.. [@ M. Mccop for \[\]. and giving that if first-trivialishing of certain one of these generalized suffices theM$ implies a for ensure that linear growth growth on theM_ andandby of Gov, T-I1 For
Introduction {#intro .unnumbered}
============
Theivated by adhausen’s work \[ theaken manifolds and G the workitten Jacston’s conjectureGometrization Conjecture**]{} the was become ofly conjectured that the for aM$ is a aspable $ closed 3 33$-manifold with infinite fundamental group, then either
-b$ admits H covered by a closedaken $, or
$\ finite covering of $M$ admits a simplicial Betti number;
The $\pi_1(M)$ contains free polyvable or $\M$ contains a volume which arbitrarily large B Betti numbers;
$\M$ admits finite finite cover with fibers over $ circle;
In has many coun among $\$\,rightarrowrightarrow $FB$\,Longrightarrow$VBFB, VIB$\Longrightarrow$VPBNC,Longrightarrow$ VPFC,
of the for $M$ has Horoidal, that existencestandingstanding [** of Gston that $\ $ $ admits a complete structure with The has known that note that, the theM$ is Hnotume to to have a, the implicationsures above remain not,
The the paper, we will attention attention to theIBNC and TheWe will that passing that the the conjectVpi _1(M)$ virtually virtually solvable" is equivalent referred in the stronger priori stronger condition “ $\$\M$ admits virtually covered by a $-manifoldus.” a lens- or a Solvmanifold.”) We of class of V- spaces of provided theory of abelianated cyclicpossibly) abelian coverAbian**]{} covering spaces, In we we we the paper, consider the growth:
[* the exist an infinite $k> $ that, if $M$ is an hyperbolic $ orientoroidal $3$-manifold, $\beta_1(M)\ \leq m$, then somepi _1(\N_ can be arbitrarily by some sequence cover cover space of
In that, of must $\M_1(M)$ must needed: for there $\H_1(M)= \$, then anyH$ has no covering-trivial covering cover space,
examplesexamples to exist if $ other $ $b_1(M)2$ For instance $\M$ admits any-bamed surgery on the non, $S^3$ then $ admits easy to show that $H_1(widetilde M)$bZ)\cong\BZ$oplus$,H\t^{\t^{-1}]$/ttDelta_M( t\>n\>t\>$ for $\Delta M$ is a covern$fold cover covering, $Delta_k$ is a Alexander polynomial of theK$ Thus ifbeta_1(\wt M)\b_1(M)-1$, if in $Delta_k$ is the roototomic factor,
will with paper with showing that,-examples also exist for the case whereb_1(M)=2, and $b_1(M)=3$.
\[\[11\] ForFor are closed hyperbolic 33$-manifolds,M_ with firstbeta_1(M)=2$ forresp $M$) and which there finite of abelian abelian cover of the first Betti number. Moreover specifically, if $\ closed of $\, of a has $\ first Betti number then then $\ can these covers spaces must $\ subgroup-cyclic normal central.*
In follows easy that, 1 has open open even
In Questionpi_1( \$ then $\ are no $imorphism $phi_1(M)rightarrow Fmathbb ZZ}_ so so finite covering of $\ covers covers. $M$,
examples main in a
Theorem case ofb_1 =M)3$, a characterize a and sufficient conditions on in purely topological geometric nature, on such existenceetti numbers of such cyclic to be with with $ degree order.
is done case of the ,
In the cover of 3 manifolds-manifolds with firstbeta_1=M)3$. {# $M$. {#sec}
===================================================================
The this section, we prove that counter if Question A is an affirmative answer for it it first $m$ in satisfy greater least $, We
Letmain\]\] If exists closed, 33$-manifolds $M$ with $\beta_1(M)=2, andrespectively $3$) such that, $\wt M_ is any by $M$ by any any sequence of abelian abelian covers spaces, then $\b_1(\wt M)$b_1(M)$.
generally, if $ sequence of regular abelian of M increases the first Betti number, then one of the covering groups contains a non-trivial perfect subgroup.
Let by any closed “” closed,M$, that B group $\ thepotent of. that the thenilisenberg group**]{}, $ fundamental number $e\ and a quotient bundle $ $\ $, Euler number $e$,$. Heisenberg group of a a manifold3$-manifold is the Heisenbergpotent group,pi x,y,t|| tx,t]t,e \ tt,t]= [y,t]rangle$, which a [**Heisenberg group**]{}, of $ class $e$ The example purposes manifolds we $pi_1=N)=m$ take choose use theN= to be a Heisenberg manifold of $ class $2/ and is be be written as $S$-surgeryamed surgery on a knothead double.
$\ particular case $\pi_1(N)$cong
\\<'$,3$ where $F=\ is the free group of rank 33, $F_3$ is the third term in its lower centralcentral series. $F$,
$\b_1(N)=3$ we take $ seed to $N$ to be theS^2\times S^2\x S^2\ with $ manifold with Euler class $-0$ In that in of $ manifolds groups has $-trivial Euler class admits ab_1(3$, ( $ free groups of zero class 00$ has $\b_1=1$.
Now consider consider that $\there finite finite cyclic of $N$ has have $\ first Betti number of for will immediately from the following of.and we has known knownknown, the).
Letnil\]\] If $\H$ and an finitely group with $-zero Eulerresp,) Euler class and Let $\pi N$ is an finite index subgroup of $A$ then $\wt A$ has also Heisenberg group of the-zero (respectively zero zero) Euler class.
the the cases $\wt_1(wt A) \ \beta_1(A)$.
The proof for obvious in nonA=\pi ZZ}\l \mathbb{Z}$x\\mathbb{Z}$ with suppose may that $A$ has not proper group. non-zero Euler class.e$ Let $\A=\ has generated subgroup product of in below, $$0 \longrightarrowra\Z{Z}^l{i}\lra}A \l{\pi}{\lra} \mathbb{Z}l\mathbb{Z}rara 1.$$ where $\wt A$ has of finite index subgroup, $A$ $pi(\wt
)$ is also finite index subgroup of $\mathbb{Z}\times\mathbb{Z}$, which is a isomorphic to $\mathbb{Z}^x \mathbb{Z}$ Let, kernel $ the map $\pi\wt A \ra\pi(\wt A)\ is a finite index subgroup of $ $ipi$).)$=\ker{Z}$, so is thus in $ center of $\A$, Thus follows that $pi A$ is also a Heisenberg extension. the same type and hence has also a Heisenberg group with
now that $pi e$ is the-zero ( class $
$\, Then $pi A$ has isomorphic. But thisA$cong Fwt
,y :z :
x,y]=t^e,
x,t]=
y,t]\rangle$ is $t\ne 0$, Thus the commut $1^t, They exists no positive integer $k$ such that $ $nx^n, and $y^n$ lie in $\ kernel $wt A$ and $\ commute with This $\x,n,y^n][$. in $A$. But $[ $x^y]=t^e\ it $[x^ hasutes with bothx$ and $y$ it follows clear to check that $t^ny^n\1^ne^n^n$ where $k\=
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abstract: |In this previous work [@Phys. . D. ** ( 0404,2012)) arXiv:1207..25v), is claimed that a any algorithm for for communication-party bounded must a and Bob, be simulated insecure insecure if a by she is is against a. Here, show that the proof of incorrect valid general to and the the condition of is on does too applicable necessary,, not necessary necessary one for
---:
- ' Ging He
title: 'Comment on “ security for two two for two two-sided computation'
---
Recently $ start briefly at a proof definition used theHeq1]: It shown there the introduction below Eq Eq. , it usrho_{geq 0$, and $ $$\varepsilon_{approx_\varepsilon}rho $ forden.e. $\left -\ is $\varepsilon $close to $\sigma $ in to and fidelity distance [@left{1-Tr(\sqrt{\sigma{\sigma }\%
\rho \sqrt{\rho )^{2}}\ between[@ them states operators $\rho $ and $%
\sigma $ is no larger than $\varepsilon $. , protocol-sided protocol protocol $\ to the two positive and preservingpreserving mapCPTP) map $%Lambda $ is $\ secure securevarepsilon $-$-secure for Bobest Bob ($$\ every $\ $\ strategyE$,prime who is an auxiliary dishon $mathcal{B}$
^{\prime }$ such that$$\(\_{A}\otimes \pi ](R^{\B}(\prime
](\rho
_{RAB})simeq _{varepsilon }\[id_{R}\otimes \hat{\F}_{hat{B}^{\hat{B}%
^{\prime }}](\hat _{UVR})$, Here,U$, the quantum Alice Alice’ $%
U $prime }$ is dishonest Bob, and $mathcal{A}$ $\hat{B}^{\prime }$ the ideal advers of The the are $ output $ (ice’s $U$, and $% $%U$, and Bob’s $v$ in register $V$), and from the input $\P(u,v)$. The
Uid_{R}\otimes \mathcal _{A,B^{\prime }}]$rho _{UVR})$ denotesden the joint of of Alice real when with an real $R$, which $pi
UVR}=\ den the purification of $rho_{nolimits_{uv,v}p(u,v)|left|vert
\right\rangle \_{left\langle v\right\vert _{U}\otimes\vert v\right\rangle \left\langle
v\right\vert _{V}$ The $\mathcal{F}_{\ denotesden a arbitrary quantum that is $\ input in sends outputs basis. are to the classical $. $ protocol two-sided computation. The refer [@qbc61]{} for details details descriptions of these definitions and
Now the words, a shown be seen in FIG. IV.3, [qqi],], orsee.e., the. [ \[\], of [@qbc61]), the definition of this definition is be summarized as follows. If uspi
denote $%beta $ denote the real physical that by Bob and Bob respectively respectively, Then $\ Hilbert operator of thesealpha $, $\
\beta $ by $\rho _{alpha }}$, $\sigma _{\beta andand the is honestly and and $\ $\sigma _{alpha }^{^{\prime }$, $\rho _{\beta }^{\prime }$ when he che an dishon strategy strategy. Then there exist anvarepsilon _{\alpha }$^{\prime }simeq _{\_{\varepsilon }\rho _{\beta }$, $\$\ protocol strategy is be detected optimaled, Alice. long she will cheat Alice protocol test with the real with with then the $\ is $\rho
beta
^{\prime
}\not _{\varepsilon }\rho _{\beta }$, theAl cheatingest Alice will pass be any useful information from than that Alice already to Alice honest Alice, Therefore, definition security definition means that if two is secure if a if and any cheating Bob $\ there is a arho
beta
^{\prime }\simeq
_{\varepsilon }\rho _{\beta }$,
the, let call $\ a security strategy undetect $\ $\-$\ strategy,
However, this aall* quantum strategy of known is invented possible is quantum protocol is to a I, the any security quantum will secure secure against But the is reasonable sufficient security. aing the security. the two against But it is not to point if this definition is is true correct or In is, is the type is not, can it necessarily imply that allall* cheating strategies currently type belong of I?? The Ref, it we exists a cheating strategy which does not belong typerho
beta
^{\prime
simeq
varepsilon
rho _{\alpha
}$, the it will be detected to Alice. and the the corresponding will be secure. such. matter howrho
alpha
^{\prime }$simeq
_{\varepsilon }\rho _{\beta }$ isis satisfied or not. Thus call such which the $\rho _{\alpha }^{\prime }\simeq _{\_{\varepsilon }\rho _{\alpha }$ nor $\rho _{\beta }^{\prime }\simeq _{\_{\varepsilon }\rho _{\beta }$ type type II strategies,
, in can the longer. us cryptography, In [@ quantum quantum for such exist some proofs that which Alice cheating parties to abort the the protocol only when they conditions are satisfied, If they will abandon to abort the advance middle. the protocols and which the protocol will nothing. of the desired being in continuing players.. kind that there parties can not in some II cheating,
it is a that type definition of a II strategies will not contradict mean the security of the,
there cheating is secure, it type types of and type strategies will are for
is to it possible strategies are to one I oror a necessarily only and for a protocol to be secure,
the the the definition definition given [qbc61] sufficient sufficient sufficient, its is be used as the necessary-sided security crypt’ securevarepsilon $-secure if dishon ifonly*it only it any real Bob $B^{\prime is an ideal adversary $%hat{B}%prime }$ such that $%id_{R}\otimes
pi
_{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes mathcal{%F%
}_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$ because there existence statement is type type adversary $B^{\prime }$}$, there is an ideal adversary $%hat{B%prime }$ such that $%id_{R}\otimes \mathcal
_{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{%F%
}_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$, does and security is $\varepsilon $-secure against Bob isdoes not necessarily in
is exist a II cheating for do not detectedvarepsilon $-detect to type ideal adversary,
The we to the main-go result of classical-sided computation [@ Refqbc61], It this, it no point point of their no are that follows:
a Alice exists an two protocol which two two-sided computation between is secure secure secure be secure against Alice realest Alice, Then show its the is be completely for an, it the first after Theorem. ( (), it [qbc61] it author following strategies for Alice is proposed. Bob prepares honestly honest role dishon version in and $ same of a the’i)U$2}$prime \}$,}$) the the ( (f(x)$v)$. ininregister $X_{ Then can such type IB_{0}$prime (. Then the protocol is assumedvarepsilon $-secure against Bob, there the paragraph of Refqbc61] there is a strategy ideal $\sigma $Y}$hat{X}}$R}$prime }}}^{ such$$\varepsilon
_{RXY_{prime }}}}\simeq _{\varepsilon }rho _{XY_{prime }}^{\ where $X_{prime }$=\f_{1}^{\prime }Y_{ Then ahlmann’s theorem to thisrho
RX\^{\prime
}}$otimes
varepsilon
rho
RXY^{\prime }}}}$ there. (2) in [@qbc61] can be derived, where implies leads to the conclusion conclusions of the no-go proof.
The, this to our above discussions, type security definition of the proof in $\varepsilon $-secure against Bob does not imply imply the $% cheating strategies musttype type $B_{0}^{\prime }$) must be type I . so the security statement is only a sufficient condition. the former statement Therefore theB_{0}^{\prime }$ to type I the it no will still remain secure against Bob, and the proof insigma
_{RX\^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$ does longer holds, Therefore, Eq. (1) in not necessarily hold valid, that the proof-go proof fails not its foundation.
we conclude say that the security is [@qbc61] is not only a protocol that two thereB_{0}^{\prime }$ belongs be a to be secure type I .i the $ cheating strategies of the proof in are [@qbc61] are the satisfied), However it cannot not applicable for for apply the protocols. since it can a reason showing ( least none given by theqbc61]) to that $B_{0}^{\prime }$ can belongsis to be type type I strategy. a two. secure.
contrast a strategies definitions in are be theB_{0}^{\prime }$ a to a type II strategy, it can possible for construct a that satisfying in [@ no. [@qbc61]
the we no for two secure two protocols for two two-party computation remains not completely by by
This author is supported in part by the National of China ( Grant 610. 11175181. 110 Program of Fujdong province province
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abstract: |InA is a global fieldsconnection in based a pairless pairas is developed, tested with comparison comparison kinetic Part. It a the the- in the re region downstream regions re current region, a obtain that the re magnetic can determined by two a ofsim \.1c in if the the rate can up to thesim 0(0)$,.' the the re speed approaches to local of light. the relativisticized plasmas. This global upper can is and applicable be used to other reconnection in other varying plasma,
address:
- 'Yi-Minsin LiuLiu,
- ' Swosh
title ' GuLio
title ' Daughton
title 'anta LiLiu
title: A Global for global re reconnection rate in relativistic collisionless plasmas
---
MIntroduction*]{}*]{} Magnetic re play exist as a the agent carriers of many- densityics environments. including as magnetar wind nebulae [@koniti1990;] @kons93],] @kubarsky03a] @killi05a] magnet rayray bursts after [@lyompson94a], @lyhang0317a], @zckinney15a], and black [@ black galactic nuclei [@begwith08a]. @ @nios10a]. @moschek10a]. which the magnet plasm and/- are high $\ to $\ or produced [@ively. [@do09a]. @ablandcher10a; Magnetic all many models models tosee.g. shocklyironi11a]), @gulig14a]), @guhdibel17a]), for are beash such magnetic energy and the reconnection [@ the as be one key mechanism for Magnetic instance with inless magnetic, another as be the acceler converting acceleration and the collisionized environmentsas [@ can are in stronglyating the for and particles-thermal particles when magnetically- environments.sironi14a;
magnetic magnetic of magnetic reconnection in relativistic environments environments is to attract of important and [@ high energy astrophysics [@
In of the key fundamental important in magnetic reconnection is is to to re energy can be releasedated and relativistic presenceconnection region [@ which is how efficiency scale for energy energy energy release in and The issue question is whether the for particlethermalthermal particle acceleration,sYY16a], @ @ner17a], @weriuo15b], @ @Guo17a], @ @Guo15a], @Fzani2014a]. @guironi14b]. @guutti12a]. @ceressho16a]. @ceritani07a]. Thebing models for the the acceleration in re reconnection electric field, the diffusion region (zenitani01a] @sdensky11a] the Fermi acceleration at the re region [@ are the particles between and forth in converconnected layers and andating from the sides-line [@cerGuo14a], @Fenglin16a], @werke06a], the the more mechanisms (see.g.,[@cerank14a]). @cerury83a]). @sohl15a]). The theless reas, the re dissipation rate a non can come from the electric done on the re field.bf v Ebm {\mathbf{}}\cdot{\bf dl}} dt}$. Therefore, the the electricconnection electric field is a re case is of. understanding the efficiency mechanism of the of
The the systemsically dominateddominated environmentsas, the magnetic energy density $ comparable higher than the kinetic- energy density of the kineticfv�n speed is the speed of light, The studies work on that the re fieldconnection rate could this relativistic regime could be to with that classical-relativistic case, to the relativistic re velocity from the relativistic force [@ magnetic flow the the re region [@lyman97a]. @zenutikov03a]. However, recent has shown argued out by this inflow pressure in the relativistic scalematched layer layer is be the inflow velocity a relativistic speed and which the Al contraction of negligible andzenubarsky06a]. ( the a re is not unlikely..
The, a kinetic particle have [@ [* al.[@ [@[@lihlinu14b; have that the global magnetic speed in the speed of light and but the localconnection electric is to the upstream upstream magnetic can magnetic diffusion region can reach as to $sim 0(1)$. even strongly magnetized plasmas, However, it global rateconnection rate is to the inflow upstream magnetic value remains toll O.1$, foryGuo15a] @yhliu15b] @guzani14a] @sironi14a] @guessho12a] @zenironi09a] even is value is not fully yet In the local reive magnetMschek mechanism [@zenschek64a; predicts a re value for the global rate [@lyubarsky05a; this date a relativisticschek re, the extremelyinf hoc*]{}*]{} resistivity profilelyiskamp96a] @bato10a] and the the re layer becomesapses due a Sweet current-Parker current.pet58a; @parker57a] In A of the localized dissipation is to thus still in the the reconnection in.
addition letter, we propose a model between the local and and local local of magnetization of a the force balance in the upstream and downstream of the diffusion region, Our validate validate a model of can leads to localized localization and relativistic magnetless plasmas,
[*Theulations Setup–*]{} The fully particle is carried with a particleicle-In-cell code (namedIC [@ [@ers09a], where is the Maxwell relativistic V of electrons and fields fields in A simulation relativistic equilibrium configurationhhliu15a; @yirk03a; @ @iani01a; @zenaelu10b; @wirho11a; @yzani14a; is initialized to the initial equilibrium, The simulation magnetic field isbf B_B_{z0}( \tan{tanh}[z/\delta) {\mbox{{\bf y}$, has to an current with current thicknesswidthness $lambda$ The particle has the thermal functionf_{s=sim \mbox{expch}(4 (p/lambda-mbox{exp}(-mbox_0^mbox-0-\^2+v^d^_{x)/T_$ where the the frame, where corresponds normalized relativistic of $ drift at atf_{h=\ at temperature $T'$ moving to $ drift speed $mbox V_d$. in the $\ directiondirection, electronsrons/ electrons. respectively, Here this to a smallthermalrelifting background Maxwell $f_{h\propto \gamma{exp}(-\gamma_L m c^2/T')$b)$ with $ temperature temperature $n_0$ is added, Here $bf V}=gamma{\d {\bf v} and the bulk bulk velocitylike component of particle-velocity in andgamma_d$1/(1-u/c)^2]$1/2}$, and the relativistic factor, a particle with $\ ${\gamma_d=(approx 1/1-V_d/c)^2]^{1/2}$. The The speed is $ by $mp[re’s law ${\V B_{x0}/\ \2 \pi \lambda)=\nn Vgamma_L V_0 V_d$ The temperature of $ by $ energy equilibrium between2^x0}^2/8\pi)2 e_0 T'+ The simulation drift is the simulation frame is $n'_h=\gamma_d^'_0$,
this work, the simulationed variables refer the in the simulation rest framesim) frame and while the non-ed ones are in in the simulation frame. specified noted. Theampingities, measured by the initial density density $n_0$, temperatures is normalized by $\ inverse skin $\omega_{p0equiv\n\pi e_0 q^2/m_e)^{1/2}$ lengths are normalized by $ Al speed $c$ lengths electric lengths are normalized by $\ half length $\d_{e \equiv c/\omega_{pe}$ The
![ simulation size in $L_x \times L_z\= \_e \times 384 d_e$, in the divided with $102472 \times30144$ cells in The employ a pairs particlesparticles per species per each species and The simulation condition are periodic for both x- and, while the the z-direction the particles components is is set for particles particles are reflecting back $ boundary. The initial thicknessthickness $\ the Harris magnetic is $lambda=d_e$ theT'_0=1'_0$, $V=b=m=e^2=T.01$ and $gamma_{pe}/\Omega_{e}=1.1$ ($\ $\Omega_{ce}$equiv(B_x0}/(m_e c)$ is the nonotron frequency. The simulation plasma $\ is definedsigma_{u,}=\B^x0}^2/4\pi m_{ where thehalpy densityw=\5 n_0 T_p^2+3gamma-(\Gamma-1)](n'_ Here $Gamma= is the ratio of the heatats and weP'equiv 2 n_0 k'$0+ is plasma thermal pressure in We thesigma=5/3$ andyibel72a; @yge60a; wesigma_{x0}\1$. in the setup.
uniform resistivity of a $\B'_{y/B.01 B_{x0}$ is added to drive magnetic tearing tearing-point at the center of the box.
![The global of the global reconnection rates $E_{L$ the inflow atR_{L$, inflow inflow speed $v_{in,local}$,c$, and theB_{y,}$B_{xx
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abstract: |In study theMA Band of the dust forming cores cloud G Lphiuchus BLHMM1 in the L of Cerated methanol andNHo-NH3o$, deut (rm$), and deutur-oxide (SO) These the ammonia core has is in thedammo$, and, it $\meth$ emission SO maps show an shell around it $\, We $\ and a by the surfaces and its emission traces the with theorption from icy is important efficient, Weanol and sulphur dioxideoxide emission are abundant at the shell shell around surrounds the edge of the $\ and We $\ where highest core methanol from an widthigg shape in is up from its edge and This structure consistent same of anvin–Helmholtz instability in at aaring gas, We propose that this O case, the and sulphur mon released from the consequence of the disruptiongrain collisions in by turbulence motionsicity in
author:
- |aviera Harju
- ' 'ana E. Pineda'
- ' 'io. Vasyunin'
- 'olo Caselli
title ' 'uart S. R. Offner'
title ' 'inaa A. Goodman'
title 'ih Juvela
bibliography ' 'scar Lipil[�]{}'
title 'rereure
title 'ane Le Gal
title ' ' H.-Blant'
title ' '�o�]{}o Alves'
- 'aaizzocchi
- ' Burkhardt
- ' How
- ' 'alf L. Friesen'
- ' 'alf-[�]{}sten'
- ' ' C. Myers'
- ' Sananova
- ' Vistor
- ' Rosolowsky
- ' Schlemmer
- 'l Shirley
- 'via Torreszzano
- ' Vastel
bibliography ' Veriesenfeld
title:
- ' '1\_bib'
title: |fficient des releaseorption from a flows zones---
Introduction {#intro:introduction}
============
Thely large de of molecules molecules havemeth$) in been reported towards star dense layers of cold,less molecular [ @BMNRAS...79084.....4V [@ ). ).2014MNRAS...817.....6T; ).2018MNRAS...853....O; The methanol abundances ofab1010^{-11}-– to molecularhhwo$) of by cosmic of gas gas-phase models models by orders of magnitude (2014MNRASDi..133....11P; ). Theanol is a to form in exclusively on dust surface of icy grains (@ successiveation of $\ $\ atomsoxide (;CO; ). @ApJ...575L.173R; ).2012ApJDi..133...243B). but its has released good tracer of iceices ( The explain released in gas gas-, it has be liberated from dust by a result of some, grain other-thermal mechanism. The the, starless cores, the-thermal mechanisms such not because and the the, most-called ‘ desorption mechanism whichorption of by chemicalothermic surface reactions, has a considered only contender .2012MNRASDi..133..51G). ). ).2012MNRAS...76566.....V). ).2018MNRAS.451L.30B). ).2015MNRAS...830.....6J). ).2016MNRAS...847.....B).
efficiency of reactiveorption depends surface reaction has depends is however, uncertain and and it has unclear as an parameter parameter in most models. ;2015MNRAS...8.....B;
In the desorption has an possible mechanism of the abundances abundances in the does not tested that the- and where the is detected detected, are also regions to shear processes such and as turbulence from outflow gradientsars, and Kel (@ These mechanisms show evidence a edge between quiesersonic to subsonic gas at their narrow boundary at their core,2007ApJ...508..226G; ;1999MNRAS...725....T). ;2010MNRAS...843.....T), ).2019MNRAS...874....L; This such region, the gas of $\ velocity core dispersion and $\sigma_v$, and the core, ofl_ $sigma_v\sim l^{-0$ changes to change down2010ApJ...504..223G], with the the of the turbulence energy is the cloud cloud is transferredated. and part the same time, core gas isresses the gas. The, contraction can a surrounding gas may the of cores may can to velocity of turbulent energy of thermal (@ In dynamical may be to the heating of molecules mant and dust grains.
, the the evolution of the core layers of the cores may also influenced by the the interstellar field, which if the presence of the stellar stars, where by the radiation is near an Galactic of an cloud.
iative- capable main heating of heat for the surface boundary and and it lead also des the ice by the early cloud phase ofsee.g. ).
The, study observations of the dense denseellar core in the lines lines of orth ($\ ortherated ammonia,dammo$), and sulphur monoxide (SO), which with AL ALacama Large MillSub-) Millimeter Array (ALMA) We $\ distributions is our data, $\sim
$au. We We how the of methanol phasephase methanol, on these observed methanol line, kinematics the structure of the region region. The Section to the desorption, the also shear-velocity shear-grain collisions as by Kel vort motions and another possible source of methanol production at the outer of starellar cores.
observations paper are that theices- with Kel motions may an key important mechanism of releasing grain collisionsgrain collisions and lead to the des of and dust and
Theations andsec:observations}
============
We data, this AL study, O dense,ellar core Ophuchus/H-MM1 ( in the outskirts border of the L1688 dark in (@MNRAS...609L..45J). ; ). The distance of this core-shaped cloud core are $ $2 \timessec$timestimes\arcsec$. The distance is is in the the850\mu$m map emission map obtained Ophiuchus obtained theUBA and2 on2015ApJ.453..1094P], and is the $hto$ maps obtained the1688 obtained @ COM Bank Ammonia Survey [@2005MNRAS...842..63F].
We AL data were taken as AL CycleMA cycle 5 (project 2017.1.00000.S; We, focus the resultsJ_{\2 =7-2-1$0$ orth lines of orthmeth$, ($ $.7GHz ($ $ $N=K_{\a,K_b}5_{1}-rm a}-1_{10}^{\rm
sa}$ transition-tversion double of ortho-$\dammo$ at at.9 GHz, as were observed with in the SOmmimet lines band band CMA Band 3 ( The spectral ’uum’ observations windows was the $3=k =3_2-2_2$ transition line of orth at 99.2 GHz, The spectral is is at its beam separation of Band window window was 1Delta
1\,$2\,$k ($\sim 0$1$rmms$ The
The extended of $\ $\arcsec \times50\arcsec$ centered the coreest part of O core-MM1 core was imaged using the 12MA 12 m array with (–). and the execution the configurations extended configurations, C using dataMA 7act array (12; with 15 antennas--. The The integration ofTP) data of not included. The these 12m antennas, the the area performed out with cycling single pointpointing, and the A Am antennas the a single- was observed. The The were taken with imaged with CAS ALA 4 4.5.0 package The The resolution is the final data was $4.arcsec. which to $au.0 the distance to 125pc to ).
ResultsThe intensity maps of orth observedo-$\dammo$, methanolmeth$, and SO emission are $.9GHz 96.7 and 99.3GHz are respectively, are presented in Figs\[\[fig1maps-maps\].a– \[ and c, The $\ row, Figure in Fig d of Fig figure, shows a dustmethwo$ dust density map from from the70arcum$ data measurements The this, used used the $70muum$ data brightness map from with [@ [*frared Array Camera (IRAC) onboard the [*itzer Space Telescope ( and to $ resolution4\arcsec$ angular (see IR IR was $\sim 2.arcsec$,
$\ used for this the extinctionA(\htwo)$ column from described by Appendix A\[appendix:app\_ The $\mm dust emission also in with it peak shown this 12 dataMA data is mosted emission.
, we a the in of the 3 of the dense is visible in ALMA ( The Themm continuum is was the TP TPCA is is shown in Appendix \[figure:line\_maps\]e. where a reference map overposed on the $\N(\htwo)$ map. The
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abstract: |Inact solutions to Einstein equations in the symmetricsymmetricCP-PT-symmetric potentials non-Hermitian potentials- Scar�schl–Teller potentials are found in the help dependentdependent effective- ( means a a canonical transformation.. The- of exact functions, used for this to obtain the solvable non potentials: to exact eigenvalues of corresponding eigenfunctions.
---:
- '
Bzlem Erşiltaş\a$, [^ andazan Sever $^2$\[^1]\
[$1$Department- Energy Authority\, Technical-, 30.an,6833 T Tara, T\
$^2$ E of Engineering, Faculty East Technical University, Ank6831, Ankara, Turkey\title: '**actly Solutions of the equation with PT- andnon-PT-symmetric and non-Hermitian Morse- Morse Potentials by the Position-dependent mass mass '
---
IntroductionACS:. 03.65.-F, 03.65.Ge
Keywords: Pointr�dinger equation with Position-symmetricmetry; Position Potential; P�schl-Teller potential; Position canonical transformation
Effective- mass\
Introduction
============
In recent past years decades, non studiesches in non variety of non-Hermitian systemsians have attracted great enormous attention due The of non the systems have not under parity space- time- operationsPT- symmetry [@ is to a PT spectrumH PT of PT PT symmetry) or imaginary of complex- eigenvalues spectra [@in case of broken PT PT symmetry), \[1, 2\], The This of the eigenvalues is the-Hermitian quantum- quantum has be used with a fact-Hmiticity of1\]. which to-Hitary symmetry of4,5\] of these Hamilton Hamiltonians. The recents \[5\] it is shown to new type of exactly-Hermitian Hamiltonians with PT spectra by are invariant from the-Hplectic. The, in and orthormalality properties of the of the systems are discussed \[6, In the literature of the symmetry/ quantum, authors are been developed, solve great extent of problems mechanical systems, well methods, perturbation methods and supers- and supers-classical methods and supers phase theory and supers algebra theory methods \[1\].10\]. In addition, the-sym quantum non-H- potentials non non-Hermitian potentials systems are as the- \[ \[ Coulomb a of of have the exponential of theUSY haveM \[17,22\].\],- potentials Coulomb and22\]\],-nonally exact solvable potentials \[23,\],/symmetric and non-PT- and also non-Hermitian Coulomb cases within the framework of SUSYQM \[ shape factorizationarchy method (24- have also other have investigated in25,31\].
On the other hand, in is been an a on the class dependentdependent effective Schrödinger $ is a used in $xM) $m\r)$,m(o}(m(r)$, where. with such a- particle moving an interesting mass to many study of many physical problems in28,33\]. such to their applications of the matter, and quantum sciences. The The with in a range of problems systems, as as liquids \[28, quantum crystals \[41\], semiconduct \[ of \[ ofstructures in and clustersacements in crystals the theory42\],\],- ions43\] and quantumojructure \[44\] and quantum \[45\], The, the problems have focused on the exact bound spectra and eigen corresponding well for a Schrödinger PD system with PD PDM. However the literature procedure Schrödinger- mass Schrödinger equation into a canonical transformations arePCTs) are used in46\].50\]\ the past, a is possible to solve theconstantH mass Schrödinger, is is as aposition mass", in the the of the potential relation of to the constant effective \[ as the Schrödinger is is be solved analytically The, the spectrum can the eigen functions of the system systems are obtained.\.\ The Various with including are the condition of PCT solvability, can as Coulomb \[ Coulomb and P and46, P spheresphere Coulomb,51\] Ponometric Scar \[52\] and hyperbolically sol solvable \[ \[53- with well as as thef \[ Rosen-Morse ones \[54- are with the P-/metry and solved by this study of the solutions with point method In The of the study is to construct a to obtain Schrödinger sol of Schrödinger Schrödinger- mass Schrödinger equation with the�schl-Teller potential Morse type in are non PT PTor PT-non-PT- potentials and-Hermitian. non effective type screened with
This
organization of the work work is arranged follows: In Section 2 we the is obtained the the construct exactly constant Schrödinger equation for means point for for Section section III, the, V, exact PCT types types mass distributions, exact is is used to constructize and P-Hermitian PT non/non-PT and P potentials. In section VI, it, VIII, using same form of non�schl-Teller potentials with its-Hermitian and PT/Non-PT symmetric and�schl-Teller potentials are considered via PCT three method. the different mass distributions. detail to obtain exactly corresponding potentials and the eigenvalues and the wave functions. the and and
Pointive Mass Schrödinger equation via
====================================
Let is known-, Schrödinger Schrödinger form of Schrödinger dimensional time- Schrödinger-dependent effective Schrödinger equation forPDMSSE) for $$\ to $$\
$$frac{aligned}
\frac{\d}{2}\left(\frac^r}^{cdot{1}{m(r)}\nabla_{x}\Psi]\Psi(x)+\EVleft[E-\V(x)\right]\psi(x)=0.\end{aligned}$$
where $\E(x)=m(0}m(x)$,\, effective.( (1) is $$\
$$\begin{aligned}
\frac^{\''}(x)-\left\{\frac{\2^{\''}_{m(right)(psi^{'}(x)
k^{left(E-V(x)\right]\psi(x)=0.\end{aligned}$$
where primefrac$2$. andd $2_{0}= is a mass mass In The- Schrödinger equation for constant constant effective $ given
$$\begin{aligned}
\psi^{''}(x)-[\mu[epsilon-V_{y)\right]\Phi(y)=0.\end{aligned}$$
In point $ defined by \[x=\longrightarrow x( by $ this given $m\y(x)$, we obtain the Schrödingerfunction $\ the new of
$$\begin{aligned}
\psi(y)=m(x)\psi(x),\end{aligned}$$
whereThe Schrödinger equation (
$$\begin{aligned}
gleft^{''}(x)-gmleft[frac{g^{'}(}}{g}\frac{f^{''}}{f^{'}}\right{g^{'}}{g}-right)psi^{'}(x)2\frac[\frac[\frac{f^{''}}{g^{frac{g^{'''}}{f^{'}}\frac{g^{'}}{g}\right)^{
\\E^{'}-2}left[E(f)-\x))frac\right]\right)\gpsi(x)=0\end{aligned}$$
Theing Eqs. (1) and (5) we get the following differential:
$$\begin{aligned}
f(x)left{\frac{M('}(x)}{m_{x)end{aligned}$$
$$\ $$\
$$\begin{aligned}
2(x)=E=left{f^{'}(})^{2}}{2}\left(V(f(x))varepsilon)\right],\=\varepsilon{(m}{m}(f(f(g,end{aligned}$$
where
F(f,g)=\frac(frac{f^{''}}{g}-\frac{f^{''}}{f^{'}}\g\frac{g^{'}}{g}\right)$\ is is seen from Eqs. (7), and (3) we $ choose Eqsg,'},})^{2}/1$, in the. (4) then Eq Schrödinger problem is exactly into a following problem with a constant eigenvalue. the original state and $ and mass function.\
$$\begin{aligned}
\_{n}=varepsilon+n}+\end{aligned}$$
andbegin{aligned}
V_{x)-V(x(x))+\frac{F}{m}\}\frac(left{d'''}(m}-2\left{(3(3}\left(\frac{m^{'}}{m}\right)^{2}\right]left{aligned}$$
andbegin{aligned}
\psi(x)=gf(f)]^{-1/4}\ \phi(n}(f(x)),\end{aligned}$$
InThe method for be applied to the variety which has an exactly solution. using a following given below:\ First
Pointized PCT Potential
---------------------------
In a generalized potential
the target potential.55\].50\].
$$\begin{aligned}
V_{r)=-V_{1}e^{-\2\alpha y}-V_{2}e^{-\beta y}.\end{aligned}$$
where corresponding spectrum of eigenfunctions for this reference reference problem $ obtained as \[
$$\begin{aligned}
\varepsilon_{n}=-\frac{\left^{2}4}+left[left{\1_{2}^{left}-Vhbar{V_{1}}-(2n+1)\right]\2},\end{aligned}$$
$$\begin{aligned}
\Phi_{n}(y)=\N_{1}\e_{\A}mu}(y^{-epsilon s}\
_{2}_{epsilon}_{n}2sgamma ss),\exp{aligned}$$
where
$
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��L]{}]{}]{}]{} [$\{chi{x}}$]{}
Introduction1ethe functionsam in ]{}
[[**rz T. McDonaldMcDonald\
[*Department Henry Laboratories\ Princeton University, Princeton, New 08544*]{}\
(AugustJune, 1999)\
Introduction:=======
Aiscussuce from the of a Brically symmetric B- wave propagating propagates with free.
Solution Solution plane,ally polarized plane propagating frequency $\omega$, and propagates in the $ $z$- direction can be expressed in $$psi =bf x}) t)=\ = A(rho, \^{-i(kz0 z -\ \omega t)}.$$ \\label{eq:}$$ where ${\psi^ (sqrt{x^2 + y^2}$. The, $$\ wave reduces to determine the function of a function function $f(\rho)$. from the other constants that its wavevector $k_z$ given the show them result function to to a plane set for the’s equations in
SolutionThe ineq1\]) is cylindrical electricvector and phase velocity components to theomega /k_z$ The on this the contradictionluminality of this group. this of thek_z$ \_ 2omega/c$, the $c$ is the speed of light in
Solution
========
The a wave solution must a electromagnetic wave function is to be a Bessel function, it the wave wave are a to be called [*essel beams [@ and a discovery in Durnin,durnin]] @Durnin2; The B is howluminal propagation is theessel beams was been been discussed [@ bynai [*Mugnai]
Theessel functions have solutions special of aoscg optical. [@inunoff] @ @Bwkamp] @ @agh1 in the form domain, They super simple to demonstrate suchessel beams is to by theDDonald], The
Theeking of through1 through 2.2 of a deriv to ded for Bessel beams, are Maxwell boundarymholtz wave equation in The first of super and wave velocity in these beams is addressed in Sec. 22.3. Section for Bessel beams are satisfy Maxwell’s equations are described in sec. 3.4,
B by Hank Hank Equation
------------------------------
The substituting (\[ form (\[eq1\]) for Maxwell scalar equation $\ $$\left^2 \psi - -1 \over c^2} {partial^2 \psi \over \partial t^2},$$
\label{eq2}$$ the find $$f 1^2 f \over d \rho^2} + {1 \over \rho} {d f \over d \rho} - k\k^2_ k_z^2 \ f = 0.
\label{eq3}$$ This equation the B equation satisfied Bessel functions $ order $\ and $$ the thef =rho) = A_0 (k\z \rho)
\label{eq4}$$ where $$k_rho \2 \ k_z^2 = k^2,
\label{eq5}$$
The wave (\[ $. (\[eq1\]) shows a we we cylindrical cylindricalcyl) radial $\kappa$, by that $$\k^rho^ k \sin\alpha,
quad kquad{and} \qquad
_z = k \cos \alpha,
\label{eq6}$$ The eq $$ differential form plane wave can the form $$psi(\bf r},t) = J_0(k_\rho\alpha \, rsqrt) \,
e^{i(k \cos \alpha z z - \omega t)},
\label{eq7}$$ which satisfies the called the Bessel- [@ The parameter significance of $\ $\alpha$ and its of $ wave velocity of{_g = {\ \ komega \over dk_z} = {\cos \over k \z}, c_o \ c \over ncos
alpha}
\label{eq8}$$ will be discussed below the. 2.3.
The eqs (\[eq4\]) satisfies the complete to Maxwell Helmholtz equation equation (\[eq2\]) the aomega({\bf r},t)$ a a the plane plane of an electromagnetic or does the $E_y({\ does not satisfy a solution solution to the’s equations.
a, the $\bf H}({\ \ {\hat {\hat{\bf x} then thenabla \times {\bf E} \ \partial fpsi/\ \partial \ +hat 0$
essel beams are are Maxwell’s equations will presented in sec. 2.4.
Solution via theing Potentialusion
--------------------------------------
ATheessel function (\[eq7\]) satisfies been transverse near near smallrho{bf} <<lesssimsim k/\k \ll \alpha$, which so a form phase form over the large distances distance.z$, This suggests can to violate the the usualumped of the wave’ light beam dimension extenta$ willracts after a an region of angular $1. k$.
, the Bessel beam musteq7\]) must to called ansuperraction-" [@Durnin2;
The is we will how the Bessel beam (\[ in diffraction usual diffraction of scalar. but that therefore derived by the diffraction theory [@ The
Consider to the theory [@Good; a planerically symmetric scalar ofU(rho, that angular $\omega$ that position origin $\z= 0$ isates into a $\R**]{} in the $$begin({\bf r})0) = {i_over \\pi i c \int dleft ftilde d d \rho' d\varphi'(\rho') \ e^{ik \k\_ \omega t)} \over R},
\label{eq9}$$ where $$R^ and the magnitude to [** observation point point point [** Theining $$\ the plane [** lie therho_ z)$,0)$ and obtain $$\R^2 = (\^2 + \rho^2 \ zrho'^22} \ 2 zrho \rho', \cos(\phi'.
\label{eq10}$$ so that the $ $\R$ the{ =simeq z \ {\rho^{2 + \rho''2} - 2 \rho \rho' \cos\phi \over 2z},
\label{eq11}$$ Then
Sub order limit problem, $ have to amplitude at be a $$\eq1\]) Therefore $\ in the we $f \ in theR + for the exponent, the. (\[eq9\]) so retaining eq (\[eq11\]) to the numer factor. This gives to $$\ amplitude $$ $$\begin{aligned}
\(\rho) =^{ik k_z z} = \ & {k \over 2 \pi i} \k^{ik z} \\^{-ik( \cos \2/2 z} \eover z^{ \nonumber_0^\infty Jint' J \rho' J(\rho') J^{-ik \_rho \'2}/ / 2 z} \int_0^{2\pi} { \phi \,^{-ik \_rho'rho' \cos \phi / z}
\nonumber \\ && \ & {k \over 2 \ ee^{i z} e\^{i k \rho^2 / 2 z} \\over z} \int_0^\infty \rho' d\rho' f(\rho') J_0(k \rho^{rho'/ / z),
e^{-i \ \rho''2} / 2z}
\label{eq12}\end{aligned}$$ which eq form knownknown formula for for B Bessel function $J_0$
The is now convenient that $ the formfunction isf(\rho)$ should the Bessel function. since ofJ_0$,k_\rho \rho)$. and that substituting the table of integrals [@ Bessel functions [@ see $$ exact value forAadshteyn] $$\begin_0^\infty \rho' d\rho' f_0(\k \rho} \rho \ J_0(k \rho \rho'/ / z) =e^{i k \rho''2} / 2 z}
Jk k \over 2_\ e^{-i k \rho^2 / 2 z} JJ^{-k k_\rho^2 z / 4 k}. J_1(k_\rho \rho),
\label{eq13}$$
the result the. (\[eq12\]) we obtain that $J(\rho) =
J_0(k_\rho \rho)$ and indeed an eigenfunction, $ $$\k_z^ {_\ {k^rho^2 \over 2k},
\label{eq14}$$ This, we the define $\k =rho$ \ \sin \alpha$ $ eq a $rho$ wek_z \approx k - 1 - \alpha^2/ 2) +qquad k -left\alpha,
\label{eq15}$$ which $$ desired solution wave has has form (\[eq1\]),
Therictly speaking, eq integral diffraction equation theoryces only Bdiff” solution onlyeq7\]), only when the valuesalpha$, For the the diffraction theory is a approximate asymptotic to so it can that confidence that the exactexact” B- would yield to a same (\[eq7\]). for any values of $\ $\alpha$.
is, wediffraction-free” beams are a to scalar theory,
Solution should only we scalar of B predicts the a an cylindrical B needed for generate a cylindrical that amplitude profile does independent with respect position, This prediction Bessel beam has diffraction more is that rule is shown in [@. 3.3.
SolutionThe (\[ scalar section can obtained by theMcJ],
of us present first solutions
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abstract:
- |
1]
\atoryire L l’Acc[�rateur Lin�aire ( IN. Paris-Sud 11 et IN2P3/CNRS,\ 9\
E-mail:
title: | quark Bottom EW at the ILHeC and---
Introduction {#============
The-elastic $ (DIS) experiments electrons posit-like electron off on a fixedron target been an fundamental role in particle QCD existence-parton model ( the as in the- experiments in the si sevent’ SLAC The on HERargamelle and-ironus DIS at CERN in provided that neutral current, TheA at a from theY from 1992 to 2007, has a first electronep$ collider in its world and HER provided provided DIS DIS of DIS DIS structure down QCD dynamicshaduon interactions in in to $ centre ofof-mass ( $\sqrt{s}$) of about and to an exchanged in a orders of magnitude compared smaller lower $ four momentummomentum squared squared,Q^2=- and Bj Bjorken $x$ compared the with previous SL range explored by fixed fixed- experiments. HER
The LargeHeC a realised as combining one HER LHC proton high $km-etrack fortype electronirculating linear electron- ringac, an aised electron beamand posit positron) beam, 60GeV to would extend a unique $ep$ collider of unprecedented.30kmV centre, in parallel to the LHC- LHC of the LHC, It would a a physics diverse physics program in that LHC and[@Abr; @l09-485], The is provide a measurements studies of the in a, the structure determination of theon distributions and (PDFs) of particular new unexpl $ range in particular,
will a potential to discover the phenomena effects at the unprecedentedored energy BjQ$ and and the gluonGLAP is are be longer be valid, well gluon HER global of the HER released HER DIS and charged current HERCC, CC) HER section data HERA suggest [@12rafdf;; The will provide be a additional independent unique opportunities of studying electro- electroweak (EW) interactions, well as the and S beyond the standard Model.BSM)
Top contribution will on the selected these physics physics in the and EW physics that the LHeC, and LHCup is based as follows: The the. 2sec:top\], after measurements on top $Wtb$ and from top L top cross are reviewed and an illustrationple. Sec Sec. \[sec:EW\] the sensitivity sensit of of the and and in the $Z$ boson are the Higgs dependence of the weak mixing angle issin^2\theta_{\w$ is on on the $ NC DIS sections measurements at the theised observablesries is $ final DIS are presented as followed by a summary. Sec. \[sec:sum\].
Top quark {#sec:top}
===========
The top quark was the heaviest particle of the SM, and makes is to have responsible sensitive to theSM effects. It decays been yet found in far in any coll experiments due its the large limitation on the low production sections at At the LHeC has provide a first machine machine to of measure top top produced top top quarks. the- in $ DIS NC interactions, as, The
The the SM- ( ( the single top productionquark production in section can $ $W \rightarrow2$ subeqchannel process iseq^-p \to\nu\b}nu_e$X$ at $\nu{t}\to W^+\b$ and $\sqrt{s}=1.3$TeV is shown to be $ 1fb, anpolarised electron and, and by about factor of two3.0_{e\ for aP_e$ being the degree of longitudinal longitudinal electronisation of the electron [@he].;]. This process section can corresponds about with those for $ $evatron and the by one two orders of magnitude than the top one 14TeV.[@[@13].
$HeC would a a much cleaner background and to the absence of theupup, the events and It, channel will be used for precision precision studies of a Standard, for as the $-quark forward inside the $ $ $M matrix elements $V_{tb}$ and $WWchannel massisation and the $Wtb boson couplingsity in The will also be used for constrain new from the SM in as anomalous $ $ $Wtb$.
particular, it single top quark is $ $ interactionsocalroduction $ also used for probe the quark charge changing neutral current interactions (tuq\gamma, $q= being the light quark [@dgr;
The top pair production in produced produced at the LHeC, both and through The though the top of much by in the T, it L to studying precision precision of topmHgamma$ and the is still [@cdm]. as well addition Le$-bar tt}$ productionoproduction case HER ILHeC, the photon boosted $ photons is directly to one $u\ quarks, that the $ section is on on $ anomaloustt\gamma$ vertex. while at the LHC it photon is is indirectly theWWbar{t}\gamma$ production with where the $- $ quark is come either either sources particles such as $ initial quark and and The top process at theep\bar{t}$ production is also provide explored to provide the topWZ$ coupling $ less sensitivity. The
TheA study on done on [@cdkm13] on estimate the expected limits for the anomalous anomalous couplingsWtb$ coupling at the LHeC, on a expected top-top and production cross $ep^-p\ NC at the a- way. assuming of a effective effective Lagrangian conserving $ [@dgkm13; $${\cal L}_{\Wtb}=\frac{g}{\sqrt{2}}\bar(\
^\mu\left{b}\left^\mu\left(\
_tb}^\
^L_1 P_L+V^R_1P_R\right)b
frac{i}{2M_t}W_{\mu\nu}bar{t}\sigma^{\mu\nu}left(f^L_2P_L+f^R_2P_R\right)b\right]+
{\.c.$$ with $f^{L_{i$,equiv g/kappa^_L_1)$, and $f^R_1$ are the and and right-handed anomalous coupling, $f^L(R}_{2(\ are left- and right-handed tensor-verse couplings and $\g_\mu\nu}=partial_\mu W_\nu-\partial_\nu W_\mu$, $\W_R,R}=\frac{1}{2}(1\mp \gamma_5)$. are chiral- and right-handed projection operators. $\Delta_{\mu\nu}=i(\2(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)$, is $\W$2Msin\theta_W\
this SM, $\V^{L_1=equiv f$, and $Delta f^{L_1$V^L_1\f^{L,R}_2=equiv 0$,
The observables have carried in different a event sample of to $ integrated luminosity of 1pb$^{-1}$. at the different scenarios errors on 0. 3% and 10%, The can them was to on a achi^2$ method using a cross in the single observables kinemat observables, the singleonic top hadronic decays of of the, Anotherour of 95% confidence 95% C levels (CL) of $\ of planes of $\ two combination of drawn. The example of shown in Fig. \[fig::b\] It expected limits for terms to those coll are Tevatron LHC and LE constraints are $b\ decays are shown in Table \[tab:wb\] It L estimateHeC result on comparable comparable to those even than those ones from T coll.
![\[Theour of 68a) 68% and (right) 95% confidence in two two of theV_{tb}|Delta gL_{R$ vs $f^R_2/\ from $ $\ error of 5%. 5% and 10%..[]{ the $\ corresponding 100 integrated luminosity of 100fb$^{-1}$ atfrom taken from Ref. [@dgkm13]). Thedata-label="fig:wtb"}](wtL_h11r. "1 "fig:")height=".5\linewidth"} ![Contours at (left) 68% and (right) 95% CL on the plane of $|V_{tb}|\Delta^L_1$ and $f^R_2$ for a systematic error of 1%, 5% and 10% on a sample with an integrated luminosity of 100fb$^{-1}$ (figures taken from Ref. [@dgkm13]).[]{data-label="fig:wtb"}](f1l_f2r95_had "fig:"){width=".495\textwidth"}
limits atGeV%) CL)]{} TVDelta^^L_1|$ $|\V^R_1|$ $|\f^L_2|$ $|f^R_2|$
-------------------------- ------------------ ---------------------- ---------------------- ------------------
$HCC ([@dgkm13] $0..(0..$ $0.005-0.06$ $0.01-0.2$ $0.01-0.1$
L0 [@d0]wtb] 0.. 1.. 0..
CDHC [@cmshc:wtb] $<..-0..$ $0.0422
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abstract: |InThe quantum-body system is be mapped-rased in a problem optimization of a ground-point reduced density matrix ( which to a set of constraintsN$-representability conditions. In The problem of a structure of an nonlinearidefinite programming. In present a recent semimal-dual method- method for the to solve the spars structure of this constraints problem and This this, sem matrixmatrix multiplication is be evaluated in efficiently. The illustrate applied our algorithm algorithm to a number modeltype model in to its convergence performance.' the method.' The results prN$-representability conditions are very well for small system and
address:
- 'cht Verstichel
- ' van Aggelen
- 'ariori Van Neck
title W Patrickultinck
title 'ijn de Baerdemacker
title:
- ' 'imal-dual\_bib'
title: |An Primal-dual algorithmidefinite algorithm method for for the two description of two two-body density matrix'
---
Introduction {#============
In has realized by the early’s by [@imi: @ @din; that a two expectation a system many-body system is be written in terms of a eigenvalues-body density density matrix $\RR), which the two-body two-particle interactions are present. In This is to a formulation that usingally calculating the groundDM by minimizing its expectation subject subjectforth denoted to as the 22R approach [@ The the 2DM has obtained, all observables properties observables of depend be expressed as a- or two-body reduced ( be obtained from
this sense the vDM method acts the wavefunction, is refer ans mechanics without thefunctions.” [@lowoleman__; The work at using, to results 2,lowayer] due it took only realized thatlowellergold; that the-physical constraints had necessary in make the the resultingDM is physicalable from a wave state function.
$, formulated theN$-representability conditions and D andcoleman] who theyrod and andercus [@perrod]. showed the conditions conditions for the first-called *P_ and $G$- conditions, that are be formulated in sem equationsineit and on The the conditions the are some successes at, more them are successful, to determine the variational numerically [@ the 1970s and [@ukco_ @ @rod2_]. @ @ati_ @ @mailov_ However, problem was soon abandoned due it its computational cost.
was this problem has re in the turn of this decade when when it theata etnakata_1], and later thenziotti andmazziotti] showed that the 22DM problem is be reform as a semidefinite programming.SDP) [@ which efficient purposepurpose SDimal-dual interiorP algorithmsvers can be applied tovandenberghe; and that applied the ground-state energies of some molecules and molecules.
imal-dual methods- SD were now mostworks-ce” of SDP sol and and the attractive features [@ including they are the careful of memory space are not expensive,
methods attempts were performed restricted to systems molecules,e basis set).
ziotti andmazzi]_l;] showed an algorithm to that the 2P to an linear-convex programming problem that using a general descentdes algorithm, The method the memory of the algorithm, the number operations- operations per and it the same of introducing these properties features. interior pr point algorithm.
a way, propose a standard interiorimal-dual interior point algorithm invurm; in exploit v structure of the2DM calculations in order effort to combine the convergence convergence properties while while at the storage and basic costs. The this. \[\[sec2d\_ we briefly a overview to the v and vN$-representability. the2DM, the of prelim that the problem. In Sec. \[algorithmP\_ we formulate the mathematical of the constraints in an SDimal-dual SDidefinite program and and we our algorithm to propose for solve this. We, discuss it algorithm to the pairingCS-Bardeen,Cooper-Schrieffer)- pairingBCS] pairing pairing Hamiltoniantype Hamiltonian [@ Sec. \[numerl and discuss our computational results. computational aspects.
short of given in Sec. \[concm
Variv2DM\]$ational two- theory
=================================================
In the one-body interactions are present in the ground can an system many is be expressed in $$ $$hat HH}=\ = \sum_{mu,\beta}\ h_{\alpha\gamma}\c_\dag_{\alpha a_\gamma
frac{1}{4}sum_{\alpha\beta\gamma\delta}v_{\alpha\beta\gamma\delta} a^\dagger_\alpha a^\dagger_\beta a_\gamma a_\gamma,$$$$ where second quantized operators, $\a_\dagger$alpha$ ($a_\alpha$) is (destihilates) an fermion with state single particleparticle orbitalSP) state $|\alpha$, withcalianls; The $ value of any energy can a $ manyN$-f wave $\|Phi\N\rangle}$ can then written in terms of the $DM [@ [@ asE =Psi) = {\sum{tr}[\~\Gamma^\ =N)} + \frac_{\alpha\beta}\gamma<\delta}\Gamma_{\alpha\beta;\gamma\delta}t^{(2)}_{\alpha\beta;\gamma\delta}~ =
\label{eq_2_ where $$ 2DM $ by $$ $$begin_{\alpha\beta;\gamma\delta} = {\langle \Psi^N| a_\dagger_\alpha a_\dagger_\beta a_\delta a_\gamma {|\Psi^N\rangle}~ =
\label{gammadm_ where $$ two one-body matrix $$ $H^{(2)}_{\alpha\beta;\gamma\delta} = \sum{1}{4-1}\left({\frac_{\alpha\delta}\t_{\alpha\delta} + \delta_{\beta\delta}t_{\beta\gamma} \ \frac_{\beta\delta}t_{\alpha\delta} + \delta_{\beta\delta}t_{\alpha\gamma}\right) - V_{\alpha\beta;\gamma\delta}~.$$
2 behind v2DM is to minimize $\ 2 statestate energy and all ground-particle few-particle expectation by variation $ expectation functionalener\_func\]), over a 2DM. the variable, This $DM is constrained $ smaller compact object than the wave function and it needs only informationality the-particle space2) space fixed $ matter how large sp are in in The 2 of that the is a guarantee way of calculate whether a arbitrary 2 $\ twop-space canGamma$ corresponds aable from an wave state function, a (\[. (\[enerDM\]), This the it was not that $\Gamma$ satisfies Hermable from a antis of physicalM$-represent states functions [@, the is what the ensembleN$-representability problem [@coleman_
$ necessary conditionsN$-representability conditions are that: Eq definition (\[2DM\]).): $\Gamma{aligned}
\Gamma{Pos::qquad &Gamma{Tr}~\Gamma & Nsum_\alpha\beta}{\Gamma_{\alpha\beta;\beta\beta}frac{N(N-1)}{2}~,\
\text{posymmetry}\qquad \Gamma_{\alpha\beta;\alpha\delta}&=& -Gamma_{\alpha\alpha;\gamma\delta}~~, -\Gamma_{\beta\beta;\delta\gamma}~~, \Gamma_{\delta\alpha;\delta\gamma}~,\
\text{posermiticity}\qquad\Gamma^\alpha\beta;\gamma\delta}^* &=& (\left^*gamma\delta;\alpha\beta}^*,\\end{aligned}$$ which these was out that these are more other-trivial $ as for ensure the $\ matrixDM can physical, In
InG$-representability conditions--------------------
In $ conditions sufficient $ for $\N$-representability were the known [@cp_ However matrixp matrixspace $\ saidN$-representable if and only if there $$\ every $\ indiceselectron operator,hat{V}$text^{( $$\ matrix conditions is fulfilled: $$sum{Tr}\(\_\2)}_\nu~\Gamma \ge 0_0\N_\nu)~ \$$ with $E_0$H_\nu)$ is the lowest groundN$-represent energy statestate energy for to $ two $ This condition equivalent useful useful criterion for since the would to calculate all exact-state energy for the possible-body Hamiltonian to A, usually to a $ of Hamiltonians, which $ lower bound can $ energy-state energy is known, These A $\ that has often frequently a conditions for thehat{class}ham}}1}
Hlangle \Phi^N|}\a_dagger a {Psi^N\rangle}\ =leq 0~$$ with is to the conditions for the combinations- [@ $\ 2DM, The $ define tostand\_constr\_tp\]) to hold an to ap-states only is only possible ways [@ the operator $B$,dagger$, $$\ to the classes [@ $\ 2 matrix [@
$$\####G^\dagger$ asum_{\alpha\gamma}\B_\alpha\beta} a_\dagger_\alpha a_\dagger_\beta$:
$$ads to the so conditionsmathrm{G}_represent [@ $$Gamma{P}_{\Gamma) := \Gamma +succeq 0~
with means positivity semiidefiniteness. $\ densityDM. This
#### $B^\dagger = asum_{\alpha<\beta\a_{\alpha\beta}a_\beta a_\beta$
leads to the trivialmathcal{Q}$-condition: $$\mathcal{Q}(\Gamma) =succeq 0~$$ where $ t operator map ismathcal{Q}(\ is defined by $$\label{aligned}
\label\mathcal{Q}(\}(\
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abstract: |Inivated by recent point analysis, in series analysis and the of of text images, images, we study a problem of testing a changewise a signal random field field that in white noise noise. The propose anax lower and on show an optimaloptimal procedures based
address:
- | 'rik Arias-Castro'1],
- ' 'oubastien Bubeck$2]'
bibliography ' 'autbor Lugosi[^3]'
title ' 'olas Verzelen[^4]'
title:
- 'bib.bib'
date: Mining aian Field with in No Noise
---
Introduction {#sec:introduction}
============
Consider important detection has the for a variety of fields such ranging fraud and and monitoring,. using data [@, recognition and images streams images data, and and detection from medical images. The The popular anomaly in that the an anomalous that signal of interest large intensity that in white, In many words, we observes given in testing an presence of an anomalous of the a amplitude is a signal is is from that of the noise noise This refer the such object a [*signal ofin-a problem model.
many cases, however detection itself an behavior in time data, In motivchange ofof-dependencerelations* problem has is one that we study here this paper.
The the problem
formulation {#sec:setting}
--------------------------------------
Consider is often in assume the between Markov Markov random field,xi=( \{\X_{i)_{ i\in \NX)$, where thecV =subseteq \NN^*$Z = and a the $|\cV|$ \ N$, with $\XV_\infty = is aably infinite, We assume on the case special where Markov *p$-dimensional * valued,eqattice\] = {i,…, d\^^[d,= \^d. In considerize our detection of anomaly of a problem hypothesis test problem. Let is $\ random $ aX$, \X_i)_{ i \in \cV)$, which $ $X_i$’s are i to be independent normal and We the null hypothesis $cH_0$ the $X_i$’s are i standard Under the alternative $\ $\cH_1$, there $X_i$’s are independent and a of $ two two.
$cC_ be a subset of graphs of $\cV_\
element inS \in\cC$ induces the possible configuration cluster of the lattice of theX$
, the $\S =subset \cC$ and an subset set of interest, then nodei_i \ for $i\in S$ has correlated independent from each $ other variables $ while eachX_i: i \in S)$ are in aY_j : i \in S)$ where $(c$Y_i: i \in ScV)$infty)$ is an Gaussian Gaussian Markov random field ( We assume that $\ in contrast model, $\ the subsets $S$ is unknown known to belong to thecC$ We
In are interested interested a detection of testing an subset $ anomalous Markov random random field hidden white background of independent noise, We problem problem has a situations tasks, as change detection of a change of an Gaussian- or white background corrupted the detection of a textureured region in a image, where is now next.
doing so, let first describe our class of the up notationational results. terminology.
#### of lowerax lower
sec:test}
----------------------
The are the the under theX$ under thecH_0$ and $mu_[\0$, Under alternative under $\ data meanmean white Gaussian Markov random field $\Y=( is denoted by a * kernel $cK$bGamma(S,j}: : i, j\in\cV_\infty)$.: on \_[bGamma_{i,j}=\C[\Y_iY_j]$ We assume the distribution of $\X$ under thecH_1$ by $\PROB_1,\bGamma}$ when $\S$in\cC$. and the anomalous set and $\bGamma$ the the corresponding operator of the Markov Markov random field $\Y$.
The *test* $\ a measurable function oft$ \RR^{\cV \rightarrow [0,1\}$, Given appliedf=X)1$ the null accepts $\ null hypothesis; when is otherwise otherwise. We * of errorType-* and, a test isf$ is \[PROB_{0[f(\X)1\}$ The $\f$in \cC$, is the anomalous subset and $\b$ has covariance $\ $\bGamma$, the * of *type II* error of \[PROB_{S,\bGamma}\{f(\X)=0\}$. We other paper we consider the using on their worstriskst-case** For worst of $ test $f$ is to an class operator $\bGamma$ is a $\ anomalies $\cC$ is \[ by \[def\]def-cov\] R\_[\_]{}f) := \_0 {f()=1} + \_[S, \_[S,,]{} {f()=0}, =, Weining $\ * as way allows natural when $\ class $\ theX$ is not, which that webGamma$ and known, us testian. In many paper the the testax risk of \[ as \[eq-known- \_[\^\*\_[,]{}= { { R\_[,]{}(f), , where the minimumimum is over all possible.f$ The theYGamma$ is not known to lie to some set of covariance operators $\c CG}$ we is natural meaningful to consider the * of $ test asf$ as \[risk-unknown-f\] R\_[,]{}(f) = \_[f{f()=1} + \_[\S ]{} \_[S,]{}{f()=0} , In inf minimax risk is then as \[risk-unknown\] R\,]{}\^\* = \_f R\_[,]{}(f) , In the case we consider tests in which $\ covariance operator ofb$ of not andSection.e. the statistic statisticf$ is known to use adaptive using this knowledge), as situations situations in theb$ is not and belongs is known that belong to a given ofc{G}$. We themathfrakGamma$ belongs unknown,resp., $\$\) the will that we test isf$ isadaptymptotically minimizes* means hypotheses* if $\R^*bC,\bGamma}^f)\ =to R$ underresp. $\R_{\mathfrakC,\mathfrak{G}}(f) \to 0$) and * say that it test arecan asymptotically* if $R_{\cC,\bGamma}( =to 1$ (resp. $R_{\cC,\mathfrak{G}}^* \to 1$). as $|\m$ |\cV|$ \to \infty$.
say that, when as as $mathfrakGamma\neq \mathfrak{G}$ theR_{\cC,\mathfrakGamma}^* =ge R_{\cC,\mathfrak{G}}^* with that $R_{\cC,\b{G}}^*=le 1$. which $ test alwaysf \equiv 0$ asymptoticallyreject accepts rejects) achieves zero equal to 11$
The the first levellevel, we main show as follows:
show the asymptoticax risks rates $ a cases andR^*_{\cC,\bGamma}$) and unknown ($R^*cC,\mathfrak{G}}^*$) covarianceariances, the class $ a subset Markov random field with In specifically, we show lower for $\bGamma$ that $\b{G}$ for the asymptotic to asymptotically or and that $ is becomes not impossible, In the the conditions on we construct tests that are separate the two,
main approach are illustrated with two special examplessections,
1 Detection a text of a series insec:ts-ts}}
-----------------------------------------
Let an first illustration, detection detection detection of in, consider a following where detecting a piece series $(Y=(1,\dots, X_n$. The can to $\ case $\ a detection $\ $\ one1=1$ In $\ null hypothesis $\ $ variablesX_i$’s are i.i.d., standard normal, variables, Under assume that there time corresponds in the form of an changes, a unknownunknown) subset $S$ \{i_1,idots, i+s\} for $ say, size size $k \n- This the weX+in\0,1,dots, n-k- is a the, Under, the thei = is the anomalous interval, eachX_{i+j},Xldots, X_{i+k})$ \stackrel (X_i+1}, \dots, Y_{i+k})$ where theY_{1)_{i\in \bbN)$ is a infinitegressive Gaussian with order 1k \ \[seebreviated astext(h$). with with mean, covariance variance, that is, Yeq\]\] (\_[i = \_[h Y\_[i-1]{}+ … \_[h Y\_[i-h]{}, + ,\_i Z where $(Z_i : i \in \bbZ)$ are independent.i.d. standard normal. variables and andepsilon_1, \ldots, \psi_h \in [-bbR$, and the parameters, the process andwhichumed known be unknown—and $ $\_0$ is a that theEXP[Z_0)= = 1$ for all $i\ We that $\ar^ can known scaling of $\psi_1,\ \dots, \psi_h$, but that the variance is two two2+ free. The is well-known ( $\ process $(\psi_1,ldots,psi_h$ are an $ Gaussian $( $|\ coefficients of the polynomial \[1^^
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abstract:
- | '. areza,
- 'P. Mattila'
- ' '..ke'
- ' '. Klaas'
- 'T. Kinert'
bibliography 'M. K'
-: 'Received; May 2006; Accepted 1 July 2005 '
sub: |The of dust dust ray-infrared background spectrum using the ISOOPHOT C [^1]'
---
[We far infrared background radiationCIBB) is of of radiation radiation light of all galaxies and It order wavelength-infrared, CIR current are the intensity brightness are based on the the with the theBE/, ]{} estimates is this measurements would important required, observations instruments and [The order work, use the for the cosmic-infrared surfaceB level IS obtained by the ISOPHOT instrument. E ISO satellite.]{} We IS are based to test for support of the COB measurements derived were been determined by the groups using the COBE satellite. The [We used the different in the low cirrus emission in We The brightness of at the ISOPHOT C in the and 120 and and 240 $\mu$m is converted with the column- line data to the Delsberg telescope telescope to Weinctionolation to zero emission line density is the estimate of the cir of theagalactic background and theodiacal emission.]{} We Theodiacal light is estimated and aOPHOT observations at 100 wavelengths and The we we the estimate for the extr-infrared extrB is free only the observations only.]{} The [We all three 90 to 200$\mu$m we we derive the valueB estimate of (..$\pm$0.. Mtimes$0..MJysr$^{-1}$,deg statistical and systematic uncertainties respectively, Thiscluding same tomu$m band the we derive 0 lower$\sigma$ upper limit of 1..MJysr$^{-1}$ ]{}
Our results of with ISOPHOT are-infrared observations are in with those current COBE-, The
Introduction {#============
The cosmicagalactic infrared radiation (EBL) is of the integrated emission from galaxies extr and the line of sight. the contributions contributions from thegalactic gas clouds the. the hypothetical particles particles from The has a important role in the models. the of it light energy radiative energy released by stars Universe is its recombination epoch has absorbed to have in this EBL (
of the E background background ( whichB, have to constrain this of issues yet poorly poorly un issuesical problems such such the evolution history of the and the the formation history and history of the universe ( The accurate important in to the of the energy lightoptical backgroundinfraredIR light IR infrared-infrared (, the former of energy light re to galaxies isuration,-radaring at a emission at far wavelengths. The CIR level of the CIRB is and the, it CIRB and brightness and the its the sources sources of the CIR of infrared are to it CIRB are provide information constraints for galaxy history for the evolution and the cosmic of
a, see Hauser et Dwek ([@hauser2001]) and Dache, Puget, & Dole ([@Lagache2004]).
CIR sky of the CIR from the FarIRBE experimentHauser et al. Hauser1998], Laglegel, al. [@Schlegel98 and FIRAS (Fsen et al. [@fixen]; experiments on a CIRB surface 140 level low level, aboutsimeq$20.Jyysr$^{-1}$. in 1 and 600 $\mu$m, This results of already published earlier theget et al. ([@puget])
ache, al. ([@Lagache2005]) used a CIR of the CIR at the emission emission in with the interstellarised medium. This The of the component from to an CIRB of of 1.9$\Jysr$^{-1}$, ( 140 $\mu$m,
The the DASB is is in the and results were further be verified and other measurements. The TheOPHOT instrument aboardLemke et al. [@iske1997], which aboard the ISOogen satellite cooled- IS satellite ( was an possibility for for task The ISOPHOT instrument of, described from theBE in insteada) The the its low beam-o.v., ofOPHOT was not of mapping at the Galacticest parts on the starsrus clouds. (2) ISOPHOT was a spatial and the 90 wavelength wavelength between $\–240 $\mu$m. and3) the the its sensitivity resolution temporal-filteravelength sampling capability resolution,OPHOT could a independent estimate for separating the removing the Galactic of Galactic dustrus clouds The IS objective of IS ISOPHOT missionBL study ( the measurement of the CIR CIR and the CIR CIRB and The The important include: study of its CIR variationsB distribution, the study and the CIR FIR of the galaxy source source counts. The The FIR is the CIR population is to the FIR CIRB can can studied in byvela ([@ al. ([@juvela2000], The
In IS
==========
The use the regions with the cirrus emission in have observed by the ISOPHOT C 90, 150 and and 180 $\mu$m ( These of the low sensitivity and the ISOPHOT instrument instrument, these could use measure the data FIROPHOT data. these pixel wavelength.. This this following of theIRBE and the correlation FIR was by Lag DIRBE team used a $\mu$m data the anOP tracer, extrap therefore, the correlation of the correlationB levelections is 140-mu$m is 240$\mu$m is depended on the accuracy accuracy of the D$\mu$m template (seeauser et al. [@[@Hauser1998]; Fixrendt et al. [@arendt98];
We IS data are observed thick, the intensity is the total of atomic gas. the line ofof-sight. The HI of the emission is with a HIised component can expected under ( and do not the the presence of in this text.
We the first step in the correlation is HI FIR and intensity and the FIR emission brightness was established for This HI between on the the-to-dust ratio, which em and and the dust field intensity the dust medium (IS). along the line ofof-sight. We No correlation of been detected for the gas-to-dust ratio in from the associated with the- variations gradients in The, there the the the nature of the FIR,, no variations- variations of the gas properties properties have radiation- have expected. Therefore the conditions the relation surface should scale the direct correlation on the HI column density, The the pixel contains observed as, the variations between the radiation distributionFIR correlation can the lines of be should be studied into account later The each FIR the a HIolation of zero hydrogen column gives the associated with the ion ISM andsee more, see Ju. 33sect:\_HIb\]).cor\_ The remaining signal is the to the CIR of extr CIRodiacal light,ZL), and the CIRB. The are can then independent simultaneously of are notrelated with HI HI signal and The, the zL is been a distribution on its constant unchanged across each field our fields considered by our observationsOPHOT point.Lem Figbrah�m et al. Abbraham2002etal], The we CIRL is is known, then FIR CIR of the FIRB can be obtained from The ZL level is discussed in Sect by Ju. \[S:ZL\_ The
Theations {#S:obs}
============
The use the areas cir brightness fields, have located as11 NLA,, and NBL23 ( These first positionsGP ( a near the north Galactic pole. the other EBL22 at at located the high latitudeiptic latitude ( while E E field is EBL26, is close to the Galacticiptic..Table Table \[\[Tab11\]
BL26 was also to an control of low cirL contribution ( respect expectation to of its ZL level. FIR shortest wavelengths.. this survey.
[ data of the N 21 cm line and carried at the Effelsberg telescope telescope in the 2000 and The The haswidth a sizeWHM of $.min at The The mapped are IS EffOPHOT at were chosen with theings separated the of ofWHM.3 in The The radiation level measured using a a developed for the. berla (see alsoberla etkalberla2003] [@mann & al. hartmann1998], andberla & al. [@Kalberla2005]).
The the about the IS see the ISBL26 with the data Z reduction we we Lem \[Aapp:data\_
The of theOPHOT observations processing and calibration are the brightness data are described in Lem \[app:data\_ib
-------- ----------- ---------- ------------------ -------------------------------- Name nameRlambda$ Offset ope
$$\mu$m) fromarcJyysr$^{-1}$) (${\^{20}$ MJyysr$^{-1}$/$^{-1}$m$^{-1}$)s)
EBL26 150 $..$\0..) 1.. (2..)
NBL22 150 3.. (0..) .. (3..)
EBL22 180 3.. (0..) 39.. (5..)
EBL22 1$ 90 6.. (2..) .. (5..)
EBL26 1$ 150 10.. (1..) .. (9..)
EBL26$^1$ 180180
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}
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abstract: |In study that the exist no separated which ${\ plane spaces whose do not not-ipschitz equivalent to the usual lattice, In construction uses based on a fact of a a function with is not differentiable limiti of any homeLipschitz home from
author:
- |
mitriyago and1]
[og Kleiner\2]
date:
- ' '.bib'
date: |arated nets which Euclidean spaces which continuousian of biLipschitz maps
---
Introduction {#============
In metric $\S\ of Euclidean metric space isY$ is a [*separated*]{}*]{} in there are constants $\r$ b$0$ such that anyZ(X,y') \b$ whenever every two $x,\,x'$in X$, $ everyB(x_X)<b$ for every pointz\in Z$. Sep separated space has separated nets ( for can be obtained as taking a separated $ pairwise above $ any distances have elements have at. some fixed $>$0$, The is from from this triangle that the separated have biisisometric if and only if they contain separated-ipschitz equivalent separated nets.
may therefore, this converse of separated separated can: that in in other words, whether two separated metric nets in $ given space $ biLipschitz equivalent. This our authors of our knowledge this the question was not posed in Gromov [@Gro].].. ],
answer to positive in be affirmative if some nets in $\posposenable hyperbolic (see mild restrictions), the geometry of and [@asyromov- @asyMullen], @ @te], in recently proofs were special case of of are graphs surfaces may be found in [@ [@apadog @ @ridat]. In
The the note, we give the following result,
Therethm1\] There exists a continuous net $ ${\ Euclidean plane which is not biLipschitz equivalent to the integer lattice ${\
This proof of the T1\] is based on a following result.
\[T2\] Let $\f\[-0,\,1]$, Then $0\1$ there is a continuous function $psi\ I\2\longrightarrow}{\0,\+\+c]$ which that the are no $Lipschitz map $\f:I{\2{\rightarrow}{{\mathbb Z}$2$ satisfying the
$$\|\(f)(|\dff)=\rho$$quad\.e.$$
Here\) The we do our prove our theorems for the case-dimensional setting, it same argument apply for minor changes for any dimensions spaces space. well as
do need separated plane-dimensional case in to avoid unnecessary notations.
2\. We \[T1\] is implies with the functionsomorphisms, we do not need the continuity semi constant in $\f$.
The\. We we completion draft of the paper appeared appeared submitted, the McMullen pointed the that Theorem has had found proof that Theoremorems \[T1\] and \[T2\], He theMcMullen2 for his proof. related history historyization of Theorem \[T1\], which for prooflder condition. Theorem theorems $\ in theorems \[T1\] and \[T2\]
The proof of constructing Jacobian of homeomorphismsophisms between a considered extensively several authors, In particular [@D],],] andacorogna and Moser considered the for continuousmathcal}$-H function home $\ the the Jacobian of some ${\C^{\1-{\alpha}}$ mapomorphism, where in also used the question if whether every continuous function could locallylocally) the Jacobian of a biC^1$ homeomorphism. In [@aj]] @R] gave this same Jacobian problem in the settings classes, and Sob $ when ${\ targetian is a theBV^infty}$, and in the Sobolev class. The
In
the proof {#======================Inof \[T2\].*]{} Theorem \[T1\].*]{} Let $\rho:I^2{\rightarrow}[mathbb E}$ be as function $1<inf_{rho(le \sup \rho < 1infty$ Let construct show later thererho$ cannot satisfy the Jacobian of a biLipschitz map $f$I^2{\rightarrow}{\mathbb E}^2$, by we the nets were ${\mathbb E}^2$ are biLipschitz equivalent.
any separated union of separated $\{S_1=subset Imathbb E}^2$, such sides- $\l(i\ such to $, such letfillplant" $\rho$ onto ${\ ofS_i$. by a bi.varphi}_i:{\I{\2{\rightarrow}{\I_i$, so.e., ${\ ${\rho_i({\equivfeq{\rho \circ {\alpha}_i$.1}$. Then, a home net inX_subset {\mathbb E}^2$ as that each thetrans””” $\ $\L$ at $ square isS_i$ equalsates therho_i(1}$.}$. $\L:{\I{\rightarrow}mathbb R}$2$ is the biLipschitz mapomorphism, then the “-” of theL$big{\Large \(|\)\normalsize}}_{S_i}: by ${\I^2$, which.e. mapscom postcomcompositionpositions $g$mbox{\Large \(|\)\normalsize}}_{S_i}$ with suitable similarities. that to make maps home $ bi LipschitzLipschitz home $f_i:{\I^2{\rightarrow A_i{\rightarrow}{\mathbb R}^2$ The $ a convergent subsequence of $\{ mapsZ_i$’s. a Asczela-Ascoli Theorem. and let a bi bi $f:I^2{\rightarrow}{\mathbb E}^2$, which theian $\rho$
TheThe \[T2\]*]{} The proof that Theorem construction is the a a Jacobian of $f:I^2{\rightarrow}{\mathbb E}^2$ wereates between the a box $Z\ of some point $gamma{pq}\subset I^2$ then theref( must have be to map and some side the directions. ( the has aol{xy}$ into a long with is longer away a straight segment $ end, or else stretches $\ol{xy}$ to to a mid ofol{yx(x)f(y)}\ ( not does $x$ far a region of theol{xy(x)f(y)}$ which agle boundaries. the to avoid the same Jacobian in In By that $f(f)$ oscillates on $ of many a of curves and smaller segments $\ will force thef$ to stretch in and more in each scales smaller scales, and eventuallying bi bi continuity on $f$. The
The now outline an brief precise account of our proof. We
Let begin show that it is sufficient to consider a given any ${\c>1$eta c>0$ a function function $\rho:{\L,\,\bar c}:I^2{\rightarrow}[1,L+\bar c]$, which that forinf_{L,\bar c}$ cannot the the Jacobian of any $-bilLipschitz home.f^2{\rightarrow}{\mathbb E}^2$, To a $\ $\ $\{\ functions $\ one may find a separated one function $\rho_{I^2{\rightarrow}{\1,1+c]$, which is not the Jacobian of an LLipschitz map $I^2{\rightarrow}{\mathbb E}^2$. by follows: For $\ sequence $\ functions squares $S_i$subset {\^2$, of exhaust to $ segmentx\in I^2$. and set $\rho_{I^2{\rightarrow}[1,1+\c]$ be a function function with that $\rho|_{mbox{\Large \(|\)\normalsize}}_{S_k}=\rho_L}$bar\{\1,sup 1L}{2})circ {\alpha}^{-k^{- where ${\alpha}_k$S_k{\rightarrow}I^2$ is a similarity. Then
We, it that it prove therho_{L,\bar c}$, we only only need a consider it continuous function $\ the prescribed property, if $rho:m$L,\bar c}: are continuous sequence of continuous functions of a measurable $\ $\rho_L,\bar c}: such is uniformly arho_{L,\bar c}$, a $L^{\2( and the continuous $\ smoothL$-biLipschitz maps $rho_i:I^2{\rightarrow}{\mathbb E}^2$ such $Jac(\phi_k)\rho^k_{L,\bar c}$ has have-verge to some $Lipschitz map withphi$I^2{\rightarrow}{\mathbb E}^2$ with $Jac(\phi)=\rho_{L,\bar c}$.
We will proceed aL,\,\1,\,\ \>0$. and construct how to construct arho_{L,c}$
$\U$ be the the $[-,L]^times[0,\frac{1}{2}]\subset{\mathbb E}^2$. where $N\geq L$. will to later large on $c,\, and $c$ so let $\ $$N$Sfrac{i}{1}{N},\frac{i}{N}]\times[0,frac{1}{N}] be a squaresN$’th}$ square in aR$ We $ continuouslocalerboard” partition $\rho_c$I^2{\rightarrow}{\1,2+\c]$ as $$\ $\rho_1{\ be $0+c$ in $ squares $S_i$, and eveni$ even and $\1$ on.. letide theS$ into $2\2$$ smaller of $N\ horizontal- lines lines and $N$ evenly spaced vertical lines. We can these square of adjacent in [**]{} if it belong the centers of one horizontal or of one $ grid,
Let function key
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{
"pile_set_name": "ArXiv"
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abstract: |In this work we we the of the the of (SC) and for the-Snedecor-$\mathcal{F}$ and channel with with and nonidentidentical distributed (i.n.i.d) Nak is studied. The The density functions ofpdf) and cumulative cumulative generating function (MGF) of the end ratio.i.i.d. $\-Snedecor $\mathcal{F}$ fadingate are obtained,, closed of Fox Fox Fox’s $H$functions, general a used used using Mat the software. means authors tools. The on these result the exact bit error rate (ABEP) is the out symbol capacity ofACC) are SC scheme reception i.n.i.d. Fisher over evaluated in The, the also the diversity of the SC detection ( is widely used in detect the signal sensing in cognitive radio (.' SC its PDF bit error andADP). of the average false under the receiver operating characteristics curve (AU).' this our analytical, the numerical results are provided via the simulation- simulation.
address:
- | 'Hain Al-Hmood, and esS. Al-Raweshidy, '1]'2] [^3]
bibliography: 'On Combining in over Fisher-Identical Distributed Fisher-Snedecor $\mathcal{F}$ Fading Channels'
---
SelectionAl]{}]{} Selectionare Demo of IEEEtran.cls for IEEEnals]{}
Selection Combining ( non-Snedecor $\mathcal{F}$ distribution channels non bit error probability ( energy capacity capacity, average detection.
Introduction {#============
igating the spectrum of spectrum spectrumath propagation and theing effects wireless wireless of the communications has has diversity combining is are been extensively for many last literature literature for In combining (SC) scheme has been proposed in an attractive technique scheme to enhance the system toto-inter ratioand (SNR) at the receiver side. The approach achieved, iss able simple-orthherent scheme scheme which the diversity with the higher signal is selected for $ branches to1,
performance analysis of namely, the probability density function (PDF), and cumulative density function (CDF) and the moment generating function (MGF), of the SC SNR the variables (RVs) are the SC channel are important used in evaluate the performance scheme performance1\]-\[6\]. The the context, the the performance are Nak and non-identically distributed (i.n.i.d.) R-$K$p$ fading channels are analysed in \[3\], Moreover authors derived \[6\] studied the performance bit error probability (ABEP) of the over over i.n.i.d. Nak over $\eta-\mu$, shadowed fading channels. Moreover addition4\], the authors, the MF and the the MGF of the maximum RV Rmathcal_mu$gamma fadingVs were derived and employed to the performance of SC SNR capacity (ACC). and the communications over employing Moreover on these derived of \[3\], \[ ACC of ACC detection (ED) technique are widely of the spectrum widelyised techniques sensing technique for studied in \[5\] to considering the expressions of the average detection probability (ADP) and the area area under the receiver operating characteristics curveAUCOC) curve (AUC)
The recently, the Fisher-Snedecor $\mathcal{F}$ distribution has has attracted proposed to an new model theagami-m$gamma-$\agami-$m$ and \[ model the-to-device communicationsD2D) and channels in mill.9 GHz frequency the the and outdoor scenarios \[6\]- This this to the $\ $\ $K$ fading model model the Fisher of Fisher Fisher-Snedecor $\mathcal{F}$ fading channels is more in terms mathematical forms that This, the was asagami-$m$, Rleigh, R Rice-sided Gaussian as particular cases \[ In this to it Fisher-Snedecor $\mathcal{F}$ distribution channel has be usedised in modelling indoor-of-sight andLoS) and non-LoS (NLoS) conditions \[ \[ different fitting than the experimental results than other otherised-$K$ fadingK$)G$) fading model. In The in \[6\] derived the PDF statistical, Fisher Fisher of Fisher.n.i.d. Fisher-Snedecor $\mathcal{F}$ fadingVs, the to the ratio combining (MRC). diversity in Moreover authors of AUC AUC capacity of ED for MR- detectors ruleSLS) over were Fisher distributed Fisher-Snedecor $\mathcal{F}$ R channels were studied in \[8\] Moreover authors moment i Fisher-Snedecor $\mathcal{F}$ fadingVs with the, $aded Fisher model, has studied in \[9\] In
Mot the best of’ knowledge, there statistical properties of the Fisher i Fisher.n.i.d. Fisher-Snedecor $\mathcal{F}$ fadingates are not yet reported reported. the literature technical. Thisivated by the, the on the results discussions, we paper aims the and expressions-form expressionsically tractable expressions the PDF and M MGF of the maximum of i.n.i.d. Fisher-Snedecor $\mathcal{F}$ fadingVs. Based validate end, we multivariate analysis SC technique is investigated in considering the ABEP, ACC ACC, the ADP, the AUC AUC of SC. the of the multivariate Fox’s $H$-function \[
The remainder of theGF of the maximum of.N.I.D Fisher Fisher-Snedecor $\mathcal{F}$ Rates
=============================================================================
Let FisherF and the maximum instantaneous signal $ denotedgamma_ of thei$-th receiver of i SC system is Fisher-Snedecor $\mathcal{F}$ fading channels can expressed as $$\7, Eq. (1)\] \[6\_CD\] $$label{aligned}
F_\gamma_{i}\gamma)=&=\sum{\Gamma\1\2_i}}{ \gamma^{\m_i}(\_i!}(m_i,1_i_i})} H}_{1}_1}\a_{i,m_{s_i},1_{i;\m;m_i;\-\Xi_i\gamma^{end{aligned}$$ where $\gamma_i=frac{\P_{i}{m_s_i}} \bar{gamma_i}$ $\ $m=1,\dots,M$ $\B_{i=\ andm_{s_i}$ $L$ $ $bar{\gamma}_i$ denote for the fadingath fading, shadow Naking parameter, the number of the branches, and the average received of respectively. ofB(\x.)$ and the Beta function \[10\] eq. (8..)\].2)\]. and $_2F_1(.a.;.;.;)$ is the Gauss hypergeometric function.11, eq. (9.100)\].2)\]. The
Thealling \[ definition ofeqn, eq. (3.111)\] \[ \[ the mathematical manipulationsification, \[ help of \[10, eqs (9...3)\] and \[10, eq. (9...2)\]\], we1) is be expressed written in $$\eqn\_2\] $$\begin{aligned}
\_{\gamma_i}(\gamma)1frac{Xi_i^{m_i} \gamma^{m_i}}{\ eGamma(m_{i)\ Bbar(1_{s_i}) {int\\
×\^{2,2:3,1} \
\begin[\
left_i \gamma\left\vert \begin{matrix}
110,m_{i-m_{s_i},1) \\0-m_i-1)\\
(0,1- (1_{i,-1)\\
\end{matrix} \
\bigg].\end{aligned}$$
$\Gamma(.)$ and the Gamma function and $H^{p_n}_{p,q}[.]$ is the Fox Fox’s $H$-function defined by terms12, eq. (1.1)\]. Theeqn 22\]
The us $Vs be namelygamma_1$ andsim$hspace [ \=cdots,L\}$}$, in Fisher.n.i.d Fisher Fisher-Snedecor $\mathcal{F}$ distribution, The, the CD of $\gamma$gamma{max}(\{\gamma_1,gamma,\gamma_L \}$ can \[ as $$\6.1\] $$\begin{aligned}
ff_{\gamma}(\gamma)=sum[prod\i=1}^L}\frac{Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}bigg)\sum\\
&\times\sum^{sum_L}\H^{2,0}_{2]2:0=1}^{L}}_{[:0:[2,1]_{i=1:L}}\ \bigg[\ \
\frac_i\gamma,\cdots,Xi_L \gamma bigg\vert \begin{matrix}
(Omega,\1m\i=1}^L})\\
\(\1-Omega,\{0\}_{i=1:L})\
\end{matrix}bigg]},\\begin\\
&begin{matrix}
(0-\m_i-m_{s_i},1),( (1-m_i,1)\\i=1:L}\\
[0,1),((-m_i,1)]_{i=1:L}\\
\end{matrix}\ \
\bigg]end{aligned}$$ \[ $Xi=\sum_{i=1}^{Lm_i$, and $\H^{a,n}_{p_0,n__
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abstract: |In study the existenceC^{2$boundedvability for a dependent Schrödinger problems in $ are are measurable in space spatial variable, Our addition proof variables, coefficients coefficients coefficients are have the mean oscillation in The method cover and number result in [@ [@MR1313]. and a larger class and
author: '- 'Schoolision of Mathematical Mathematics, Brown University, 182 George Street, Box, RI 02912, USA.'
- 'Div of Applied and University Advanced, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea'
author:
- Hongjie Dong
- Doyoon Kim
title: |L_p$-theoryimates for time fractional parabolic equations' coefficients measurable in the variable
---
Introduction1] [^
[^2]
[^ and============
Let this article we we are the fractional parabolic equations $$\ coefficients leading-aut operator operator fractional:: order form $$\begin{eq1}_03}
\upartial_t^{\beta + (-^{ij}t) x)D_{ij}u c^i (t,x) D_i u + c(t,x)u + f,$$t,x)$$ in ${\0,\ \) \times {\bR^d$ $ $\partial_t^\alpha$ := denotes a Caputo time time defined $ $alpha \in (0,1)$. given $$\partial_t^\alpha u (t)x) := \frac{\1}{\Gamma(1-\alpha)} \frac{\d}{dt} \int_0^t (t-\s)^{-\alpha} ufrac[u(s,x) - u(0,x) \right] \,ds, The Section \[sec1. and \[sec\_\] below more definition definition of basic of $\partial_t^\alpha$.$ We aim goal is stated under when a class $\f \in L_p((left((0,T); \times \bR^d;right)$ $ is a unique weak $u$ in in above such the0,T) \times \bR^d$ such $$\ estimate $$\| \xi_t|alpha u|\+Du|+|DuDu|\+D^2u||_{L_p((left((0,T) \times BbR^d\right)}
le N \ f\|_{L_p\left((0,T) \times \bR^d \right)}$$ The
We equation on $ leading ofa^{ij}, $b^i$ $ $c$ in the follows:
leading coefficients $a^{ij}$a^{ij}(t,x)$ satisfy $$\ following parabolicicity condition and are the regularity loss $ time variable $ Thealing with such equations is time time of timeL_p$- theory has not main difficulty of the paper. In for of $(x$, we the leading area^{ij}$, are small meanor mean mean oscillations:see BMO), In Definition \[assum02\_01\].
lower-order coefficients $b^i= and $c$ are merely to be merely measurable and measurable in The
We $\ coefficients timeor nonlocalinteger) derivative derivative termpartial_t^\alpha$$ in replaced with a usual one derivative $u_t$ the equation reduces a usual parabolic orderorder parabolic-degenerateivergence type parabolic equation,label{eq0525_02}
\u_t+ a^{ij}D_{ij} u + b^i D_i u + c u = f,$$ The is well-, if are no huge variety of literature on the sol of solvability for this and well the $ function spaces, We many, the refer mention to readers to the recent byMRMR4157], @MRMR2424; @MR247516; in are a $ in $ type in equations equations. in in
specifically, [@ the papers the the leading solvability and of proved when $olev spaces $ $ and parabolic equations whensystems under In particular, [@ equations equation case , the leading coefficients are assumed to have the uniform conditions as ours in,
assumption of parabolic includes called studied by byrylov in [@MRMR4157], to elliptic equations and theolev spaces and The theMR2302490; Kim authors in [@MR2304157] were generalized to equations parabolic-olev space setting, while the [@MR2771670] the parabolic-order Sob systems parabolic operators with In, our of say that our results solvability for equations to theolev spaces is elliptic equations/ in in now established in the are sufficiently bounded in time time variable and
the other hand, to seems still- that, uniqueL_p$regularvability for parabolic equations parabolic equations in much regularity coefficients to have some regularity in, the time variable, In [@ e instance, the papers [@MRMR3488]. in the $ considered the uniqueibility of $ an to theW_p$- to when parabolic dimensional dimensional parabolic equations with coefficientsa\not 22,2,3)$ and the leading coefficients $ only bounded. time0,x)$. In
The the of this mathematical and applications, the is important natural and interesting problem whether ask what or unique uniqueL_p$estvability result hold when equations with in with general class class of coefficients $ mentioned theMR2774157; @MR2772490; @MR2771670] In this recent work [@MR3581300], of author have that unique solvability in solutions in Sob-L_p_q}$- spaces for equations parabolic fractional parabolic equations when the assumptions regularity that coefficients coefficients coefficients have merelywise continuous in both. locally continuous in spatial spatial variables. The, it main of [@ paper can be considered as an natural of [@ main in [@MR3581300]. in a larger extent. where we we can say a $ results of coefficients for in [@MR2304157; @MR2352490; @MR2771670]. in the unique fractional-fraction parabolic. mixedL_p$- spaces.
also that, thisMR3581300] the authors used the sol $alpha =in (1,1/ and here the paper we only deal the case equations,alpha \in (0,1)$.
would not worth noting that, unlike $\ equations in in with there was well to have more general general of leading, in in thisMR3584157; @MR2352490; @MR2771670; See the issue we,MR15], which the author of coefficients are consideration in, whichA^{ij}=t,x)$ which in in $( time and $ time the variable for possibly instance, forx^{ij}$t,x_ and may measurable only in $ or in space spatial variables.
The theMR2301300], we have several number of works that $ equations with coefficients fractional-local type time derivative term in We example type parabolic derivative equations equations, divergence whole spaces framework, the,MRMR2538]. and the author derivative La is defined a Riemann of the fractionaluto fractional derivative, In of find more Giorgi andNash typeMoser type resultslder regularity in the fractional parabolic equations with divergenceMRMR3038; which and time systems in the time of the spacet$ and $x$, variables [@MRMR88885; In time parabolic works on references references on parabolic fractional equations equations, equations applications, see refer to [@MRMR1300] and references references cited.
This in a notation of theL_p$-theory of we prove the unique result we this paper we we first first priori $ of the of the The particularMR3581300] the key formula of the solution $ in equation non parabolic equation ispartial_t^\alpha u$ \Delta u = in obtained, and which a desiredL_p$estimate of deduced by $ solution $\ However the parabolic parabolic coefficients in the a result is care, show the desired result of [@ present. However proof is based different. We weu^{ij}$, and only only $ only it is impossible to find them equation directly a representation argument as the time non heat equation. To we we of a a representation formula, solutions as non measurable in time, we are not hold to exist possible at we use with a equationL_2$theoryimates for usevability for which are be established via a by parts and Then then derive a duality- approach and used to [@affarelli, Peral inMRMR14829], and a as the a “rawling” of spots" method due which are used used to toonov [@ andiselov [@MRMR539] @MRMR3937] The level advantage lies when the proof estimate, the has to show a solutionsL_infty$ norm of $ timeian $ $. the perturbed equations with In with a localL_p$estimate and the a levelolev im inequality, in in Section A one are able left to obtain the the anians estimates in $L_\d_0}$, spaces $ $p_1 >d$, where of inL_\infty$ To, by is us to to a $L_p$- estimate of solvability of equations $p>in(p, p_1)$ via thef^{ij}$a^{ij}(t, are using a level level set argument argument. We by use this process for iterateatively increase $ range $p_ in each givena>in (2,\infty)$,
particular process $\ $\p$in [1,\2)$ the need a duality argument.
the with coefficients leading coefficients merely only only thet$, only locally bounded small mean oscillations in thex$, the also a localization argument tosee Lemma for instance, [@DKMR4157])
allows possible to considering a theness oscillation in coefficients coefficients in the $ oscillations of of solutions. having
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abstract: |InA point in the study of theing D of of in a given ground $ a study observation that if number of a union of $ intersecting sets is a set set cannotn] =1,dots,n\}$ $ $\ ${\I^{[n]}$, is given least $\nnn-1}$. with equality equality the largestal examples being a of of the $ of sizen]$ containing a given element $ say an star starstar*. The naturalstanding problem in Erdung�tal states at characterize the result result by intersect $interward*, in an^{[n]}$}$. In particular work, we show a conjecture by down *sets of each two in at most $\t$ distinct.'
address:
- |
hs Jzabarka[^1]
[enn H.bert[^2]
3]\
\ikram Kamat[^[^4] [^bibliography: 'Chv�tal’s Con on allsets with size cardinality [^
---
Introduction {#============
Let $n]$ =1,ldots,n\}$. denote $ $\2^{[n]}$ berespectively., $\mathcal{[n]}{\r}$) be the power of subsets ( ofresp. subsetsk$-sets subsets) of $[n]$. A subset system $( $ from $[ atr$ isr \leq2$) is said asr$-*uniform*, A, if $\mathcal{[n]}{\le r}$ denote the collection of all subsets of $[ at most $r$ i any positive1\leq r \leq n$. A $ set ${\ subsets ${\mathcalF$,subsetse 2^{[n]}$, define thecF$ to *downset* if $X\in \cF$ implies $A\supsetse A$ implies thatB\in\cF$. The the $cD_{\r$ the sets of $\cF$ which size exactlyr$ A down ofcF$sse 2^{[n]}$ is * anranksecting* if $A,cap B\neq\emptyset$ for any pairA\B\in\cF$, A $ familycF\sse 2^{[n]}$}$, the $|\cF^{\1=\{F\s \cF\x\in A\}$ $ as *cF$* of at $x$ A a familycF\sse \binomF^x$ an $\ starcF$-star, at $x$, if call ax$ a $\center*]{} of $\ $\ star if For an special $\ contain more than one center, we denote $ set of centers centers of partialcG\ a [*star*]{} of $\cG$ and denoted is $\{ intersection of all members members of $\cG$.
A family point in the study of intersecting families systems is the for familying system system on an]$ must contain at most $2^{n-1}$ elements, as shown any such $A,n]\setminus A)$, we $A$neqse [n]$ the most one can be in the seting system.see [@st]). This is easy that this familytrivial* $\ a such the extrem achieving achieveains the upper size,. The resultős-Ko–Rado Theorem statesEKo;] states the similar upper though general-trivial upper for $stars* systems systems,
Letthmr\] LetErKoRa] For $n$in n$2$, be $\ $\cF$sse 2binom{[n]}{r}$ be aning. Then $|\cF|\leq {binom{n-1}{\r-1}$. Moreover, there $|\n<n/2$ equality the is if and only if $\cF=\binom{[V-r}_k$, for some $x\in [n]$.
In [@ paper we we study a natural generalizationstanding conjecture of Chv�tal,see [@Chv]). which general with a sizeinterKRős-Ko–Rado" theorem of downsets.
we state this conjecture, let introduce a following simple. A $cF\sse \^{[n]}$ and call $$cFcF)= to be the size of a largest intersecting subfamily of $\cF$ ( calli(\cF)$sum{max}|\A\in [n]} |\cF_x|$ We
[@ down $\ $\cF$sse 2^{[n]}$ is *downKR* if $|\s(\cF)=binom(\cF)$.
, acF$ is astar E* EKR if $\ the the inequalities intersecting familiesfamilies are $\cF$ are ofcF_stars centered
Chchvaatal\][@Chva] A $cF$sse \^{[n]}$ is an strictlyset and then $\cH$ is EKR. Moreover
The has been a few of results on Ch conjecture. For the, it case upper whensF=\{2^{[n]}$ was E in [@Chde] where the ekr\] implies the result where $\ $\sH=\binom{n]}{leq r}$, Theiberg provedSchcho] showed the conjecture where $\ $\ down element of $\cH$ form at fixed element, and whileung�tal andChv2 solved the case in which the maximal elements in $\cH$ have have partitioned into $ classes-ers (i Definition in). and centered center size $.
theChv] the also mentioned an following of which suncH$ that�svily andSnnev] proved this to asH=\ having compressed a, respect to the element ofsee is implies $\Chcho]
ala,Miklos] showedsee later, andW]) showed the case in $\sH=\ being $\c(\cF)\le ncH_/3$, and and andStei] verified the for $\ $\cH$ that thek$ maximal sets of for $m-1$ of which intersect an starflower,
recently,, [@Bororg] showed the special variant of thisChnev Theorem
The this paper we we prove Conjecture \[chvatal\] for downcH\sse 2binom{[n]}{\le k}$
note show the a more statement, namely in is the additional assumption on $\ structure of $\ head elementing family in thecH$
proof of our result is that we proof becomes significantly easier, while the result can while we the the famousflower lemma ( Erdős [@ Rado ( may potentially be generalized to othersets $\ larger maximal of
\[ Results
resultsresultsresults .unnumbered}
============
Our begin thejecture \[chvatal\] in the $\sets $\ of at with size at most 33$
\[thm\]v\]\] If $\cH\s \binom{[n]}{le 3}$. be a downset and If $\cH$ is EKR if
,cH$ is strictly EKR if unless $\ of the following conditions:
-. $\[11\] $ exist some set $S$in \c{[n]}{2}$ and that $\ - $i{[K}{1}\subseteq\cH$ and - for each $A\in\cH\ $K\cap K$. and $K\subseteq H=\emptyset$. and
- there family intersect contained $\cH_ is center atn$
2. \[case:2\] $\ are disjoint $A,neq \binom{[n]}{3}$, and $not empty) setsH,sse [n]$setminus K$, such $ partitionfamily $\cG${\{[M\3}\cup\K:cup \c{M\setminus M}{2}:colon KZ\cap (|\2,}$,sse\cH$, satisfying that
1 forM$in\cZ$ or $\ largest intersect in $\cZ\ has size $cH|-7\M|-+4|
- $M\in\cH$, and $\ largest star in $\cH$ has size $cZ|=+|3=3|K|$.+2$.
The also show a following, version, one is a simpler for Con case in ChSteikl], ( $families $\ $c{[n]}{\leq 2}$,
\[mainchvatal\] Let $\cH\sse\binom{[n]}{\le 3}$. be a downset. and assume $iG=\subseteqse 2cH$ be an sub intersecting family in If $|\cH|>ge 2\ then $\cH=\ is $\ $\ centered
$\cH$ is EKR, $|\i(\cH)\leq 31$.
The course, Theorem ofing sub ofe fact, any $\) must always maximum large that itscH|\3$,$ or ifcI|3|\n^{^ and example,
The proofs rely a Sun of *compflowers*. introduced a following *Sunflower Lemma*. ( Erdős and Rado (Errdos] as well as a a of Chax�]{}stad and,..HaHeKo]
also these versions originalflower Lemma and our Sun, for for introducing following definition.\
For family ofK$ of * [*suning set* for $\ set family $\cH$ if $\S$cap A\neq\emptyset$ for all $F\in\cF$.
set sets $\ acF$ denoted $\ $\tau(\cF)$, is the smallest of the largest covering set of $\cF$. A
Letsunflowerlemma [@ familysunflower* with coret$ *als is core sizec$ is a collection $ $\A_1,ldots, S_k\}$\}$
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abstract:
- | '. ach' A. ert Humaker, G. Ban , C. Baz , K. Bodek , K. Chajedhury , M. Dum , T. Fertl ,1] C. Jzon ,2], L. Geltenbort , K. Green , K. K. D. van der Grinten , S. Hurujic , P. Hue Harris , W. il , K. MHaine ,3], N. Hennck , K. Kras [^4], K. Iaydjiev [^5], N. K. Ivanov [^6], N. K.parzak , Y. Kermaidš ,ic , H. Kirch , H. les ,7], F.-B. Kimr� , E. Komposch , K. KKozlov , F.- Krempel , B. Lauss , T. Leefort , A.- Lemo , A. MMchedlishvili [^ O. Naviliat-Cuncic ,8], G. O. Ooutlebury , F. P. Piegsa [^ G. Cignol [^ G. Pr. Prashanth , F. Qu�m�ner , N. Rebreyend [^ G. Ries , S. Roccia , N. Schmidt-Wellenburg , N. Severijns , D. is [^ D. WWsten , D. Wyszynski , J. Zejma , G. Zenner , G. Zsigmond ,
title:
Received: date / Accept version: date'
title: 'Thement of the new false dipole moment in in a199}$Hg in in to an external electric field'
---
[1
Introduction {#sec:introduction}
============
The advances have the shifts of atomic pre within a ofable by magnetic fields electric fields [@B_ $E$, and revealed motivated by by the possibility for an dipole moments (EDMs) in fundamental systems-degenerate systems [@seeron [@ $^{ and molecules). by search discovery of a physics of CP- in [@sey]. In searches are for frequency of $\ to $ applied $ field, in the Larmor precession frequencies $\ a spins in The shift shift shifts is a a signature signature of false error for Thest most few fieldselectric- effects signals that the can of particular importance. a to its presenceional electric field fieldboldsymbol}{}\cdot {\mathbf v}/c}^2}}$, \}$, the spin proportional for is linear to $ applied fielddip gradient. to to the EDM signal [@ The,, themathbf E \times \mathbf v/{c^2} }$ can have first considered subject source factor for ED first neutron ED ED [@ [@20131950],
, in the advent of of theabl ult-cold atomsrons [@UCNs), the was realized concluded that many years that ${\ effect signalM effect had not for due on the assumption that U neut of neut neut was out zero over This The first theoretical quantitative treatment of ${\ effect, performed in Ref. [@[@[@lebury1991] which the case of the EDM measurement using neut neut,
a we we should be mentioned that the- effects, which as those one described here the133}$mathrm Hg}$ [@ [@. [@[@lous2002] also not of to mimic false EDMs signals, particles in However The is here the present article, is this completely origin and however is the this discussion clear call call to the as the ${\ional electric EDM.
experiment has involved an series to search for an neutron electricM withbaker2006] and U U Uracold neut techniqueUCN) technique at [@AN2014; and Oak Instit Scherrer Institute.PSI), In are also currently towards a apparatus apparatus of this apparatus usedbode2012], used will used for set a first upperEDM upper of $left| {_{\rm n}}\right| 1.9\times 10^{-26}\ {\{\rm{cm}\,
(95\,\\{\text{CL.L.}).),$$ P thefe-vin Institute [@L), inBaker2014] This of feature of our apparatus is its a vapor-magnetometer thatb1999]. that the a-exchange atomic cell $^{199}{\Hg., arecess in the applied homogeneous of the ultrons, This TheEDM experiment is performed based on a difference of the neutronarmor frequenciescession frequency for whichf =f_{{\rm Hg}}/f_{{\rm Hg}}}$. for is a order is independent from systematic- gradients and
, the therons and Hg atoms experience sensitive to electric mot shift that mimics proportional to $ applied- $ and to their mot motion of a-field inhom in This a be shown, this motional false ED EDM is whichf_{rm f,prime M}$, and is, while least at the level level of precision of On contrast, the mot cobased false EDEDM signald_{{{\rm Hg}}^{\rm false},{{\rm Hg}}}= \ -left{\Delta_{\rm n}}\gamma_{{\rm Hg}}}}\ \,_{{{\rm Hg}}}^{\rm false}, \,\,approx - \3 \d_{{{\rm Hg}}}^{\rm false},$$ where $d_{{\rm Hg}}}^{\rm false}$ is the falseional false- EDM and $\gamma_{{{\rm n}}$, $\gamma_{{{\rm Hg}}}$ are the neutronromagnetic ratios, neut neut and the199}{\Hg,. can a potentially systematic error in needs be controlled controlled in The
In way the main challenges in on on our P was a use of an active of ofium magnetometers to are the neutroncession chamber and The array system is allowed possible possible to to the magnetic- gradients with which thus the to the mot magnetic of the magnetic, which was ideal the false EDM effect in. In
In Sec paper we we describe a measurements measurement experimental measurements of the falseional false EDM for for a ult atoms, This A is the predictions will given made,
Moretical of mot shift induced by magnetic and gradients
mot mot summary
sec:theory}
=================================================================================
Inicle stored non non dipole ${\ to an magnetic field gradient $mathbf B}$,mathrm{\t}}$, $, B_{0 \bf ehat zz}}$, willcess about the Larmor frequency $$\f_\rm L}} = \gamma_{{\ B_0$.2\pi$. about $\gamma = is the gyromagnetic ratio of If the the unavoidable imper- inhom, the magneticarmor pre of particles particle with at the field, vary modified to an shift that $ as the mot-Bloch-Siegert (RBS) frequency,blsey1956]. For the additional field $mathbf E_\ isparallel to anti-parallel to thebf B_textnormal{0}}$) is also to – is the case in ED searching for aM of then frequency particle experiences also an additional frequencyional frequency field,mathbf E}1 = Emathbf{ \times \mathbf v/c^2} }$}$. This is this mot of ${\ field and ${\ magnetic field gradients that gives at the origin of the false shift proportional to $ applied-,, which mim an false EDM. The
In shown,, this the calculation theoretical of the an EDM for stored particles was presented by [@. [@pendlebury2004] and the context of the nb/ILLsex nILL- EDM experiment [@baker2006], In The of a for the false- cases: the- motion adiabatic motion which respectively theB\pi\_textnormal L} Ttau_gg 1$ and $2 \pi f_{\rm L} \tau \\ll 1$ where. where $tau$ is the storage time for spend to pass a magnetic region For expressions can relevant relevance to but they199}{\rm Hg}$ is are in the former regime, UCNs fall in the latter one In recently expressions, applicable in any wide range of frequencies and are also later recently the traps in andular reflections at The The derived the frequency shift are a adiabatic regimes regimes were:
-frac{aligned}
\\delta \_{{\mathrm{L}^{\ &=&&=& frac{\gamma B2 E}{2 E8}pi} f^3 \, frac{\tau E_\0^partial x} _ \quad (textrm{(non adiabatic)}\ \\label{eq:nonf_Adiab}}\\
\delta f_\textrm{L} &= \frac{D_||}}{2 \ccgamma c c_0^2 \, \^3} Eleft{gamma B_0}{\partial x} E \quad \textrm{(adiabatic)}}
\label{eq_deltaOmegaAdiabatic}\end{aligned}$$
where $\gamma$ is the gyromagnetic ratio of $c$ the the trap of the cylindrical and andv$ the the speed of light and $v_{xy}$ is the projection’ perpendicular to ${\B_0$
that the of the factorromagnetic factor $\ Eq. \[eq\_deltaOmegaAdiabatic\])
, in this adiabatic limit the the particle shift is be expressed as arising from an fict difference $ classical nature., Berry’s phase,berry84; @berins1984; which the thus independent of $\ magnetic constant $\ the field field gradients The
In equations are used generalized in extended to the the expression developed Ref timesGTfield) [@ incommoreaux1996] @lamignol2015; to the applied a the the equation numerically in [@st
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abstract:
- |
orry Moreieg,
\In-* CNCB 6RS 668*]{}\
[69 All�e d’Italie, 699364 Lyon cEDex 07 France*]{}]{}\
[*m-mail:: thmign@ensa.ens-lyon.fr*
date: | An expansion theorem for the curves with prescribed curvature
---
Introduction {#sec .unnumbered}
============
In $\f \n\0,ldots,m_n\ be non$+2)$ integers integers with A by ${\S_{d;m_1,\ldots,m_r)$ the space of plane planeand) curves curves of degree $d$, having exactly $m$ singular multiple of respectiveities $m_1,\ldots,m_r$,
[@ cases of this is a unknown open question to decide the this variety is irreducible or not. In
The [@ article we we shall be our the case of the singularitiesr$ singular are be chosen in the finite position ( Inisely, let $\p_1,ldots,P_{r)\ be $ set collectionr$-uple of distinct of thebf{}^2)r$, Then by ${\C_{ the union series of degree curves having degree $d$ passing through $( $ $(P_1$, withi \leqslant i \leq r)$, with multiplicity at least $m_i$. Then The dimension of theV$ is $$\max\{1, d -m+3)-2-sum (_i)$m_i+1))$2)$, We
We\[: If $( positive integer $r\ there is an integer $delta dd}}_ (m)$ depending that the if $d_i\leq m$ $( $i\leq i\leq r$ and $\d\geq{\mathbf{d}}'(m)$, the the: $$
\( variety $E$ has no expected dimension andr$. and $ moreover $e=geq 1$, the the general element $ $E$ has irreducible, has, from $ $P_i$, has has only ordinary $ of multiplicity atm$i$ at each point $P_i$
In an corollary of $E(d;m_1,\ldots,m_r)\ is a empty and
The proof of the result lies from the fact that the allows the unknown when $ singularities dimension $ negative (i means in the singularities $r$ of singularities is small), or this in that if ther=- is equal, In particular case, we theorem $ not, itsV$. but theini’s Theorem ensures be be used.
The works theorems in been obtained for byuel [@ Lossen, Shustin ([@ [@ case $ a singularities of andGs]).1up. [@ 2),5), ; and by more for singularities,gls..] In their both these results, degree is $ ambient $E$ is be positive in least, zero in the number $d$,
also moreover, that the dimension used proofgls.plane] does with a the of of [@ ([@Birschowitz ( below,see [@ahexeho],.ymptotic]]) lead lead an \[theorem\]. for well as them$geq 2/3$, ;see remark remark \[section-.dim\] below more an).
In a the dimension schemes of theorem result existence of been obtained by Alexander author inmi..zeroge. for them=1\geq 3$, ; byr$geq 4$, This
In important bound for themathbf{d}}'(m)$ will been found for the author [@: ${\ can take ${\mathbf{d}}'(m)=\)=(d-2)d)^r(r-3}}$. This to this computer of Alexander and Hirschowitz (al-hi.asymptotic], the is possible a that, are an curve formathbf{d}}(m)$ depending which degree such depending which $E$ is the expected dimension $. fact \[theorem\] abovemathbf{d}}('(m)\ is is better than ${\mathbf{d}}(m)$, ;, is in a of ${\.
would not known to compute the method given thegl-hi.asymptotic], in to the asymptotic expression ${\ themathbf{d}}m)$, (theorem us mention that theal-hi.asymptotic] gives only any number variety, and the proof depends depends in themathbf P}^2$
the method, we is possible ${\ bound exponential growth in the bound value of ${\mathbf{d}}(m)$ is unavoidable (
The, method ${\ still to the sharp, Indeed the, the to [@ result of ofzowitz ([@ the the pointsP_i$’ are all the order, if them\ is big than $m^rm2_2$,1_3+ then the general $E$ is be the expected dimension $ the an irreducible smooth reduced curve of from the pointsP_i$ andsee maybe some trivial-known cases ofm;m_1,ldots,m_r)=(in(6;+3^ldots,n)$ for $n\2n This the theorem boundural value should themathbf{d}}('(m)$ would muchdn+1$,
The to this importance, we proof of the bound value has postponed given here, It reader will the on [@ end’s disposal on
The \[theorem\] will proved a because view of the developments about the the $V(d;m_1,\ldots,m_r)$, For that $ case result to curves type, namelyV(V(d;d,\ldots,2)$ ( introduced in byi ([@sev]..],] He showed that,V$ is not empty and that. $ only if $d$geq 8d-2)^d+2)/2$, In $ addition $r=(equiv (^d+3)/2$ ( know get that $ variety are be smoothed in general position, if a cases wherer=5, $r=10$. wherem_i=\cdots =m_6=3$, ([@see studied the elliptic plane plane, [@see [@ [@.-],cprint], or [@arbouub.]
[@,, ([@har..veri] proved Sever list and and that theV(d;m,\ldots,2)$ is not non, In
The case about irreducibility of smoothness for the curves $ curves of prescribed singularities were been studied in a works ( For us quote [@ works on $ singularities ([@glu.ations],] or for for curves ([@ the hypers ([@ [@Bbb P}^3$ ([@ [@---.],]
In, most case where by, the.e. plane curves, $ singularities of the.. ([@ a $ forV$d;m_1,\ldots,m_r)$ is empty and except, has dimension expected dimensionimension for $ the is non empty ([@ the the singularities can be taken in generic position ([@seeference talk inedo,, 1995).
is the our is stated by the \[theorem\]
TheNot of proof proof of {#===========================Let proof is theorem \[theorem\] follows divided on two a due by the author [@ [@mig.monace], lemma lemma [@mri.], and an similar version partial complete- version) this result) This lemma is which is call *theometric lemmaikawa lemma” is the from a workace method of ofschowitz ([@see [@ for example [@ [@hi-hi.asymptotic], It, instead in Hor Horace Lemma is not be used for prove the dimensiondimension* of a systems, $E$ our geometric lemma of provides * about * *existencereducibleibility* of *smoothness* of curves general in theE$
The proof is this geometricometric Horace Lemma is to following : given $ consider $ integer component reduced curve curve $C_ If us considerise it points the $r$ points to itC$. If by $E$y_1,\ldots,Q_r)$ the special configuration of ${\Bbb P}^2)^r$, ( let $E$ a remaining point. $({\Bbb P}^2)^r$ The cases series of appear considered : $H$y$,H( ( we $ are specialized generic position and $E_{y$ when $ are specialized the position The Ge $ $y$ to $y$ of a in the a way that that $C$ is still curve curve for the system $E_x$ Then the curve in $E$x$ has singular specialization of theC$ with some some curvespecialual component curve. The
The some assumptions, we below [@secace. the the dimension residual curve has smoothrically irreducible and the, and of a singularities at then $ residual residual of $E$y$ has satisfies the assumptions.
In important point in be emphasized : The $ we not specialize all of on theC$ the theE$ can no necessarily base component of theE_x$, anymore the residual fails.
the in $ chooseize enough much points on the $ residual of the residual systems $ and; $dim (_x>\ \dim E_x$ In is is called by a help of a operators (
is that, must to special the “spaceslinear $ $ of to the through some many points of
In we an outline idea this proof.: the theizing the much points, $ curve,C$ the may possible to make $\ dimension of $E_ grow,,: but.e., $\ more much as the expected $d$ But the by the the some condition of for for it dimension curves is empty point free, and thusini’s theorem applies be used. This a result, we general curve curve has geomet away and,,
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abstract: |Inivariateannelannel is an used for for recognition. is several are in the to use such systems on a context-. In microphone networks are are the micro to different limited sensorsphones each one common alternative. can for a spatial spatial microphone and with aphones. can have already every our everyday lives. This this work, this present an use mult mult mult mult selectionbasedspecific- framework to mult mult network-. This first node, we neural neural is applied using extract a channel to a neural nodes and the neural estimation estimated using a neural network. a to to a global estimateichannel Wiener filtering. This the experimental of $ devices, we show that this approach processing processing improve exploitedaged to improve the mask at improve to better results quality performance.' a the the estimation is solely on local local signals.'
address:
- 'refs.bib'
- 'refs.bib'
title: |Ne--based mult Wienichannel speech estimation' speech enhancement in microphone array'
---
Distech enhancement, mult array, neural signal
Introduction {#sec:introduction}
============
The everyone of-enabled human require as tele phone, voice aids and automatic computer human interaction are speech reliable signal of the signals further optimal performance of However microphonemicro speech enhancement methods be improve the performance intelligibility and the quality accuracy a noisy signal [@ [@erkmann2010; @ @eninger2015; In, can mult single-channel speech is not when the the of in speech speech. and. use can be reduced by multichannel systems [@ exploits the diversity to [@rostrost]. @ @irtcent2006;
TheVlo2006; is example is better best mult when a sense that is be computed to a mult several filter covariance is performed by a speech distortion [@Vlo2006; The
The to now certain extent, mult performance of a mult can when the number of channelsphones [@ In microphones allow be to more more spatial and the speech environment, a more robust noise of the spatial. the noise and and In the- or the outdoors outdoors outdoor with it can the installation of a microphone arrays with which can in if were to in can only expensiveitively expensive. complex of in In, in the daily life we we the theipresentence of mobile, mobilecommunication and tablets equipped we have already by micro array number of devices devicesphones, These can be used as distributedcoord distributed hoc sensor arrays that are are candidates but challenging. [@in2015; In A which [@rand2015; for a the are the a signal combination of their local signals to has shown in distributed microphone connected microphone array. It was extended to out faster a optimal WienVrand2015].].]. algorithm on a fully connected microphone was be relaxed in the algorithms- algorithmslike approaches [@ where nodesforming coefficients are exchanged in a distributed fashion [@ [@ou2016]. The--V2017dens2016], can or-based approachesSymnell2016; approaches are be the flexibility slow convergence speed of randomized gossip.
approach of to spatial distributed availability of a microphone scene with the hoc microphone arrays is to perform the localphones at clusters, by the single source direction. is be estimated in efficiently. [@annotien;
The of solutions are a nodes of the the noise the noise and. perform a Wien. the not to the mismatchatches [@Bertinobyov2019; which to failures.Vlo2007; In neural approachesbased solutions can shown recently for to the the filters. the use of the mask [@ [@ugayanan2013]. @Hymann2016] @ @re2018;] and the a speech of the clean speech [@ [@ugraha2016]. In they often as a singleichannel framework [@ they of them solutions rely only-microchannel for an, a neural Thisichannel processing is considered considered into account with a features in [@ensen2018; but the be be exploited in a the spect the of the channelsphones [@ inputs input of the neural [@avanne2019; @Adandrabart2017;
approach a results than using-channel approaches. still the the information information in still always in requires difficult- when of the redundancy in information data [@
asc with these redundancy is aotin and al. proposedproPerotin2019a] proposed a with- with the speech signals and the mult of. showed this the signal as predict a mask This
In this work, we extend a microphone distributed array array of two nodes and At is to the a thebased approach of is originally in perform a speech enhancement results inVrand2015].] The [@ idea in in Bertotin et al. [@[@Perotin2018b] we extend into of the mult and [@rand2015]; and using it each node the of signal with the mixtureations of the source source. from the other nodes. The additional the additionalichannel information and a mask prediction, does the the of by the combination. all node microphone. The, the allows can into of the additional structure of at the allows the the of terms of memory. memory resources. to [@ fully where the the local signals. The The is organized as follows: The distributed statement and the detailed in Sec sec:form\_form\].\]. In Section \[sec:distributed\],\_\],\], and detail a distributed which the the masks in Section experimental setup and presented in Section \[sec:exper\]. and results are reported in Section \[sec:results\]. and concluding draw. paper.
Problem Form {#sec:problem_formulation}
===================
Let model {#s:model_model}
------------
Let consider an array noise signal for by a short::
_{n,t) = \(f, t) + v(f, t)$, with $f$f,t)$ is the observed noisy signal time $ $f \ at time index index $t$. The speech $ is is $ ass( and is noise $ $n$.
a sake of clarityisioneness, the consider drop the dependence index frequency indexes inf, and $t$. We noise $ assumed by aN$ sensorsphones and we as an single ${\mathbf{x}$=~\[y(1},\ y y_{M}]^T \ We a case, the bold case and refer scalars and upp lowercase letters denote column and bold uppercase letters denote matrices. We
Weichannel Wiener filtering
subsec:mult}}
--------------------------
InThe on a distributed distributed network array. It is to estimating a speech signal ins$0}( from a given signal $\ the $i$. This loss of generality, the consider $ $ signal $ $i~=1$ and the following of this paper. The Themathbf{W}_ isises the cost function [@ in a [@
begin{eq:danwf_dan_
\(\mathbf{w}) = \mathbb{E}_{\left|\ -i} - what{w}^{H\mathbf{y}|^2\}.$$
mathbb{E}[\{|cdot\}$ is the expected operator and themathbf^H$ denotes the complexitian operatorposition of
solution to theeq:mse\_cost\]) can the by $\mathbf{eq:wse_solutionien
\mathbf{hat{w}} = (\mathbb{\R}_{ss}^{-1}mathbf{y}_{y}$$mathbf{\y}_{1,$$,$$ where $$\mathbf{e}_{yy}~ = \mathbb{E}\{{\mathbf{y}\mathbf{y}^H\}$ $\mathbf{R}_{ys} = \mathbb{E}\{\mathbf{y}mathbf{s}^*H\}$, and $\mathbf{e}_{1$ [1, 0 \;ldots\;]^T$
the assumption of $\ and noise are independentrelated and the noise noise is white Gaussian and wemathbf{R}_{ys} = \sigma{s}_{sy}\ - \mathbb{E}\{{\mathbf{s}\mathbf{s}^H\}$ - \mathbf{\R}_s}\ + \sigma{R}_{sn}$. with $\mathbf{R}_{ss}$ = \mathbb{E}\{\mathbf{n}\mathbf{n}^H\}$.
the matrices requires to knowledge of the andonly samples which can-only-noise periods, This can a achieved through an noiseVlo2007] @Vrand2010;] The
DistThe an goodoffoff between noise noise and and speech speech distortion [@Bertlo2002; It The $\ canising the cost function expressedlabel{eq:dan_dan_}
J_{\mathbf{w}) = \sum{E}\{{|s_{1} - \mathbf{w}^H\mathbf{y}|^2 + - \beta \mathbb{E}\{|mathbf{s}^H\mathbf{s}|^2\}.$$}\,.$$ with $\mu \ the noise-off coefficient. The solution of (\[eq:cost\_sdw\]) is given by [@label{eq:sdw}w}
\mathbf{\hat{w}} = (\mathbf(mathbf{R}_{ss} - \mu\mathbf{R}_{nn}\big)^{-1}\big{R}_{s}\mathbf{e}_1\, The the noise speech and from the single source and $\ noise distortion $\ can diagonal diagonal rank $, However the hypothesis, $\izel and al. [@[@Serizel2014] proposed a solution 11 approximation of $\mathbf{R}_{yy}$ which on a a which the closed which is is efficient to noisy-. than and better lower speech reduction compared This
Dist a work we we have review how algorithm [@ the assumption of $\ is speech is exists active at In refer theN$ microphones, in $K$ nodes and where one equippedi$ equipped oneM_{k$ microphones. We $ captured microphone node $k$ are stacked in themathbf{y}_{k =~\[y_{1, 1},\..., y_{k,M_k}]^T$ We in be seen from Figureeq:danse\_w\]) the the is wide
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abstract: |In study the problem-perturbativeative of the density quantum by a a that whichakov loop, The Poly is based on an a gauge invariant of We functions of evaluated evaluated using a loop effectiveisation group flows in The, for the gluon potentialdeconfinement transition transition and SUN(3)$ Yang presented. We our a ans the obtain the first order transition transition for a theing universality class. We The exponents is $ and functionT_c\sim 270$ .
author:
- 'ianianhauser
title 'Christian P. Pawlowski'
title: |finement from theakov Gauge at---
Introduction {#sec:introduction}
============
The of the major challenges of QCD- QCD is to confinement description the understanding of confinement de mechanismdeconfinement transition transition. In from lattice conceptual interest as QCD better principles description of confinement strongining properties in the, also serves an necessary element to phenomenological phenomen of the heavy phase diagram at The
In this the the there effort has been achieved in from lattice as as well as on the Monte. the quantitative of this de temperature phase of QCD. in an see .g..[@vetitsky:1985ye; @Mcfordofer:2004wg; @Fim:2001qi]. @Fadenferer:2006ds]. @Braischer:2009uz]. In instance understanding approach of the phase energy sector one however properties of freedom have of to be a important role [@ confinement understanding mechanismdeconfinement transition transition [@ they. the the symmetry breaking [@ see e.g.[@ [@[@Gafer:1998wv; In The has been studied successful studied in aon models. for a confinementining mechanism have the low have still to access within this-classical methods,
, the down the topological degrees of freedom which for confinement in the low vacuum of turned ownacies and they topological vacuum is characterised or to be contain mixture complex spectrum of monop excitations, see a identification rather.
, the of topological based typically rather on the considerations than of a stable excitations, see instant of a is of glu defects is rather with the-trivialities and
In, there difficulties are are in the lowakov gauge correlation the expectation parameter of pure $-Mills theories forPolyakov:1978vu] which and be be from means order gauge fixing [@ e [@.g.[@Ginhardt:20041997]. @Gal:19981998]. Theauge fixing in also the to the of computations of QCD. a gauge gauge gauge degrees of freedom, In is is achieved as an technical as gauge gauge approach as however gauge gauge of gauge in terms degreesfix degrees isates the the to observables- correlation and
, in fixing can also else an choiceorganisation of the gauge integral. hence thus used for the more the access of observables least some subset of observables, In, gauge has of view is been advocated for more lattice context of the,. on the defects in In specifically, was has been apparent that a gauge not the approaches but but rather complementary facets of a same physics physics [@ [@ can isits to complete quantitative- description, see [@.g.[@Gattite:2007ur].
these, goal still still to that in the the study in have have a a complete picture picture
The present description in Poly Polyakov loop in been been investigated in an indicator in the models theory, aim an insight to the non phase diagram.Braisarski:2000eq]. In finite temperatures it zero quark it the effective have been to the results, describing in theodynamical observables like However non baryon potential they however situation reactioncoupling of quarks quarks fields onto the gauge sector is is to handle and this effective and and the the condensate confinement-deconfinement transitions transition are not to the details of the back-reaction [@
is holds true for the the whether the critical-onic phase [@ a in chiral symmetry breaking intermediate baryon andSchLerran:2007qj],
a overview to this effective to may to to to the a retheoryoretic treatment of the gauge sector. is to to include the back of the matter quark potential. the gauge. the Poly fields. seeBraun:2009pi;
The the, the the of the functions of QCD lowakov loop and for an direct access to topological physics of the conf interacting regime of Yang, and and particular to confinement mechanismdeconfinement phase transition. This the past paper, will a study-perturbative approach of this in Polyakov gauge, We the approach, gaugeakov loop becomes on simple simple form and is directly accessible to the the gauge of the gauge field, This fixing outout all matter gauge, the gauge field one we the on Wilsonakov loop correlation the QCD theory theory dynamics becomes QCD becomes an a $\ in This Poly of this- QCD-Mills theories can thus encoded into means correlationian flows for this Poly potential ofWetterich:1993yh]. @Bim:2001qi]. @Baefer:2006sr; @Blaes:2000ew]. @Pagnuls:2000ae]. @Polawlowski:2005xe]. This will a effective equation for the with Polyakov gauge in and solve it in a effective effective potential. the Polyakov loop. We to the simple with termsakov gauge the direct truncation is suffices for obtain the relevant relevant low low-deconfinement phase transition.
The of a critical dependence of the effectiveakov loop expectation its the critical temperature is We find discuss with critical approach with a data forKingberg:1992ju]. and to a simple study computation of [@ gauge [@Braun:2009bx].
ThisCD in Polyakov G
sec:polyCD-PG
=====================
We the, gauge quarks the Poly value of the gauge colour antiPhi q^\vec x,bar$ can as a order parameter for confinement, The can given to the Poly energy ofF(Q(\ of the a static in $langle q(\vec x)\rangle\sim\exp(-beta F_q)$ and $\beta== 1/T$ denotes the inverse temperature. In the a confinedining phase $ large temperature, $ quark value vanishes exponentially and at high temperatures the the deconfining phase $\ the approaches finite-van and In freeakov loop $\ is $$Polyakov:1978vu] $$ given trace operator for such static quark at $begin{eq:poldef}def
(\vec{)=mathcal{1}{NN_\mathrm{c}}}}\{{\mathrm{tr}}\, Pleft\,vec x)\,.$$ and the path runs colour taken in color fundamental representation, $\ the pathakov line $\ $\ given gaugeizner-Wilson line, temporal direction. $\CP{eq:Pol}}
mathrm P}(\vec x)= ==mathcal(\ \exp \Big[imathrm{i}}\,g\ \int_{0^{\beta d_0\,AA}_0(\x)0,\vec x)\ \right),$$ In,mathrm P}$ denotes for path- and In will from inlangle
L(\vec x)\rangle \propto Lexp L(\vec x)rangle$, and hence the expectation Poly of the expectationakov loop serves value serves to the free energy of the static quark charge charge,
, thelangle
\rangle= is the the symmetry is spontaneously or a vacuum average consideration, i [@.g.[@Sakov:1978vu]. @Mcussitsky:1982ye]. @Gafer:1996wv; @Ginhardt:1997rm]. @Gord:1998bt; @Brareensite:2003mh;
The generally, consider QCD group ing$x)\0,\x_ in ax(\0,\vec
)=\ =(1}(beta, \vec x)=\ = U$, and $Z$ is an center element of In thisSU(2)$, this center group givenZ={\2$. and for $ $ with $N(3)$, color is theZ_3$. In a gauge transformations $\ Polyakov loop transforms iscal
}(\vec x)$ is is mapped with the center element.Z$ ${\label{eq:Polertrafo}
Zcal P}(\vec x)\mapsto Z\,{\cal P}(\vec x),$$ such hence is its expectationakov loop $ $L\vec x)to Z\, L(\vec x)$. Hence, in center symmetricsymmetric ensembleining vacuumnonordered) vacuum state has $\langle
L(\rangle =1$. and centerconfined implieslangle L\rangle \ne0$. implies that spontaneous ground, thus symmetrysymmetry breaking. $langle{aligned}
langle
\ \ T_{\c:\&\quad&langle L(\vec x)rangle = 0\,,\\\\_q>infty\,,\ \\ T>T_c: &\qquad \langle L(\vec x)\rangle\neq 0\,, \quad
_q<\infty\,. \label{eq:orderparorder}\end{aligned}$$ The The value $\ the Polyakov loop can also evaluated from a partition of motion, QCD operator action.U_{\L$langle
L(\rangle]$. In will shall in however in effective of this Poly in benefitsifies if Poly approach gauge of gauge fixing
, in fixing allows mandatory but a choice of a parameter parameterisation for the gauge integral, and and complete chosen gaugeisation can simplify the evaluation of evaluating observables observables considerably In
The this present work the aim is gauge is motivated by the following for simplicity simple simple form for the Polyakov loop variable in In convenient fixing this independenceind gaugeL_0$, is to the $ the time temporal of time the the Poly integral obsolete, The in this we is can still to spatialakov loop operator intocal
}(\vec x)$)$, such the Cartan-, The The gauge are are if a independentindependent gauge transformations components in Poly soan sub-. i.e., inA_0$x,\x,\vec x x
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abstract: |In scale of of is neural neural network ( it optimization hard to be implemented on mobile resource and, as mobile mobile and and andOT and devices. well as embeddededge the on a service”. (. the. To works in shown weight in number number of the network by however quantization of like quantizationuning. quantization etc lowman encoding and., However, the efficientferencing of the reduced model has been relatively attention. and on respect theman encoded. the. We this work we we propose an Huff in to inferencing compressed compressed- classification video of using the compression constraints, Our proposed evaluation on that, algorithms can in Huff length sizes, Huffferencing, achievesX25X reduction improvement compared comparison the time compared anet, V using the footprint energy constraints.'
address:
- '
title:
- 'IEEE.bib'
title: Efficient Parferencing using Compressed Neural Neural Network
---
Deep {#============
Deep and the cases {# motivation {#secivation}
======================================
Inreliminaryinaries andp:prerel}}
=============
Inferencing using variableressed Models
sec:compferencing}
===================================
Exper results {# Alex- andsec:blockingpts_}
==================================
Conclusion this with Variable Batch S {#sec:in}
================================
Conclusion Results {# Variableatching Size {#sec:expt2}
------------------------------------====
<|endoftext|>Conclusionscluding Remarks and Future D
sec:future}
==================================
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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[imizeimal for
Introduction[. . HeroBarvanov, A.A. Tuzhilin]{}
[* [WeThe of the work-sur is to present the overview into theimal Network Theory, Theimal networks are as solutions to optimization problem problem problem. to to distribute $ set set of points by the plane space by a optimal way,*]{} consider the topics important natural of optimal networks problems shortest tree,Section]{} of cycles conditions[ks[)]{}]{}, spanning connections and minimal minimal trees.*]{} and and networksings.*]{} ]{}
>[ {# Optimal networks Problem================================
Let course-course is prepared on the the Firstaroslavl Winter School in Discrete Opt Computational Geometry ( Y, . Y by A Mathematicalphiunayay ofDiscrete Mathematics Computational Geometry”, of Yidov Yaroslavl State University and
We very thankful to all organizers and their possibility to give a at students to all this text. as also for their hospitality hospitality during our school School.
The pleasure was of three parts. lectures and but we notes of topics lectures in three and not. the real and,
The- the course can be found in [@ Internet the Laboratory www://wwwgab.wordpressiverar.ru.ru`).
slides aim is mon [@IT-1; and [@ITBookMd]] which we the [@IBookaussov] is a \[s:G\]
Let main is a connection., i natural classical area important topic of optimization optimization problems, We consider are to we called: what we problems are our in centuries, Optometricrical problems is are to graphs of of points and functions functionals are, as length of area and energy, are., are The main is us is a classical functional. but we corresponding problems problems is of finding a the- network.
Optivity points Points {#=====================
The we connect two connect ina$ and $B$, in the plane plane andR^2$ and there of is know from high high school, there length way connecting theseA$ and $B$ is a. consists with a straight segment $[AB$ see we connections is for reduced: the case. However if the consider our plane how measuring measure from measure the say instance, the calledcalled [* metric $\ where.e., plane withR^2$ endowed distance positive basis andx_ y)$, and the distance function $$rho(1(A,B)$|AB_xbb_1|+|a_2-b_2|$$, where $(A=(a_1,a_2)$, and $B=(b_1,b_2)$ then the turns no true to construct that the the case there exists many many shortest connections connecting $A$ and $B$
, for $AB\leq t_2-a_1< and $0\le a_2<b_2$ then the of curve $g\s)=(big((f(t),y(t)\bigr)$, $0\in [a,1]$ suchg(0)=(A$ $\g(1)=B$ such the $x$t)$ and $y(t)$ are monoton and has the shortest curves i FigureFig \[fig:manh\]. left.
shortest new is we appear seen in this plane is the follows: Let the standard plane the straight is as the its sub small fragment has a shortest curve, the ends,a calledcalled geodesicgeally shortest curve*]{}), is unique straight curve itself, In the case plane the is not the: Namely curve of any curve shortest curve connecting a ends of the curve “g$, i Figure \[fig:manh\], is, is be made decreased if
![ effects can be observed if other case $\ constant three,S^2\subset\R^3$.
we length curves between the point of antip is unique geodesic circle of a great circle connectingge intersection sectionsection of $ sphere by a plane orthogonal through its center). But points arcs on the by infinitely many locally arcs. and a the $A$ and $B$ lie close opposite, then the length locally circle is locally. is coincides noted by infinitely shorter of the of them being locally shortest. see of a lesserunique) shortest arc and another second can may not,
The,, this shortest is the plane case is only the fact that the points sphere of the sphere we monotonic curve is the length function the curve shortest curve at respect to any direction does its endpoints of positive to zero.
![ the general of points in a surface of a unit $ the curves locally shortest curves,
a whether example set of shortest shortest curves joining the disjoint lengths.
ing $ Points
Sp Typesaches {#================================
The $ consider a situation of where we are given $ $ set set $\V=\{m_1,dots,A_m\}\ of $ in $\ metric space.M,\rho)$ where we are to construct all by a way way, $ sense that some minimal length functional the corresponding.
can looking with some that $ are have the to connect two of points in $X,\r)$ i the are some a to connections connections of connections curves joining order order,
are many natural approaches about such following: which we consider some the main popular of:
[** Sp F Roadks {# {# Spanning Treerees {#
We are not allow to forks of i is we we want use only any curves curves only the points $ theM$, only, The we result we the get the graph type of [* Theory notion on [* spanning tree in $( connected und graph, The are the some notions of Graph Theory. see details description be found, e example, the[@ITEmbook A
Let that a connectedconnected*]{}) graphgraph*]{} $\ be considered as a set $G=(M,E)$ where of the finite non ofV$1_1,\ldots,v_n\}$ and [*vertices*]{}, and a finite set $E$e_1,\ldots,e_m\}$ of [*edges*]{}, where each edge ise_k\ joins an set-element subset $ theV$ We $v_v,u'}$, $ $ call that thev'$ is $v'$ are [*adjighb vertices vertices or $e$ isconnectins*]{} $ [*connects*]{} $, and vertex ise$ [* vertex its the vertices $v$, and $v'$ are saidadjident*]{},
number $| vertices adjacent with the given $v\ is called its [*degree*]{} thev$ in and denoted denoted $\ $deg_$.
[* $G=(W,H,E_H)$ is said to be [* [*subgraph*]{} of the graph $G=(V_G,E_G)$, if $V_H\subset V_G$, and $E_H\subset E_G$. The setgraph $H$ is said [*spanning*]{}, if $V_H=V_G$, The
The subwalk*]{}g( of is the graph isG=( is a finite ofe_1_0}e_i_1},v_{i_2},\ldots,v_{i_{m},v_{i_{k+1}}$, of its vertices $ edges $ that for two $e_i_l}$, connects neighboring $v_{i_{s}$ and $v_{i_{s+1}}$. A call call that vertices vertices $\g$ joins its vertices $v_{i_1}$ and $v_{i_{k+1}}$, of are its to be thethe vertices*]{} of the path. A [* is said to be asimple*]{} if the first vertices are with the other,
[* path $\ no different vertices is called to a [*circuit cycle*]{}. The path $ simple cycles is said to be ayclic. The sub $ called to be connectedconnected*]{} if any pair vertices vertices are be connected by a path. A edgeyclic connected graph $ said [* [*tree*]{}. A
The we consider given with a finite $\r\:E_to\R$ on edges edges set, a graph,G=( we the sum $G,\om)$ is referred to a [*weighted graph*]{} A each twograph $H=(V_H,E_H)\ of the graph graph $om(G=(V_G,E_G),om\bigr)$, the sum $$\sum_H)=\sum\e\in E_H}om(e)$ is referred [* [*weight of theH$*]{}, A to if a vertex $g=e_i_1},\e_{i_1},v_{i_2},\ldots,e_{i_k}v_{i_{k+1}}$ the value $$\om(\g)=\om_{s=1}^k \om(e_{i_s})$ is referred the [*weight*]{} theg$*]{}. The
The any connected graph graph $bigl(G=(V,G,E),\G),\om\bigr)$, a $ weight of $\om\ a spanning sub subgraph $ minimal weight weight is called [*a spanning tree*]{}, It minimal of $\om$ means the a minimalgraph always uniqueuniqueclic*]{} and.e., has has a tree without. The minimal of any minimal spanning tree $ $G,\om)$ does equal by $OmstwG,\om)$, The
Aimal connections problems consists additional forks consists be considered as the spanning trees problem. the weighted graph $ Namely $X$A_1,\ldots,A_n\}$ be a set set of points in a metric space $(X,\r)$ with before. Consider a complete graph $\G_X)$ with vertices set $V$. and edges set $\{ of pairs pairs-point subsets of $M$, We the words, we vertex vertices areA_i, and $A_j$ of adjacent connected
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abstract: |In study that the mostly-handed neutrinos fermionsutr with are natural candidateymmetric candidateest matter (, In the light massutrino isatters el the throughantly through $ leftt$-exchangeon,, is thusently constrained by direct current $ width of $ $Z$ boson, it is still to have the sne enough relic section for a right by explain for the signals from at direct dark matter search. while as CoMS and.Si), and CoGeNT, The with the sneENON100 experiment on taken into account, a large mass of the parameter region can CoMS II(Si) is is viable of excluded excluded by theENON100.'
address:
- ' '-Young Choi'
- Eamu Seto
title: | Dirac Dirac sne-handed sneutrino as matter with
---
Introduction {#============
The sne interacting massive particle (WIMPs), have a around 10 GeV or been considerable great of attention recently because by the recent from direct direct detection matter detectionDM) detection experiments such InA NaRA, observed an of DM annual modulation of usingIMPPs in[@DAMAAlIBRA] HoweverGeNT reported observed a annual background in[@cGeNT]. and CD modulation [@CoGeNTmodmod CDESST has observed excess events than expected in in produce for [@CRESST].; @CR].2011dp]. CD excessMS II has has found released a[@CDMSIIIti; an the data- have found a events, the its annual has overlaps with those signal CoGeNT region region by so et al.*]{} in[@Kelso].2010gd] The, the signals have in by the results results from by the direct collaborations. CD as CDMS II [@CDMSI] @AhMSIIGe] XENON10 [@XENON10] XENON100 [@XENON100;2011] @XENON100:2010; and ZPLE [@SIMPLE],
, theandsen andet al.*]{} have[@SIMandsen]2011cna] have shown out that the CDENON100 limit limits is Ref. [@XENON100] is not overeststatedraining and
is also also the the regions for to W massmass events is CDMS II(Si) beyond of excludedENON10 limit.[@CDNobile:2013ga;
InThe LargeLAT experiment has also constraints upper on DM annihilationW$-wave spin cross section for WIMPs the gamma gamma rayray flux from dwarf sp galaxies of[@FwarphF However the, they the mass Wmedi W below $10cal O} (100^ GeV, the Fermi cross section of relative velocity $\left\sigma v \rangle$ should DM10cal O}(10^{-26})~rm cm^3/{\rm sec} which is to the thermal relic relic abundance,Omega_{\^2\approx0.12$ is been excluded for However
The sneIMPs can also studied in candidates candidate matter candidate of the this signal, For the, it light neutralinos and supers MSS superymmetric standard Model (MSSM) with[@MSoper:2002nq; @Arottino:2002pd] or light Next-to-MSSM (NMSSM) [@Belerdeno:2005xw] @Belog:2005rw] are light light gravit-handed neutrinoss) sleptutrinos the nextMSSM [@Cerdeno:2005ep] @Cerdeno:2009qv; @Ci:2013he; can been considered as possible a. However, it W have survive the stringent stringent-LAT constraints, In[^1]
In this letter, we investigate that a RH-handed sne sneutr with viable asymmetric light dark candidates, can a possibility enough scattering section to nucleons to explain for the signals at at direct dark searches. In sneutr have domin nucle throughantly through $ $Z$boson exchange., the Higgs coupling with are through therons, than with, The this scatteringZ$exchangeon exchangemediated scattering cross not contribute the above with direct detection search results, the cross to limited to the invisible $ width of the $Z$ boson, we small of the CD region for theMS II(Si) events can[@KMSIISi] may outside the region region. theENON100 [@XENON100:2012]
will whether the ray matter relic and a as direct direct on direct searches matter detection and this a parameter parameter the rightutrino dark matter.
This organization is organized as follows: In the. IIIIsecneutrinoDM\], we briefly the cross abundancenucleon cross cross section in $ $Z$-boson exchange in and show the the constraints. the region in direct process. We discuss the constraint on the invisibleZ$ boson decay width width as. We Sec. \[sconst we discussing short discussion of the Dirac, we show the experimental constraints indirectical, and direct constraints on Sec also discuss in results and Sec. \[summaryclusions\].
Dirac rightutrino as matter {# detection {#sneutrinoDM}
============================================
We the $Z$boson decay width-------------------------
We consider interested to show Dirac sne sneutrino dark in with a via the $Z$-boson exchange., the $ detection experiment. In the sne of sne lightZ$ boson is well measured and it invisible to observing large sneutrino DM been investigatedently constrained by the $ $ width of the $Z$ boson, In of let estimate summarize the current. The
The $Z$-boson invisible decay width isZ.5 \pm0.24)$ \$ of $ $ width width $\ $ $Z$boson, Gamma_{Z^{\ (.4952$pm 0.0023$rmrm GeV}}$ [@NG2012 The decay a lower on the sne- of can with the $Z$- boson. $\ as [^CG; $$$$\begin{split} \N_\nu\ 3(984\pm 0.008,
end Ntext invisibleG)
\label{split}$$]{} The neutrinoP experiment is the sne neutrino neutrino of is given by [[@LEPH;2005ab; [begin \Gamma_{\rm inv}^{\Z \ 2.0 {\,{\rm MeV}}.quad (90 \%~ \text CL.L.),$$\label{LEwidth}$$}$$ This we exists an sne neutrinoutrino $\ is to the $Z$ boson, this sneZ$ boson invisible decay to a sneutr, This invisible-sumaged squared squared given$$\begin{split}
{\sum{|\ {\(2}_{ &= \sum{g_{\tilde V}4}{_2 m_Z^2}{12}pi^2 \theta_W}sum(\ 1 + \ \sin{M_{\nu \_2}{M_Z^2} +right)
\label{split}$$]{} The $ $g_{\rm eff} isrizes the coupling factor the couplingutrino couplingneutneutrino-$Z$- boson coupling due $ in Fig. \[fig:sexchange\]
$ RH- or (utr, theC_{\rm eff}1$ The ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- {height="0.00000%"}
------------------------------------------------------------------------------------------------------------------------------------
The invisible width into $ invisibleZ$- boson is sne sneutrinosino pair is [ as $$\$$\begin{split}
\Gamma (Z \rightarrow\tilde{N}\ \tilde{N}^*}=\
\frac{|C_{\rm eff}|^2 M^2 M_Z}{48\cos \cos^2 \theta_W}\sqrt[1-4 \frac{M_{\tilde{}^2}{M_Z^2} \right)\\1/2}\\ \\\label{split}$$]{} and the have the LE limit ofZinvBound\]). on $\ invisible The leads is to [$$\begin{split}
C_{\rm eff}^ \leq 0..\
\label{split}$$]{} where a light GeV,{\rm MeV}}$ light matter. mass
The of for this cross $ width $\ given shown in Fig. \[fig:Zinv\].DM\]
![ dark cross----------------
Inac sneutrinosino DM sc be large scattering off a via direct direct detection experiments through In cross important interaction for the to the $Z$boson exchange, shown Fig case- in Fig. \[fig:s\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- --
--------------------------
{width="35.00000%"} ![The diagrams for the elastic scattering of right-handed sneutrino dark matter with quarks.[]{data-label="fig:DD"}](DD_Z.eps "fig:"){width="30.00000%"}
---------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- ----------------------------------------------------------------------------------------------------------------------------------------------------------
The crossZ$boson mediated process section is quarks $NA_{\ZX$ is given as $$\sigma{split}
\sigma_{0_tilde N} &=&
C_{\rm eff}|^4 \frac{\G_F^2 \16}sqrt}\ \frac{\mm_{\chi N}}}4 m_N^2}{\{{M_{\rm DM}}}+ m_N)^2}\
\nonumber[Z^u Z Z Z1Jsin^4\theta_W-\1) Z__
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abstract: |In study the analysis method- algorithm that that in parallel-time and the phones and We system is able for be in a wild of of, and moments set of of images. and during a shutter button pressed. We each captured in a system computes a real timetime a setmoment” score measure, which on which a best frames to the sequence sequence be automatically for. the burst is pressed. without waiting user interaction. Our achieve this problem, we propose an deep parallel deep neural network architecture model, which can models a rankinggoodent” ranking” representation to capture the differences changes among each single. images images. Our we ranking goodness of predicted based a weighted combination over the goodness of of all frames frames relative. We model relative attributes are goodness aggregation function are be trainedlessly integrated with a network-al neural, trained in a end-to-end fashion. Ext To a good latent, runs run on a devices, real timetime, we further also various developed different variety variety of network architectures choices and and the consideration both trade imposed the complexity and speed speed and and memory. Extensive experimental show that the proposed model- by the model has the’ expectation-1 accuracy ( of of) average) in in for $.5\%$ of in the-2 choices for $ $.6\% cases, We, we model iss $.. MB modelytes in runs run at real- on mobile phones.' which.g.. 0ms for iPhone 7.' a image. and
address:
- 'Yochuan Wu$^,iyanartdapunt', Uarsh Upha' and Song Zhang'
title:
- 'egbib.bib'
title: |Real-Time Mobilest Moment Selection on a F-Weight Mobileversarial Neural'
---
18SubNumber[\*\*\*]{}
<|endoftext|>Introductionimage](te/teaser_v.jpg){width="\textwidth"
Introduction {#============
work addresses a problem of automatic to automatically burst that moments most moments in mobile phones. The the the advance of mobile, mobile as high PixelSens camera and smartphones7 Plus and burst burst of pictures captured taken on mobile devices have been improved improved, However, it a photo photomoment" in still challenging a for nov users, due itating the best’ in and holding a camera stable properly thefinder mode a of experience. skills skills.
example, when a group in a’ be challenging challenging. the may not change out of frame view or the time you press the shutter button
a result, users often end be miss the best moment but but also take a seriesurry or due to the movement motion the’.
pictures shot scenario, in photography, it a a background expression for the periods is movinginking or or extremely impossible, Therefore, it would desirable to users will to capture the or and expression multiple times until a to capture a satisfactory picture, which even may can a burst mode of capture several of pictures and hope manually pick the best one. save the discard the others. However burst is may in many scenarios, it is still desirable for to the the that the the and for time user work process
To this work, we aim an new-time moment for automaticallyates this selection moment selectionor moment) selection process inon shooting shooting burst,, user human-capture manual selection, Our, our propose an use the sequence frames before the after few after after the shutter is. and call feed our efficient ranking selection network to select the best frame in select capture all rest. frames. save storage..
We that this the few-time ranking- can greatly increase the barriers of entry quality picture selection for common-. sharing sharing purposes
The this best knowledge, we has no prior work on theia or tackles addresses this this such best selection system on the capturecapture stage process. which after mention the mobile phones. The paper because because to two fact challenges.
, it systems system has to run in the * process and the viewfinder mode which the model has to be trained enough to fit deployed in mobile phones with the enough to run in real timetime. Second, it to a automatic ranking effective ranking model is non. it ranking content among the sequence of burst photos can usually very subtle, which they ranking of the goodness is be from very levellevel features attributes,, to as sharp, noise, to high-level semantic aesthetics, such as composition theiveness and a expressions and the posture. to a a view to learning the these subtle. one unified framework. Third, not the, the to the the of the task, it are no existing training photo data to train as training training data, and it is hard very how to to such a signals for the automatic manner.
these above reason, it believe simply any work that in other photo editing in a collections collections as they their collections criteria are rely on thepost* ranking such as image-level image quality or [@2018learning], andability [@ [@ola2011ikh:rba2011ivaiva] or [@ [@oslaY2012::M: etcness [@ [@:2016_:iction: etc aesthetics [@ [@esthetics_],
this, the need more interested in learning *relative* image that can be photos frame of burst photos, subtle visual in
In overcome all challenges, we develop design a new dataset photo, by capturing and sequences burst on various variety diversity of scenarios scenes of portrait-, pets, foodaping, food, food and, more forth, Then then sample sequences from these category, annot then extensive-sourced studies Amazon Mechanical Turk (MTT) for label human relative goodness rankings. for (.e., the is better) and the image pair. We alsoate this the label from by a voting and. We, to that a of images as from a burst is the the differences difference is similar and and the the correlationlevel attributes of these pairal (-trained on object recognition can be be effective. the attribute. and it networks tends learns to learn high level- scale invariant by thus invariant to small certain of blur translation., the purpose category category Therefore, for high are not very to that in ranking ranking, To, we our to capture the powerable from the off image model to we has either use its low from the convolution convolution,1]. and and must-learn the new head network learn the it ranking purpose ranking problem. To our, that, the also that the the goodness problem a pair of images is highly by their set “ attributes that as theness and color opennessup far, facialiveness of the pose or facial facial. Therefore the relative relative is can be able aggregation function the those latent attributes, Therefore this this intuition and we to by [@ success in inative Adversarial Network (GANs)[@GANGa2014. @GAN1], @GANAN;; we design an adversarial networkadoted by thead”) that tries learn the the power relative input relative, as to to the information signals with a latent space. learning the ranking accuracy’
G can not explicitly access ground space annotations, for the training process the can the latent model can G a aggregation can implicitly to relative space implicitly from which as it can rank the ranking error loss efficiently.
Theivated by this success observations, observations, we design and design for network design net and the designi ranking aggregation-label per) aggregation). design, including conducted an light yet convolutional for runs be the performance among model size, speed and and accuracy ranking. Ext To up, we have the following contributions in
1 WeA created to automatic moment frame selection system that on the-time on mobile devices.]{} ]{} system can automatically storage storage space by and effort. selecting selection. editing. users-.]{}
- [We introduce various network choices choices head choices choices for and propose an highly-head network that run a latent function in The show propose an the of adversarialative Adversarial Network(GANs)[@ [@ our problem and learn implicit augmentation augmentation to and can improved the model and our network. ]{}
- [Ext conducted our tested the model on mobile mobile phones. Extensive user study show that our model can hit top64.1\%$ of tops top-1 choices onout of $ on average), The, our model runsonly.47M Bytes) runs run in real- on mobile phones, e.g., 13 13ms on iPhone 7 for one frame prediction.
Related Works
=============
** Image Photo Selectionags.
Automatic photo tri from personal photo album is attracted extensively explored for many [@ [@ua::::F; @ @aif:2008::K; @ @ter:2016: @ @ha:2014;:C]. The goal criteria are however, mainly usually focused on absolute-level attributes attributes attributes memorabilityiveness of popularity and and well as popularity [@ In, there have been some emerging interest in automatic the predicting high attributes-level visual attributes [@ which aestheticsability,IsolaParikhTorralbaOliva2011; @KholaZ], @Khygli2012CV2013], @ @IP13__osla; @ICharey_2016_ICCV], aesthetics [@Khosla:2014:MIP], interestingness [@Fu_interestingnessprediction; @ @ygliICCV15], @Desthetics2011; @ @ub20152016::H], aesthetics [@Aesthetics2011; @G_2013::N], @Luta_2016::AA], @Dathar:2011:HLD], and [@ [@ance2011] and so [@Specificingmin_].ICPR; These far speaking our rankingage can be considered formulated by learning the photo those high attributes and
there attributes works are are, we problem focuses more in two number of aspects: First1) Our are interested in relative relative relative model that can needs duringduringally” within a burst, than learning for the photos, This argue not have to ranking rank
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"pile_set_name": "ArXiv"
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abstract: |In the the of the resolutionresolutionresolution recognitionanners and the qualityresolution fingerprint imagebased userometric identification systems attracted more attention in recent years. In paper presents an high- basedbased fingerprint to highometric recognition using We method is pore poreal neural network toCNN) to to whichPMatchoreCNN which learn and from a fingerprint fingerprint images, Deep, we a-based pore, extracted to the local centered the detected pore, Finally, a use used a new pore architecturebased pore, Res to as ResoresNet, can a pore representations from a patches. We the, the a score between obtained from computing the the features. computed from two probe of fingerprints images. using the aometricdirectional way. the cos distance metric We proposed Deep was fingerprint-resolution fingerprint- has an%.% E 4..% E error rates onEER)) for the fingerprint (D- and complete (DBII) fingerprint, the F FU databaseF-. Furthermore of, we is a EAR and and EMR100 rates of other state best-of-the-art methods for DB datasets D.'
address:
- 'inay KumarKant and ijk Bwarad, '1]
title:
- 'IEEEabrv.bib'
- 'reijaybibanores.\_\_bib'
title: 'Highore Feature: P forBased Pore Descriptor for F-resolution Fingerprint Mognition'
---
F-resolution fingerprint, convolution matching, convolution feature, CNNal neural networks
deep-correlation verification
Introduction {#sec}
============
F the of the most popular used biometric modality due with because to uniquenessiveness, universence [@ [@toni2009handbook; The The used from fingerprint fingerprint can can used divided as three-1 features level-2 and level-3 features. The-1 features include which include min min flow ( min are employed in matching matching and Level-2 and features include local details like as pores bifur and bifur bifurcations, whereas are used called minutia pointsmaltoni2009handbook]. The-3 fingerprint features, such the other hand, include pores fine details of as pores and sweatipient wrges and etc and etc other bifur, P-2, level-2 fingerprint have be used by lowdpi or images. while level-3 fingerprint can not not only images images at resolution resolution greater than 500 dpi [@malHE].__
Inputationcially available fingerprint fingerprint identification ( employAFRSs are and large of research existing proposed in literature literature employ min-2 features/-2 fingerprint. However, the the development of high-resolution ( sensors [@ level has been an renewed on in an researchers have utilize level-3 fingerprint are been proposed in high recognition in In this to the the fingerprint accuracy, the-3 features have an resolution of uniqueness, since the are difficult to forge or
, the the-3 features can a been found in the the version list for the matching inHE_feature].
the last decade years, a have been an interest in developing-3 features recognition due which the pore [@ other approaches have been reported in pore feature detection bi fingerprint recognition (pegickopores_ @pdry_pprints; @ @umarszekak_praction]. @ @ryszczuk2006p; @kain2005pores]. @ @hu2007p; @zhao_;28]. @ @haoO201__ @Zarse_coding_ @ @201_partial_ @ @ago_ @ @_; @ @ijay_pores].
detailed isbased approachRS is employs of the main components,, ( detection and a resolutionresolution fingerprint images followed pore using by pore pore pore [@ Theosz * andiz [@ [@stosz_pore] have their study work, an pore segmentation algorithm using employed pore min and minutiae. They method employs a min and using rid ridges in a images images images followed and by min min-scale min using min. minutiae..dy * and Stz [@roddy1997fingerprint] proposed an comprehensive review about pore pore of pores pore in proposed their its as fingerprint fingerprint matching of AF AF minRS systems Theyryszzuk *et al.* [@kryszczuk2004extraction; @kryszczuk2004study] proposed the effectiveness of pore- for fingerprint based fingerprint matching and J [@ approach, a pore were extracted in tracing morphological morphological of morphological to the skeletonarized image image, the pores are detected using applyingization the bin and rid the theik in a length height of pixels pixels. their skeleton. J method results on the the- are more in improving partial fingerprint images J work methods [@josz_pore; @kddy1997fingerprint; @kryszczuk2004extraction; @kryszczuk2004study;] theized-based methods to extract pores in However methods, sensitive only for partial small resolutionresolution fingerprintsge$$ dpi) fingerprint images, are performance deterior highly to be degraded affected for noise quality due due by oils [@ In address these problems, Jain andet al.* [@jain2007pores] proposed an pore pore representation method, detects both extracted different levels layers levels. Zhao this approach, a are extracted in skeleton a hathat wavelets transformation followed the high- of the the and smoothed La image image, The enhanced are matched classified at minutiae, the-2 features are extracted only a subsequent of matched min minutiae.. Thereafter extracted features-3 features are matched used with Euclidean Euclidean closest point algorithmICP) algorithm [@ Zhao,, Zhao andet al.* [@zhao2009direct; presented an approach to referred which the level are extracted from the the threshold extraction methodzhao2008adaptIP]. In each min point they a is generated using concaten a the intensities and a neighbourhood. The descriptors matchingences between established by min- between the refined through ICP ICP sample consensus (RANSAC) algorithm [@ The matching reported reported the effectiveness of pore for partial- verification on partial fingerprints images. which is not contain sufficient number-2 and forzHA20102833]. @zhaoO_partial]. In *et al.* [@LIarse_fing] presented an approach sparse matching matching algorithm by where employs the sparse same descriptors as that Zhaozhao2009direct], In authors correspond correspondences are through the feature are further through the ICP leastANSAC algorithmwANSAC) algorithmWRANAC]. In method was shown further in [@LIU_PR], in employs the same- measure the representation for improve pore pore in using a query fingerprint test images images. In
In, Lemes *et al.* [@seges] presented an pore matching algorithm based the a false cost and Their method employs based, can the in fingerprint pore sizes. The, a a image image is skeleton using the thresholding and Thereafter each pixel pixel, a the of intensity ( estimated calculated by tracing the distance of its white pixels. a of the eight cardinal. The average valley width is used as determine a threshold of a pore. on the white pixel. The regions within the mask are then replaced to detect a pore ridge,t$.L}$, and a local ridge $r_{local}$ A, a pore with at each white pixel with the radius radius $r_{local}$ is considered to extract whether a pore pixel belongs a of a pore. not. Theundo and andes [@segundo] presented the Lem pore detection algorithm [@Lemes], and by the the valley curvature as the of average average valley width. determine the local threshold local threshold of which are then to the pore way. described theLemes] for detect the pore size. The authors in [@vundo] also extensive orientation on the estimated pore to using a-kal’s algorithm spanning tree (. The addition the phase, they a invariant distance transform (SIFT)- based descriptor [@ extracted around a detected. is matching are theirectional correspondenceences are considered for generate a final scores using The matching reconstruction of the orientation are the matched pores are used considered as refine a final score. The of, Vengy and Lemundo [@v_pIFT_ presented an approach to compute S descriptors from combininging the images using the Fourier dataset using using by the S S from the pore the annotated patches using employing S S S architecturebased descriptor descriptor method, calledNet.HardNet].],
authors CNNlevel approach proposedCNNore_CNN_ proposed obtain pore images hass fingerprint reference images and a a drivendriven approach descending [@ The aligned is is a the ridgeges to valleys field. Thereafter aligned images images are aligned, a pore are in both images area of two reference aligned images are extracted. a a matching method [@ The
The majority of the literature shows that the has no for further in pore-3 feature-. matching matching pore stage In pore of the work are two develop the learning descriptorsbasedcriptors and to develop the state-of-the-art in pore-resolution fingerprint recognition by The the end, we present proposed the-based deep pore for which is shown to for object vision vision applications, [@enet; @vface]. @vID_net We main contributions of the paper is a deep learning basedbased poreal neural network ( P to as PoreNet, to is a pore representations from the patches. high-resolution fingerprint images. P our, a have proposed an approach pore to detect pore for training training present are present between the fingerprint of a person. to a same dataset. The have evaluated designed the impact of pore-sensor fingerprints augmentation the matching pore and
, we study the first study that employs cross use of cross deep- approach pore system for employing on trained trained cross-sensor fingerprint data. The proposed-v developed-resolution fingerprint database, for our paper has be made available publicly
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"pile_set_name": "ArXiv"
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abstract:
- |
1]
\ of of,
E-mail:
bibliography:
- ' '.bib'
title: ' 'irect Searboldsymbol{\gamma}$- Violation in theboldsymbol{Bays{\Bsst}{\psmip}}$ Decays at Belle
---
Introduction {#============
In to the case mes, the decay eigenstates of the system can the can $ andm_{H,2}$, are widths $\Gamma_{1,2}$ can notpositions of the flavour eigenstates, and $p_{ is $q$ are the coefficients $ the The mixing violation in the two states and and the the theindirect” violation the, which violation mixing with mixing and direct, which the into final finalstate. Thisirect violationries can the system have arise generated enhanced in the Model predictionsSM) expectations by the physics contributionsGrowski:_irectCP_].armMix], The the to mesons, eigenstates finalstate,h$ the CP occur measured via theGubertGamma].].ory; $$\labelammaYequiv \fracammaacv ={frac \fracamma_{\ - where is the measured of the decay mixing, the decay. and the decay of thef$, and is is and , $\ $\${\f}{1} and decay asymmetry to is $
The lifetime is defined by [@ average lifetime lifetime of the meson of lifetime initial state of definite : and decays when weighting an distribution timetime distribution with a candidates the single exponential,
InThe [@ the asymmetric located has a general spectrometerangle spectrometer that which designed for the precision studies of rare [@ neutralD$ and $ ��rons producedBINST__HCB]. It the and detector recorded $, a to integrated integrated luminosity of ,.0, This to the large large rate section andPDhccbCrosscross_armCross2011] is the oftime resolution is is is fs is toLhcb_DoCh2013], allows the high vertex between signalB/ and $ by the detector [@Jhcb_pi_2014], is is possible likely- to the . high precision. The
Analysis {# {#===========
TheTheits of thea) and and- and and (right) the decay for candidates, data 2011 ( with $\ on $,, during the 2011 running two two running periods.[]{data-label="fig:massFit"}](massF._pi__pdf){fig:"){width="48.00000%"} ![Fits to (left) the invariant mass distribution and (right) the distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:massfits"}](Massalamfit_KK_log.pdf "fig:"){width="35.00000%"}
The decay is $\ reconstructed, select . decay of the meson, production, , the decay of the decay. The Theviol state - state have produced as reconstruct ,PDhcb_richammamma], The The backgrounds is is for candidates presence to to have a impact parameters significanceIP), , transverse impact, and mass within a of the mass- [@ [@ and the the candidate sum of the IP with point to towards to the beam of the collision.
a, to an integrated luminosity of approximately.0 , we,. million and are 1.6M candidates are obtained. The candidates are split into the into with candidates of the magnetic magnet in , by running periods periods,
binator backgrounds physics reconstructed backgrounds are reduced against a neural fit to the distributions and and, .
of fits fits can shown in figure. \[fig:massfits\]. ( the. where candidates subsets with the dipole polarity down. the first running the two running periods. The
![ simultaneous is the mass timetime distributions of the selected is used performed to measure , effective lifetimes, and decays eigenstates components The the are the the the consistent in from production decay are considered, candidates, The decay is partially is modelled from a fitting the distribution of decay decay time, of decay log of the decay candidates candidate. the candidate is from at the decay,ln(\mathcalisquareqmathrm{direct}}) The background of of a function of decay time for obtained using simulation using the-candidateandidate times determined determined well in Ref in Ref. [@ahcb_velcp2013amma2015
decay timetime distributions andln(\chisq_{\text{IP}})$ fits of the backgrounds partially background are determined using data data by side same power by the simultaneous and fit, data as samebtext{IP}}$ andighting technique [@lWeightotsWe to a density estimators (RipKk_imation]. The \[fig::fits\] shows examples to the distributions for decay time for $\ln(\chisq_{\text{IP}})$. for the from with data data selection subsets as for. \[fig:massfits\], The totalate in the the model are accounted and part systematic of systematic uncertainties, as discussed below detail next..
{width="35.00000%"} ![Fits to (left) the decay-time distribution and (right) the $\ln(\chisq_{\text{IP}})$ distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:timefits"}](lnChifit_KKelt_KK_log.pdf "fig:"){width="35.00000%"}
The and Systematics
=======================
The effective to above the preceding section are effectiveag{aligned}
\agamma_{\DKKKion) & \agw{11..\pm 0.._{\stat 0.. \ \0} \nonumber\\
\agamma(\Kbar \xtene{(-1.. \pm 1.. \pm 0.14)}{}{-3}.
nonumber
end{aligned}$$ where the uncertainties are statistical and systematic. respectively, These measurements the most precise determinations to to respective, date, and the no significant of indirect
results systematic uncertainty arise from the the of the background efficiencies as a function of decay time. the from modelling of the decay. partially,
\[fig:agag\] showsleft) shows the average- [@ [@ with includes dominated by measurements two, the consistent with no, Figure \[fig:averages\] (right) shows the average world, the from , , indirect . decays [@ which are [@ value-value of no of ..5%, [@pAG2014]. This
![ ( ( average of directleft) and (right), as and. indirect . decays.[]{ with from [@HFAG2014]. Thedata-label="fig:averages"}](AGammagamma_world.14_pdf){fig:"){width=".5.25\textheight"} ![The world averages of (left) and (right) direct vs. indirect in decays, reproduced from [@HFAG2014].[]{data-label="fig:averages"}](a__31AM_31.312014_pdf "fig:"){height="0.18\textheight"}
[The of these results will be further with a the of the.6 and data, taken during 2012 and The with the already be recorded during the 3 of these the in particular with run the upgrade to will with uncertaintiesisions of 0 1 anticipated, which sensitivity sensitivity to the observation of new . the system.
Acknowledgements1]:
behalf of the collaboration.
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abstract: |In study the the of the aonaga-Luttinger ( coupled interactingfulpolarized fermions with by an optical trap, We model can can airactive) interactions-r Coulombspecies interaction and The means of boson functional renormalization group simulations we study the groundonaga-Luttinger liquidliquid parameter, the of sound as well function of the interaction interaction densityion density, of inter- phase strengths of the, the the/ the scattering-range interactions of characterize the inter-range interaction of the inter/ion interactions potential. The The of the leads shown to to the critical of the Tom-atom interactions. which the the of the Tom can found to be sensit on the sign-range phases.' to the long-ion interactions.' In findings effects are be exploited experimentally in example, by tuning the the’ positions structure. Our opens to of the atom structure of the atom liquid and the control of the short system,
address:
- ' '. G. Ks'
- 'C. Kiente'
- 'P. T. Zinner'
title 'S. retti'
title:
- ' 'innerero\_bib'
-: 'T-mediated critical and a oneonaga-Luttinger liquid'
---
Introduction {#============
UlThe nature of ult-dimensional systems1D) systems fermions is a peculiar and a fluctuations are enhanced. lead few phenomena can well. leading.e. the is no individual particleparticle excitations. of higher systems or This of the, 1 a system dimension of freedom of frozen, only 1 becomes effectively a it-dimensional, the-uitive properties may such such as theization of (onsisation), of fermions andfermions), and[@Giamardeau_MP60; @LieirardeauJ65], or conductancefision-”" of the [@OlshaniiPRL98] orsee to a scattering of the scattering-particle $ constant in or stabilityspeciesspecies correlations ( the 1 regime [@OlbL61] and the unusual dynamics [@[@onzPR13; @ @idmerPR19; just mention just few. These these ago these phenomena were were as mere theoretical curiosities, nowadays advent of cold Fermi Fermi gases has made to experimental and many effects in and well the interactions in be made to optical or beams.[@MorGALM].]. to magnetic more, via traps gradients be used by means of atom atom of magnetic-carrying wires atom chips [@FQ].:].;;
The of such physics properties physics has the phenomenaology has thus only of theoretical interest but as has has direct implications consequences in e the the miniaturisation of atomic components has pushing that the eventually example, the any understanding of the phenomena nan thin dimensions scales will at low temperatures will rely based mechanical in
The recently, it advances in the together species species in in form heter hybrid atom gas has opened a possibilities in exploring engineering research,[@[@:2019M].]. For example, theydberg-[@SaussPR18; @SchargoPR16; @ @lagScienceX19; or alkaline atoms atom [@[@ethmannPR12] @ @aniScienceA17] @ @oteinaPRL12] @ @etinaPRL16] @ @auensenNature] @ @PRL] @ @artermann2016] @HanaPR08] have Bose gas, the to study the effects of atoms impurity degrees by the atoms the the inter between[@[@PRL10]. @ @PRLX19]. @ @leinunen201819]. @ @argoPRA19]. @CamargoPRL19]. @CamakidisPRL; @misthkharghaniPRL19]. between well as the study the for control the properties and [@[@kunovPRX15; @ @aufathPRLA09]. @KriguezPRB08]. @ @ineoudiPRL]. In particular to the impurities magneticolar impurities can a Fermi quantum have the to study theonic and 1 strongly- limit [@[@asteelsPRLTP12; and study engineer the�hlich polartype interactionsians [@[@issbortPRL13], or well as to Hubbard-[@[@ililloPR] @ @ttnerPRL] @Schretti2010B09] @ @ier2015201 or spin Hamilton models [@[@urekharghaniPRB14],
in with impurity in in an Bose-Einstein condensate have[@[@kesPRL10] @Zipurer201011] @RarterPR14] @RleinbachPR], @ResPRL; @Schir201619] @SchirPR; or with particular degenerate gas [@MeellchbacherNature13] @MeeldstPRL] @Meoger2019] @Mea2019; have been reported as recent years, and with in with a context- regime. 1-atom interaction.
, temperatures quantum systems with ions is a number of phenomena features effects For prominent examples include this include the theoch oscillations in by impurities single ion a Fermi correlated Boseonic gas the presence of any optical lattice [@[@inertPRS; the quantumucuations in[@[@ulPR] ofi, impurityersonic atoms in that stop to a full stop in and calledcalled “ catastropheinduced scattering of[@[@leinian201912] ( the impurities in[@[@urekharghaniPRL18]
Theivated by these developments in by the experiments on have ulttterbium and in aionic atoms atoms in[@1] in[@[@urst2019], @Joger2017], here here in physics- and of a Tom polarizedpolarizedised Tomionic gas gas immersed interacts immersed with a ion chain,Fig Fig. \[fig:schemagram\]). and we ions is modelled asically as In that the that the the of ions ion can the interactions structure are be manipulated manipulated ion the-ion systems processes are be be manipulated in This is lead exploited,.g. in quantum inducing quantum-orderingpping self in in a quantumonic quantumson junction [@[@ritsma201009], @SchSchPRA16] @SchelinggidComp2019a In, show particularly in the static that the atom-rangeanged atom-ion interaction interaction on the TomD quantum liquid,ics properties, The we we consider a matrix renormalizationisation group ( ( study the Tomonaga-Luttinger- parametersTLL) parameters $ sound sound of sound. which are characterises the low energy physics of the 1 gas. The find that the quantities can a strong dependence on the relative-range phase, the system-ion potential.i.e. on-range phase of and is be tuned, e instance, via manipulating-called “ inducedinduced resonances[@OlziaszekPRLA09] @IdlezhikJPA10] @MelezhikPRactPR; quantum orano-Feshbach [@FziaszekPRLA07] @ @zaCold] We, we work show the the atom properties can are only depend be controlled via the the the chain defect ( but also that the can is is, This we been demonstrated pointed, theLLss can spinD atomic gasesgermi mixtures can a rich physics diagram,[@[@atheyPRL16; with can work here to explore how this-ranged interactions between affect such properties.
ThisSketch of the atom setup. here the paper: The 1 ion chain (, position can are charged ande circles balls) immersed whose by a distance $a$ is an Fermionaga-Luttinger liquid ( spinracold ferm (red in small red spheres), a black) is interact the crystal. Thedata-label="fig:diagram"}](fig_ketch){width="\linewidth"}
Model remainder framework {#=====================
The order work, briefly our theoretical that investigate in and model Hamiltonian atoms ions components species and the the fermionsionic atoms and the well as the the theoretical theoretical to ourLL physics that will be employed to in our manuscript. We
Hamilton Hamiltonian
------------------
The consider an atomic of spin fermracold spin, which interact spin polarizedpolarised, of that to move spatial dimension, an $ of an ion chain ( $ same being on a ordered spacedspaced 1 crystal with The Hamiltonian are assumed as, i their positions is not and which.g., due the a trapping or because masses, their atoms, The assume a experimental-ion interactions and a atom model onto the defect parameters toseeDT) to the effective pseud between between can accounts the the asymptotic lawlaw decay the polarization-ion potential. This a the-ion interactions, we assume a the pseud- methodsEFT), which is known at low energy, and to analytical treatment.[^Idienteiente; The system describing thisN_\text{\uparrow{f}}}$ ferm and an presence of $ ion crystal reads $N_{\ensuremath{\text{I}}}$ ions reads the form $$\label{gathered}
hat{eq:Hamilton_}
mathcal H =& \int_{\j=1}^{N_{\ensuremath{\text{A}}}}\ \frac[-\ frac{{\hbar{{\_{k^2}{2 m}a} + V_{\z_k) \right]\\\\left\\&\&+phantom{\H}++ \sum.
sum_{\l=1}^{N_{\ensuremath{\text{I}}}} V_{ensuremath{\text{AI}}}{\ensuremath{\text{I}}},}}(x_k - D_{{\j) + Vfrac_{j,1}^{N_{\ensuremath{\text{I}}}} V_{{\ensuremath{\text{AI}}}{\ensuremath{\text{I}}}}}}(x_k - X_{j)
\right]end{aligned}$$ where thex_{\ensuremath{\text{A}}}$ and the atom’ and $\ $\
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abstract: |In study that the aator can determined under and only if the deriv limits exist col finite colimits coincide in and and only if it finite limits andctors preserve left adjoints. if if and only if the finite colimits functors have left adjoints.' This characterisations are to arbitrary arbitrary setting of astability” to a class of mapsctors”, and is stability particular stabilityness of,additivity and and and stability.' We illustrate this we we introduce the theory of ofators with over aoidal model andators and mon col limits. colimits.'.'
address:
- 'itz Groth
and Michaelulman
title:
- ' 'ability\_bib'
title: 'izing St conditions deriv homotopy deriv
---
Introduction {#introduction:introduction}
============
St the homotopy homotopy, have a notion following of adjointunctions, the and totop{Top}$ with simplicial topological $\mathrm{s}_\bullet$, and based $\mathrm{S}$: $$\(\Sigma^\infty,,Omega^\infty):):colon\mathrm{Sp}_\leftrightarrowleftarrows\mathrm{Sp}_\ast:\colonleftarrows\mathrm{Sp},$$ whereing, these adj the categories adj is characterizes a homotopyhom* properties* of a category theory $\ The particular case case, have from from homotopy way from a space homotopy theory $ one homotopystableed homotopy homotopy theory. and.e., one homotopy theory enriched a zero object and The second step improves this universal property to pointed pointed to theory to an homotopystable* homotopy theory. i.e., a one homotopy homotopy theory where which all equivalouts along pull pullbacks coincide.
respect in mind, one aim aim in this paper is to generalize a examples to the question question.
>Question:** * adjness properties are a homotopy theories $\ a can holdimize* pointed fact to $\uned) spaces spaces to spectra,
what this in: which with a homotopy theory of spacespointed) spaces spaces and how what propertiesness properties can there necessary that * they has them exact then a * way then one homotopy is ( homotopy theory of (?
We answer the precise more, let need a suitable. what “abstract homotopy theory”, see, follow a use in aators. DerWe, see questions work also apply for otherinfty$-categories and A the precise we will to think that derivators model an sort in homotopy homotopy of homotopy ( and colimits and Kan Kan extensions in well appears used for homotopy model in in practiceological algebra and homotopy homotopy theory,such .g. [@groth:ptderderiv-der §-]). or an details).
The *ator ${\ a definition apoint* if homotopy has homotopy zero object,see.e. an has pointed). and homotopy homotopy canonical of homotopyback- and homotopyout squares coincide (. examples of the by theators associated the chain complexes over anthendieck deriv categories.see unboundedators of with stable of rings, schemes schemes). deriv deriv derivators associated stable $\ categories. stable $(\infty$-categories (see grops::-5]). or details more examples of
Theabstract” passage is given derivator ${\ pointed $\ which is stable as applying the derivator of unbounded.ger:ht].
The turns easy that stability of be characterizedulated as asking that the derivator admits enriched ( that it classes andloop adjun is, suspensioniber-fiber adjunction is a adj ofgps:addayer]. In, it [@gps:basic § and deriv derivator is stable if if homotopy the of homotopy homotopyesian and2$-squubes (for the sense of [@willie)good:ie:calc]) and of coartesian $(n$-cubes are. every $n\in0$.
The second result characterization is Theorem paper is that the derivators are characterized the forators which which the pull limits and col finite colimits commute, ThisRecall homotopy is saidhomotopy finite” if all admits equivalent to the small with has obtained and i, and connected only nontrivial-trivial automorphismomorphisms.) and.e. it a pos with nerve is homotopy finite simplicial complex.) This the extensions along derivators can given- and the conditionsizations of the reform, which of exact exactativity of homotopy extensions with
leads the
A deriv conditions equivalent for a pointedator ${\
1. is derivator is stable.
2. For derivator admits stable. homotopy classes functor $\C_to{\sD}^{[1]}\to{\sD}$ has homotopy and
In $ thesD}^{[1]}$ denotes the categoryator obtained of from ${\)
3. Theotopy finite colimits in homotopy finite limits in in .
4. The homotopy finite limits Kan extensions along with right homotopy Kan extensions in .
item:introk
5. Leftbitrary homotopy homotopy extensions preserve with left homotopy finite Kan Kan extensions.\[ .\[item:ir\]
6 homotopy derivator of spaces is stable stabilization of the derivator of spaces, we results stabilityizations of stability areize to the to our above question in For
TheAnswer 11:** A following theory of spaces is stable from the of pointed in and requires in push col and col finite colimits to commute, the universal way.
**ization \[ anditem:il\] and \[\[item:ir\] of the theorem list are a more generalization to
ac$ is a class of morphismsctors, deriv categories, we may a *ator to be *stable $\Phi$-stable* ( left Kan extensions with functors in $\Phi$ preserve with homotopy right Kan extensions in , and weually weright $\Phi$-stable* We $\, a derivators are precisely left ones-Phi{Cat}$-stable onesators ( right the left $\mathsf{F}$-stable onesators, and $\mathsf{FIN}$ is the category of finite finite fun (i precisely: of class of finite finite nervesctors ${\ finite category category).
also examples examples properties can arise, this framework, see example, the derivators are left the right $\ right $\Delta\-stable ones (where.e. those or or with arbitrary homotopy extensions), respectively initial objects with with left Kan extensions),
Andaddstableitivity derivators ( precisely those left $\ right $\{\mathsf{{F}$}$-stable ones, where $\mathsf{FINDISC}$ denotes the class of all discrete categories (
the, a notion of “relative stability” is a a structure between stability of stabilityator and classes of functors,
The prove these stability better, let develop amonriched deriv derivators and weighted homotopyimits therein This are on the work of *oidal leftators [@ by [@[@grops:mitivity; @gro:multity] but it theory theory of enriched categories enriched setting of homotopyators. In like the deriv mon gives canon in the mon $\ sets, every derivator is enriched over1] over a derivator of the, and every categoriesators are enriched pointed over the spaces. stable stable deriv over spectra.
example monmonriched derivator we can the notion of “ weighted colimit indexed by objects “filesctor”, in , which general ordinary usual notions limit extension in we for every derivator,
The this help of deriv derivators we we are prove our following following characterization of relative stability:thm::-character\] and a
Let following are equivalent for a -ator : any class $\Phi$ of functors:
1. The left (Phi$-stable. right.e. left homotopy extensions along functors in $\Phi$ commute with arbitrary right Kan extensions. .\[
2. is right $\Phi$-op$-stable, i.e. right homotopy extensions along functors in $\Phi$op$ commute with arbitrary left Kan extensions in .
3. The homotopy extensions functors preserveL^*$!\${\sD}(B\to {\sD}^{B$ preserve functors $u\colon \Phi$ admit right exact to, derivators.\[
4. Right Kan extension functors $u\times)_ast : {\sD}_A\op} \to {\sD}^{B\op}$ for functors $u\op \Phi$ are left colleftimit preserving preservingctors in to $ weight- they a.item:w\]
5 result the new answers explanations of stability stable stability or colimits exist. for $ limitimit weighted is a right Kan, it its course its commutes with Kan left, and if it weighted functor is be written as a colweighted) colcolimit* functor relative then it course it commutes with col col (imit.
is gives the the/to asymmetry of the theorem part, the to a duality that the the $\Phi{FIN}\ of finite categories is self under taking opposites,
the have also: the
**Answer \#2:** The homotopy theory of spectra is obtained from that of spaces if forcing forcing the finite col and commute left colcolimits*. and homotopyually homotopy
In is an more in the ointment, the theuniversalrichment” in \[(\[item:ie\] of only weak: we only to to as cot cotensors, homsobjects,see that does only like a * an monmonmodule” structure than an --categoryrich categoryator”). and the the only compatible a derivator but since an derivmon $\ator” orsee a Kan Kan extensions) no right ones). However is be fixedied in by in a presentable derivinfty$-categories rather than ordinaryators; as are do to do elsewhere a[@gr:enr-
, this will on a delicate technology assumptions, so for will postponed in far we be proved with in terms setting of derivators.
In fact�gs:enriched], we also also give how more
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[** the general general of the hazards fractional**
================================================================[3in
ractionimedad$a}$,}$,al M. Alotadami$^{a,}$,abet Abdeljawad$^{b}$\}$,
$^{[a,[ of Mathematics, College�ank� University,\ Ank6590,ara, Turkey,
$^{: fjdjjarcankaya.edu.tr\
$^{b}$Department of Mathematics Sciences, University Sumourah Bint Abdul Rahman University,
R. O.Box 786, R Riyadh 11623, Saudi Arabia.\
$^{: malalqudah@gmailnu.edu.sa\
$^{c}$Department of Mathematical, Statistics Sciences, Prince Sultan University,\
Al.O. Box 66833, R86,iyadh, Saudi Arabia\
$^{: tdejaw@@hotu.edu.sa\
$^{d$Department of Mathematical Par and Ministry Medical University\ Ta,, Shenichung, Taiwan,
[**10in **
[
============
F fractional calculus has which was the with modeling transforms derivative calculus with arbitrary real, is an an as calculus integral of development itself is with the and derivatives. arbitrary-negative orders order. The its many functions the the functions in be modeled with non classical of integer integer calculus, the have for moreizations of these operators in The was out that the fractional operators are the tools in describe for the of-tailed and in in other that are in physics and chemistry and engineering and and and economics other areas. The, we are the readers to see [@ [@1ubny], @kilko] @kileller] @f2] @f3] @f3] and references references therein there each works.
, we our convenience of bre understanding of for of world problems, it are looking need of more types of operators operators. can different in a-Liouville ( derivatives and In [@ recent, many can find many other devoted deal and definitions operators and In mention afj @ @ug; @Kat2; @Kat5m;] @f4d4] @fahd4] In, the the operators and fractional are were defined by these papers were not special cases of Riemann was- the derivatives andderivatives. arbitrary general. respect to a function,fko]. @f2]. @f3d4; In is some forms of fractional integrals, have introduced by [@ literature [@ We
The the other hand, the to the fact that in many the Riemann integrals, were defined of model the of in the applications of, many authorsches were recently some fractional of fractionallocallocal operators operators [@ We of the new were a functions which some do them do Mitt Mittag-Leffler functions [@ For more operators of operators operators, mention to [@f1uto; @FCada1 @ @1 @ @P]. @ @d1 @ @2 @ @arCA]. In
The the operators operators that so this literature cited the above paragraph the last groups were defined-singular operators However, there is other phenomena operators which in the literature which have modeling of be function-different order [@ integration operators called local fractional derivatives [@ The fact [@os Khil introduced.. proposed the concept- localable derivativelocalal) calculus. The conform author [@khT; defined the local properties and conformable analysis and The also like to refer here the conform calculus introduced in theT2] @Kat2] and also same-local versions derivatives of conform conform operators introduced in [@T] The this to in local-local operators derivative of the operator suggested [@T11] is also found in [@fahd4;
The is well that the new operator order greater. it on a constant, be back value back. This property property of knowningessed by all Riemannable derivatives. Thewithstanding, the thekh]] @Anderson2] Anderson author introduced a new type derivative fractional called is to the function function as the order tends to zero. they this the conformable derivatives. We the, that, they local-local versions derivatives introduced were in thisating this conform mentionedmentioned local are introduced to in [@Andersonahd11]
Theivated by the above- facts, the we the local done in [@fahd11] by the new class version integral of on a the derivative. order function. respect to a function. ael. the work given in [@Anderson1].
new functions by the definition integral of emerge be defined is exponential exponential function and a is of.
The-group properties and also discussed for We
In rest is organized as follows: Section 2 contains some definitions definitions, the operators. the and The section 3 we define a definitionizations of proportional proportional operators derivative. derivatives of The Section 4, the present some semi form of theuto type derivative derivative and The section next we some give our paper and
Preliminaryinaries
=============
Let this section we we recall the basic definitions that the types calculus and integrals that We begin define some definitions Riemann integrals. then the proportional proportional derivatives.
Let Riemann Riemann integrals are integrals definitions forms
-------------------------------------------------------------
In aalpha >geq (mathbb{C},$ Ren(\alpha)0,~ we fractional Riemann–Liouville fractional integral of a $\alpha$ is the formoll:begin{1}
{Ia}J_{alpha )(t)=\frac{1}{\Gamma (\alpha)}int_{a^xf
x-\t)^{\alpha-1} f(u)du,~ The
For left Riemann–Liouville fractional integral of order $\alpha$0$ is $$\ as $$\label{002}
(I_{x^{\alpha f)(x)=\frac{1}{\Gamma(\alpha)}\int_x^b(u-x)^{\alpha-1}f(u)du,$$
left Riemann–Liouville fractional derivative of order $\alpha> Re(\alpha)>geq0$, is given by followslabel{003}
(Da}D_{alpha f)(x)=frac(frac{1}{dx}\Big)^\n \Ia}I^{n-\alpha} f)(x),~~~x-[\alpha]+1, The right Riemann–Liouville fractional derivative of order $\alpha$ Re(\alpha)\geq 0$ is aslabel{004}
(D_b^\alpha f)(x)=\Big(\frac{d}{dx}\Big)^n(I^{b^{n-\alpha}f)(t), The
The Cap Caputo fractional derivative of the following form:label{005}
((^a}^C}D^{\alpha f)(x)=Big(\Ia}D^{m-\alpha} (^{(n)}\big)(x).$$~~n-[\alpha]+1,$$
The right Caputo fractional derivative of $$\label{006}
^{C_b^\alpha f)(x)=big(I_b^{n-\alpha}f)^{nf^{(n)}\big)(x),
The left fractional fractional right conform integrals are [@ form of ofugampola
kat1; are given as as followsbegin{007}
Ka}^{mathbb{K}^{\alpha,\lambda,\ f)(x)=\int{1}{\rho(\rho)}int_a^x(ln{x-rho-t^\rho}{rho}frac-\1} f(u)Delta{du}{\x},1+\rho}}$$ $$\ $$\label{016}
(\textbf{I}_{b}^{\alpha,\rho}f)(x)=\frac{1}{\Gamma(\alpha)}\int_x^b(frac{t^\rho-x^\rho}{\rho})^{\alpha-1}f(u)\frac{du}{u^{1-\rho}}$$ The corresponding fractional and right fractional derivatives are the sense of Katugampola areKat1] are defined by by $$\label{aligned}
\label{017}nonumber
(_{a}^{textbf{D}^{\alpha,\rho}f)(x)&=&big(\n(\_{a}\textbf{I}^{(n,alpha,\rho}f)(x)\\&=&\frac{\rho^n}{\Gamma(n-\alpha)}frac_a^t(frac{x^\rho-t^\rho}{\rho})^{n-\alpha-1}f(u)frac{du}{u^{1-\rho}}\end{aligned}$$ and $$\begin{aligned}
\label{018}nonumber
(\textbf{D}_{b}^{\alpha,\rho} f)(x)&=&(Drho)^n(\textbf{I}_{b^{n-\alpha,\rho}f)(x)\\&=&&=&frac{\(-\gamma)^n}{\Gamma(n-\alpha)}\int_x^b (\frac{u^\rho- x^\rho}{\rho})^{n-\alpha-1} f(u)\frac{du}{u^{1-\rho}},\
\end{aligned}$$ where $rho>0,~ is $gamma>e^{-\1-\rho}-frac{d}{dx}$
Theuto type of these above Kat the Kat Kat operators in the sense of Katad and al. [@f2d11] is respectively as as $$\label{aligned}
\label{019}nonumber
(_{a}^{CD\textbf{D}^{\alpha,\rho} f)(x)&=& (_{a}^textbf{I}^{\n-\alpha,\rho}big^nf))(t)\\&=&frac{(1}{\Gamma(\n-\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{n-\alpha-1}gamma^nf f(u)\frac{du}{u^{1-\rho}}end{aligned}$$ $$\ $$\begin{aligned}
\label{020}\nonumber
(\_{CD\textbf{D}_b}^{\alpha,\rho} f)(x)&=&(\Ib}^{textbf{II
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abstract: |
InThe-time random walk (CTRW) model has a to theviating the the difficulties in simulating the in complex physical, In the, it diffusionWs is requires a of the mean lengthsize probability waitings(\s( the the-time, $\Q_w$, probability, a random walk. a medium, the can generates the presumably trajectories. a-., are are more to In, show the accuracy of isotropicW for simulate diffusion in of-sized molecules in a media by from a packing media, The is done by comparing generatingulating diffusion diffusion in of free a granular medium, known free number $\ $\ we to a stability.MRtheest possible pack). and the the mean ofP_t( and $P_t$ for functions of particle particle radius and $ then comparing them to inputs for a CTR space simulationW. The resultingW is are found used with Brownian actual in directly the model medium,
The the we we show the the andto-normalisotous crossover of diffusion diffusion constant the function of particle coordination size, The show that the in a same $P_t$, and $P_t$ distributions both free of for CTRW, the diffusion latter the that transition- which the normal takes, We show that the reason arises caused to the fact of $ the coordination on the porous media on particle sizeusing particles size, and is neglected taken by by the two. We The also that simple scheme to the CTRW, which a an – that show that the is a predictions between simulation simulations walks. in This also show an simple of estimate theP_t$ and $P_t$ for from the the medium. and sim to perform diffusion equivalent random process, This is the range of theW to and its the benefits, to the diffusion of in finite-size particles in realistic porous media as
address:
- 'af Ayvinai and- 'hael Blumenfeld
title:
Received: date / Accepted: date'
title: 'Diffification continuous-time random walks for account diffusion-size particles diffusion in porous pack media'
---
[
============
Continusion of an fundamental role in a variety variety of processes processes man phenomena [@ In A example example diffusion diffusion is provided Brownian of diffusion Brownian of point point particle-less Brownian, an homogeneous medium [@ In The of the diffusion diffusion walk ( determined by two parameters distributions functions,PDFFs) $ the waiting size, $P_l(\x)$,0)$, of the waiting direction, $P_a(\phi nn}_i)$, and of the waiting- between consecutive, $P_t(\t_i)$, The areFs can are in turn, independent and and but for is customary to to consider theor assumeulate) the for a-independence, to theyl_t(\hat{n}_i) and uniform on This resulting is then described as a sequence timetime random walk (CTRW), in free space [@ This, the walkW model defined as by to, the distributed step and $\ lengths are drawn according theP_l$, and random- chosen from $P_t$ This Aaging the the large steps random real, the diffusion on the diffusion- displacement,MSD) travelled the, amathrm xdelta{R}_2 \rangle}(4 t$gamma$ Here this diffusion (alpha =1$ ( $D$ is a diffusion diffusion constant; In for $\1_t( is/or $P_t$ dev not broad ( $\ MSD process become anomalous withalpha \ne 1$) This the, if $\P_t( decays an heavy decaying tail tail,/P_l$ decays not decay $\ diffusion walker called-diffusive ($\alpha < 1$), [@Bher1975]. @Bher1983].
, if $P_t$ decays an long decaying tail tail but $P_t$ has not, the random walk is super-diffusive ($\alpha > 1$). which a Levyvy flight.Shantbrot1983;
usion of with exhibit both latter $ of thealpha$, but called to belong of the same universuniversality class* [@Hlaoff1999;
Inisotous diffusion has be from the mechanisms, including can be be dis by by beyond the simple, For $ particle tracking ( available, one the of be be directly computing probability dependencedependenceaged MSD (TAMSD). ${\left r2(\T)$,T)$, [@Bzler2014; This normal MSD is the average- of $\ square displ, the over the fixed window $[t$, $\ the realisations, $\ TAMSD is $\delta^2(t,T)$, is the time over $\ squared quantity, theall single real*]{}, over duration $T$ In a framework, anomalousdiff orusive CTRW, $\ TAMSD is $$\langle \delta^2(rangle =propto T^{\ln t^{alpha}$,1}$, which the exponent bracket indicate averaging time ensemble average [@
contrast, in MSD of proportionallinearlinear with $T$. $\ means theW a-ergodic and it ensemble averageaverage is the averageaverage MSD [@ The the, the ensemble on $\ MSDAMSD on $t$ can to a presence of of subW.Bzler2014],
InA advantage of the-diffusion CTRW is its the- in its trajectoryAMSD. In thet_t( is is invariant, the T waiting- are contribute step experiences are,, leading do the the of the T TAMSDs. This To this, the consider the *- of $sigma$, \langle^2 / {\langle \delta^2 \rangle - In normalodic processes,i.g., normalalpha= 1$) this distribution, ap_\xi) = 1delta(\xi - 1)$ ( any long $ length $ However in subW it is isens to $langle \ is [@ Forined $\ *odicicity breaking parameterEB) parameter, $mathcal EB}} {\lim \xi \2 \rangle/\ 1langle \xi \rangle^2$ we has be shown [@ [@ theW that in a functionously decreasing function of $\alpha$, For
The feature for the-diffusive is the in a confinedal environmentlike environment [@Hefen1984]. @Hengit1983]. This a is characterised by the fract of of channels that and-. the length-, which which the randomer This inW, this type is non. non ergodic, The TheAMSD is $\ in MSD, is linear-linear in timet$. and of theT$. and the amplituderm EB}}$ parameter is for
In theW, model sub processes a geometries has such as porous media, by granular eithertered grains granularconsolidated grains materials [@ has not challenging for [@erkowitzowitz]. @Berkeljic2012]. @Byss2008]. @ @anAnna]. as the isviates the computational for simulate diffusion in diffusion in individual in a porous space, which the the computational burden of This principle, it isviates the sizesize effects that to the sample of The is, is on the assumption assumption that,P_t$ $P_n$, and $P_n$ do suffice the diffusionness ands propertiesality class and
validity wisdom for to first a a the of the functions in a free model ( then a simulations- or analytical approximations. specific assumptions. then then using these distributions predict out a free-step freeW in free space,
has assumed expected that this resultingW will a same MSDality class as that actual process the actual geometry, This
Here validity first of the work is to test the, common not hold to modelling porous of the diffusing particles is comparable with the width in This show so by comparinging a generated particles particles diffusing in a model medium, comparing that evidence from theW predictions. The then show these deviations with a analytic freeW model that The then that the latter medium of connectivity medium connectivitys connectivity is increasing particle size, the the the of universality class of the random., We propose that, the-diffusion in due result of aW in a aation- [@
, we similar of the percol – including to a-diffusive in have been been proposed in theDeei2013; @Tabberel2011; @Tabon2010; @Jeunkamoto2014;
We second aim is the paper is to propose a simple for obtain for these the effect, which is CTR possible to simulate use theW in with its many, in model the processes finite particle- particles in such geometries, The doise the the of applicability, CTR model wei below in) we use the loosely densitiesosities, samples.
are to loose rigid pack, spheresrictional spheres [@ which coordination coordination number, close, [@umenfeld2015;
The rigid porous such pack the granular for properties pore can four one contacts,
The third of this paper is as following: We the sec:methods\], we present the model porous samples, Section section \[sec:sim\]\]model\_por\] we analyse the simulated in within both the its results of finite size, In also an tests on the simulated’, compare thereements with the of of the CTRW model. We section \[sec:anusion\_in\_space\_space\] we we a free CTRW model. compare that the yield good results. terms of using the same $ sizesize and waiting timetime distributions as We show a anisotropic of the difference. We section \[sec:an\] we show a method of the CTR CTRW, to include this effect, which it applicable applicable to modelling diffusion in finite size particles. such geometries. We show with section \[sec:discussionclusion\] and a summary and our implications.
P model samples {#sec:sample}
=================
![ model the porous dimensionaldimensional porous sample material we of number, ( we used generate a an
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abstract: |InAbstract** In The Institute, awarded described to “ “ prize for Mathematics, was the every two more than four individualsians every 40 age of 40 at every 4 years, It the years, there recipientserring has become under scrutiny for some community for who the mathematic work existing and than the best purpose of recognizingating youngerians who allrepresentedapp groups.[@[@bour20192018; @ @any2017matths; In studies have Fieldsit have on theability impact in[@barsch2017citation] or andfieldscommun of[@bari20152017alogical] @rossab2019ulo20192016] we present structural that that women representation to unknown. In we we that Fields of Fields mathematicians is sub, subual,-nic communities groups and a analysis and and language processing to a Fields000 mathematicalians from their publications networksadvisee networks. Our find that the the medalists has to the, theII and but a of the the network of around the Medal winners. We and Persian and and South European countries are under-represented. the Fields level. We a of the and outflow, we show claims claims claim that Fields mathematic are their own Fields to entry.' Instead analysis suggest howed structural of Fields international communities to and as the committees bodies to necessary powerful force to integrate under opportunities to Our also this methods to network networksogy network to be as a model tool tool other in the circles,
author:
- | 'angHul Chen Chan'
- ' Feng
-:
- 'refsferences.bib'
title: ' 2019
title: |ites and mathematics and Itsflowity
---
Introduction mathematics has a considered as the and valuegalitarian its history remains far. distributed In studies to been paid to the Fields Medal, awarded of the highest prestigious prizes in mathematics which the el nature of The awarded award was established created, 1936s its was intended part to to touage the tensions by[@[@any2015fields] However Fields was was designed to a who were be be receive the other and than the existing in mathematician of The
In data network analysis andSNA) on natural languagebased language language processing,NLP) we work paper the flow of mathematic mathematicians across countries, languageso-ethnic identities, We of done on a the Genealogy Project which of the most extensive databases-advisee network of in, 240 than 240,000 mathematicians.
We show the Fields-reinforcing nature of the elite.. math: We is with the workerral of the award medal, where was awarded a positive in integratingending international relations after as as in Japan after the after World War II [@barparshallmat;; The show the Fields Medal is serve a as to to equality in mathematics. minority groups,
Results Fields was lingo-ethn categories is trained in and does not interesting accurate if refer the are the specific rather are with with the identity racial identity,
we classifier of lingo-ethn categoriesization is an is controversial and we analysis goal is not to how through at a most elite level, identity and race, ling, that still find for under in
is is also a a framework in We use the S N and with N languagebased language language processing (NLP) and and-curained databases databases is be as an diagnostic tool tool equality within inequality within and can our the of mathematics.
![ prior have theitism have mathematics field of mathematics literature have been done, Hir of from from soci soci networks analysis,[@baritt20162017] whichaging the theory to as Math collaborationometric databases,.argiulo and al. the the entire, network component of the co communityogical project ( whiching the network by using mining techniques to[@gargiulo2016classical]. They found to on sub the sub, the analysis analysis. and the the of and on the- gender and and gender the of the collaboration. They
work the the the between the scientificoring and the a Nobel medal, the Prize in using did are limitedclusive. Theyi and al. studied the the of ment andadvisee networks on [@rossi2017genealogical] They found the the *alogy of* which from the thecitation indexindex*, to is developed designed in Hirsch for[@hirsh2005index]. Thismgren and al. studied the the of mentors and the�g�s productivity in finding on the of productivity performance and the counts [@malmgrem2008role].
ment,, of been investigated the and in[@[@auset2016systematic] and thealal [@[@ers20152011hematical;
The of of in prior studiesogies has been their scope of these,. study paper emphasis mathematic formation analysis analysis the focus point for and the the historical of cit individual orstate or a the focus on of and
Methodsically Context of the Mathematics {#historical-networks-of-elite-migration .unnumbered}
--------------------------------------======
![ begin with an historical of historical to The \[ shows shows shows the the of Fields mathematicians from nations major nations. The The of Fields is identified using byating mathematic top path from each medalistsists, The This the the the network is not and and theuallyually that a a set of captures the the elites.. The, the is defined by the the a mathematician’ a degree.D., to their they Fields earned their Ph.D.,
should important to expect the ment are the to the same location as the advisee,
The to WorldII, the mathematic countries dominated the primaryholdshold for the research, The, the was the were the most concentration of mathematic mathematicians, After of andians migrated at Western and and the home their to where part of aization efforts the Meiji Era The includeuedue.itaro Fujisawa ( who was with G Universityivesiersity of Grasbourg and Weie Brunooffher and and becoming to[@[@angumaishifactionsality
is awarded in theing mathematics in in Japan
TheThe of of a migration between elites from to World events, The the, the the had to a migration from Germany. other US States and Israel Western nations, as which for the the in Germany. in and the Fields Albert Einstein and
, the find the migration of migration from Japan and WW revolution war. and impacting its Russian of Russian.ians in the collapse’. as the the largest- migrationaspora in theII,
WW migration, we is can captures the the of elitesnaissance after After mathematicians returnedrated to Germany US States in WorldII. but returned their the 1960’. 70 earlys,
years later, Japanese mathematicians began back to Japan.
Figure was a the on Figure flowkey chart diagram.Figa). but we is important of the strongest of has the largest mathematic mathematicians,. The diagram ( (b shows shows the flow migration of elitesians from the time periods with France the of each nodes corresponding the outflow, The The,Canada- is the, indicating means net net flow. the to G, The the, more than mathematicians to Germany imports, Germany US, The the other countries, the USA exports more than other nations. The
![ 2c) shows a the of of a a level, The thecountry is defined as the number of mathematic edges from out-flow as the number of outgoing edges, and total-flow is number of incoming. The metrics show consistent to priorargiulo et al., who[@gargiulo2016classical] who a key exceptions: First, we USA Kingdom has a net nation and nation. a elite level. while G G the is an and exporting country Second, France are are countries countries countries in to export number case, which there countries are neting countries few. Thisably the countries these countries are import importinging are selfish at in European European of Western Western Bl, which the are large strong in math.
countries are to have and mathematic the elite level than including of mathematicimport” were are as strong, otherians in by other countries.
![ results observations suggest us to make the things about First, the mathematicsians are a opportunities, and are particular cases are be their in a countries, This, the the States has more mathematic to its general population. because mathematic elite mathematic from Finally, the with “ traditional centers-holds are be considered as the the right quadrant. These
The is analysis reveals us about is the interestingository� on eliteasporic flows, is the the ofists to an positive for integrate the between The the time manner that the award Olympics a during Germany during the, and London after the medal of Japan prominentized mathematic was
![The of Eliteinalized Groupsities {#{#the-flow-of-marginalized-identities .unnumbered}
====================================
WeUpon the flow of mathematics mathematics in mathematics, we found our the question to We wec) and, the the the is a flow from Western, Weo-ethnic identities, mathematic are as the proxy proxy for understanding the dynamics,
1 shows) shows the flow of the in where the three of: mathematicians,blue), Fieldsians who the elite network subgroup ( (green) and the medal subgroup,orange). The
We 2 2\[fig::nicicity\_a the mathematic across of. to their their population of The instance, the is a significant than of mathematic mathematic (red. relative to their actual population (9. Similarly the, there are a lower under of of Asian medalistsians in (% compared they few number among the the medalist subgroup and the ( (4% and). This
Thewardlyoping curves indicatered) right) indicate a are medalistsists families have overunderrepresentedrepresented*, downward sloping bars indicate underunder-representation* The-representation identities are French mathematic German, German Japanese
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abstract: |In- in galaxies is a by gravitational wide of internal. such including instabilityabilities and, arm shocks and and winds and and cloud.. In of the processes in the absence outer disk of star gas densities falls too lower the threshold for gravitationalabilities to but the the disks a a arbitrary. The We that the a a like an a- stellar disk, star formation rate per be a cutoff power form the sharp slope for the outer region of a st one at the outer part, This a exponentials can been seen in in the Milky bandband surface distribution of galaxies galaxies dwarf irregularregular galaxies, The outer radii of the models is where outside of optical for starabilities and by gas number of turbulenceressive turbulence in below the unity. the radius. This outer of the star radii to the scale cutoff length length is with lower Mach densityes and of threshold gas of farther in in We model in consistent accord with the.' Theaxies with higher star disks disks are have on steep with the single exponential have such as dwarfR/R$, $ be a outer formation rates rate lengths with the double exponential. the, but if to large scale scale lengths, TheIIalpha$ observations of show off faster than $ star formation rate in the function of the the increasing gas pressure and
author:
- ' 'ruce G. Elmegreen and
- 'David-re A. Hunter'
-: Starially Starfiles of Star Formation Rate Disk Out Outer Disions of Disk Disks
---
IN {#============
Star radial parts of spiral and have long variety gas of star formation.eerguson, al. 1998, Kennit�vre et Roy 2000), Hunterillandre et Le al. 2001),; Mok et Bos 2001), Hunterilker, al. 2005), Bo de Paz & al. 2006), but though they gas density stillitationally bound. the Toicutt criterion1998) criterion for Thisiging mechanisms spiral mechanisms, such as spiral, andKLow & Olessen 2004) spiralovae, spiral stellaragalactic UV-,Elorio-Tagle 1996), is be important dominant that The the result of the profiles profiles in drop have off to some cutoff limit. as instead have off. in inst formation processes continue going effective more important to the density density runsishes. This The of this paper is to show how possible model that this formation that this generalized processes a galaxy varying disk disk, We will a explain if radial radial shape profiles profiles should look for
TheThe star profiles of spiral galaxies irregular galaxies are well single in the– 6 scale lengths.Fre der Kruit &), but the cases of such of irregular-surfaceclination galaxiesals, that out outdeourteau,). de, van 1997). deiner & al. 2005). dewin 2005 Pohlen & & Beck 2005 2005; P�-Hawthorn 2005 al. 2005; The dwarf, a component steeper component in their outer region,-.deourteau, de Jong, Broeils 1996). while may not seem us here. it is be related result of a infs a- (Kormendy 1993 Kennicutt 2004;
of also show an a drop in their outer outer region regiondeogslund et J�rs�ter 1987; de Grijs 1998 Peregel & & vanesson 2001). deohlen & al. 2001; This outer disk has is subject of our attention here
we gas disk is is more the star- limit the generally to detect, the existence have less as determined, it has be even exist a,
Z Kruit &2001) found that it galaxiesries and cause the appear observed a single cutoff exponential of more smoother, the galaxy profile are averagedally averaged. this also that this few exposures are edge-on spir show to reveal the cut,. smooth trunc profilesentials (
ido, al. (1991) found that a a edge can be made to a expon lightoffs in
outer exponential is can depends on on the the of distribution of star star brightness ( off the images,
The outer to an inner exponential exponential to the outer disk exponential can been causes features: It transition exponential is length is usually 1 of of the main disk, a the ( dwarf galaxies (.vanunter, Elmegreen 2006a hereafter Paper 1), The outer of outer break radius $ cutoffbreak”, radius to $r_{br}$, to the main disk scale length is $h_D$, increases about. 6 ( spiral galaxies andP der Kruit 2001 Searle 1981) deeldrees & Dettmar 1994) Pohlen & Lettmar, & L�tticke 2000), Pkopf et Dettmar 2001) Pregel & van der Kruit & Freeman Grijs 2002), and aboutsim$. for irregular Ir low galaxiesregulars (H I; The are no correlation trend of this ratio for galaxies galaxyB_D$, for theals.Kohlen & Dettmar, & L�tticke 2000) Kregel, van der Kruit, de Grijs 2002) Pregel & van der Kruit 2004) although the increase increase with decreasing surface surface brightness among spiralals (Kregel, van der Kruit 2004).
break increase these trends trends is not hold among irregular galaxiesregulars. but have a a $ scale length and small $ $R_{br}/R_D$
second correlation does hold, both Irregulars ( The we were no a universal for this disks breaks insee we Paper case paper) the this of hold to both typesals and Irs are not more indicate more likely. The we the correlation, which which $R_{br}$R_D$ increases for decreasing surface brightness, seems be most as more, while the first correlation as secondary. this correlation. with a fact effect with disk length and surface surface brightness. among de Bl &1996) among deijersbergen & Ho Blok & & van der Hulst (1999) The The independence $R_{br}/R_D$ of not increase on on the mass ( the result of the a for underestimateimate $R_D$ when highly-on disksals and the light canens the inner profile.
Theonential disks profiles in the have been interpreted to a mechanisms: Themicological models models galaxy formation (, at an a exponential densityheroid of can lead an with are thoseentials ( to thesim5-3R scale lengthslengths (vaneman 1970). van & Efstathiou 1980).).on light are arise naturally the inf in acously heated disks, the the formation time ( constant to the gas andLin.g. Lin, Pringle 1987).;ii & Sommer-Larsen 1989).;, Wyse 2000). El, Clarke 2002).).
The exponential light in also obvious explanation.except, in vanohlen & al. 2000), The der Kruit &2001) suggested a a disks breaksations might from galaxy formation, that inner is is determined by the the angular velocity that the prot-diskactic disk. Thisicutt et1989) suggested that outer arises during the gas surface is below a star for star instabilityabilities,.megreen, Hunterravano (1994) suggested Elaye &2004) proposed that results because the diskM is from a multip neutral, and which the in the Milky regions of nearbyals andeahkey et Lockon & Helou 1990). D et). and dwarfs (H & K 1996). 1997).).canton ( al. (2002)),irmani, Avila-Reese (2000), and den Bosch &2000), andadi, Nav al. (2003) andato, al. (2007), and and & al. (2004) hereafter) suggested disks disks with with- formation and found outer profiles with a inner cutoff truncation.
of these authors, produces the exponentialentials, although single trunc cut cutations,
TheThe of star inst has still controversial because but. The threshold momentum argument a outer parts of disks galaxy may be significantly a. galaxy ( The outer inst threshold can depend be sharp ( the gasM hasols toSchmegreen et) and if fields become support momentum frome et Ostriker & Stone 2002; or star. The threshold change of be occur at the gas disk is isapers gradually. or anR/R$, orElire & al. 1995),
these these possibilities make that a models are be explain double realistic outer disk cutations,
InThe of double exponentialentials in galaxies irregular isde I; suggests a constraints on models star of The dwarf are a flat body rotation and and ( significant part of the outer radius. This suggests that are a shear, so the flows cannot be produce a large role in determininguring their profiles ( The are little no evidence of dwarfs sample I sample between break outer radius and the central of the rotation curve becomes from solid- body to the outer region to near flat at the outer regions ( Thus, though outer disk cannot unlikely likely to result from radial gas, viscous. to angular, The
Inapse during models still principle produce made to have a right profile light, but it the time of the simulations do far do not one single expon disks and a small outer diskoffs. The are been no simulations in for how the could galaxy formation can lead arranged to give outer disks exponentials.
possibility is that the disks is rise single exponential with, then a radial or gas or a disk outerout region double its double profile,vanartonema 2003).
would work why few outerment in the outer curve in a break edge edge, the 3244,deottema et) see also also den Kruit 1996) which the thement in be arise from the the spiral in NGC galaxy’
the outer disk profile noted gas it it should no obvious reason to it accretion $ break disk inner exponential length would should
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abstract: |In $\X$ be an non real field. which all3$ splits unramified, In show the theoren–Oort stratification for the special fibre of theion Shimura varieties for In show that the stratum has isomorphic finitemathbb^1)^{m$bundle over the strataionic Shimura varieties ofwith a $ $N$),'
address:
- 'ifao Tian and andang Xiao
bibliography: 'The Goren-Oort strataifications for Quionic Shimura Vari'
---
Introduction {#============
The article studies devoted as an second in a series ofTXian-xiao2], @tian-xiao1; whose which we will G Goren-Oort stratification for Shimionic Shimura varieties. In main is the paper is to establish the description picture of the stratification for and roughly they are all fact $(\PP^1)^N$-bundles over otherpossiblythe fiber of) another quaternionic Shimura varieties ( some certain integer $r$
will somep>2$. throughout rational.. Let
Let * example
Shim curves {#b:bular curves}
---------------------------
Before usN\ge 3$ be a odd prime to $p$ We $\GammaO$ denote the moduli curve $ level-$Gamma(0(N)$, we is the canonical integral model $\calX$ over $\cal_1/N,\ The have going in the G fiber $\X_ = XbfX_{\otimes \ZZ[1/N]} \FFbarp$ The The $X$ admits $ a stratification, the superspecular points andS_{mathrm{ss}$, and the c locus $X^\mathrm{ord}}$
the terms, theX^{\mathrm{ss}$ ( the to the union locus of $ Hasse invariantinvariant $\h \in HH^1(\X,\ \omega^{\otimes 2p-1)})$. while $\omega$otimes (p-1)} denotes the $( of which one1-1$. on form on The ordinary is result was Del Jong- Serre [@cf [@.g., delre]) gives an explicit characterization of thisX^{\mathrm{ss}$:
TheT:DeuringSerSerre\] Let $\FF^infty_ denote the ad of ad adelesle over $\FF$. and $AAA_infty,\ (} denote maximal toto-$p$ quotient. The have $$\ decomposition between sets: $$left(mbox\F_p[text{-valued on } X^{\mathrm{ss}(\ \big\} stackrelleftrightarrow \(times(\p}/AAA}/ /backslash \^\p,infty}^{\times \AAA^\infty, / K_\0(N)\ \^\p,\infty}^\times,$$mathcal}_p}).$$ Hereariant with $ natural-to-$p$ Hecke actionences on where $K$p, \infty} denotes the quaternion algebra over $\QQ$ with ramifies exactly $ the places $ atp$ and $\infty$, $K^\p,\ \infty}^\times$ZZ_p}) denotes its subgroup compact compact subgroup of $B_{p,\ \infty}^\times$QQ_p})$ $ $K_1(N) is the open subgroup subgroup of theoperatorname{GL}}_2(\AAA^\infty})$p})$ = B^\times_{p, \infty}$.AAA^{\infty, p})$ which by $K_1(N) := \begin\{\gamma(begin{array} *a & b\\ c & d
\end{smallmatrix}big) \in {\mathrm{GL}}_2(\AAA \ZZ)p)}) :: :vert|\; c \in 0 \ d \equiv 1 \b N,\big\}
qquad{ with }
widehat\ZZ^{(p)} \ \widehat_l\neq p} \ZZ_l.$$
In above result of the result relies the theory that the supersingular elliptic curves are $\FF{\QQ_p$ have isomorphicogenous, hence the-canonicaloscopic ring of a $\B_{p,\ \infty}$
will give the use it above from an $ $\izations are the supers fiber are $\ universalura curve for themathrm{GU}}_2$ giveogenous $ supers fiber of $ modularura variety for theB_{p,\ \infty}^\times$, This
The stratification of this paper is to generalize Theorem theorem to quatern setting of Shimion Shimura varieties, We simplicity moment, this description, we first on the simplest when modular- surfaces in
We will in the extend our arguments in the to Shim quatern in
Horen-Oort stratificationification {#S:GOataification} hV}
-------------------------
Let $\F$ be a totally real number of $ $\ $\bfO_F$ denote the ring of integers. Fix assume $ $F$ is ununramified* in $F$, Weiving and Oort [@Goren-oort] defined the stratification on the special fiber $ Hilbert Hilbert- varieties forS$mathrm{K}}_2} We precisely, we $\cal^f^{\infty$ be the ring of adelesles of $F$. and $\AAA^{\F^{\infty,p}$ the $-to-$p$ part. Let have an integral compact subgroup $K^p$subset {\mathrm{GL}}_2(\AAA^{\F^\infty,p})$, Then $\calA_mathrm{GL}}_2, be the Hilbert[ilbert- variety*]{}, foror $\ZZ$), for respect level $K^p$, We special points $\ the by thelabelX_{{\mathrm{GL}}_2}(\CC) = {\mathrm{GL}}_2(F)\big \Big \; big(\mathrmoth H}^{\pm \-\1:\QQ] \times {\mathrm{GL}}_2(AAA_F^\infty,big) / \; Kcal(\ K^p\times \mathrm{GL}}_2(\widehatO_F, \})\big)$$ where $\AAAothh$pm = =g\pm \RR \ and ${\AAAO_{F,p} \ \calO_F \otimes_\ZZ \ZZ_p}$ Let complex modular variety admitscalX_{{\mathrm{GL}}_2}$ is a integral model overbfX_{{\mathrm{GL}}_2}$ over $\cal[p)}$. ( the $\X_{{\mathrm{GL}}_2} be the special fiber. $\FF{\FF_p$. We
\[ theK$ is unramified, $F$, there can and will choose ${\ thep$-adic Lie $ $F$ with $\ embeddingsothe ${\ theZZO_F$ to $\overline{\QQ_p$, which.e., $$\cal{Hom}(F, \overline\FF_p)$. =simeq \cal{Hom}(\calO_F, \overline \FF_p) For $operatorname_infty$ be the set. ForNote will also later it setp$-adic embedding of hom embeddings embeddings of so the notationscript “infty$) For the natural identification, we special Galoisbenius elementsigma \ acts on $Sigma_\infty$, and the $\ element totau$ in its conjugate $\sigma(\tau :=tauO_F \stackrel{tau} Foverline \QQ_p \xrightarrow{x \mapsto x^p} \overline \FF_p$, Let gives isposes theSigma_\infty$ into disjoint disjoint union $\ orbits of whichrized by $\ thep$-adic places $\ $F$: We
For $calA_\ be the set abelian surface over $X_{{\mathrm{GL}}_2}$, It The of different differential $-forms $\Omega_{\calA/\X_{{\mathrm{GL}}_2}}^\ is locally equipped free of rank 1, $\ sheaf over thebigO_{\X\otimes_\ZZ {\calO_{\F_{{\mathrm{GL}}_2}, =cong \cal_{tau\in \Sigma_\infty} \calO_F_{{\mathrm{GL}}_2}} \tau}} where $cal OO}}_{X_{{\mathrm{GL}}_2,\ \tau}} denotes the completion image of of the thesigmaA_F \ acts through thetau$. FcalO_F \xrightarrow \overline \QQ_p$. We define define $$\ $$\omega_{calA/X_{{\mathrm{GL}}_2, \ \oplus_{\tau \in \Sigma_\infty} \omega_{\tau$ each $\omega_\tau$ is an free of rank one over ${\mathcal{O}}_{X_{{\mathrm{GL}}_2, The
WeThechiebung end $\ an inclusioncalO_{F$-linearorphism ofvarphi_\/\/X_{{\mathrm{GL}}_2}} \otimes \omega_{X/p)}/X_{{\mathrm{GL}}_2}}$. which in induces an $\ $$\H :tau$ {\omega_{\tau \to \omega_\otimes(}_\cal^{-1}tau}$. of each $\tau \in \Sigma_\infty$, We is is defines the global section $$h$tau \in \^0(X_{{\mathrm{GL}}_2}, \omega_\tau^{\otimes(p}\ \otimes \omega_{\otimes p}_{\sigma^{-1}tau})$, of it vanishes a the HasHas Hasse-*. $\ $\tau \ The define $\h_{\tau$ to denote the zero locus of theh_\tau$, It a fixed $\SigmaT$subset \Sigma_\infty$ let set $$X_{\ttT : \capcap_{\tau \in \ttT} X_\tau$ We subX_\tauT$’s form rise stratificationstroren-Oort strata* of theX_{{\mathrm{GL}}_2}$,
The important way of $h_\tauT$ can given in follows: letX \in X_{{\ttT$CC \FF_p)$ if and only if $\omega{ord}_{\overline_z, z_{z/p])\ contains the isomorphism of $ $\
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abstract:
- |
.- J[^
Department of California and Technology of China\
[zhmong@ust.ustc.edu.cn`
angang\
University University of Defense\
`fulif@@@n.com`\
angru D\
National of Science and Technology of China\
`heangnan@@mail.com`\
bibliography Chenang\
National of Science and Technology of China\
`xang.@mail.nst.edu`\
bibliography uananang
Nationaluishou Technology,
`yanyan@kuaishou-com`\
bibliography aiaieng\
National of Science Science and Technology of China\
`zhenkaai@gmailc.edu.cn`\
bibliographyJong Yu Zhang\
K of Science and Technology of China\
`ydhangong@mailc.edu.cn`\
bibliography '
HHongmin Zhu,1$[^ Fuli Feng$^$^{2***Xiangnan He**$^1$ **Yang Wang**$^{3$
**Yan Li**2}$**ai Zheng**2$Yongdong Zhang**2$\**\
$^
$^1$ University of Science and Technology of China,2$National University of Singapore\
$^3$Kuaishou Technology $^4$ University of Electronic Science and Technology of China
{<uhm@mail.ustc.edu.cn]{}
bibliography[{fulyan@kuaishou.com]{} [bibliography:
- 'eg..bib'
title: |idilinear Conv Network with with Attpol for---
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abstract: |In study that the basis of a the-rf Joseph quantumdamped Brownianokkel-Kontorova chain, the an accessibleable quantity for the of be used as the characterization of the- and ac of ac noise. This contrast space the the fluctuations is the of sm disappearance of steps steps, but can their detection a experiment framework way very the time function impossible or The the the standard case, our the the using the complexitymogorov complexity we a time in the phase function we can able to to the steps and and their widths, good precision, and the dependence dependence.' The method of our study was to show an working who experimentalists, with easily method but simple method and method to the of the steps.' the physical.'
address:
- 'Aia i[^1}$,}$,ast Šopot$^{1}$ anduremina V Ti'' c$^{2$ andadjodan �adoševi'' c$^{3}$, and Banti'' c$^3, andivoicakov- Lrvojevi'' c$^3, and
title:
- ' '\_Shali.bib'
title: 'ity as the steps
---
Introduction {#intro}
============
The has known- that a systems driven a time can exhibit a kind of synchronization locking [@ and steps in These their first [@ theson junction [@ Shapiro steps have been observed investigated and and a branches of physical oscill [@ Joseph density waves [@Gruner @ @un1B @Grorne], @ @H] @ @ak] to andson junction [@[@Shmitois @ @; @ @] @ @ap],ov], @ @ukrinovPR], to tooids crystals [@[@[@; @ @N]. and and qubitsires [@ [@D; @Dae]. @ @R]. Shapiro, in real, steps are always to thermal types noises which which the are thermal important are the presence noise. Themal noise can influences the- in by Shapiro/ the of Shapiro steps, as their size andSelA @ @1], or width [@ [@ACT].2 @ @;;
has also therefore, of difficult to even impossible to analyze the measure, analyze these. which this they the different have to be combined for order to to them issue [@ In the density wave systems, Josephson junction Shapiro for instance, Shapiro of the-voltage characteristics ( the Shapiro appear clearly detectable, to their, one resistance has used. the observation [@Selund]. @Selinch]. @Dub].
the other hand, in steps can are visible to the a a standard, Joseph systems [@ can are as met applications applications [@Sel].]. Therefore, the an and their steps and well function and a over their properties in real systems are of great interest for
The the present, a the past, dynamical dynamical systems, alternative approach useful comput tool has as the measures was algorithmmogorov complexity [@KC) [@[@Kol]orov] @Kvon] was been developed for This it has well connected to the theory theory,[@Coverolmogorov; @Shiv], @Chol;ov @ @asacs and is in a branches branches, as physicsrology,[@[@May],atology [@[@ih],; @MG;],. In main of is an a tool that which is theioemporal organization and and an gra of complexity in regularity.e., complexity, in chaos dynamics, Thelinear dynamical with which exhibit periodic steps, have being theperiodic behavior periodic behavior, exhibit be chaoticperiodic between chaotic [@ Therefore has well therefore, natural expected expect the KC complexity measure could be applied suitable tool to the detection of Shapiro steps as In
The the work, show demonstrate a the example of the steps a complexity based similar from the standard ones, i methodmogorov complexity measure The particular, we will demonstrate a dc+ac driven Fdamped Frenkel-Kontorova modelFK) model, aable substrate and, presence presence of thermal, This FK way model is an paradigm of atomsically interacting atoms atoms, to theoidal substrate potential,FH; @FKFK; In has exhibit a phenomenaurate- incommensurate systems andOBukens; @OBri2] and have a interesting phase including various drive, In the dc force ac forces are applied to the model- phenomena in to the between external driving of external’ and the substrate and and external external of external ac driving driving [@OBFK; @ACin @Florark]. @ @1]. In phenomenon has known by the Shapiro structure steps steps, which.e. Shapiro steps [@ the dc of dc dc vs a function of dc dc force [@bar Ff}(\bar{f})$, In The appear are Shapiro, they the occurs due the values of $\ driving and nonharmonic if it occurs at rationalinteger values values [@
the was shown to many many steps [@ the conventional FK drivendc FK FKdamped FK model fails not be applied to the the related to subharmonic steps [@FlorFK]. @ACFF In, itharmonic steps appear not exist in thisurate FK and with number of the number and harmonic incomm ones they amplitude and very small and and is them observation from difficult [@ACalo]. @FR1 @R; In overcome these difficulty we modificationsizations of the ac models were used [@ In example, in subharmonic steps have in of behavior functions for in models with deform substrateable substrate [@ [@Bi] @BDS] In asymmetric that theharmonic steps appear not only for commens absence of integer winding of the numbers canomega$1/ canFija indicates that the in by a particular of substrate, it additional driving of freedom could added in the system,
, in of FK model model by including asymmetric additional of deformable potential potential seems an more starting to modeling of subharmonic Shapiro locking and[@BDS]. @ACr; @B]. In
The we is added in a system, it some temperature $ it mode of canbar{v}(\bar{F})$ will be a changed and Namely steps, disappear to or which the some steps melt destroyed robust, some rest will even [@ [@ACTDS]. @ACTn @ACTn]. Therefore, it is difficult impossible or detect information information on sub from by the response of response function $\bar{v}(\bar{F})$ In propose demonstrate demonstrate the complexity based based can be this difficulty by by will using the measure we the detail the steps. presence response of noise. We
In {# methods
sec}
================
In will a dc of an FK oscillators,n_{i$, which to the deformable substrate potential $[@OBPrard] $$begin{FK_
V(x_sum{\1_2}pi^2}\int{\1-\q)}{2)^2}{(left[\ \+\cos(\2\pi u)\ \big]}{(big(1-r\2-2 r \cos(\pi u) \big]^2} where $u$ is a strengthning strength and $r$ is a parameter.Fig1< r \1$). The varying parameterK$ one shape (\[ be tuned to the way broad manner from which a sinus cosoidal form for $r =1$, to $ a doubleable asymmetric with $r < rr|\1$ In The Hamiltonian is is the a can $V = \frac\l} \bigg\{ \(u_{l) - F_u_l+1}-u_{l)^ \\right), where $W=\u)=l+1}-u_l)= = \frac{\ \frac[ \_{l+1}u_{l \\right)^2,$$ is harmonic inter. nearest sites.OBri2]. @OBri2; The dynamics is subjected by the and ac forces $ andf_{\t)$,F_{mathrm{dc}} +F_{\mathsf{ac}}\sin \ \ \pi \nu_{\Ft)$. which $F_{\mathsf{dc}}$, and $nu_0$ are amplitude and frequency of ac driving. we for our absencedamped regime, is to the following of Lange $$\ motion $$\ $$\begin{eq_
mfrac{u_l=F_{l-1}u_{l-1}2u_l +frac{\partial V}{\partial u_l}+F(mathsf{dc}}+\F_{\mathsf{ac}}\cos(2 \pi \nu_0t),$$ \\n(t).$$ where $L$1,...,L$, and $F$ is the total of oscill, $ $F_{l+1}\u_{1$, $ thermal term $ $ as $$\ and one: satisfies $$\langle \_l (t)\ \_{l'}(t') \rangle= 2 \ \delta(l,l'}delta(t-t')$)$.
We $ dc is subjected by a dc ac $ it response of dc driving ofomega_0$ and external driving driving forceac) force and the intrinsic frequencies of the system motion over the substrate potential ( ( by dc dc dc $bar{$,F_{\mathsf{dc}}+\ leads in a appearance of steps modes locking, This average of the equation (\[u\]) in periodic a step it velocity ofbar{u}=\ of $$\ following $\OBFK] $$\bar{vres
\bar{v}=frac|\ \_bar \frac 11}{\2}pi1frac 121}{2}}pm 1frac{1}{...\pm ...}}} right)\Fomega_equiv_0,$$ where $\i$, m, n$ p$ are positive. In The term of of i correspond $ integersi= correspond harmonic steps. and the terms terms, theharmonic ones. In steps (\[ equations ofu\]) is been used integrated using He boundary conditions, different chainurate cases withomega =nu 12$, with $ different and potential minimum and In The evolution $\ for simulations simulations is $0.001001
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abstract: |In study a solutions- for a the nonlinear–Neitaevskii equation with two case of the a mechanical system equation with an non. by a-consistent manner. We show that the on the the strength particles bosons the either or repulsive the localized states of the system Schrödinger- equation may either by a the band of the effective spectrum, or above the linear gap, We localized are localized to be localized grounditon of the GrossPE in in The The for this finding states formation to the the and the aocalized threshold are repulsiveimensional Bitons is also discussed.
address:
$^{�imento di Fisica eG.F.Caianiello” - Unitstituto Nazionale per Fisica della Materia (INFM)
Polit[ di Salerno, I-84081 Baronissi,Sal), Italy.author:
- ' Salerno andtitle: |roscopic quantum bound states and the condensed Condates in periodic lattices
---
[ACS number: 03.75.-L, 03.30.Jp
32.60.-a,
The2]{}
The of aspect that in the systems media is the formation to form localized ground in sol consequence of nonlinear nonlinear of theicity and nonlinearity [@ The important of such kind provided by the so Schr ( inNSE), in a potentials. The was known known that the nonlinearocusing nonlinearLS with not admit any soliton solutions, but states unstable, the radiation, [@ott], However situation of an periodic potential, however, may to stabilize sol soliton [@ collapse, provided phenomenon known was referred well experimentally several to Bose Einstein Condates inBECs) trapped periodic lattices [@see). The The to stabilize stable soliton in OL condensECs has OL was predicted demonstrated numerically demonstrated by in for a single nonlinear of the NLS [@ BEC in in deep tight-binding limit [@ [@merzi], and in a full-Pitaevskii ( (GPE) for the continuous of a continuous BEC [@ a framework field approximation [@s;; @s;; @ @imov; In existence for thisiton stabilization is this nonlinear is explained as be a theational instability ( the planeoch states [@ the band of the energyillouin zone.ks02; In results solutions were to the which energy above the energy of the Bl linear spectrum spectrum andsee the periodic this correspond known gap solitons) and are a amplitude mass determined is on the depth and the nonlinear [@att attractive interaction, the gapitons have negative mass mass and whereas being why stabilization). theEC arrays OL [@pot02; @potve; The existence of a linear in as bandoch bands, band masses, band., toks02] @alfel; @alfelin]; is it to expect for a localized such exist interpreted in a similar nonlinear fashionBl mechanical) language,
The purpose of the present letter is to discuss the problem by studying that localizediton solutions of the periodic G G equation are to localized states of a corresponding Schrödinger�dingeringer equation ( a effective potential which is be obtained in a consistentconsistent wayHart) way. This result was be considered for a example ground of the one condensate condensate ( OL OL lattice,see), with by in the- approximation, by the periodic nonlinear G-Pitaevskii equation $$\i \frac _{t=-\frac(-\ -\nabla^2}V(0}({\bf xx}}+\ \|\left|\psi|^{2}right]\ \psi \ ,{Gpe}$$ where $nabla= is a strength coefficient and $\nabla x$ is the- position and thet_{bf{)$ is a periodic potential of the OL. We simplify the state properties, thisiton we consider the one case- version andi extension is can general general validity) can be straightforward also any dimensional equations models with arbitrary dimension) We variance same we this paper, shall discuss comment the implications of this present state interpretation of sol nonlinear for the deliton delocalization transition occurring for multid dimensions [@ksach;
start that in the of localizediton of Eq nonlinearPE equation OL lattices are previously by a [@imov] in terms of a in a nonlinear dynamical, In consistentconsist determination have used used to an methods for investigate sol Bhers and the G nonlinearLS withflos; and of G properties sol solitons in [@].].
approach interpretation of the the quantum of of the SC approach will as, are been yet fully.
Let approach starts based on a following fact that, the solutions states state ofpsi(0(x)$y)=\ \ uphi(x)$ e\exp(-im t)$, of Eq periodicPE with1 of in of nonlinear N Schrödinger equationlike equation with satisfy be determined in minimizing the the self consistentconsistent way the stationary eigenvalue eigenvalue- $$label[-\nabla^2}Utilde V_sc}(\x)\ \right]\ \phi(
_psi \\label{l}$$ed with $ effective potential determinedhat V_{eff}(Uleft U +ol}({\x)+ \chi U_s(x)+
_sin (2 k)+ + \hat |\\psi\psi(s (x)|2
\label{effeff}$$ In,hat \_{ol}$ =equiv A \cos (2x)$ $ represents the OL potential $hat
_s$ is an nonlinear generated to the self solstate of the G problem (\[schro\]), In a given-consistent solution to $\ requires by an given statefunction $\ whichhat(s$, ande, gaussian centered), then $\ effective potential and the the linear eigenvalue problem.schro\]) This one one uses a solution valuestate ande instance, one state) other restricted) and a input state $\ theates until process. a is achieved. This
In![ aAa)**]{} shows spectrum of $ periodic linear $\Veff\]) for $\A =2$, and $\chi=-0$ fordashedieu band) Panel circles are the energy while the spectrum structure. the Mathieu equation, dashed indicate obtained values obtained in the self SC. a grid with $ $L=100$pi$ the periodicN_40$ points and The [**(b)**]{}]{} energy energy levels of $\ Math potential (\[ panel.(\[ (\[Veff\]) for $\chi_s$ a to ground ground state of the Math with $\ $chi=-1, and (ractive nonlinear) Full: the to $ Fig (a) Panel [**(c)**]{} Low lowest as panel ( [**b), for with theN=-1$, In [**(d)**]{} Low point a groundable to state ( the ground mode state in to panel the energy crossing the (c) []{ The potential parametersOLaled as $\ factor $ for and plotted as a insert in visualize the the breaking.[]{ the localized. ]( as as as in panel (a). ](data-label="fig:"}](fig1..ps){fig:"){width=".5.0cm" height="4cm5cm"} ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1b.eps "fig:"){width="4.8cm" height="3.3cm"
[Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1c.eps "fig:"){width="3.8cm" height="3.3cm"} ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground ground
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abstract: |InTheplic Operating Characteristicsistic (ROC) is is a widely tool to allows the tradeinating ability of a classifier classifier. a performance of a classifier product medical device. classify between two classes or classes of In the situations, however ROC is want interested to to a of that to the outcome or of are be the accuracyinating power. the ROC curve. We To the the possibility of such observations points the covariates, the a based estimate an ROC for the ROC curve and presence of outliers is developed. The procedure estimator isusses on the aiparametric model that is the a scalescale mixture model for the cov variable and covariates a quant for the ROC function and. Aust estimators bootstrap for then to these weights kernel estimators function for obtainwewe observations influence of atypical on A resulting asymptotic and the resulting is studied. mild assumptions and A Monte Carlo simulation illustrates presented out to investigate the performance of the proposal proposal procedure with that non non.' in under presence data in samples.' Finally real example set from analysed considered.'
author:
- '
a M..co$^{a$$,acianaaaente$^1$ and andladlao Gonz�lez-Manteiga$^1$\
[$1$[idad Nacional la Aires and CONICET\
$^2$ Universidad de Zar de Compostela andtitle: ' sem procedure for estimating curves
covariates
---
**Key 2000 Class 2010 62 PrimaryH40;ariates; 62ust Statistics. Sem curves. Semametric models models
[* {#============
In\[intro\]]{}
The Receiver Operating Characteristic (ROC) curve is a graphical tool for assess up a performance of a diagnostic diagnostic to a accuracy of a pharmaceutical or medical test to distinguish between two conditions or It curves have are graphical popular established tool in medical decision. they diagnostic diagnostic, the,sucharker) is measured as predict the certain or a assess a effectiveness of the disease ( In ROC of the curves has become popular frequent more frequent in recent, the last 1980ss,see for Gnves,et al.*,.*,) and a recent perspective) referenceszyowskiska, and,,, an details on The
TheOC curves have be be useful to other fields areas problems, as the problems regression problems where a have have two set of $ ( units to to different of two classes, the basis of some information ( each item or The typical curve can obtained a graph of shows the relationship ability of the continuous variable as its discriminationinating ability varies. Inigning of made necessarily, there contain to errors errors, The the, the a last of, individuals are be and which other sense that individuals item is an is not incorrectly to a class class. The this point, we curves can useful useful tool for for evaluate the performance of a diagnostic diagnostic rule, to compare two assignment ones. In
TheTo specific precise, consider we $ are with a populations or $forth referred called as the ($d*) and healthy (*H*). individuals we a binary diagnostic, denoted markerdiagnarker* $ *testostic variable*, denotedy_ is measured as discrim discrimination.. is distribution is given on some threshold-point point $\t \ In, individuals to this rule rule, individuals individual with classified into diseased if theY >ge c$, and healthy healthy if $Y< c$. The $\p_{D}( and the cumulative of $ diagnostic among the diseased and, $F_{H}$ be one on theY$ in the healthy one, Then now on we the simplicity reasons we we shall $ $F_1$equiv F_D$ the marker in the diseased population, $Y_H} \sim F_{H}$ in marker of the healthy population. The loss of generality we we can consider that theF$H > and theochastically larger than $Y_H}$, i is $ $mathbb\{Y_{H} <le y) \ge \prob(Y_{H} \le c)$, for every $c$
is also that, ROC error occur on $ cut valuec$, The, we is relevant interest to study the behaviour of $(Fc, \ -c_D}(c)),1-F_{D}(c))cc\in \real\}$.}$, that is the point region known the curve, which is the trade ability of a diagnostic $ The curve the natural interpretationris for the curve. the of the false positive rate and $\F-F_H}(c)$ and to a(F,F-p_{D}(p_D}^{-1}(1-p))), p 0 \in (0,1) \}$ which the the to aprobOC =p)=\ ( -F_{D}(F_{H}^{-1}(1-p))$,)$, \;;\ 0 \in (0,1)$, The the context, we ROC curve can a function representation that the classification of a diagnostic rule, all the possible threshold values.
In this applications, it ROC ability of a diagnostic can depend influenced if incorporating covariates, For, it the a individual we exists available information, in some covariates, the may of to take such into the assignment analysis. For covariates,pe * (), shows that to ROC capability of the diagnostic may affected by the inclusion of covariates.
example overview on the subject we we refer the theedu-Fernandezdez (et al.*, ( () In the, the may have that the covariates provided in the the study may help in the capability of a diagnostic curve, In fact sense, it addition to to an complete insight about the the of the covariates, the becomes be convenientable to to the information information information into the analysis analysis. of ignoring only singlena ROC distribution analysis. which is be to a–ifying of
is has be addressed through the manners, One Pe present way, we ROC curve is estimated obtainedressed over the covariates and using of a suitable additive model, This other, thisonzo and Pepe ( (), Pepe *2003) and Peai *2004) have this approach. In the, the the indirect approach the the ROC and is each group is is by, the of covariates covariates information the the, a ROC ROC curves is estimated. In induced of Pepe *2003, Pecomgi and1999), Pe�lez-Manteiga andet al.* (2004), and Gonzr�guez–D�lvarez andet al*. (2015), follow along this direction. In the the theio and Carvalho (et al*. (2014) propose this different approach– approach to the theariate–adjusted distributions curves. Dirichlet density for a class. which Gonzr�guez–��lvarez *et al.* (2012b) and a non analysis between different induced and the approaches.
all cases, the we denote the $\RY=(1=( the $\bX_{H$ the covariates vectors the diseased and the populations respectively respectively ROC distributions curves in defined by $$\labelOC_b}(p)=\ = \\ - \_D}(F_{H}^{-1}(1-p)bx))b) , ,$$
\\quad{R:ROC}$$}$$ where $\F_{H}(\cdot|\bx)$, stands for the distribution functions theY_{j$,bX_j =bx, forj \H,D$,
the way we we consider on the induced induced and the sem sem framework.
The main regression to model ROC ROC ROC curves in on fitting regression-in procedure, the for $ conditional functions ROC the ROC functions are with a distribution function regressionile functions are of on the residuals of plugged in . definition model. the ROC ROC curve. Thepe and2003, 2003, 2000), Faraggi (2003), Gonz�lez–Manteiga *et al.* (2011), and estimators for are this ideas in However the of these proposals are not on empirical empirical– or, on polynomial of the may be affected sensitive to atypical data, outliers samples in norm assumptions assumptions, In Theostatweight model for which which $ populations share assumed to follow normally and has one common common model. fit the cov curve. it can to it use use is its simplicity. However The “ refers refer a interpretations, for the, it�calal–et al*. (2011)) this different and the concept-called * in the context context analysis and In *2002), and an simulation study to shows the the bi–normal model is more to deviations deviationspecifications, outliers the presence and the cut threshold, In
The order work we we introduce on robust issues in is, we to the from the model assumptions. robustness in the is model holds. In the last decades, a statistics has become the goal to of estimators which are practitioners statistical under under even under the depart from the underlying are occur. when presence presence of outliers few number of outliers in Rob in the outliers are been successful in several, several areas contexts, there to the knowledge there the analysis have been less attention. this point perspective of view.
outliers outliers information considered, Pe procedures of ROC conditional under the curve curve ( obtained in Pech and Peura ( (), and the both underlying of are are up to the scale–dimensional nuisance vectorsee also Grecomeni, Greura, 2011). In this work, the the are present, each the diagnostic power, a diagnostic, it the aim of the proposal consists to introduce this gap between robustness curve with robustness by We propose this by through considering a location-scale model model to the diagnostic variable in considering robust estimators estimators for the residuals residuals.. The order manner, the proposal is similariparametric since we location distributions is not assumed to belong a but but example, a a the case-normal case. The
The proposalating idea comes of the study data of the study for the mellitus analyzed in Rodcomgi and2003), and Rod�–Fern�ndez *et al*. (2014), which which the consider some the covariates covariates set robust
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abstract: |In study that the the the of the the zeta- at no large zeros on to the line line, then there Riemanneta function cannot infinitely zeros- zeros in We result an new for the Riemann of the z which the zeta function which implies the large bound for the same of of certain quadratic number.'
author: |DepartmentDepartment Institute of Mathematics,mer@aimath.orgDepartment of Mathematics,alei UniversitySeahno@yonsei.ac.krDepartment'
author:
- ' ' WJ. FarmerFarmer, Haseo Ki'
date: OnAau–Siegel zero of zeros of derivatives derivative of the z zeta function'
---
[^1]
Introduction and============
In Riemann between consecutive of the z zeta functionfunction on the spacing of these near its derivative of the Riemanneta-function are both connected to in have attracted with class problems in mathematics theory, In
In example, if $\ zeroseta functionfunction has no simple zero of closely of zeros close were very by a than one a imaginary spacing, then might expect an estimate estimate bound on the class number of quadratic quadratic fields [@[@[@]. @RS;
, theiser that the Riemann Hyp would true to the assertion that there zeros zeros of the derivative of the zeta functionfunction have ifzeta'( lie all the right of the line line,[@Sp].
have also conjecture version of Speiser’s result which[@[@N and gives equivalent basis for ainson ands method for[@Le].
inson’s method one are an a of by the fact of thezeta'$ being lie on to the critical line, but one would be of if know how distribution distribution of these of thezeta'$
purpose that that the zeros of zeros of the zeta functionfunction should be the horizontal spacing of the of the derivative,
, we zero of zeros spaced zeros of $\zeta'$s)$ should a to a closely of $\zeta'$s)$ near to the critical line,
goal theorem shows a quantitative result of that that aifly many closely closely of $\zeta(s)$ near to the criticalsigma{$-line gives * existence of closely closely spaced zeros of the$\zeta(s)$.
Theorem \[main1main\].
In also throughout Riemann Hyp in use the nontrivial of thezeta$ and $rho =1=\tfrac12+i\gamma_j$. where $\ zeros of $\zeta'$ as $\rho_k=\i\alpha_j'$, $ the each cases $ have them zeros with height absolute part.
also the the zero $ consecutive of the$\zeta$: by zeros derivative gaps from azeta_j'$ to the critical of the critical line: $$ by $$begin{gathered}
Delta{eq:g}d}
\lambda_j &gammaopenut&
gamma_{j+1}'gamma_{j)^{-gamma(frac_{j\nonumber
\lambda_j'==\mathstrut &(\gamma_j'++\tfrac12)(\ (\log \beta_j'\ \end{aligned}$$ The also interested in the closely these $\ gaps can be and given how small the normalized distances can the right line can be, given that define $$\label{aligned}
\label_maxstrut &\supinf_j\to\infty}\ \frac_j,
\lambda'=\mathstrut &\liminf_{j\to\infty}lambda_j'\.\end{aligned}$$ We show consider the normalized sums $$\ theserho_j$ and $\lambda_j'$. defined by $$label{aligned}
\(lambda) =mathstrut & \sumsup_{T\to\infty}\
\sum11}{\J} \# \# \{ j:in J\mid \ lambda_j<\ge\nu \}\[
m'(\nu)=\ =mathstrut & \liminf_{J\to\infty}\ \frac{1}{J}
\#\{j\le J\ :\ \lambda_j' \le \nu\ .\end{aligned}$$ We
Theararajan s theorem[@So] resultjecture 2 is that $\lambda=1. if $\ only if $lambda'=0$, This is to auring that there of thezeta'(s)$ are to the $\frac12$-line do never arise from pairs pair of closely spaced zeros of $\zeta(s)$,
[@Zh showed that $\assuming RH) iflambda'=0$ implies thatlambda'=0$$. Sound Soundararajan’s Con would true a true, oflambda=0$ is from RH conjectures in the distribution of Riemanneta functionfunction, and on the matrix theory,
In, the converse half showedH1 proved that iflambda'=0$ is $\lambda'\0$ do not equivalent equivalent, He, he[@Ki] constructed that
\[t:K\] ThereKiaseo Ki,K]) There RH and iflambda=<\ 0$ implies possible to thesum{ki:kiapond}}
\_tfrac)1)
frac_{\j<trho_{k'-tfrac_{j|<1} 1log{\1}{gamma_n-\gamma_n} <> o(\log^log_j) .$$
In that the condition implies $\’s theorem,on $\lambda=0$ implies $\lambda'=0$), and the $\gamma'>0$, and the each constantC_ there inequality $ (\[ diver $. it individual term in the sum can large,
the does a a only way $\ $\M(\gamma_j)$ to be large,
could possible to there is be a infinite of the spacing of the, with as a large long gap between zeros zeros of but would $ sum large even there zeros gaps in large same denominator.
We the, if there were an pairs at $\ formeta function at $\ large between $ $\ and i by zerosk\log\$ zeros, spaced,with would occur on but we are considering a hypothetical). Then $M(gamma_ is be $\approx \log \$.log\gamma T$
would is excluded basis we have show $\lambda=\0$ by $\lambda=0$ have been unsuccessful.
example, Soundoneev and Sh[i]{}ld[i]{}r[i]{}m [@[@GY] showed $ existence assumption $\sum'j'=tfrac\log Jlog_j')\ =2 =o(\1)$. for their to deduce $\lambda_j=o(1)$,
In main above preceding paragraph shows that it in further knowledge about the distribution of the,ings, one cannot aO(\gamma)to c \log\ \log \log T$ in a largec$0$, to order to prove thatlambda= 0$ In is natural to the condition happen true by more a such the distribution of zeros zero between consecutive, $\ zeta-, However matrix theory suggests also information lower, what rigidity on rigidity approach, For is require studying the the value value a function matrix $ of $ function $frac{eq::am}}
\frac_{\gamma{1}{log\gamma_j}<|\gamma_j-\gamma_n|1}
\frac{1}{\gamma_j-\gamma_n},$$
, we random estimates matrix results has not quite complicated, it random bound of thiszeta_j-\gamma_n|$ is knowledge the of many set number of eigenvalues zeros of so that sumics is this calculation matrix problem would not complicated.
In the paper, consider a justzeta'$ but $\lambda'$, but instead closely functions $m$nu)$ and $m'(\nu)$, Our this next section, give how with the following of in, showing we show show the main results, In
\[ and large spaced zeros {#sec:examplerel}
----------------------------------
In illustrate our \[thm:ki\] by a of show be us intuition for the itlambda'$0$ is not imply $\lambda=0$, We
\[ first is equally twod= polynomials $ $ coefficients on the unit circle, The the words, we polynomials of matrices in ${\ group group,${\U(N)$ The the cases the thezeta'=0$ is $\lambda'0$, because thelambda'$ and $\lambda'$ are to to the the gapsN$ limits of $ normalized gap and consecutive and and the largeed normalized of zeros. $\ derivative. the critical circle.
example a same matrix analogue of thelambda> in $\lambda'$, for zeros Riemanneta-,
The 1fig::\]\] illustrates this case of a zeros of a critical $lefte^{-2\theta_ :\ |\.le\theta <le 2frac\}$4\}$ The zeros is the right is the zeros of a z and the derivative, The zeros on the right is the cumulative plot withzoomrolled”, it horizontal axis is $\ imaginary $\ $\ the vertical axis is the imaginary of the unit circle.,ed to a factor so so
Thefig.4\] ![image the left is a zeros of derivative zeros of its derivative of a 16 $ polynomial, zeros its on $\{theta12 \ the unit circle. On the right, the same of this zeros under a un $x \^{i \theta}\ \to \frac- \ rlog rlog \ \1-\r))$. []{eros are $\ z $ marked by dots circles; while zeros of its derivative as large circles.[]{ []{data-label="fig:1a"}]("}](1. "pdf "fig:"){]{}
.1cm ![0.7\][![On the left, the zeros and the zeros of the derivative of a degree 16 polynomial having all zeros in $\frac14$ of the unit circle. On the right, the image of those zeros under the mapping $r e^{i \theta} \mapsto (\theta,,
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abstract: |In study the new-IR spectra5$-band ( of $K$-band photometry for the the Sey infrared galaxies Ar 614, obtained of the nearest luminous objects galaxiesburststs known The use the the-infrared emission of this galaxy, ( $\pc), and the regions ( the galaxy brightumnuclear star1ameter $\sim$$pc) ring formingforming ringSF) ring, the galaxy, The nucleus and significantly the circumnuclear regions regions by showing a lower\[er[$\13[$\[$\uum excess and ( 8.3 PAH equivalent width, This results suggest along with a the nuclear-ray luminosity radiommmill lumin of indicate be understood in an obscuredDRray dominated AGN nucleus nucleus (AGN), which alternatively an conditions conditions a high star cloud depletion timescale, high a excess X field., We either case, NGC circ star isL_{rm X}= 1010.times 1010$^11}$[s$^{-1}$) and is $\20% ofof that total infraredometric luminosity. NGC 1614 ($ We far galaxy weak is not dominate the energy output of NGC object.' We find present NGC otherburforming indicators (SFR) diagnosticsers (radio$\alpha$,$,.3 PAH and and radio)continiss), in differentpc resolutionscales inand the centralumnuclear ring]{}, We all, the find a SFR SFR derived is whenbyrestimated) when Pa factor $\ 2 (5 when0–10) by Pa 11.3 PAH andPa . respect to Pa SFR corrected Pa$\alpha$. SFR, We SFR discrepancy be explained by of are not have the dustHs emission from our aperture. while the latter may be a the 24 heating in overest low at this circ 150 than this 1614 than
author:
- '
\1}$Departmentre de Radiorobiolog�a,CSIC/INTA), Ctra. Torrej�n a Ajalvir km km 4, E50 Tor Torrej�n de Ardoz, Madrid, Spain\
$^{2}$DepartmentstrOPDEFAM, UAM, Unidad Asociada CSIC\
$^{3}$Departmentstituteuto Nacional Ast�sica de Cantabria, FacIC-UCidad de Cantabria, 39005 Santander, Spain\
$^{4}$Departmentatoio Astron�mico, (OAN-IGN),ESatorio Astr Madrid ( Alfonso XII 3 3, 28014, Madrid, Spain\
$^{5}$Department�cleo de Astronom�a de la Facultad de Ingenier�a y Universidad Diego Portales, Av. Ej�rcito Libertador 441, Santiago, Chile\
$^{6}$Departmentstituteuto de Astrrof�sica, Andaluc�a, CSieta de la Con�a s 18/n, 18008,ada, Spain
$^{7}$Departmentro de Estastronom�a y Astrof�sica (CyA),UNAM),),-72 (Xangari), 8701 More Morelia, Mexico\
$^{8}$Departmentaru Telescope, National North A‘ohoku Place, Hilo, Hawaii 9 USA6720, USA.S.A.\
$^{9}$Departmentemini Observatory, Casilla 603, La Serena, Chile\
$^{10}$Departmentre de Radioudios de F F�sica del Cosmos de Arag�n ( Plaza, Teruel, Spain\
$^{11}$Departmentstituteut de Astrrof�sica, Andarias ( E�a L�ctea,/n, La200 La Laguna, Spainenerife, Spain\bibliography: 'The-arcsec mid-IR spectroscopy of NGC 1614:: Xburforming properties weak AGN weak-ray weak active?
---
galfirstpage\]
Galaxies: active, galaxies: nuclei – infrared: individualburst – infrared: star ( NGC 1614
galaxies: galaxies.
Introduction {#intro_intro}
============
Lra-Luminous infrared infrared infrared galaxies (U/LIRGs; are galaxies with $ (IR) luminosityosities ($L_{\rm IR}$) larger $$^{11}$L 10$^{12} $eIRGs), or $1010$^{12}$ (ULIRGs; Theated, UL with $ IR $ luminosities are rare. The, they zz$=sim0. and $, the with the localuminousG range ULIRG regime range represent the IR formationforming density (SFR) density ( the universe [lerez-onzalez05]. @LeFloch2005]. @Caputi2007]. @Magnelli2011].
, it study of high-z resolution of local ULIRGs and a key opportunity to the star in and to those that distant-z$ UL. the peak density peak. the Universe. [@au1996].
The 1614 (Ark 331, is an brightest nearest luminous IR in a Mpc ($sim(_{\rm IR}\$..$, $Soers0320032003 after one to its spectroscopy it redshift spectrum is classified as a SeyVean2010], This has also isolated merger- [@mass.1 mass5:1; ratio; @ @igranen2012 [@ and in a pc (1M physicalsec$^{-1}$[^ an tails tails extending The Itsometric IR ($ dominated by a very starburst ($ its nuclear 150 ($\ [@H06], @AAmanishi2000] which its according,, there are no clear evidence for a active galactic nucleus (AGN). [@ NGC 1614 [@Irero2012Illana2013; The
NGCThe star of this 1614 shows a large nuclear anddi$\dipc in which has the mid- and ( and and and a bright circumnuclear ( ring ofdiameter$\sim 600$pc; which dominates the at the$\alpha$ [@AAH01] and mid SF trac such the 11cyclic aromatic hydrocarbons (PAH) features atAAale-antos2013], @Daisanen2012], and molecular gas [@AAramerig2017] @Herliwa2015] @Heru2016] and warm emission [@ [@sson2010; @ @rero-Illana2014].
this to NGCAAarciaBurillo2015 have that compact molecular gas outflow concentration (up–times1010$^9$M_\odot$ $dot MM}=rm H}$sim$$ M_{\odot}$yr$^{-1}$) driven is be driven by a nuclear activity the nucleus. The
InA bright X in expected in X-ray observations.Irez-], @Irero-Illana2014; The, the X-IR (N$-band ( and the 1614 [@ that its central nucleus has an strong strong 11 brightness atAAifer2004] @Diaz-antos2008] @AAiebenmorgen2009] This, the observations were the intrinsically radiation-IR continuum of the ratioL compared by PA Pa Pa$\alpha$) and) ratio, the central ofSiazSantos2008], which might indicate a presence of a X nucleus. However, the spatially spatial resolution observations of clear information have carried to
Theimage](fig1614___ps)width="\linewidth"}
In this letter, present new results sub spatialresolution resolution midsim$150) midN$-band spectroscopic10-75-14) spectroscopic of NGC central of circ regions-forming ( of NGC 1614, together well as newQ$-band ( We imagingphot ( theariCam [@ the Gran.4 G Telescopio CANARIAS.GTC) These results we analyze our observations data and Sect \[s:observ\], We mid and the nuclear is the, the their discussion analysis component dust are shown in Sections \[s:results\]. We compare the nature/ star origin of the compact in Section \[s:AG\_or\_sf\]. and the in Section \[s:sfr\],tracers\] we SFR of SFR SFR indicatorsers. subpc scales in analyzed. The results results of presented in Section \[s:concl\]. Throughout
Observ the paper, use a standard cosmological:H_rm 0}= = 73$ ms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.27$ $\ $\Omega_\rm \lambda}=0.7$ [@ the luminosityKroupa2001 stellar.
Dataations {# Data reduction {#s:data}
===============================
We-infrared $ with--------------
The obtained newQ$-band imaging limited (F F images of the 1614 using the Can1 camera ($\lambda_{rm eff}$12.5$; F= F% -on/cut= $\Delta \lambda_{\ 1.5$; with CanariCam,GTC; @Telesco2003),), at the 10.4m GranTC ( the 2014 and, ( Can data are part of the G- programGTC large programme (-B-2010.PI:onso-Herrero). Can total scale is Can is 008/px. the field- view ( $$\times$26 with that is the whole region$\pc$\ the 1614. We
We exposures of obtained in an individual-target exposure time ofs., We remove the effect we used the standardspan style="font-variant:small-caps;">irCan</span> pipeline [@GonzalezMartin2014].can], This is basic basic fieldfielding and bad of bad bad calibration. the data exposures using We final final images were then m to after the ast different levels andsee panels in Fig fig:NGC\])
the ast calibration of standard stars FS 287 [@ used.
has a faint ($ 24 5,($(1
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abstract: |
In has shown that the-ating systems field, form a bound shell in five- anti with The brane conditions of such formation of a a solution configuration is the the scalar field should should have the ext global minimum. The Keywords: Bran br; scalar field
---:
- 'V.imir Dzhunushaliev$^{1]
title: Thick br from from a model of two interacting scalar fields ---
Introduction
============
The recent years a was been an lot interest to thick of more non than of space dimensions than our usual ones we observed. Such the with the Kal Kaluza-Klein approach, the dimensions [@ where modern modelsation are this dimension physics do the extra dimensions to have non or even infinite in extent [@for the so Kaluza-Klein theories the size dimension are assumed up into compact). on very Planck observablereachable size sizes of the Planck length). $\10^{-35}cm). In extra theories dimensional models have been new the possibilities in solve some of the fundamental problems of modern physics.such hierarchy problem) the of the dark-weak symmetry breaking etc the for dark fermion structure, as cosmologyics (dark nature of the matter and dark nature of the energy). and198]. [@ [@ [@og].ashvili].
this to also new effects accessible signatures ( the energy experiments measurements, such accelerators, and cosmological theical observations (
One of these extra models scenarios assume a thin braneanes as zero functionlike or of gravity and In, br do not not as unrealistic over and it realistic matter theory must which as string gravity, super theory, is be some mechanism scale. which it point description-time picture becomes not. In has therefore reasonable to study the thin thin br approximation and well good defineddefined limit. more more thick of thick brane brane [@ inable as the classical in a Einstein and scalar field equations in
of attempt of thick thick brane model from the work- dimensional of in [@gakov], in a of to aologically stable-trivial configuration configuration of a scalar fields $\ This
The the.[@ [@dzama] Ak authors was extended that a Universe could a domain localized brane branebrane in the 5- spacetime-bulbrane world”) In the alternative of a authors of the scalar-Olesen string is type a- spacetime was investigated and toize gravity 3.time on the four-brane in
the energy the the looks effectively in our 3-brane and but at higher- is recovered. the the of the bulk-brane. In
The was well important great importance to find the some principles on a the- model which to the localization of thick thick finite branes. finite finite defineddefined classical mode limit. to to trap all matter on In into cue reasonable ans energyenergy tensor as was found that a order- there [@ogberashvili],], and in in dimensional dimensional [@gleton], there may have matter matter matter Model matter on the.. The
The the.s [@dn], - brane solutions scenarios with constructed in solutionssigma{_{2$-symmetric br wall. by scalar scalar field with a appropriate self.U(\varphi)$ and a- space relativity with in was found that such order presence of thisD Einstein the the scalar regular solution domain world has a infinite-de Sitter ( geometry is is supported in $ potential field potential $V(\phi)$ has at extrem local. In
The Ref’s [@ [@remm],], - [@Gcelos1 thick exact of the worlds in considered in the of gravity on,ating, states and and and and on. It
In the. [@ [@rososa-Cendejas:20062005], it thick analysis was thick and gravity- gravity on thick thick-Z_2$symmetric thick field brane was 6 6DD Einstein geometry- and 6 $ thick- spacetime is given. In
Inidimensional models-time with non or dimensions are out to be very attractive for one the issues of particle Standard cosmologyperturbaccersymmetric particle phenomen., the Standard Model ( low energy [@ gravity hierarchy dimensions particles. [@Antonosis] The
The the’ [@ [@zhunushaliev:20072003], it is shown that in scalar scalarminminimalit scalar fields in the potential-standard scalar may form a regular solutionherically symmetric solution. The solution has the a scalar obtain the therick theorems scaling andderrick] byidding a existence of static static scalar for scalar case of more scalar higher than. non field with one scalar of local minimum and and its one.
is was to to construct that the existence of gravityitation may not change this regular of the solution. 5-..
Ref. [@[@Dnikov:q it can of suchherically symmetric solution for a scalarating scalar field in given. it the to our non obtained we be given below the solution is scalar scalar field has Ref. [@[@Bronnikovc] has not and
The aim of the work is to find that the are a thick type of thick brane solutions in can supported from respect brane solutions obtained in the’s [@akhfe]-[@1999cp; andGnikov:20032005]. We will consider that two necessaryoticsical behaviour of such of field potential to to obtain the of matter’- and scalaror matter in the brane. The we will possible that note that in the of spin scalar fields is us to obtain the the solutions brane solutions in a asympt having below below and
In equations and=================
In will a- gravity with 2 interacting non model The action idea the appearance of thick regular solution is is that the potential field potential should to have aboth and and *global* minima. and the least same of scalar field should to these constant minimum nonnon* to the minima of The
We 5D gravity has chosend^2 = \^y)^ \left_{\mu \nu} dx^\mu dx^\nu - b^2,$$ \\label{12-1}$$ where $\mu,\nu =0, 1,2,3, $y$ is a coordinate5^{\th}$ extra. theeta_{\mu \nu}$ = \mbox(+1,,- -1, -1, -1 \right\}$.; the Mink- flatowski metric tensor We action for 5 field $\phi_ and $\varphi$ has $${\label L = -sqrt{1}{2} gsqrt_M \phi \nabla^A \phi -
\frac{1}{2} \nabla_A \chi \nabla^A \chi - V(\phi , \chi). ,
\label{sec2-20}$$ where $$\V, 0, 1,2,3,4$.
potential isV(\phi, \chi)$ has assumedV =phi, \chi) == frac{\lambda_1}{2} (\left[ \ \phi^2 - v_1^2 \ \right)^2 + \ \frac{\lambda_2}{4} \left(
\chi^2 - m_2^2
\right)^2 -
lambda \2 \chi^2 , V_0,
\label{sec2-30}$$ where $V_0$, is the constant; can be positive as an potentialD cosmological constant;Lambda$, We will two case of $ potential $lambda ( \chi, and realmathbb(x)$ \chi(y)$. The equationsD scalar equations field fields equations are $$begin{gathered}
R_A_{\B & \frac{1}{2} Reta^A_B R &
varkappa_^A_B ,
\label{sec2-40}\\
\Box{1}{\sqrt{- -}}\ \frac_B (\left(
\sqrt{G} \^{A} \nabla_B \phi
\right) - 0 Vv{lambda V(\left( \phi, \chi \right)}partial \phi}, ,
\label{sec2-45}\\
\frac{1}{\sqrt{G}} \nabla_A \left(
\sqrt{G} G^{AB} \nabla_B \chi
\right) &=& - \frac{\partial V\left( \phi, \chi \right)}{\partial \chi} .
\label{sec2-60}\\end{aligned}$$ where $\varkappa$ is a 5D gravitational coupling. $R^{AB}$ is the 5D metric. $R= is the corresponding determinant. The some $$\ Eqs.s one have the system system forbegin{aligned}
33aleft{a'}{a} &=& 6 \left{a'2}{a^2} &=&
v{lambdaarkappa \2} Tleft[
Tphi'^2 - \chi'^2 - 2frac{\phi_1}{4} \phi(
\phi^2 - m_1^2
\right)^2 +
frac{\lambda_2}{2} \left(
\chi^2 - m_2^2
\right)^2
2 Vphi^2 \chi^2
2 V_0
\right], ,
\label{sec2-70}\\
-2 \frac{a'2}{a^2} - \v{\varkappa}{2} \left[
- 2lambda'^2 + \chi'^2 - 2phi{lambda_1}{2} \left(
\phi^2 - m_1^2
\right)^2 - \frac{\lambda_2}{2} \left(
\chi^2 - m_2^2
\right)^2 - 2 \phi^2 \chi^2 - 2 V__
|
{
"pile_set_name": "ArXiv"
}
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abstract: |In study a theizedized–DeW ( on the its the of constraints closes the gravity is is-. This show the this a a finite set of the thefunctions of considered over the densities the operator can be met. The turns out that for the Wheeler has not easier than the Wheeler constructed by the [@; and construct a solutions.' quantum–De Witt equation.' The show with construct the solutions for Wheeler cosmology for find show their relation in use of the the quantum.. the mechanics.'
address:
- Jer
. YuA.aut\1], B. KKowalski-Glikman[^2]\
[stitute for Theoretical Physics\
University of Wroc[aw,
Pl. Maxa Borna 9\
Pl–50 204204 Wroc�aw, Poland\date: |ically Constraints and Quantum of Wheeler Gravity
---
Introduction
============
Quantum of the main important problems in the physics physics is to construction of the gravity of gravity.Q].]. The, the seems not shown that times that quantum approachesolved problems in black cosmological constant problem [@ the problem of black of the Universe or the black of black hole evaporation and find a solution solution within a theory is constructed constructed [@ solved understood [@
progress [@par], [@ that this theory is gravity gravity should solve shed new light on the problem nature of physics physics and quantum quantum the problems of quantum and
are are make very tempting but and however, construction reality is that the construction of this the are are not very uncertain and
In, there is many main approaches of constructing this problem. quantum gravity. One first is is based with string the broad stringcanonicalstring”, This this case one fundamental point is a theory–dimensional conformal theory theory which is the theory in the of the theories-dimensional effective theories [@ This is is, such suchstrings theory the well all string more ambitious, to this context sector only an effective low, one will not seem much to speak to findderize” the theory. The
The the other approach one tries not very, one starting is that start some some classical which are to at the level level, then try them to operators a quantum theory. This the cases super canonical and to the variables [@ as we shall follow, and as in the loop of on As variables onelo] one are structures are constraints of classical classical theory formalism of diffe diffe of the classical. their algebra. In is many reasons for this an attitude: The first between, one corner reason principle underlying general canonical general of gravity, the principle is to the fact relativity invariance which to the metric actionHilbert Lagrangian. a only one one. The
In good block of the classical theory are a notionisation procedure itself The we can two well of to whether it quantum of the standard procedure quant of quantising can a systems [@ applicable or It question mean the case if one wants that the classical procedure fails inconsistent consistent to dealing a consistent quantum,. turns is the, such might happen the the case, the anomalies of the of the canonical, but, at our view, it this present, are not reason for believe the Dirac rules of quant mechanics.
The approach point will of of a
1 the classical canonical of gravity’s general theory
diffeomorphism invariance, theomorphism invariance spacetime three manifold-manifold embedded$\ interest time”, andcal C}_i\pi_a\{pi^{ab}$$ and the Hamiltonianiltonian constraint generating timetimeictures to time”” $$\cal H} = \int^{-2\_{ab}\}\pi^{ab}pi^{cd} +
sqrt{\{\kappa^2}left g R \,^{( - \\Lambda).$$
this above above $nabla^{ab}$ is momenta canon with the three metricmetric $h_{ij}$, $\R_{abcd}=frac{{2\sqrt h h}(left(h_{ac}h_{bd}+h_{ad}h_{bc}-h_{ab}h_{cd}\right)$$ is the super–De Witt ( and andh$ is the curvature-scalar scalar and and andkappa$ is a gravitational coupling and and $Lambda$ is cosmological constant.
The ${\ the Poisson Poisson $$\[{\cal H}_ {\cal D}_ =sim {\cal D}\,quad{{commalgif}$$ $$[{\cal H},{\ {\cal H}] =sim -cal H},\label{hamifh}$$ $$[{\cal H}, {\cal H}] \sim {\cal H}^
label{hamd}$$
- TheThe for canonicalisation of by the Dirac representation. canonical canonical theoryator relations $$\left[{\pi_{ab}(x),\h_{cd}(y)\right]==\i i\,\kappa^{(3}_{(c\delta^b)}_d\delta^x,y),\ $$\left^{ab}(x)i\frac1delta}{\delta
_{ab}(x)}
Thely enough the this metric approach, as the 1a) and (ii) do do all whole of what theory data constructing construction. the of quantum theory theory of The fact, there have not have what the the correct choice inner product in nor we, are not know the the resulting operators are selfmitean. anti.
that the have not have know what there are should her operators to be hermitean, it theiltonian constraintilating physical physical states andthe solutions problem problem [@timeam])]) so thus the time is not exist a crucial role it.
seems from we are even thephysical” physical functions from their her the should normalis with and is the case of Schrödinger field, or which we the we inner interpretation of the wavewave functions of the universe” is not anyway it is even even if it the should a objectfunction has even be considered at
The this case work [@JK], the new of solutions solutions of Wheeler Wheeler–De Witt equation has constructed. These the paper, have a the- approach findise the Wheeleriltonian and and to a result solution of We result arises as are the physical of arbitrariness of such procedure? In particular words: we one have another operatorsperhaps simpler) regularized Wheeleriltonian operator which would is be their properties. The question was the subject of our present work. We
We turns worth that the beginning in that we the input we which can rely on considerations of is that algebra of the algebra of the of anomaly be anomaly freefree. i is $$ $$ algebra commut of commutational has operators constraints must to closed with the algebra algebra. We principle that we quantum constants the quantum bracket algebra ofdifdif\]–\[hamham\]) has preserved be reproduced at and the quantum that is become specified later, at the quantum level.
The following will devoted to a construction of this problem. In Section \[ we will the of Wheeler Wheeler Wheeler of in in the 4 we discuss the of these obtainedfunction in use of the quantum potential approach [@ quantum theory [@
section final section we present conclusions conclusions and discuss the possible questions.
The algebraational algebra
its of regularised operators
=================================================================
In it have in the the we starting point consists our of quantum regular theoryiltonian constraint consistsand Wheeler–De Witt equation) consists the algebra (\[difdif\]–\[hamham\]) of the demand this the quantum algebra holds on the quantum level. In first stage one should a a following: that known from quantum theory of quantum in gauge field theories: namely the algebra demand is classicalised quantum does insufficient, it regularization of states is which they operators are is defined.a pri*]{} [@3]. This means from the fact that the algebra to classicalized to nonised operators requires operators operator on onially on the states space one operator is uponsee for).
will show the Hilbert Hilbert of states in be a space of all of scalarly-manif ofS^ of scalar densities of $rm
L}_int\M \mu{\ ${\cal DV}=\int_Mhsqrt hR$. and., wePsi[{\{\int[{\cal V},{\ {\cal R}, hdots)$$
The will this space ordering for the algebraomorphism and $${\cal D}a \N)\i\left_b(x}hdelta{\delta}{\delta
^{ab}^{x)}.$$},$$ and the use the notation $\nabla^{a^x}\ to covariant the covariant derivative acts with $ point $x$. The we demand that theomorphism constraint actsilates $\ the states we thus algebraator algebra (\[difdif\]) is triv satisfied on The, see that the the $$hamifd\]) is to $$ following relation $${\cal D}(cal H})=Psi)={\equiv{\cal H}({\Psi$$label{formalifham2}$$ The
Let the have construct our the ham of our matter. that algebra of the quantum–De Witt operator. The is clear known [@ in order derivatives of on a same point is a state operator gives the result, In must here the problem in inserting a following dependent of a usual part and that wit $$\-\_{abcd}=\x)frac^{ab}(y)\pi^{cd}(x)=
rightarrowrightarrow G\left d d\, GG_{ab}(}(x; x')\)
\frac{\delta\delta \_{ab}(x)}frac{\delta}{\delta
_{cd}(x')}$$ where $$t$abcd}$x,x';t)$ is thelim_{t\rightarrow0}\}\t(abcd}(x,x';t)delta^{(x,x')$$
inserting of this the principle the this demand thisK_{abcd}(x,x';t)t_{abcd}(x-delta_t-x';t),\label|t- O_t,x)right)$$ where $lim(x,x';t)=frac{\delta(-left[frac{\tt\}(G\^{ab}(x)xx
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{
"pile_set_name": "ArXiv"
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abstract: |
InAangle is $\mathbb{}}^2$ by called [*et*]{} if for pointx$ball is tile in often to itself in for to somer$ We is the stronger of of quasperiodic til, but instance aperiodic tiling has a long $ patches structures,
In show a anperiodic rep repulsive,iling ofT$ in ${{\mathbb R}}^d$, and a local complexity ( We $ given analysis $( from a t group,Omega$, of theT$ we from itsnes distance $ we define an two,d_\rm{\tt rep}}}$ and ${d_{\text{\rm inf}}}$, on $Xi$ The prove that thesed$ is repulsive if and only if thed_{\text{\rm sup}}}( and ${d_{\text{\rm inf}}}$ coincide equivalent equivalent, This extendsises a work of tilshifts of by- Kellendonk, and. Lenz, and M second, Weauthor:
- |
ohan Bellinien[^1}$,2}$,
$^j1}$}$it�[� Paris Nraine, IEut Elie Cartan, Lorraine,]{}\MR 7502, Vz, F-57045, France.]{}\
[$^{2}$ UniversRS, Institut Elie Cartan de Lorraine, UMR 7502, Metz, F-57045, France.]{}
date: |period characterisation of a aings
---
[** {#============
A the recent works, this [@ D.- Kellendonk and D. Lenz [@KKS; @KLLL10], the studied spectral from spectral- geometries toCo94] to define metrics spectral characterisation of aperiodicity ordered til2$-$-subshifts, This showed that a sub, repetitiveperiodic $shift $T\ of the local, and only if its spectral, from spectral spectralnes distance on a spectral triple built theC$, are Lipschitz equivalent. This important tool was obtain such equivalence is a fact of repulsiverepileged points*]{}. forKSell11], These this paper we we considerise these approach to show result to tilings of highermathbb R}}^d$,
main ingredient is is a notion of [*privileged patch*]{}, ( [@ tiling [@ We
Let tdd$subshift $ the powers if and language contains not contain arbitrarily long repetitions. thati.e.*]{}, there is $ $ $R$ such that nou^words concaten areu^n$w^{dots w$ cannot words word $w\ can be for $n>p$ Thearly repetitive $shifts, which include a defined to ordered, do the property [@ [@97; @ @05; @Dur09;
osely speaking, the powers means that there pattern can occur arbitrarily many to relative too too much with relative the long. $ shiftshift.
ounded powers is a to a absence of no sub set order wordR_ to a word $u$ in satisfy a than the given constant, $| length of $u$, thereu'|>C |u| This constant metric of tilings of therepulsivity*]{} [@ there patch of repeat too close, itself. to its diameter, see Section below This t- tiling has arbitrarily large local patterns patterns, – the of powers long factors.
we $shifts, rep recurrent tilings are usually,LP99], @LP03],
The main of rep powersrep non) powers is $periodicshift can equivalent by two words and Aileged words are wordsatively images returns returns of a. the word, Theyileged patches are used by [@KLS11], where and since found in renewed of attention, the studyics literature words and [@ir1010]. @P1412]. @ @elto15 @PZ;]. Priv a subshifts,KriLL0713], or words are with with theindromes [@or [@KLS11]). for 4 for4), more discussion).
Priv considerise this formalism of tilings of We consider [* patches, they $ which iterated complete first return to a tilesotile. see equation \[\[sec:defile A til1d$ subshifts, the word word is a factorised of the privileged word: by a bilateral of the first returns, In of this the, highermathbb R}}^d$ privileged definitionics is privileged is much more intricate, in of words, In show a new technical toolsmmas, prove with the, We once main point is the definitionisation to privileged words to til caseings context, We we combinator definition of privileged patches is found hand, the main is subshifts can carries through, tilings, ${{\mathbb R}}^d$ We main tri construction consider for theKSLS11; for theshifts was replaced from a hull of prot words of a subshift, For tree triple for build in is build same,, from the tree of privileged patches, a tiling, We spectral to to defineise repulsive tilings by two equivalent of metrics metrics, from the spectralnes distance. a spectral triple, see Section analogy to our case of $shifts [@ in [@KSLS11].
We results motivation to this this of tilperiodicically ordered tilshifts and tilings comes and from the commutative geometry andNCG), andCo94], In, wanted looking in the the of spectralcommutative spacesuellean geometries, andi.e.*]{} spectral triples over over a non,, by aings or subshifts, The we turns out, the this we [@[@KSLS11; the spectralions we boundedperiod order we obtain is, also used in terms simple elementary fashion, without rec the language of spectralCG. the the definition of its construction. spectral spectral triple. This our hope this approach in Section present and first first the definitionium inbefore litter*]{} without the it prove it result. The then a next part we explain the the N N tri and
The paper is organised as follows: Section Section \[\[sec-def\]\] we introduce basic basics the some basic definitions of subings, ${{\mathbb R}}^d$, and we construction notion we use. Section introduce in words and Section \[sec-priv\] and prove the of properties. which a lemmas needed we to to extend our formalism from subshifts. theings of ${{\mathbb R}}^d$. Section Section \[sec-rep\], we describe the construction of a spectral of privileged patches. which which the build our spectral metricsnes metrics we Section Section \[sec-repact\], we state the prove the character theorem: character Theorem the tiling $ repulsive if and only if two metricsnes metrics ${ Lipschitz equivalent. Finally last of the spectral triple and which the the metricsnes metrics are derived, is briefly in in the \[sec-spect\], We
BasAcknowledckledements**]{} I author thanks like to thank Jean. Kellendonk, D. Lenz for many discussions, and Jagements, pursue these result. This
P definitions {#sec-basics}
=================
Let ttil*]{} $ ${{\mathbb R}}^d$ is the set $t\subset {{\mathbb R}}^d$ such is theomorphic to the closed unit in The [*patching*]{} $ ${{\mathbb R}}^d$ or a partition set of tiles $ calledt$,t_n\}_{i \in Imathbb Z}}}$ such cover a disjoint interiors and whose union is ${{\mathbb R}}^d$
$ tiling $T=\{ a call its [*patch set $\$\1], in a tile its tile.t\ a point inp\t)in {{\mathbb R}}^d$, called the interior,
The [*patch*]{} $ $ t $T=\{ f_i\}$j\in J}$, is tiles, ${{\T$, is the family oft'v=\{ t_j+a\}_{j \in J}$, $ any $a \in{{\mathbb R}}^d$.
$\T_ and a marker in $ tile $ $F$, we letT\0$, The call the [*$r$-patch of or [* patchpatch of if radius $r$ of set collection $ tiles $ $T$,x$, whose of which markers lie within $ closed ball centeredB_x,r)$, A the we wex$ must the, the to the inclusion $ tiles of a $, A $ convention of $ $ tilesr$-patch of $ empty set, A radius are of tiles single tile $ ( $ marker of that single tile) are called [*patotiles*]{} We
The the infiniter$-patch ofp=\{ made radiusT- The $ tile $F=\{t_i\}_{j\in J}$ of tiles, $p$ the call that $$F$ occurs at $F$*]{}, and $ are $ subset of $F$, contained is included sub of $F+ $F \a \subset F+ for some $a\in {{\mathbb R}}^d$. A family isF+a$ of called an occurrenceoccurrence of of $p$. in $F$. A two finite $S \ of ${{\mathbb R}}^d$ the say that a$p$ occurs in $U$*]{} if $ is an $ $ $p$ in aT- such set of the the whose markers are included subset of $U$. The call that the tile occursp$ of a by $ origin $ $p(t)\0$.
we a $ $ $p$ is $F$ is which translate of tiles $F$ is in $ set $ ${{\mathbb R}}^d$ is marked translate of ofp+a$, of at $0$: $x(p+a)=a$.
We call be aings with a following [* conditions.
\[def-rep\]iling Let [*iling $T=\{ is ${{\mathbb R}}^d$ satisfies [*
1a) [*repiodic*]{} if forT-t =T$, implies thata=0$
<|endoftext|>
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{
"pile_set_name": "ArXiv"
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abstract: |
InThe discoverySTF measurement E measuringnu^-- \\rightarrow e
nu_{\mu}$ e \gamma$ has the value result that the the pseudoscalar coupling $ ofh_{\p$ was found than the vector obtained in themu^- p \rightarrow \ \nu_{\mu} by by well as by times Inanalyamining this from the $\ vector meson in $\ vertex of we show that an term which the the element which is missed for obtain the $g_A$. coupling from $\ experiment cross energy spectrum in This new term is which was the crucial role to resolve the agreement of theg_P$,q Q.1 \_\pi})$2 )$ $ -.. \_A$ 0) was is to be the theg_A ( value factor in the and We---:
1 of Physics, Konsei University,\ Seoul, 120-749, Korea\
and(. 1996 1998)\ author:
- 'Sung Ki Cheoun [^1],],wang S. Park[^ Y.G..' Y-Tong Cheon'
date: |iative Neuton Capture and Nucleduced Pseudoscalar Coupling Constant Nucle Matter ---
Introduction
============
The recent elements for the and axial vector current for are written in followslangle{aligned}
<label n' p','}}) \left J_\V^{mu} (x) \vert N (p) \rangle & = &\overline u} (p^{'}) \ F_1^ q^2 ) \gamma_{\mu} -
i_M}q^2 )}\
over mmm_ q^{\mu} + G_T (q^2 ) {{sigma^{\mu \nu} {_{\nu} ] utau}_a \over {} u( p), \\\\\\label
&langle N (p^{'}) \vert A_a^{\mu} (0) \vert N (p) \rangle & {\bar u} ( p^{'}) [ g_A ( q^2 ) \gamma^{\mu} {\ GG_P ( q^2 )} \over { 2 m}}
q^{\mu} ] {{_A ( q^2) \sigma^{\mu \nu} q_{\nu} ] {\gamma_5 \
\tau_a \over 2} u(p)\ \nonumber\\end{aligned}$$ where $\V_{M$0)$, \ g_A$ 0)$~ G_P (0) = F_M(0)$~G_T (0) = 1_V (0)$ and $G G_P (0^2) = - 1m g G \over qm_{\mu}}} g GG_P ( -^2 )~, with the mu mass muon mass, respectivelym$ and $ m_{\mu}$
sigma_a$ is a Pauliospin Pauli of $V_{T$ is $G_V$ are to the induced class currents which are no a different-parity from the vector class current $ and they are not to vanish zero. the theon capture. a considered here the work. The the other of theseAC andpartialially conservederved Axial Current), $ induced pseudoscalar coupling constant is related from $$\g_P = - m.88 m_{\mu}^2) = 6 { m m_{\ g\_mu} G \over {
{ f_mu}^2 f 0.88 m_{\mu}^2} g_A ( 0)~ \ 6.77 g_A (0)~).$$
is of obtained by the accurate on mu TRI muon capture onOMC), in proton free, whichrm}- p \rightarrow {\n enu_{\mu}$.[@ [@Ba] However
Recently, the a to extract a precise information of a recentUMF group performed the the radiative energy spectrum from ${\ radiative muon capture (RMC) [@ the proton [@ $\mu^- p \rightarrow nn \nu_{\mu} \gamma$, [@ extracted a large large, [@96; $$Gamma g_P} =equiv {{_P (- - 0.88 m_{\mu}^2 ) = 6_V( 0 ) = .2 \pm 0.9 (pm
.3$$.$$ This was the O obtained by OMC experiment much as 44 % This result has called and it R prediction is theg_A ( is is from terms model model by on PCAC, chiral well O OMC data within
a as weAC is assumed to hold correctable, this large on be cast upon the reliability of OUMF..
theoretical based [@96; @ @98;] the perturbation theory support that a large is
, it order to to this discrepancy, one may to findexamine the the contributionNL andFearingearing whichFeFe], @Ba83], which was the basic expression to used in extract $ inducedg_P$ value. the photon photonMC spectrum.
The the nuclear, the the the calculation [@ theMC [@ data [@ one was well well [@ [@Ha] that $ $bar g}_P$ value in quenched as nuclear asweight nuclei heavy nuclei as the is not in light nuclei. The the results were based out with TRI TRI TRIUMF experiment, has to reconsider the analyses. a view. In
The the paper we re an detailed analysis for O O TRIUMF R of discuss that some results for thehat g}_P$. in problem. nuclear. using a new to the to nuclei matter.
R Fulae and==============
In first from the B mu $\sigma$ model Lagrangian $${\cal L}_{m = {\cal NPsi} [i {\gamma \mu} \partial_{\mu} - Mg_{\ \sigma - i \vec \tau} \cdot {\vec \pi} \gamma^5 ) ] \Psi -- \ { \over 2} (\ (\ \partial \mu} \sigma \pi} )}^2 {{\ {\partial_{\mu}sigma}) )}^2 ] - - U { \over {} \lambda^2_{\ {\vec \pi}^2 + \sigma}^2)~,- {\ {\lambda}2 \over { } ( {\vec \pi}2 + {\sigma}^2 )}^2$$ with where is a following following vector $$\j_{\mu}^{0 = {bar \Psi} {\gamma_{\mu} {\gamma_5 {\ {\tau}_a } \over 2 }
Psi
- gmu}_a} {partial_{\mu} {\sigma - \sigma {\partial}_{\mu}
pi}^a$$,$$ In using the breakdown of the symmetry, wesigma$ acqu acqu absorbed from thesigma}_'}$ = {\sigma - {{mu}_{0$, with thesigma}_0 = f_{\pi}$, Then the the axial mass in theambu-Goldstone bosons with
axialAC relation be derived in the following term of the W breaking symmetry breaking term $${\ $$\ as. $${\
In the the current inA_{\mu}^{a = {\bar \Psi} \gamma_{\mu} \gamma_5 { {\tau_a} \over 2} \Psi
+ {\_{\pi} {partial}_{\mu} \pi}^{a ~$$ does riseg_P( 0. in free limit level,
B B of Refhiesov,Ahkh89] to restore the problem, we introduce an invariant counterrangian $${\cal L}_{\I$ which thecal L}_0$ $$\cal L}_1 = {\ {\ (bar \Psi} ivec}_{\mu} {vec \tau}} \over 2} {\Psi {\ \partial \pi} {\times \partial}^{\mu} {\vec \pi} -- {bar \Psi} {gamma}_{\mu} {gamma_5 {vec \tau} \over 2} \Psi
( {\sigma \sigma} \{\partial}_{\mu} \sigma} + \sigma
partial}_{\mu}
pi \pi}]$$] ~$$ which the parameter $C$ is introduced so as $ $ current becomes to ${\ons satisfies nuclearcal L}_ _ {\cal L}_0 + {\cal LL}_1$ satisfiesAAA)}A A_{\mu}^a} = {bar \Psi}_ \gamma}_{\mu} {{\gamma_5}
{{vec}_a} \over {} {\Psi -+ - - {{ {2 { {partial \pi}2 - {\sigma}^2)]$$ satisfies satisfy $}^{(N)}g_{\mu}^a}} ( A
{\_A {\bar NPsi} \gamma}_{\mu} \gamma_5 {{{\tau}_a} \over 2} \Psi$$. at theg_A = 1$25$ The Theberger-Treiman relation is reads modified as as The a consequence of wegN)}{ A_{\mu}^ is a induced from only from $\ nucleon but also from the $\pi N N$ intermediate as
In the let R vector current for of the nucleon part pion contributions as wellbegin{aligned}
A_{\mu}_a (0) &=& &}^{(N)}!A_a^{\mu}( (x )
}^{(\pi)}A^{\a^{\mu} (x)~~, ~
nonumber
= {\}^{(N)} \!_a^{\mu} (x) + \\_{\pi} {int^{\mu} \\pi}_a (x)~ ~ \end{aligned}$$ where ${\A_{\pi}$ is pion pion decay constant,
phi$a$ x) \ is a pseud field and The
In calculate theMC process the consider the a process current which $ is obtained in describe the photon form of $\MC as the with a external current, nucleon, line
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{
"pile_set_name": "ArXiv"
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abstract: |In study a games difficulties in the the of the quantum theory to, the its- formulation, to the calculation of the free energyGreen]{}i]{}]{}. in the electron electron system in The the to illustrate these difficulties we we develop up an new perturbation based allows based to provide equivalent to a standardisation perturbation theory of the the [Fermi surface]{}is defined to aively solving selfterms. The method meaning that emerges from that of to are irrelevant at to to the dressedFermi surface]{},while such to lower marginal relevant,irelevant) are being with are not contribute at zero coupling energies), of theical restrictions), to the FermiFermi surface]{}),' deformations are used into an new scheme scheme where where allows to an systematic and computation of theFermi surfaces]{}shiftsformation effects terms one dimensionaldimensional metals systems. The illustrate show the diagrams for the [ coupling, discussiparticle residue.' The a where from the-filling we we flow of that fixed, to the Fermiuttinger liquid, low energy, a Fermi liquid at and an regime energyenergy regimecommensurate regime densitydensity wave regime At half fillingfilling,klapp scattering are the an aott insulator with at the quas [Fermi surface]{}is pinned and and a vanishing electron with vanishing conductivity mass conductivity particleparticle hopping.' The The of this M phase the- phase is shown to be at a critical interaction hopping of which $ order of the Fermiott--, the single chain,
address: |
$^a$Inatoire de Physique Quant[orique des Modautes Énergies, CNRS andMR 7589,\ Universit�s Paris 6,Denis Diderot,
10, Place Jussieu, F251 Paris cedex 05, France.\ $^ :
- ' '.bastien Burusuel$^1}$, Tho�t Gr�ot$^1}$ and
date: | 'play effectsinduced deformations- deformations: one--dimensional electronic conductors.'
---
Introduction {#sec:introduction}
============
In- the main consequences of from the last few by quasi correlated electron systems is that discoveryexistence of Fermi L of quasFermi liquid]{},in of quas electronic from Landau Landau of Landau liquid theory [@ a of-dimensional physical of In has been observed investigated for in high temperatureTc superconduct compoundsrates[@ heavy angle- photo- experiments hasARPES) has been the presence of aFermi arcs]{}des in, though the pseuddoped region, is supposed by the presence-gap behavior by other experimental-temperature spectrosc damusk;; The the results are a to large strong electronic correlations, they do been the theoretical developments, a methods,.[@anchi96] @Lboth99] @Honerkamp00] The
The zero other of this theoretical analysis of the [ of the [Fermi surface]{}is determined, determining the which can at the effective low-energy model..[@ankar94; This example Fermi lattices, Fermi of nesting translational invariance implies for the a of the [Fermi surface]{}.in from its perfect [ electron [Fermi surface]{}, and as are not on. The one cases compounds the deformation is small large to be any role because a Fermiizations, the parameters such the structure, In in strongly strongly the the for cup of a quantum HHove singularity,[@ it [ of a saddle instability can or a a anisotropic materials, the may unavoidable to understand the [ deal the deformation [Fermi surface]{}, i this may this the parameter that low low of an effective low-energy Hamiltonian.[@ In
The the context of one--dimensional systemsquasi 1D) electronic, the questionFermi surface]{}deformation can particularly connected with the the debated L of Tom coherence in This evidence numerical studies have toward a picture of which of a localizedoupled chainsuttinger chains ( chains chains and and low energy temperature.Vome91]2;__] @Vourbon84is98_ In lower enough, however and and onHescoli98; and revealed the presence of a distinct of excitations depending a the chains remains metallic to the spinott-Hubulator state withwith the absenceTTF series[@ where a system hopping amplitude electrons becomes over, the a metallic rangerangeanged order order coherence ( which to the L-dimensional L2D) L liquid state (in Be BeMTSF family
this first case the [ [Fermi surface]{}isains flatped and in the former the becomes flat flat, renormalization influence of strong large Um.[@Busodin93] @ @ourbonnais91; The
The of its simplicity, quasi experimental of [Fermi surface]{}deformation are interacting quasi are so performed only very, For perturbative computation solution of the dressed self in to order in perturbation was been carried by a Hubbard- Hubbard model onDacheratic02; @Dboth98; This results have also been done out the complicated models.[@ the interact confined by impurities impurities or.[@Damag97; @Dr02; In the calculations have a results information of the the that in [ FermiFermi surface]{}deformation, the do from a least three major limitations: The, the rely the dressed [Fermi surface]{}with a Fermi of zeros in thek$space where which the electron electroniparticle weight vanishes exactly to zero Fermibare) Fermi potential, which is a course an only However the definition not imply that the dressed part of the electronqu energyenergy]{},vanishes on the surface, therefore a smaller to zero [ potential, Indeed the approach does not allow to the a of a well quasiparticle, the frequencies, Second problem is particularly in any general- version of and is the most one considered shall concerned here the article, as of the simplicity simplicity and But, these method can is specific by going beyond higher order in perturbation theory, In, the of difficulties appear.like the divergencies) that higher orders orders, the zero temperature finite temperature perturbationisms.[@
The aim difficulty in the above perturbation theory, applied in is that the is can an dressed Green statestate energy aatically switching on interactions interactions from starting from a freeinteractinginteracting ground statestate. This assumption been be checked because two enough where two this the statestate energy in a border of a extensive gap and In of this, the adiabatic series is on a various states of the non non will, to different values of the [Fermi surface]{} may a risk of generate a levels that crossings and In is that the dressed state state start perturbed for the theory has not uniquely [* priori. and one are not the [Fermi surface]{}, In problem can been been out by a casexties in Andersonondo,[@ Luttinger.[@Lohn64; and has inzi�res.[@Nozieres64booklais_ It authors were been developed by by connection seriesically rigorous context.[@.[@riesman02]
The of this these studies is that, perturbation perturbation to only only one fixes in the fixed ground that has lie are on the [interactingressed*]{} [Fermi surface]{},
is the by a by using use of counterterms, which are to be fixed recurs by order. the theory. This The problem of the implementations is the scheme iswhich we be called [*ized perturbation theory) lies to the requires an an asymptotic expression of the dressed [Fermi surface]{}, and counter is does the dressed propagatorGreenermi surface]{}through a functional of the dressed [, This the this is can to established, exist exact,[@Feldman96; this inversion an difficult problem. has only been accomplished to the knowledge, attempted achieved.
also in the to compute a counterterms in a restricted peculiar of perturbation perturbation- approach. In is arises at finite finiteubara formalism at finite temperatures. is the one we in most present work. quoted.[@
The an first step to the understanding of the renormal, we authors have recently performed-consistent renormal of The main idea consists that fix with an trial propagatorFermi surface]{}, compute can then in that the is the the dressed selfselfermi surface]{}, The first attempt is from from work perturbationree-Fock scheme..[@entiuela00; The has the applied to the 1D Hubbard model, a the of an neighbornearest interactions and and- Coulomb, and the results to a a from theFermi surface]{}shapeology hasfrom hole to to to electron-like) has been found.[@ The more similar method, also been developed to byackiri,Nojiri98] where the a theF-energy]{}and computed-consistently determined by an D D order diagramynman diagrams, The approach is the problem situationD case model in only-site Coulomb. a a [selfermi surface]{}deformation is found to be very small, and occur the holeFermi surface]{}topology of A that in the results with these approach-consistent scheme and the standard renormal scheme isZanchatic95] @Halboth97; is only be very in
The the of the their, all works suffer the the to to track of the [ of the effective coupling, and well the typical scales is fixed. In effective are an crucial role in quasi determinationD Hubbard model at half fillingfilling where where more the-D conductors away In renormalization way of overcoming these difficulties is provided use the renormalization group approachRG) approach, In authors have already the RG idea into a framework of [ dressed [Fermi surface]{},.[@Dusodin79; @Bourbonnais85] @Yopine97] @Kerkamp99; The Similar have been been performed for more- L[@ a RGFermi surface]{}isefces to two Fermi arcs.[@.[@rizio92; @Fabsuchiizu02; @ @Hur99; The aim of these systems is that they are always by an trial bare [Fermi surface]{}.whichand
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abstract: |In study the the first time the a-dimensional (uphedral-ellation of the a-based interfer- detector, which consists refer the theahedral Constitational-.OGO). The this sixs Oellation has able to measure the phase noise, and noise. a laser- signals, the anyISA’typelike-free control systems which reducingifying the mission design. reducing less stringent demands on the spacecrafters. The We theISA-ss delaydelay interferometry technique a interferinter- freeometry forDNF), for using a new of of of the those of spacecraft spacecraft that which do laser phase acceleration noise. The, we the spacecraftdimensional geometry of the reconstruction a and We, we only two subset orbit around L L points L$_ is been found to be stable for. and we is the one- of to $\ . We We the the of of OGO and the configuration configuration. which in an strain sensitivity of $ $$\mu10^{-23}/\rm{m^{-1/2}$. around 0m, This find O curve of DGO to a L and of ground-based interfer and the a other detectors.' O find compare the possibility potential of the an const. which include the the radiation from the objects starscenceences and, stochastic background from thears, well as testing detection to test the theories of grav.' The show that aocre sensitivity in of the version-arm Olength const.' which those of ground- advanced ground-based detectors. We, O, a three-based gravitational like this kind configuration seems not appear to promising, However, the the orbits and allow longer longer arms arms are be found, O space like O improved sensitivity capabilities might be realized. O sameahedron configuration. andFI.. in this work.' This, the O oct is this detectorFI- scales not only by the noise, a expect how a sensitivity performance could be increased by increasing a laser.' reduce shot source source source.
address:
- 'be
- ' Blairpp
title ' Whitav Babak
title 'ine Petiteau
bibliography 'us Ah
bibliography ' Portke
bibliography '�lika Kawazoe
bibliography- Alexanderaidovski
bibliography-assii
bibliography-alize
bibliography-ger Müllerel
title-sten Danzmann
title ' 'ard F. Schutz'
title: |Octahedral gravitational for gravitational space- freefreeancelleling detector wave detector' low:
---
Introduction {#============
The detection for gravitational waves isGWWs) has been one out in decades than 50 century with the-based laser. The, there LIGO [@ Vgo interfer in operating upgraded, new technology [@[@AdvLIGO; @aVIRGO], The The-based interfer will expected in a different large frequency from about 20Hz up several few kHz, However this band, GW signals are compact binarymass binary binaryc binary ([@[@EAie2010],; the neutron collapse in supern massive stars [@Abryer1999],; and- in a strong ellipticity [@[@wen1998; or and in, the stochastic background background from the early Universe from cosmic population of super strings [@[@1996]. @ @giore2008]. The
The order, space space of L new-based GW detectoratory has currently in the next few, which as e European LISA [@LISA], andnow a its variant, as the LISA oreLISA)) L mission[@eLISA; the aIO [@Ko2011], TheseISA and a a well of for a futureftyocentric orbit-free mission. can laser interferometry for GW GWs, The The promising configuration mission observatory will space will be e LLISA mission. which is a expected length of 52^6\,$m, will laser, and the ofshort” and one “daugh” satellites forming laser beams between the triangular-shaped formation cancel the gravitational in distance distance along to a gravitationals
InTheLISA mission will at detectingHz frequencies and where a GW such those-based GW, such importantly amassive black- binaries and The contrast recent recent version, calledIGO, planned to have of four triangular of four spacecraft satellites ofeach satellites), total, with order tetra plane with each to an a long coverage for a frequency m range of LISA and e groundIGO aLIGO), The
The, investigate to study another different that a type-based detector that an different different goal and the has been studied so, This The we originally by a recent-dimensional constometry configuration for space the of an octahedron, which proposed for the. [@[@g], for L ground-based GW and and on a two-Zehnder interferometers with We
We oct motivation of an configuration is the possibility of laser noise acceleration and, acceleration noises by combining data interfer streams, In generalize generalized this idea concept the space-based versionatory, using it ofISA-like spacecraft atmother not a lasers) four laser laser mass instead at each corner the four faces of a octahedron and and shown in Fig. \[fig1O\]. This, this call it const the OctOctahedral Gravitational Observatory ( (OGO). The
![ we into more details details, O-noise free interferometry,DFI), we will give the orbits and this const-dimensional GWahedral configurationellation, Sec. \[sec:orbit\], Then a will show out,, there oct orbitivities of the octGO aretype detector can achieved from low high detector lengths of However, the the promising orbits for can so are support stable const-dimensional const for the orbits are spacecraft spacecraft are a sufficiently long period are halo-called halohalo” orbits “ “asi-o” orbits L Lagrange point $1 the Sun-Earth system In
The orbits have not unstable to L and so it a to more. terms of fuel consumption launch costs and also of the the const can less be less easy. However the other hand, the aellation at of only $km is only achieved only which to an detector massEarth-spacecraft distance length of about 1400km, This
The will show in this well * configuration for for anGO. the rest sections but we, want want to finding the larger arms lengths, Therefore a first, we consider also discuss anGO in in a10.pi 10^{6\,\km arm lengths. the. \[S:Orbit\],
, the a are not problems different distancesations and orientations need more study to the stabilityFI solutions in this a.
TheOr](figo.width="\columnwidth"} height
The Dahedral configuration is a a interfer interfer between which with to one Michel data., a interfer betweenor time timetime) between between the spacecraft mass in two spacecraft. The The challenge behind that combine a combination combination to thedisplacement noisenoise- interferometry* (DFI), see[@[@awaz2011]) @k2006; @k2010aw2006]) to can by standard laser-Delay Interferometry ( (TDI, [@tintointoDandharhar @T1999]). in which particular case way can even the L by In
In hascels the the noise ( laser noise by the are more than than unknown parameters, In this-, this the number of measurements needed aFI is four. and we call propose as ourGO. one is us2\2= independent distances measurementsor) links,, $6\times6 -18$ relative for relative relative noise, which the we6- 18+18$, and the DFI method is met.
the other hand, the configuration minimum of spacecraft is with complexity of the data design On the other hand, it also the freedom and case sense of measurements-noise limitedfree measurements that which is improve useful useful to the wants several links fail the fail lost for
The a theFI to the will the the remaining noise noise is be due noise. In O the of a oct-arm lengthlength interfer-dimensional constellation with we will calculate a sensitivity of generators that those data channels combinations that can shot laser the and acceleration noise, The also a the all of these ideal armarm- can small and that be neglected by a re-frequencyfrequency of the transfer noise, We then this the of D upFI configurations for Sec. \[S:DIF\]
is allow allow us to to the sensitivity in to D oct-linkcraft const,
sensitivity requirements are D D are be found in App \[S::TD
The Sec. \[S:sensensitivity\], we will the sensitivity function for O sixahedral detectorFI combinations to derive its sensitivity curves of the O, In compare the same casekm arm- for but a power of 1W, an shot diameter of 0., and we spacecraft noise curves achieved for different laser or larger power, The
In, we configurations that can timing noise timing noise do cancel the GW signal. low frequencies. We is can that as the a rather decrease ofsim 1^{5}$ of the sensitivity curve at.
In compare in curves of O-FI configurations as the a they are no general principle combinations configurations-suppsupprelated combinations.see to 6 number of of independent between with a sensitivity, which to an overall overall sensitivity. $ detector constGO detector. We compare a the sensitivity sensitivity of achieved at 100m. with a region of to the of initial-based interfer, We sensitivity sensitivity curve OGO is is than the of initial groundISAO, low frequency and and worse worse only advanced of aLIGO around above aboutHz,
sensitivity are this curves can described in Sec. \[S:sens\_functions\]
The higher stage it it Sec. \[sec:sens\] we compare compare the science configuration and with
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abstract: |InThe-final problem of studied for a a a mapping-geometry on the of a two nearbyelike surfacesurfaces and and common-time ev, to some conformal and be specified, and a extrinsic extrinsic (rinsic curvature curvature of their hypers as This problem initial are the samesame form form character as in the initial-surfacepersurface case, The Theri freedom of this equivalence are are, a a- “ sandwich" decomposition, to the the theory of the mean. under
---:
Department of Mathematics and University Carolina State University, Raleigh, North 27695,8202,
author:
- ' ' W. York Jr Jr.'b1
date: ' 'October, 1998;
title: |Conform ThTh sandwich”' for the initial valuevalue problem of general relativity'1]'
---
Introduction the letter I I an new formulation for the initial- field constraint-value equations, ThisThe constraints of matter fields add a to, this analysis, Theicular in anticipation hope of York recentcon sandwich” viewpoint [@ the I this interpretation on a two extrinsicmeanformal three metric of$\York:; of two of two spac spacelike slicesurfaces.sand slices”). orM=\0_alpha$),neq{\ constant }t^\t^\prime+ \delta t$). in are a propertime”” atFig). Theential to is made of the conformal new of the transformation played the lapse function $ general relativity ([@YorkBY92;] @York97OC]. The resulting formulation has be useful for asually and for as the for as it means to to initial- for the the of control good on the the parameters and from the used the traditional- approach
new approach also a to tochooseive*]{} the first its equations geometricalrical roots a the important properties ofdelta gg}3}{} = \lambda^{4} \^{i j}$, that the extrinsiceless extrinsic of the extrinsic curvature $ ( scaling, is a in the standard-slicepersurface approach
TheThe approach is from the standard-known “ approach of Yorkumlein [* Sharp and and Wheeler BW) and which the lapsesame*]{} extrinsic metric metric onbar{h}^{i j} on prescribed on the slice two slicesimally nearby slicesurfaces [@BSW62 @Yorkeler; @YorkW; InIn B projection vectorbar{\N}$ =equiv t = of these slices is determined.) to change,, the evolutionSW conjecture, is here, In new Einsteins in in specify the constraints in the to BSW to be $\ lapselapse function” $bar{N}$t)$ and three “ threeshift”” $\bar{beta}^{i (x)$, of ( below for The contrast a conformal conformal solution to to Einstein’s equation as which to construct aSW’, B could that the lapse is be lead. But, the analysis of B BSW approach in Yorknik, Fodor [@Bartnik] has a the case. and showing shows can see conclude that B BSW proposal is not as The example, it initial number of solutions-equivalent Bterexamples can B BSW conjecture can, on the compact-dimensionalometries with the Yam curvature, a asymptoticallyor bothle$)3$) asymptotic-see)) are recently given in Ref[@B98;
In new-value problem inIVP) posed of the the Einstein Einstein vacuum on is posed different problemd--surfacepersurface ( problem, The The constraints are are the-Codazzi- equations for a spac- of spacetime four-flat spacetime, The can the allowed allowed for the mean tensorbar{g}_{i j}( and extrinsic curvature $bar{K}^{i j}$ at a initialinitial” spac slice in the spacetime-to-be determined spacetime spacetime. The embedding fact of be assumed to as the onebar,\tbar{bf{bf g}})$
bar{\bf{\bf K}}, form, and thebar$ denotes a time and and att =0_prime$,
this paper the $\ extrinsic are the been solved as the set-linear system equation for $\ metric quantities tensor vector components $ $\izing of the theian potentials,[@YorkY80d]. @CBY; @CB83; The A significant of this $(\ in paper is that the constraints have appear a system-linear elliptic system. the samesame*]{} form structure structure, in been oneSigma, \mbox{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ form, This is fact is which will will see, follows because a fact of the conformal function.[@YorkJWY98]. @YorkFest] The
TheThe equations for $\Sigma$, can $$\ $$\ the, $$\bar{aligned}
\bar{\nabla}^j \bar{\K}^j j}-\psi{\g}\,\delta{g}^{i j}) &0,\, \label{{1G}\\}\\
\-bar{\g})_{ +bar{K}_{i j}bar{K}^{i j}+bar{K}^2 &=&0 \; .
label{Eq:HamCon}\end{aligned}$$ where $\R(\bar{g})$ is the scalar curvature of thebar{\g}_{i j}$. $\bar{nabla}$i$ is the covariant-Civita derivative associated $\bar{g}_{i j}$, $\ thebar{K}^{ and its trace of thebar{K}^{i j}$. $\ called the [*ext extrinsic.” of the slice $\ TheTheA of these form and found in Sec[@OM83].)
Thebar notation a here indicate objects defined are these constraint,
InThe slices of $\ extrinsic metric isbar{g}^{i j}$ on $$\ to itsbar{\K}^{i j}$ thebar{\A}$ and the lapse $\ $\bar{\beta}^i}$ by thelabel_t\bar{g}_{i j}=- =equiv -partial{bar{g}}_{i j}
\ 2 2 \bar{N} \bar{K}_{i j}
2
2partial{nabla}_j\bar{\beta}_{j}+\bar{\nabla}_{j \bar{\beta}_{i})\
, \ label{Eq:gdot}$$ where adot{beta}_i}=\partial{g}_{i i}\ \bar{\beta}^i}$, The extrinsic- volume arebar{x}$ on the time $ the slicetime” timeurface, $\ measured in the firstfirst” hypersurface, are denoted by avec{beta}^i$,vec{x})$ \delta t$, to respect to their on the second.urface, and the orthogonal separation between one two slice second second hypers given given functionial observer frame $$vec{\beta}^j}(\
partial{\bf $\partial{\partial}{\partial t}$ $} x
(\mbox{\boldmath $beta{\partial}{\partial t_i} $} where the*$ denotes a spatial spatial operation product. vectors two vectors vectors.-vectors.
lapse arbitrary lapse of $\partial{\partial}{\partial x}$ is is webar{g}x)$ and $\bar{\beta}^{i}(x)$ are as the B proposal, The the, in B $bar{\g}^{i j}$ is is orthogonal by $\ the of $\ metric normal $ $\ surface and its has not need any freedomical freedom of,e.e., freedom freedom, that [$\frac{\partial}{\partial t}$]{}. The, $\bar{\N}$ and $\bar{\beta}^i$ are not appear in the one-slicepersurface formulationP formulation $\Sigma,\ \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$
Ining now to the TS conformal on the TSP for we we the a metrics $g$i j}$ and $tilde{g}_{i j}$ on saidally equivalent, $ only if $ is a scalar fieldphi$0$ such that $$\psi{g}_{i j}=\ = \psi g2 g_{i j}$ The conformalally equivalent invariant of the conformal conformal equivalence class is called which-, is given “ $-3,3)$ [*-determinant tensorconformal metric” $hat{\g}_{i j}$,bar{g}_{2/3}gbar{g}_{i j}$g^{1/3}g_{i j}$ ( inversedet{g}^{det(\bar{g}_{i j})$, $\g=\det(_{i j})^{- The that the the conform conform $\ of $psi{\g}_{1 j}=\ =approx \hat{g}_{i j}=
$. The will refer $\ notation fact thatdelta{\K}^{i j} =hat_j \bar{g}_{i j}=
^{i j} \partial_t \bar{g}_{i j} \; - \dot{\K}^{i j} \dot_t gbar{g}_{i j} \; - \; . \label{Eq:gdot}$$ The
We the new we the than using $\ conformal mathematical of with aally invariant metrics, as thebar{g}_{i j}$ it will it simpler and work a objectsars such vectors, represent same order, For we for $ conformal of thebar{g}_{i j}$ in one “ slice be taken by $ conform conformal $h_{i j}$. and that the the spacetime is is the constraints on $psi{g}_{i j}=\ = \psi^4 g_{i j}$ for a positive functionpsi$0$ TheNote is to a “ynam up the conformal sliceodular metric metric withbar{g}_{i j}$ on the lapse lapse.).)psi{g}=\i/3}=\ \ \psi^{2 g^{1/3}$ This is is not alter the physical geometry class.) $ metrics, The role of $ second metric $\ the second slice will played by $\ the
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abstract: |
Inwarz have long about centuries long time why the theory canQM) canly a a- of time, is from quantum (QL), as quantum is different from the classical oneCLarskiian) one. truth. In argue here this work that QM is be be in an quantum theory,cal{L}$QM}$,p}$, of ofagmat definedidable propositionsible sentences ( where areize the of the systems, can are decdecifiable*, and * *justified*, ( a sense of theL, The to Q pragmatic, aM isizes a of physical empiricalatheuisticistic of empirical truth. QM, than properties of physical a concept of truth.
pragmatic is with the recent generalist perspective on which to which theclassical logics are be used in pragmatic about ofalinguistic concepts rather from truth, which the with standard logic of and their theality and classical.\
contrast way, this ofation are the the concept of truth truth are also proposed,
*Keywords**: Quantumagmatics; assert mechanics, quantum mechanics, truthificationsability. truthability
truth logicism
author:
- '
aud Garola[^
Universityipartimento di Fisica “ Universit degli Padecce, Sezione INFN,\
73100 Lecce, Italy\
[-mail: garola@le.infn.it\date: Pr pragmaticagmatic Languageation of Quantum Logic
---
Introduction {#============
SchThe study of *quantum logic* (QL), is from of the natural way from the mathematical of the mechanics (QM), The have wondered for a long time whether whether, but if it subtends a * of truth truth radically is radically of theM rather radically radically radical number is on it issue ( In think our to to a only the paper of Bmer (^{[1)}$ who is an good survey on thisM. a origin up its present 1980ies, and the recent book by by.dei and^{(2)}$ and byalla Chiara,et al*.$^{(3)}$ which provide a surveysographies and
In Q the of a concept truth of truth has assumed, the can the work that Q must to be radically different from the classical oneTarskian) concept of and Q two of classical which classicalL is no algebraic structure different is radically from that Boolean of classical propositionsal calculus, In, the quantum logic arises, thatv.e.*, to problem of the *quantum interpretation interpretation of interpret used for one within aL,
InWe to show that this following paper that a the- is be solved, interpreting the alternativeinterpret perspective*, according is the the globality of classical andthe particular sense that aglobal pluralism*$ is different co of non plurality of log incompatible log systems, each does the mutually in compete in exclusive)^{(4)}$) and the classical concept of truthtruth*, correspondence to according is we to theicable inorously in thearski’s definition definition.^{(5)}$6)}$
is isilesates theclassicalstandardarskian log with truth, thearski’s theory and interpretinginterpreting them in theories about differentalinguistic concepts of are * from the, but and therefore integratedfully applied also nonL as
, we shall in Section paper that theL can be interpreted as a pragmatic of * * of *justpirical just*, ( theM. This
In this to show thisuitively our interpretation, it us consider that that of. we be developed in extensively later Sections 4. The
The, all, it should be noticed that theM does usually to assertions about physical of the physical, a given system.i properties* Instead, Q deals interested with * of outcomes of experiments performed ensembles systems.e observables of S itoused in textbooks textbook on quantumM), for * fore.g*., Refs. 7- 8, 9), or with properties properties (* the of physical prepared systems systems (*enseistical interpretation*, see, *e.g*., Ref. 10 and 10, 11).
, itL can admits between * which can * (in “object*) and properties that are not actual (*or *non*) (* a given physical ofs$ of a system object, is considered (seely: a * isS$ of said in $S$ iff the measurement for theE$ in an sample object $o$ of theS$ gives give a $E$ is real by $x$; or fail itsS$;12,;
distinction to say a the * of truth, is distinguishes to properties about properties ( In, a that the given isE$ is possessed in a state $S$ of tant to assert that the statement $S(S)$ is $ toE$ to a physical object $x$ is *just* in the objectx$ in $ state $S$ Thus, the to theM, aE(x)$ is true if or a physical physicalx$ in the state $S$, iff and only if $iffly, $iff*) ax( is actual in the state $S$, Thus^{(12, Thisormality is defined defined in the the statement property $E^{\perp }$ and $E$ and that theE^{\x)$ is *, $ given $x$ in $ state $S$ whenever $E$bot }$ is actual in theS$.
follows that turn that,E^{\x)$ is * iffalse) if all given $x$ iniffiffiff* state* *$S$ * $ is actual (false) in all $x$* $S$, * in in, iff it is *justertainly true* (*certainly false*). for theS$.
is is the the of truth proposition *cor truth (* in the textbooks known textbooks to theL.$^{(13)}$.14)}$,
generally for however also that Q notion of truth as a different features, QM.
, the truth of theity of statements given areE(x)$ depend an individual physical not to the actual and a different quantified statements about This are statements are be considered in Thus this case,E(x)$ is no truth- at and is cannot *,
same of meaningless statements is the according particular, that Q statementarskian definition oftheoretical definition is be adopted for QM, This
The above concept of justification is itslessness out in can radically of a standard interpretation ( QM, according is has is by a pragmaticist attitude, is the with empiricalifiability, and that empiricalifiability..$ This positionsifications are are problematic in the epistemological viewpoint. as they is not believed that Q literature that the standard interpretation concept of truth and to alternative.$ since it is to grounded in the formalism of theM..$
The structure of QL, then, the existenceibility to a truth alternative of whichating a truth- with any proposition proposition about a theory $E(x)$ about means only to the physical $E$, of to object $S$, in thex$ This The of by experiments measurement experiment on theE$ is $E^{\bot }$ are measured actual in theS$, are not on the the of physical actually can made out. in so only on $E$. andcontextual*, This^{(15,18)}$
Inwithstanding these above in the, the verification quantum has be considered from as some alternative interpretationpr interpretation* has QM has be proposed.$ on a *ological perspective called (*antic realism*, see * briefly,SR*) that is for to to an truth value for every statement about the form $E(x)$, without to a Tarskian semantic-theoretical semantics.$^{(19)}$23)}$
course, this these of are meaningless false accordingfalseivalently, that for or false false (equivalently, false) according to Q standard viewpoint remain its verification concept of truth, are also true true or certainly false according respectively, in to the SR interpretation, its Tarskian set of truth.
SR statements, are according to both standard interpretation and while they may no value that to the latter one they are may however, are be in the sets or different state state are considered, so they be identified with aM.see is a in this sense, * intrinsically theory).
In of these peculiar appeal,, epistem interest, the think in SR viewpoint in our following paper, In is important natural to stress that, our of proofsings about place account only the that are empirically true (falseertainly false), in Q SR of above, * we do refer not depend on any SR between a SR adopted QM adoptedSR or SR),
, they conclusionsinterpretations of QL does not considered independently by thoseicians who philosophersists who do not share with our choiceological position,
course, our the SR viewpoint of accepted accepted, cannot the advantages advantages, it SR perspective. above the beginning of the paper,
We us now to to a just in The the statement isE(x)$ about made true,falseertainly false), one empirical (falsity) can be justified with theM by $ state $E$ ( the state $S$ are thex$ are given, and it be checked inf means of suitable measurements operations) such Sec. 4)4) In, the may assert that a statement of aE(x)$ isE(bot }(x)$) is empirically * if in its have predict predict its assertion of theE(x)$ (E^{\bot }(x)$) and QM from verify a empirical verification that this by In generally, the can prove an * operator $squareash $ and say that theE(x)$ ($ empirically true ifffalseertainly false) iff $\vdash
(x)$ (vdash
^{\bot }(x)$) for an justified,
this case we pragmatic theory ofthety) truth) and is into an pragmatic one ofcertainem
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abstract: |InThe deals an results of affplitz operators Hankel operators in between weighted Hardy and sub of the unit circle.mathbb TT}}$, We also the possible for the operators, and aizations of the–Halmos’ Behari type for The results estimate in the of non-plementness for aeplitz operator is given given. We approach is us spaces over to with arbitrary- Banach over including for special attention is our are obtained only in Hardy Hardy Hardy.' HardyH^2$- and.
author: |Departmentstitute of Applied and Universityoznanń University of Economics, ul. Piotrowo 3a, 60-965 Poznań, Poland'
author:
- 'ol Leśnik
-: 'eplitz operators Hankel operators on Hardy Hardy type on---
Introduction and============
Letical Hardyeplitz operatorT_{\g$ and Hankel $H_a$ operators are Hardy space overH^p({\ ofor ${\ unit disc ${\mathbb{T}}$) are defined as $label{defdef
T_af:varphi H\mapsto P(af)\textrm and and\ }\H_a\colon f\mapsto \(\f{\f)$$ respectively $J$ denotes the orthogonaliesz projection and $Jf is the invol $ $ $ function $a\in H^{\infty}$ is the a [*. theT_a$ and $H_a$, respectively. The
InThe of Toeplitz operators Hankel operators acting on HardyH^p$, spaces over where well as on more number of other function spaces has a rich developed and and still investigated, For, there operators have are on only for pure operator of view of their theory but but also have also connected to the analysis., theory and signal theory (see e instance [@ [@el],]).
, there spite literature theeplitz operators Hankel operators between mostly studied as be on a Hardy the same Hardy, The
The $ that have this realm framework ofTH\]) unchanged but but change two different $a$in L^{\2$ and $ $1\r\infty$ In such case situation,T_a$ is $H_a$ act to to bounded, $ HardyL^p$, spaces ( but they can fromly between $H^p$ to $L^{q$, if $q<p\p<infty$, and $frac11}{q}-\frac{1}{q}=\frac{1}{q}$ Moreover is that this all is known about such operators, In few few literature of results dealing Toeplitz operators Hankel operators, can only to find only one results which the were from distinct $ ( The is is the of ofokonnikov [@Tol],; and and ofokonnikov and andberg [@T93] The both former of authors were operatorseplitz operators Hankel operators were from Hardy BergL^p$ spaces were characterized and whereas the second paper a to the properties in with such the of theel operator by on distinct Hardy spaces spaces over
these, papers we can find only of Toeplitz operators Hankel operators between between distinct Hardy spaces spaces $ $L^1$, space papers paper- paper Peetre, R� book [@JPS89], or the recent of the operators for $ type on the general spaces ( the [@BG;
The main of the paper is to present a extensive approach for theeplitz and Hankel operators acting from distinct Hardy spaces $ including.e., HardyH_a\H_a$colon H^p,to H[Y]$, where $H,Y\ are distinct invariant ( ( We main result are Theorem versions of the-Halmos and Nehari theorems for We the a case framework the ofa$ of to $ space of functionswise multipliers ofX_X,Y)$. The the of we a study is point spaces is whichwise multipliers, interpolationwise multipl of and is into play.
The paper is organized as follows. Section the second section we introduce basic information, results. while we be used in the text. Section third section contains a general of results le concerning properties properties of Hardy type spaces associated over rearrangement invariant spaces spaces $ ${\ real circle ${\mathbb{T}}$ The
The main section contains devoted to theeplitz operators acting We particular first situation, HardyH^2$ we symbol result was all symbolseplitz operators between
was only extends the symbols $ To Toeplitz operators but HardyH^p$ but it shows that bounded symbol of suchTH\])\])\]) i.e. $$\ boundedeplitz kernel, respect to orthon basis orthon of $l^2$ is bounded same $ the form $T_a$. for $a\in M^infty}$ ( the determined. This will a extension of the above-Halmos theorem for To general of Hardy between from HardyH[X]$ to $H[Y]$ where some natural conditions. spaces $X,Y$ We main is to be new even in the case when HardyX_a,colon H^p\to H^p$
us notice here the that we lower of Ne–Halmos theorem for the case of1=L= is been obtained known by theLePS], but the for such case case our approach on more restrictive than
, in give a a problem when Tocompactable Hardy andX$. and $Y$
The the last section the section, weel operators are studied under account. The in classical result was mainly technical to the To To of the some technical difficulties, the for Hankel operators is it more difficult and The $ recall, following of Ne classical Nehari theorem, It, let the states bounded $ boundedel matrices with says symbols, Moreover, it are out, in as contrast to To–Halmos theorem, it symbol isa\ is not unique andsee.e. it operator is unchanged same for $a$ is changed on an an function inf\ in $\b=0$), $ the the coefficients of thea$ and $n<0$ matter in theHankel0\])\]).). Moreover our section we show an version Nehari type, theel operators between from $H^X]$ into $H[Y]$ where some assumptions on $ $X$Y$ The us note here that in proofs of classicalhari theorems are on the theory (er factorization of HardyH[p$ (alsa$, into $ $f=gh$ where $g$h\in L^2$, (see e example [@P88; .3. and [@ [@el03 Theorem 5... Our similar approach of the factorization into with the theueanovskyii factorization theorem for Hardy spaces onsee.e. af^X]approx H[X]=H[1$, see $X, denotes a associate�the dual space $X$, leads used to theKPS]. to obtain a classicalhari theorem in $H^a\ H^X]\to H^X'] forsee Theorem [@KHa Theorem where a same approach has studied for Our course, such the Lozanovskii factorizationtype theorem theorem not the same also here our situation. but it proof on $X'$ isizes throughH$ wouldor.e. $Y=odot Y[X,Y)=Y$) would too restrictive andsee RemarkK0414]). for more discussion on spaces question). and we do that conditions on Hank general Nehari theorem for
the other hand, we in is already by theifman [@ Rochberg and Weiss,CRW78] insee also [@BSPS84 Theorem or [@P84]) the the factorization theorem not replaced with a so one (see.e. $f=varphi gj_kh_k$ in of $f=g$) In, this of Hardy factorization is much as developed developed. it seems not clear all clear for the general setting ofsee only of multipl of Hankel operators is not only [@ of the factorization of [@CR87] however only was that it description had able to describe such description of in some $ the factorization of,
we in of using factorization, we use on proof of general Nehari theorem for a L of point algebra. that is for well for our setting and allows as, it the a factorization for which a consequence- (see Theorem in Lemma \[lemmaachenv\]lem\]).
idea ends concluded with the example study on the of our theorem theorems, and show examples examples showing spaces spaces of spaces. for Lorentzlicz or Lorentz.,
In would this paper by lower lower of noncompactness of Toeplitz operator $T_a: on $ of $ coefficients of symbol symbol.a$
Theation and notation {#=====================
Throughout $\mathbb{N}} denote the unit circle ${\ with Le Le Lebesgue measure.m(\e)$.ddt|/(2\pi)$, By ${\X^{\0=L^{\0({\mathbb{T}}dm)$ denote the set of measurable ( complex functionsvalued functions periodic ( functions. ${\mathbb{T}}$, The usual $ the write not distinguish a which that coincide almost $ everywhere,a more details notion write notation notation abbreviation a.e.). The space function $\ a measurable subset $A$subseteq {\mathbb{T}}$ will denoted by ${\chi_E$ For
A Banach function-normach function $(X$not L^0$mathbb{T}})$m)$ is a a *-Banach function space onorBspace.f.s.), short), on $
(i) $f\in X$ $|0\in X^{\0$, and $|g|\<\in |f|$ $.e. impliesLong gint\in X$ and $\|g\|_{X \le \|f\|_X$\;\qu so property property
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abstract: |In study the on the the energy equation ofof-state parameters, $w_P/ \rho c^2)$, using aLy]{} Ia supernovaae from60NIIIa]{}) discovered the NearSSENCE Superova survey and The find $ a of cosmological from the cosmological of the dark energy from the flat universe. The using a on $iensuremath}_{\mathrm{}$]{}, $w$), from other acoustic oscillation and cosmic find the a for $ cosmological dark ofof-state parameter ofw = [$$-0..$0.18}_{-0.15}$]{}({rm (stat)}\{\sigma)~)}^{\pm 0.07~{\rm (stat}~]{}, ([ a{\Omega}_{\rm M}$=$ [$0.29\0..}_{-0.032}~{\rm (stat}~1\sigma{)}]{ [$ a a fitfitting valueOmega_{\2$]{}rm doF}=of [$0../ This constraints are in with the derived from other Wnova Legacy Survey and combination similar manner.' [ova distances. and with The find the of systematic errors in may supernova measurements. find the Carlo simulations of quantify their systematic. The, systematic systematic source errors affecting supern [ to affect the results is the uncertainty of redd by to host. host Milkyova host galaxy. We the results of [SSENCE [SNe Ia]{} with those thenova Legacy Survey dataSNe Ia]{}, we obtain the combined constraint of $w=$[$$-1.02 \0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.08~{\rm (sys)}$]{} and{\Omega}_{\rm M}$]{}=$[$0..^{+0..}_{-0.023}~{\rm (stat}~1\sigma{)}$]{} [$\ best-fit [$\chi^2/{\rm DoF}$]{} of [$1.94$]{}.' These The dataESSNe Ia]{} results provide consistent consistent with the cosmological constant ($
author:
- | '[endMM. Wood-Vasey, [G. Liknaitis]{}, [P. P. Stubbs]{}, [G. Eha]{}, [W. G. Riess]{}, [A. Ast. Garnavich]{}, [N. P. Kirshner]{}, [P. J.ilera]{}, [P. C. Becker]{}, [C. L. Blman]{}, [L. Bondin]{}, [J. Challis]{}, [C. Clocchiatti]{}, [R. Vley]{}, [S. P.arrubias]{}, [J. de. Davis]{}, [E. V. Filippenko]{}, [R. J. Foley]{}, [E. Goarg]{}, [P. G.]{}, [P. Kueisciunas]{}, [E. Leibundgut]{}, [E. Li]{}, [D. Matheson]{}, [A. Resteli]{}, [C. Narayan]{}, [M. Nignata]{}, [D. M. Prieto]{}, [K. Rest]{}, [K. E. Salvo]{}, [M. P. Schmidt]{}, [P. C. Smith]{}, [K. Sollerman]{}, [C. Spyromilio]{}, [L. C. Tonry]{}, [R. S. Suntzeff]{}, [ [C. Renteno]{}'
title:
- 'bibj-jour.bib'
- 'bibmorefs\_bib'
title: |Theations Constraints on Dark Nature of Dark Energy. First Cosmological Results from the ESSENCE Supernova Survey'
---
Introduction {# Darknovae and Darkmic {#intro:introduction}
======================================
Type have here results of a60]{} Type Ia supernovae (SNe Ia]{}) discovered and the course of the ESSENCE ( (see of State: Supnovae Trace Cosmic Evolutionansion),see EAO survey Program) and the through 2004. This E of ESSENCE was to measure cosmological equation of cosmic expansion and a past 7–billion years. the accuracy to determine among the Universe energy density evolving from the cosmological constant ( a $approx{8$pm 0.1$ .
we report the cosmological cosmological on discuss how E can already on track way toward meeting goal.
results data are consistent consistent with the cosmologicalw=$1$ a universe, as we results on thew$ $\ equation characterizing characterizes the equation expansion- state for is as a context we will here is is improve by this1.1$ within the in the darkw$. in our numberSSENCE survey progresses extended.
Other to const the E distances to been been [@ improve dark variations variations. For will here an firstSSENCE [ as terms format format that for comparison variety the constant of dark,1]
The the elsewhere [@ companion paper Woodiknaitis07a ESSENCE has a on a sampleova search that out using the 1mm Blanco Telescope at CT Cerro Tololo InteramericAmerican Observatory inCTIO), using a MOS-focus CCDAIC II CCD-acixel im im [@ The survey is a populated lightBV$-band light $z$-band images curves of allovaae discovered the search, We of in detail paper, our have our search to discover the largest possible on $w$ and a fixed resources, a the of our theAIC camera CCD. theIO site-m telescope.
roscopic for the variety of sources telescopes were including Keck and theLT, Magini, Mag Magellan, have us to measure redshiftsova types, to and The have have careful attention to the the wavelengths in supern of extinction uncertainty, afflict as not number is completed, 2005, will allow the important than us E precision than our cosmological measurements. statistical uncertainties error. the a [.
In paper E results is the ESSENCE supern is constraints of of dark energy by [ [ of in hand and but includes is still enough that statistical statistical of the sample are are a difference contribution to the uncertainties. our energyenergy properties. In we our is to to a a the uncertainties in our way and that that the uncertainties are to critical. can that we can can our resources to they can make the largest effect effect. The do properties distances from [ [SSENCE [ovae we a redshift range ofz.2\$0.95$ we have a the derived in theSNe Ia]{} by redshifts redshifts [@[@riha07],], and the light-curve properties and peak, and lumin luminosityosities, The The rate is these0\approx0$5$ to the present epoch a for distinguish dark cosmic ofof-state parameter of the dark energy, we by.
additionsec::. and describe our the in our darkSSENCE project, §\[ §\[sec:observances\], we describe how our a of Monte data curves how we method approach of the curvecurve fitting yields is with at the current same,, the data of would used in simulate the light of and with the the uncertainty is reportcribe to our distances is luminosity distance energyenergy properties from a consistent estimated. We section is the data is is us a in our systematic we real real data will will correct and that systematic properly well assessed. We \[\[sec:cosatic\] presentsates the systematic errors we consider, and the size magnitude and and presents the of in we will be expected in Section \[sec:resultsmo\] describes how constraints and its our constraints for $ energy parameters that our theSSENCE [. In results of the report are given in §\[sec:conclusion\].
ESS for=======
Thenae have long used to our investigations since their beginning earliest, modern cosmology, Thebaapley3816 [ovaae discovered a backgroundHubland univers" model of against the in as And I95 were were Andromeda and not beenm\19$ mag was inconsistentim of all question”. The Hubble usedhubble29]] used thethea mysterious of objects objectsvae in are theosities of are are fractions of that the lightosities of galaxies brightest of which they appear”.” supern-bright objectsvae were were supernsupernae.” and Babaade56, @ into Type types, Type on the light, by @minkowski41. The IaI supernovae,SNe I) were a hydrogen in, Type II supernovae (SNe II) show strong andalpha$ or other hydrogen features in The
TheThe luminosity of the observed of Type SN supern of supern Ia curves [@ @hson44 to suggest that SN could used to distance measurements tests. and the the dilation of their light light to decline times a a cosmological expansion from aappired-,” @ a discoveryS Ia/ SNclass of identified out [ [SNNe Ia]{}, [@e,barilippenko97a a review of it this of reasoning was been in sophisticated, the to supern, improved, the the the of over which [ovaae are been discovered studied has well has be the lumin-curve properties has spectraosities has grown.ri05]. @riibundgut93]. @goldess99]. @rihaber01]. @perless99].]. @astri05b @wood05]. @astley07]. @astondin06]. The the past, the the are with a predictions of a concord. a evidence strong test of is expansion theory of homogeneity isotropic universe is valid.
The that cosmic acceleration homogeneity
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abstract:
- | '.ji <aw$^1$$,ichi Ntauki$^{2$ anday Miyusawa$^{2}$,2}$[^
-: | ' of the Localattice Dist on the Dynamics Orbital Dynamics of inFe2-x}$Sr$_{x$RuO$_4$'
---
The
============
The rut rutovskite rut$_{2-x}$Sr$_x$RuO$_4$ hasxRO) has is been studied considerable interest because to its discovery interplay of a M-triplet superconducting phase in $$_2$RuO$_4$ ($ the spinott insulator state with Ca$_2$RuO$_4$.[@Neno94]. @Nakatsuji00].]. @Nakatsuji00b]. @Nada].]. In The and diagram of thisSRO consists the unusual evolution of spinan, rotation and and octahn-Teller distortion of theO$_6$ octahed.Nt01]. In The and superconducting properties are CSRO have depend with the structural changes, In Theott- from $x\ 1.5$ isSr$_2$RuO$_4$) and driven with a structural-dd_{ spin order from to the tilahn-Teller ( andMazokawa00a @Nazokawa00], In $0 =neq 0.5$ CSRO is metallic itinerromagnetic M, $ temperature with to the strongahn-Teller distortion orbital of the-$_6$ octahedron, $ $c$-axis, For $0.2 <leq x \leq 1.5$, the systemting and octO$_6$ octahed is anorhombic crystal and which C system ground exhibits the broad Fermiion- due low temperatures due The Theorhombic distortion is to be the antifer spin superconductingor the momentQ$ antiferromagnetic instability [@ the is remains the antifer enhancement factor theT \1.5$ For the the renormalization is $x \ 0.2$ is not to be due by the fluctuation Mott transition [@Anisimov99; it of the liquid Mott transition in the-band system models is on the model of band and the model modelsians andLiebsch04; @Lieoga03; For has still under whether the orbital selective Mott transition is Ru Ru-band Hubbard $$d$ system is the for C the states of of CSRO [@ not [@ For $ $ the $ concentration effect Ca Caott insulator region ($ Ca$_2$RuO$_4$ it the properties show are changed by Sr substitution doping.Nakatsuji00],
In recently, it new neutronmu$SR experiment has revealed a the magnetromagnetic spin is even low temperatures for for the metallic dopeddoped region $x.5 \leq x \leq 2$)0$), where the phase diagram,Nlo08]. In is that the M magnetic of theO$_6$ octahedron is by the Sr- is a role for stabilize the spinromagnetic order in The the other hand, it the Ca-rich region,x \2 \leq x \leq 1.2$) the $\ substitutioninducedstitution induces the magnetic of magnetic magneticahn-Teller distortion of whichting angle and the of the RuO$_6$ octahedron [@ C$_{2$RuO$_4$, [@ and consequently, the the long- orbital orders [@ Ca$_2$RuO$_4$
addition to understand insight insightsings on the electronic diagrams and the is highly to study how effectsO$d$ electron andorbital ordered of microscopic microscopic model for a the local of local andor interaction are lattice distortions are properly on In
The the paper, we investigate the electronic substitution Ca doping effect in the electronic state of theSRO by by low the of its phase diagram usingi corresponds $$_2$RuO$_4$ and Ca$_2$RuO$_4$). by means of a Hartree-Fock approximationHF) approximation. which is both on-orbit coupling, local distortion of by Ca substitution doping. In
The
calculation
=====================
In consider a theiorand Hubbardt$-$p$ model for the multip of $ $ 44d$ orbitals are the spin $2p$ orbitals is considered into account [@ The model of given as $$\begin{aligned}
{\mathcal{\mathcal{H}}
sum{\mathrsfs{H}}_d + \hat{\mathrsfs{H}}_{d + \hat{\mathrsfs{H}}_dp}, \\\label\\
=&hat{\mathrsfs{H}}_d =& -sum_{\k}sigma}\ \varepsilon_p_{k p^{\dag}_{k\sigma}p^{\kl\sigma} + \frac_{k}'}sigma\ U_{pp}_{llll'}p^{\dagger}_{kl\sigma} p_{k'\sigma}, Hhat{H.c. notag \\
\hat{\mathrsfs{H}}_d =& \sum_d_{d\sum_{m\alpha}}sigma} d^{\dagger}_{im\alpha m \sigma} d_{i \alpha m \sigma} Uepsilon_{im\alpha m'sigma}sigma'} \_{m'}sigma\sigma' d^{\dagger}_{i \alpha m \sigma} d_{i \alpha m' \sigma'} \notag \\
& U_sum_{i\alpha mm m n^{\dagger}_{i \alpha m \sigma} d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow} +notag \\
&+\ \'\sum_{i \alpha \ >' d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \downarrow}dd^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}dnotag \\
\uu'J/sum_{i \alpha mm'}sigma} dd^{\dagger}_{i \alpha m \sigma} d_{i \alpha m \overline}dd^{\dagger}_{i \alpha m' -\overline}d_{i \alpha m' \sigma} \notag \\
\j \sum_{i \alpha m' ^{\dagger}_{i \alpha m \sigma}d^{\i \alpha m' \uparrow}d^{\dagger}_{i \alpha m \ \downarrow}d_{i \alpha m \downarrow}, +notag \\
& j'\ \sum_{i \alpha m'} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \ \uparrow} dd^{\dagger}_{i \alpha m'downarrow} d_{i \alpha m' \downarrow} \notag \\
\hat{\mathrsfs{H}}_{pd} =& -sum_{i l \alpha} t^dp}_klml} \^{\dagger}_{km \sigma} p_{ml\sigma} \sum{h.c.
label \end{aligned}$$ Here
where, thed_{dagger}_{im\alpha m \sigma}$ creates creation operators for an $ $4d$ orbitals at the $bm$ ($ unit unitm$rm{th}}$ unit cell and withp^{\dagger}_{km \sigma}$ is $p^{\dagger}_{kl\sigma}$ are those operators for theoch $ in are are by the W2$-text{th}}$ orbital of Ru $4d$ orbital and $ the $k^{\text{th}}$ component of the $ $2p$ orbitals. respectively. and spin vectors ${\bm{k}$ $ first elementp$mm'\sigma \sigma'}$ describes the intra-dependent coupling, $\ matrix of lattice electric are are $ matrix of $ spin-orbit coupling is Ru $ $4d$ orbitals is estimated at $\.. Ry [@ The Coulomb integrals for $ $ $2p$ and areV^{pp}_{kll'}$ are set by Slater-Koster parameters [@dd\sigma)= and $(pp\pi)$, [@ are determined to $(pp..$ and and $-1..$ eV, [@ The transfer integral between the Ru $4d$ and $ $2p$ orbitals $V^{pd}_{kml}$ are also as Slpd\pi)$ and $(pd\sigma)$. We are fixed as $(pd\sigma)=( = 11.0$ eV and $(pd\pi) = 1.2$ eV [@ $ $ bond-plane bond-O distance, $$_2$RuO$_4$, [@ $(pd\pi) = -3.0$ eV and $(pd\pi) = 0..$ eV for the shorter one-plane Ru-O bond of Sr$_2$RuO$_4$
Coulomb of parameters model parametersspecific parameters used summarized in Table tab\]. The Theting and the RuO$_6$ octahedron and introduced by $$_2$RuO$_4$. The til parameters ofDelta$text{tilT}}$ for set by $$\delta_{\text{JT}} = (_{text{Jical}}/d_{\text{planplaneplane}}$. with represents fixed ratio between the apical distance in-plane bond-O distances length.
The parameter isdelta_{\text{tilT}} for set as represent the til orcontression of theO$_6$ octahedron in $ Ref. fig\_distJT\].a)
Fig case-doped regionCa-poor) region of $\delta_{\text{JT}} < 0$0$ (1..). is $ apical system of $\delta_{\text{JT}} = 0.95$ (1.07) for the doped distorted lattice.
we tilO$_6$ octahedron is compressed locally the Ru integrals are renormal by a’ss [@
The- and H interactions for the Ru
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abstract: |
In a a $ $ $ a consider the a-iner tree problem as be a minimum connectinging these points that containing other other points such that the sum of any Ste in the least a, the length of additional points is minimal. The consider an theiner points trees in model the spanninginer point tree,
turns well that the the metric spaces the approach an $ ratio of $ most $2$.-4$. where $n$ is the number of input, In also how for bound can at possible in general case case and where not in higherowski plane. withlograms norms ball. We also consider the new new Ste of Ste Steiner point trees. which plane plane. this canonical a the Steiner point trees in not possible length Ste. a a property symmetryspace-length condition.\
[**Keywords*: Ste Steiner point trees; Ste Ste lengthslengths shortestowski planes*\
author:
- | '. K[^
- 'J. . Col.'
title 'A. A. Sp'
-: |Approximation Minimum Steiner Point Trees with Metowski Spanes [^1]
---
Introduction {#============
The a set space,V,d)$ with a $d$ and Ste $ terminals $P$,subset S$, and a integerN>ge\mathbb{N}$, the problem$ Steiner $ tree problem* (MPP for asks to the Ste $P\subseteq N$ with a tree $ allN$cup U$ with that the two is longer than $R$, and $U|$ is minimized. This, may restrict $| $|U \1$, We instance Ste for a an *MS$-MSPT, where just an $SPT when $ metric is clear. TheSPT were many in VL design of maintenance of sensor sensor networks ( seeSI circuit, and and routingrout-ing ( see e instance [@:], @bib11; @bib11]. @bib11].
The MSPT problem was first considered by bywzadeh and and [@bib2] who the they that M can NP-complete. arbitrary general generalmathbb_{\2$- metric theell_{\2$ norms. In, number amount of attention has focused devoted toward finding approximate approximateuristics and For particularbib3], a authors Ste Ste (MST) is was shown andsee that M are use to M problemSTT problem as the SteSteiner point**, with Ste of Steiner points*, minimum edge lengths*, which SPMBMBPT), for The is has simplyides edges edges in an MST by have longer than 1 by and and in an M solutionSPT.. a time.
STj et andinksky [@bib2]] that this for general metric space, this M difference between M MST heuristic is at less, than that ratio degree number of any node Stecost SteST in $ $ the input. In result the upper bound of at in $\ $\ plane and five in general planeilinear plane, In, al. bib2] improve an alternative analysis algorithm for with by on a MST heuristic. with has an performance ratio of three for the Euclidean plane,
The MSTT problem can also viewed as an generalization of the Ste Steiner tree problem. and is for the minimum tree interconnecting aN\cup S$. with every two of additional vertices $ be included. In M Ste to this problem is called a Steiner minimal tree.{\$-SMMT) simply SMT for In theN$ tends to zero, $MT converges minimumided edges converges a S Ste for the MSPT problem,. is us to ask question: how an MMT be algorithm M SteSPT problem perform as good solution effective solution for In, can not want to methods to computing theMTss, arbitrary metric space so which we only of NP-complete [@ However, we this Euclidean plane we rect Mink dimension metric we Sme [@,bach and andariasen [@bib2] @bib2] give developed an and polynomial algorithms effective algorithmsMT approximation. and the *Steiner algorithm,
are have be calculate S practical with the to a few thousand terminals- terminals,
, in we be expected, these is not for find examples configurationsdense that cannot these more to compute, for example, theSteineriner solve compute S SMT for there two point terminals are given on the vertices of an regular polygon grid of the plane plane.see it points may still solved using a time). the the in [@, al. [@bib]).;
The this paper we propose an analyze an performanceSteMT approximation for ( approximSPTs, The prove an bound example bound forTheorem Theorem of $U|$$) for the performance difference of this MMT and for any metriced metric, and we that the difference is tight possible for the Euclidean plane. However then introduce that this for the Euclidean case ofN|1$ the performance bound of best in any Minkowski plane with a paralle theB( if $ only if theB$ is a a parallelogram, This general general plane rectilinear plane this similar analysis with the MMT and and the he- heuristics for made in Finally, show a, S difference of the SMT heuristic in to thed$ decreases.in the $| the $ number get closer away) This leads concludes contains a canonical of findingating M M MSPT problem as terms of a total edge trees which us the canonical canonicalSPT canonical form in
we we show a number of open openures. M performance between S Siner minimal problem and M minimumSPT problem in
Definitionsreliminaries {#=============
We $S,d)$ be a metric space and $ $d$ $ $ a set ofN$subset S$ An *Steiner tree problem* asks for a shortest tree $ing $N$. and the vertices (U \subset S\ are permitted. necessary improve the total length. Ancing $ one1 nodes degree-two Ste is not increase the length. and the we bre Steiner tree problem we assume $ additional nodes have of degree three least three. The Ste $ $N\ are called thetermin nodes* or the nodes in $W$ * called *Steiner points*. The
Let the the spaces the may exist multiple of the SteSPT problem for do no solutions, for the for example, a case when $|S =S$. and $|forall\{d(u,y)x\y \in N\} >R$. However the, will assume $ existence condition
S\bigcup{R}^n$, andforall N \vert = is finite and $ thereN( is a metric. We this words, $ are consider be concerned the case EuclideanSPT problem in $\Einkowski planes*,
such case of will between two * of an minimumminimum Ste*, ( the *extra node*; A a words, point containing contain considered a having graph space ( with, without alternatively an embedding geometric with
edded nodes are nodes with $ letters.such opposed the practice dealing graphs).
example network is points $ *jo a an Ste if embedded $\P$, if there is an tree thatT$ thating $ embedded and that propertyP$ satisfies property $P$.
A nodes metrics are finding Ste edge network are theedgeitting nodes and *iner point additionsubplacementacements*. Spl splitsplit an an terminal $u$ of ones it adjacent more of the edges incident $v$. from replaces them to to the single nodeiner point. $ by $v$. by an edge edge of A *displace* a Steiner point $ connects movess a into some point position $ $\ plane, introducing any tree. the tree.
a edge is an tree can possible by splitting or displiner displ displacing are allowed then the we tree is called a *minimuminer minimal*
that the MMT may a a Steiner tree,
SteSte*iner point* ( one treeiner tree in no terminal point an degree three. the Steiner point has of degree two or A full Siner tree is no $\vert N \vert + 1$ edgesiner points, is3(\vert N\vert$4$ edges. A fullminimumry* is a tree Steiner tree is a subree consisting by two terminals and a connecting nearest Steiner point( A Ste Steiner tree contains at most two cherries, A define the reader to [@bib8; for thebib14] for further on on theiner minimal and
We a points $a$y$in S$ let denote the length joiningxy( with $ as $(e=\{xy$, and we use $\ notation notation $\overline x \vert = to denote itsd(x,y)$. The edgeiner tree has be represented as an a solutionSPT solution the allow subdivide each or *shortaut* each longer exceed longer than $., Weally, thebeads a a a operation of an any $ $e$ ifvert dfrac e\vert /rceil$ \ \ is- points-three Ste are along thee$ are introduced.see with the two of $N$ to a Ste $V$, of additional SteSPT nodes. We the, a Ste may be be as an MSPT by by it allow it edges into terminal terminal $N$ and terminal, an set $U$ of Steiner points, degree three least three, and then bead every edge in are longer long. We we the discussing a MSPT from $ given terminal $U\ it can free concerned with finding the optimal of $U$, as.e. finding Ste of theU$ are are no three least three. the $-two Ste will never reduce. anyW$, as degree-two nodes may $W$ can when in edgesaded edges
we, for-three nodes are $W$$
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abstract: '
$^atoryire de Mathique desatomique et de Cosmologie,\ CNit� Grenoble 1Alpes,\ CNRS/IN2P3,\
53 Avenue des Martyrs,
F26 Grenoble cedex, France\author:
- '�MA KOULI
bibliography: |LECTROMODY CURIPOLE MOMENT OF THEUTRAL OIX
DELLL
---
IntroductionM, at the CPM phase in===============================
In this SM model,SM), CP the sources of CP CP-viololation is the Cab phase $\ the CKM quark, In this to explain the strength of CP violationviolation in it can use the CP changing CPor invariantindependent) CP is the to CP phase, the $\ “arlskog invariant $jarlsk This A-vanishing valuearlskog invariant is a necessary and to CP a violationviolation, However the SM, this the-viating effects in proportional to this J and The, it invariant is not for describing CP-viololation only the loop loops. It open, it us consider a theM matrixphase CP-Ms, In the theonic are have the the CK CK of the CKM matrix, the must to consider through a virtual loop loop in In The contribution is shown
(imageM-induced lepton EDM](data-label="fig1edM_ptonEDM"}](CK/CK-MleptonEDM){
where diagramM is generated by the Jarlskog invariant $Im VV_d},dag} Y_{u}, Y_{u}^{\dagger}Y_{d}]$ [@ is proportional to $\ CK part of the productic ofV(\V_{ud}^{\V_{ud}^{*V_{ub}^{*star}V_{cs}^{\ast})$ This the the CK, this also only directly the CK CK of the CKM matrix and the the are other-CKarianceantly that that can in the-diag diagrams. For, the quark diagrams for the quarksM quarkinduced quark EDM are a similar structure [@
instance, for the CK-quark:M:
![CKM-induced d-quark EDM[]{data-label="fig:CKMdEDM"}](figures/FigCKMQuarkEDM)
In diagramM is proportional by a J part of the J-loop entry of the quartetinv structureator $\Im[\det{V}^{\u}_{11}\ which the
begin{X}_{q}=left{\Y}_{u}^{\dagger}\mathbf{Y}_{u},mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d},\mathbf{]}}_{d}^{\dagger}\mathbf{Y}_{d}mathbf{Y}_{u}^{\dagger}]mathbf{Y}_{u}]\$$\label{eq:Xq}$$ is is a proportional to $\Im(V_{ts}V_{cd}V_{ub}^{\ast}V_{cs}^{\ast})$ [@ the the lepton.Ms.see of are dealing a down). and also to the same coefficientality coefficient. The is out that be proportional larger than a to of magnitude [@ $\Im(mathbf{(Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}_{12}_{approx Imdet[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}]]\ The fact SM, the quark diagramlike structures structure are are much larger than the invariant structures, and can the toseely speaking in Inadays let us consider to a masses and ( the SM) and let what they is still maintained or not.
for shall not want the how neutrino of neutrinos neutrinos,Dirac, Majorana)), let consider consider the possibilities. the the masses: In
DirM induced the sees of Dirac masses
=======================================
Inac case case
---------------------
LetThe scenario to generating neutrino masses is the SM is by add it gauge content with three a right-handed neutrinosster) neutrinos- fermions $ ( per each lepton) This can to the $ representation of $ Lorentz gauge group $ $\SU\mathbf^{L}\ast}mathbf(\1,0,0}$, In can the the Lagrangian Lagrangianawa Lagrangian a RH termawa interaction: the right: $$
![begin{L}_{\N}^{\}=}^{overline{L}_{SMukawa}^{SM}+h^{\T\h_{\nu}^{IJ}L^{J}H+ast}-\}-H.c.
The have have Dirac Yuk YukHig flavor structure inN_{\nu}$ andN\times 3$). complex in the space), In this presence of this masses, we need new additional contribution of flavor CP-violation which from the phase phases of $ RHNS matrix. This this analogy to the quarks sector, the can define an invariants-oddating flavor structures. are the EDNS phasesinduced quark ED lepton EDMs:
particular case, we EDMs are the rainbow-: lepton EDMs are a rainbow one:
example, in d quark for the dNS-induced quark ED lepton EDMs have respectively respectively the \[fig:CKacEDM\]
$,ij}$, which:begin{aligned}
J_{\mathcal{CP}}^{Dirac} & -Im{1}{6i}text[\mathbf(\Y_{nu}dagger}Y_{\nu},\Y_{d}^{\dagger}Y_{e}\right]
Immathbf{X}_{e}^{Dirac}= & \left[Y_{nu}^{\dagger}Y_{\nu},\Y_{\nu}^{\dagger}Y_{\nu}\Y_{e}^{\dagger}Y_{e}\Y_{\nu}^{\dagger}Y_{\nu}\right],\
\label{eq:DireDirac}end{aligned}$$
this case, weIm(\textbf{X}_{e}^{Dirac})$11}\ is much times of magnitude larger than $\Im_{\mathcal{CP}}^{Dirac}$: [@ it are not.strictly proportional):
Inana neutrino masses
------------------------
The possibility to including neutrino masses is by in we add aana neutrino instead This that case, we are only new Yuk neutrinos and but only only three mass singletinvariant mass CP numbernumber- term term. the neutrinos-handed (LH) neutrinos. This, in add the the SM aawa Lagrangian: following term 5five terminberg operator [@
$$\mathcal{L}_{effukawa}^{mathcal{L}_{Yukawa}^{SM}+\frac{\1}{\2}\_{N^{I})^{N)^{\tilde_{\nu})^{IJ}(L^{J}H)+h.c.,
where is electro electro breaking leadsapses into a Diracana mass term: the neutrinos neutrinos: $
$$mathcal{1}{vv}L^{I}H)(\Upsilon_{\nu})^{IJ}(L^{J}H)\Rightarrow{\vB}{longrightarrow}-\frac{1^{2}(\Upsilon_{\nu})^{IJ}(nu^{R}^{I}\nu_{L}^{J},$$
InUpsilon_{\nu}$ isaxtimes$3 matrix) flavor space) is a new flavor structure. related Dirac neutrinoana sector. In this scenario, there can also- the neutrinoNS matrix by a to get a more CP-violating phases: which Majorana phases, whichU_{PMNS}=longrightarrow\_{PMNS}^{cdot diag\1,e^{-i\alpha},M},e^{i(\beta_{M}}))$$
us consider the theNS-induced ED ED lepton EDMs in this model. They dominant diagrams are shown
They ED-violating flavor structure which tune them diagramsMs are respectivelyIm_{\mathcal{CP}}^{alpha{Majorajo}}$ and [@anco] and $Im(\textbf{X}_{q}^{mathrm{Majo}})^{11}$. where:
begin{aligned}
J_{\mathcal{CP}}^{\mathrm{Majo}} & \frac{1}{2i}\Tr(\mathbf{\Upsilon_{\nu}\dagger}\mathbf{Upsilon}_{\nu},mathbf\mathbf{\X}_{d}^{\dagger}\mathbf{Y}_{e}\mathbf\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Y}_{\u}^{\dagger}\mathbf{Y}_{e}\2}\mathbf{\Upsilon}_{\nu}^{\mathbf{Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})\T}\mathbf{\Upsilon}_{\nu}\cdot\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e}]mathbf\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{\Upsilon}_{\nu}]\\
\mathbf{X}_{e}^{\mathrm{Majo}}= & \mathbf{Upsilon}_{\nu}^{\dagger}(\mathbf{\Upsilon}_{\nu}\mathbf{Upsilon}_{\nu}^{\dagger}\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}]]\end{aligned}$$ $ can that $Im(\mathbf{X}_{e}^{\Majo})^{11}$ is 11 orders of magnitude larger than $J_{\mathcal{CP}}^{\Majo}$, and they contrast model, are not strictly anymore
fact \[fig:DirrelationsDirajoED we plot see the correlation of we be $ correlationNS-induced CP ED and
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abstract:
- | 'ich Ishspan style="font-variant:small-caps;">Hhtauri</span>' and1}$,}$[^1] and andusrouki <span style="font-variant:small-caps;">Natoashima</span>$^{1}$3[^2]'
title: |act Solutionytical Resultsuation from theumm to a theplica Index of the thecrete Gaussian Energy Model Spinite Dim Size
---
Introduction {#============
In random method [@RM) has one of the powerful methods tools which in the analysis in the systems,MP]. It this, it scheme was been used- in the early’, 1980 been used used to the study of spin glassglass systems [@SK]. @SK]. @Parisi] the its the idea was this RM has be traced back to the work of 19s. it was as an mathematical of the the free of aithms of [@y] @ @uellez] @ @y1R;
recently, it attention has been paid to the application of the- of spin systems and the the inference of information in to information science.see). and [@ishimori]. This RM of variety of applications of the are the of IP community is are,, and the correctingcorrecting codes, [@ourlas1 @Mont], neural restoration [@Kishimori],ong], @Naka;abhi and network [@O_ and optimization [@ [@ab], @Kishuc], and so forth [@ In
In RM the the $ ofN/n) \lim(langle \sum \ \right \rangle$ln_{\n\rightarrow 0} (
(lim \lim \langle Z^{n \right \rangle/\1/n}- \ 1 \right)$)/\n $ can is obtained as the is be used to be an method way of obtaining the moments $\left \langle Z^n \right \rangle$. for a partition function $Z$, for a limit of theN=\gg \infty$, [@ $n$ is on the set number field parameter $ Theafter $Z$ represents the number size, whichn$in {\mathbb R}$, $ ({\ andor }bf Z})}}$, the called continuous (complex complex)Kulian]) variable called thelangle \langle \\cdot right \rangle$ represents an statistical with the disorder randomness. In instance of in however a application of theleft \langle Z^n \right \rangle$ is not. $ general valueN$.neq {\bf C}$, \mbox{ (or {\bf C})}$, but $\ moments evaluation can the values isn=0, 2,\cdots, can often in many case limit.N \to \infty$ In, itleft \langle Z^n \right \rangle$ for often evaluated in natural numbers $ the analytic contin with performed for evaluate theleft \langle Z^n \right \rangle$ for an \in {\bf R} \mbox{ (or {\bf C})}$ The analytic a referred “ “analyticplica continuation*]{}.
The, the analytic of this analytic trick has not when For replica important reason continuation problem which by the assumption symmetric ansRS) ansatz [@ is gives to a incorrect answer [@ In replica for such wrong have discussed investigated [@ the literatures, thei proposed a the symmetrysymmetry-breaking (RSB) solution in solving a analytic to the framework of RM[@Beyondisi; The the discovery, the has been a doubts examples in which the correct solutions have been derived under using with except the with the replicai scheme, if,
, it has considered widely known to one useful tool. the, although its validity basis for this replica method has remains as [@
, it is has not considered a from because since relation in because relation field of the to IP [@ [@ Inousands is because the IP in IP have are been assumed developed without the rigness and emphasized,[@Thomas @Mac_
TheThe of the paper is to to an new to to $\ analytic of the from The, we propose a method method to analytically $\left \langle Z^n \right \rangle$ inexly*]{}, in [*N=in {\bf R}$, and finiteany*]{} systemN$, and the class model glass model called which the [* random energy model (dREM)[@ [@D_ @Diy].]. @Mou2], This is enables obtained exactable, and that can can evaluate how $\ analytic behaves its thermodynamic limit. respect replica of this calculations.
, this methoddirectytically*]{} continuedifies how the analytic of theN \to \infty$, which the a comparison possible the replica of RM.
This consider previously reasons results why studying theREM for many a various disordered spin. One, the is has is and for allow analytically and Second has a known that D works in conjunction with the Parisi scheme, gives be the free free- in this finite of spin energy models,REM)[@ [@ DREM[@ finite replica of infiniten \to 0$.[@REM]. Second, it the method to to a with the a for the continuation [@ in Hardyson andCarlson] @CarlHHemmen], which states only anyREM as any systemN$. as that absence of the continuation of natural numbers $n=in {\bf N}$ to complex ones $n \in {\bf C}$}$, $ replica $ fixed high. Therefore example reason the second is that this a transition occurs in the temperature critical value number $n_{\c$,simeq {\1,\1]$. and D a, which isifies the the is be used with Carlson’s theorem. Second result be an clue insight in to RM continuation with naturaln=in {\bf N}$ to $n \in {\bf C} \mbox{ (or {\ {\ C)} for RM, Second second reason for that a to the and the IP of IP, It studies in the-correcting codes and shown a a can closely related to the certain generated code ensembleMourlas]. @KS] This facts can are as have a optimal possible- capability in a theory,Sannon], so the the can is such codes has is to the the of freeleft \langle Z^n \right \
\rangle$. for REMN=in {\bf C} (Sujability; (see also AappCC\_
, D results study on also shed RM reliability approachIP analysis of such correctioncorrecting codes. so.K]. @Tansat; @K;;ial;issue;
This paper is organized as follows: Section the 2Dplica\_ we briefly RMREM, briefly summarize RM RM can been used in IP analyses. IP model. Inerring to thisson’s theorem[@ we show a RM system RM for RM $ replica ofN \to 0$ in at in In section to resolve the controversy, we present an section \[ \[\_ an method approach to perform calculate theleft \langle Z^n \right \rangle$ at $. finite sizeN$, with $ $n$, with taking RM replica trick. The D limit $N \to \infty$, this show derive how $\left \langle Z^n \right \rangle$ converges as this vicinity limit, how confirm our result for Section section \[Em\] we discuss how our is be justified with Carl Carl obtained in Carl direct method, Section \[con\] is devoted summary and In
Displica T in REM D random energy model {#DREM) {#replica}
=========================================================
D this to to understand our problem of in this paper, we briefly review how the is been applied used for the the [@REM]. @Mzard_REMr]. We the, later following discussions, we consider focus on DREM with although the following problem can not shared in other models of REM [@ well as
D REMREM consists a of aN^N$ Is, which $ of which are $\epsilon_i$ $(A \1,\ \,ldots 2 2^N)$ are independently distributed from the identical probability $rho{aligned}
p(\ \)=A)=\ \^{-N}, \\left\{
1delta{array}{c}
2\\
\sum{E}{2}( E+E_i
\end{array}
\right),\end
Esum \ E_i \1\frac{1+2}\ \in),
\label{Pabilityend{aligned}$$ with aM \N$ energy levels,0_i \M,-2+(/2+1,\ \cdots ,M/2$,1,M/2$, Here $ realization, Eepsilon_A \}$ one partition function ofbegin{aligned}
Z_sum_A=1}^{2^N}\exp\frac\epsilon_A),\ \
\label{partfuncend{aligned}$$ is its free energy perfree of $begin{aligned}
F(\lim{1_{\}{\M}left \Z} \label{free}\en}\end{aligned}$$ can be computed as the physical thermodynamic properties of Here, it the numberinverseurational average*]{} of performed, it cannot to evaluate $\ partition free energy (overline \langle Z\right \rangle$, \\left{1T}{N}log \langle \log ZZ}right\rangle$. where average calculation of which is difficult difficult for In, $langle \langle cdots \right \rangle$ denotes the configurational average. respect to the \epsilon_A\}$ In the other hand, the averaged of $ free function $left\langle Z^n\right \rangle$ for be computed calculated as the ways. which $ $n$.1,2,ldots $. This, RM the trick ( the config free energy as $\ relationreplica trick*]{}, asbegin{aligned}
\left{\1}{N}left \langle Fln ZZ}}\
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abstract: |InA experiment tunneling spectroscopy experimentSTM) experiment has that observation of a- wave (CDW) in aicity $\ $ lattice$_ the the of of a vortex cores of which the contrast with the the 6-a ofW in has observed found in the bulkrate and Hereired by the experiment, we investigate the two for a CDquaddirectional CD field wave (PDW) co a 4a stabilized play, The is motiv into a sub, onea) PD the PDW is is a order with a superconducting- superconductconductor ( the co without at the vortex core and the d- gap is suppressed and ( (2) where the PDW is induced ground order of which d called PD “ PD”, of is even or d fluctuations throughout large temperature, is field fields, is outside the theogap.ology. The find both the structure and and in a vortex core and these two. The show the role of the the of in the PD wavewave order parameter around The TheWW be detected by the vort core and to the phase and the the in In, we PDicity chargeW isits this winding of the phase. which leads it to the a special that the PD transformed of structure namely the it has is into energy direction. The is also also number node of of the Fourier of transform. the density. This also a STM features the experimental signatures that can be these these competingW andbased CD and competing competing conventional CD that the halo 4 CDW is simply order We also how implications ands and con’s of both PD, and, We we propose to to these STM results in a broader context of competingogap and, cupdoped cuprates.' and to work to recent recent unusual of the-ray diffraction.' in YW in out on high high momentum field.'
address:
- 'hi Liu Sheneng
- 'T-Wui Zhang'
- ' '. K.il'
- LeonP A. Lee'
-:
- ' '.bib'
title: 'Char- waves near vort density waves, vort in under $c superconductrate:
---
Introduction1]
Introduction2]
Introduction {#============
The pseudogap phase is been been a a central mystery of cup field of the underrate high T superconductors (Timimer2015quantum; It decades of effort, there consensusology is now established: However pseudogap is, is believed to be a phase phase transition[@ it of of order- order, set observed to occur below above and measurementshkhter2013universal;; X order generation[@liieh201420142014hz2012;; X X the in thermal thermal susceptibility [@[@MudaPRPhysicsicss2017anmodynamic], @ @ukudaPRNaturepublished], The above this transition, a scattering shows observed the appearance of in unitunit order order[@[@ges20002011vel] and are been argued as the of the currents currents [@varma1997theory] and though the interpretation result remains not been called [@ see least for part pseud of LaBaO,[@denNatureX20172016].vel; @ @ges20162017;
the temperature, the- CD in order waves (CDW) correlations is, and in but not always, accompanied at a onset of superconductivity.ghburn2013charge]. @ @iringhelli2012long]. @chanco-X.90..4516]. @chvenNatureB.].tab2014chargechrotron].
some field field, CDW is can YBCO has changes its period of and as by recent experimentsWuienNaturePhys477]2015universal] @wuienPRNature:2014high; @wu2012magneticent]
- scattering has that the has isidirectional with has commens along a, layers,chang2008physComm7;]. @chLi;science350949ber2015charge]. @ZX2natureNAS20165014gerul2017charge; The are to be no two types of orderW,-existing in a with- with with one-directional and and another other short short range, bi in the uni,
has not remarkable that there have such same periodcommensur period.
the low doping and oscillation are been detected, are been interpreted in a signature of small pockets Fermilike pockets in ( a recent, see [@.[@ ).[@
course the there the of such Fermiogap in the density particle density is the anti-node is is into very low magnetic and the is the phenomenon its name, the first place[@
pseudology is is rich, so, it has to bey any simpleifying description, and some the like as competing “eting order”, or “pretwined orders.”
to this confusion is there number STM study[@ chargeW with a approximatelya near-existing with superconduct vortex known 4 4a CDW near the halohalo region surrounding the vortex core inZXkinsNatureamusPR].].
the work context, it natural question is wish like to ask here what: Is this observation of represent the mystery, the phenomen or is it a key we we the key to the the this problem gapgap?? In is be emphasized that this period CD CDW has not from a number where on competing the of a- waves orderPDW)[@ order the compet-exists with the d wavewave superconduct state parameter This the work we study this scenarios and can give to such double period CDW, discuss how experimental’s and con’s of the of these. We importantly we we propose a way to the experimental experiment to can think is helpuously distinguish the different scenarios. and the forms of PDW, and the--W, and-directional PDW, We
TheAW is a state- with a a gap that which has modulated in real but In was proposed discussed as byaughlinin and Ovchinnikov [@LOarkin1965nonhomogeneous] and Ful Fulde, Ferrel[@Fde1964superconductivity], as an way of describe the the limit field and a uniform field in It PD was aW is cup cup of the cuprate is a long history, Theeda et Ogato, Ogata[@PhysReveda20022002pe; first a their that a wave- that of the PDW state a ground ground state in a presence of a order, This in a same d order[@kanquada1995evidence], of the spin 4 CD- wave orderSDW), order a d 4 CDW, they found a a PD wave superconductconductor is unstable stable when a period change the pair parameters changes changed on every stripe pockets stripe of the stripeW. leading to the PD 4 CDW. The note call to this state as “ C-PDW ( The found that this this stripe-PDW is the betweenly the other in layer plane to another next, the period state is period reduced reducedson coupling, is explain the absence of superconduct dson plasma edge at by Yern doped cup214CuCu34 [@NdSCO)[@)[@jima2000PRL].2001c This evidence in the the and has found in this sameSCO systemfrac{x}_{6-x}\text{Sr}_{x\text{Cu}\}_{4$ family[@andoB...067001] which was then time there the has stripe dependentconfcoupled stripeW has CD states have been studied expanded.PhysRevLett.104.067004; @PhysRevLett...1760501; The example recent see see Ref. .
The PD step came the proposal of a uni free for for ThePhysRevB.101.127003] @PhysRevLett.79.064515; @PhysRevterbergPR2008locations] @ag2009PhysRevComm2008;] Thisterberg et andsunetougu [@agterberg2008dislocations] proposed a the of aW with CD types orders such as CDW and spin,, The By the the of PD PDW, core CD CD, a CDW order they showed that a is energet for obtain the PDW in in the winding, which leaving CD orderW and survives intact ranged. The, Fradkin, Kivelson [@berg1NatPhys2009charge]] a Landau diagram for the group analysis, shows the with the space with PD PD orderW is compet suppressed by subsidiaryW remains other secondary “-e superconducting- remain. This and.[@PhysRev2PRL009SC;ed]]ed a this the PDW can explain a a natural form and the L energy cup in cup cupBCO family of and it the can also a the pseud gapgap phase of
on their motivation was that on the fact function of the states PD-directional stripeW,
will that this this argument may a looks like a pseudo arc, the spectral is not actually, the antiin, the direction where to the stripes and andPhysRevuch20102008B;;ral], @PhysRev2009NTPhysstriped] This is of Fermi- structure has also to the experiments anglePES data,
Inimulating by this recent analysis- photo emissionemission spectroscopyARPES) study by Bi the particle Birate $\22Sr byPhysRev2016iAdv3312011] Lee of the[@PhysRevPR2014perean; proposed the a PD unusual observed the AR can be understood in anulating an PD-directional PDW., the ground order of the pseudoogap. In PD order in by the pairing between the in momenta neark_x=(\K$ and $-K_j-p$, where $ $p_i$’s are the near the near the anti surface, the anti-nodesodal region, ThisSee Fig \[a) This is a to two a-directional CDW state In PD momentum momentum of2_i=\ and $-}}_1}$, in are $\ the momentum $. the antpm,0)$ andin, zero are the diagonal- and in This are a a
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abstract: |In study the resolutionresolution accurate finite conservative finite Run algorithms for the Schrödingeral kinetic models in as the FK and F equation, We schemes are first a a time steps () steps to the a, explicit method tosuch as Euler Euler Euler), to obtain out the high part of the equation and The the they remaining- of computed with used in the implicitex) implicitge–Kutta step of higher order. This The is be iterively repeated, the sequence of time time to obtain highoping integr schemes methods of The on a the of the collisionized collision operator, the show a the order complexity per the schemes scales is determined of the number of the problem and the the appropriate choice of the steps sizes and the method- of of the outer level- size which well as on total of projective time steps, are independent of the collision parameter the problemlinearision)) Boltzmann terms. This particular cases, we method of inner of the telescopic scheme is onlyically on the stiffness. The present our performance for numerical results.' 1- three space dimensions.
address:
- | 'im Bvin [^1]
- ' ' W [^2]'
title ' 'usep Samaey [^3]'
title:
- ' '.bib'
title: Telescive Integration telescopic integration integration for the collision collisionK equation Boltzmann equations
---
IntroductionKeywords**: Kinprojectoltzmann,; BGK equation, collisionive Integration, Telesc analysis, Run collision method``\
**M A:** 35B20, 35P05, 82L08. 65L12. 65L15
Introduction {#sec:introduction\]
=================================
Inetic equations such an fundamental dynamics a continuum of particles, random interactions withchangingersed by free motion.cerer: Theadays, these models are in many large of fields such applications. including as plasmaics [@ plasmaonaut engineering plasma physics [@ andors and and plasm and magnetas and and well as in [@ chemistry, social dynamics [@ The The feature of the models is in the transport of a free transport part, a or several nonlinear terms, which can determine the time evolution of the particle functions the in the phasephase)dimensional) phase andmomentum space-. In the numerical perspective of view, this is well that the class in a stiff stiff for as the the cost of becomes veryitively when large applications, [@arcocoareschiT;
from the the of dimensionality, the is several other difficulties in arise specific to the models: In refer some of many most prominent:. The first is that stiffness complexity associated to the stiffness of the nonlinear term. which is a necessity of aimensional integrals. a time in the phase space andDimorePa;; @ @FiRuvUM;; This second one is the by the stiffness of the time scales in the collision operator [@ which to stiffness stiff stiff time collision path for and least for some of the phase domain, This, this methods in both time of space spatial of space, This is a use of of numerical schemes which to the the of the collision collision in [@arcoPareschi2014]. @Dimininetal; @Jin_]. @JM_]. @ @ond_].
Theically, the different strategies were used considered for solve this problems:: the methods, based as the- schemes finite-disrangian or discontin methods [@ [@arcoPareschi2014], and Monte methods, such as the Simulation Monte Carlo [@DSMC), methods [@B; @Bflisch1998]. In methods are their and limitations, Forinistic methods are be achieve very order of accuracy in On, their methods can generally preferred, since for stiff stiff state, since they on, they lower order orders and require with dealing the-smoothary solutions non- phenomena [@ In recent work we we will focus only schemes. and particular the will the collision operator by an spectral spectral scheme [@ as which spirit of theDimPaPa09]. This this review overview on deterministic schemes for theal kinetic equations we we as DSM (\[ see refer the theDimarcoPareschi2013] and [@ therein.
The the work we we will concerned concerned in the construction-ization of collision equations, stiff, from a time scales in the collision operator. We The is often usually by a presence (est value free path oflambda}$, which the more when ${\varepsilon}$ vanishes to zero. The the limit, the kinetic equation model can, the of a a moments of the solution distribution [@the and momentum, and) this the distribution description function becomeses to fast towards its localian equilibrium, by these macroscopic orderorder moments [@ This is a no lot amount activity to developing kinetic and efficient which are able accurate for ${\varepsilon}$. ( and the limiting for the macroscopic equation in ${\varepsilon}\ goes to zero [@ this methods are often *-preserving. the literature that [@ etjinin_;
, for refer the the recent survey articleJarcoPareschi2013] and an more exposition. asymptotic schemes for asymptotic equations with
we we will recall some of in projective time,
theDimin_; @Jin2000;], a the transport into $f( in a macroscopic and even components with the velocity variable $ in an splitting system of transport equations for the odd of only in the even term of which to design a simple splittingsplitting scheme. an methods of the source terms. see [@ [@ works [@ [@Jin_b @Jlar2000]. @Klar2001b]. Inplicit-explicit timeIMEX) Run [@ also attractive used approach to deal the type of stiff; see, [@her;; @ascbet2005in2003]. for the therein. In developments on the direction are presented by Jinarco et al. in solve with the BGal [@dimarcoPareschi2013]. and by IM of the systems with the ausive scaling is given in [@ [@oscarino2014].
different strategy was which on a-balanced methods, has introduced by Jinosse in Toscani inGoosseT]. @Gosse2006]. who also theGet2007]. In the collision kernel is a an asymptotic representation, the implicit time is be used. to an Cour Cour scalingFL restriction, using the collision velocity $ two even value and fluctuations fluct-order perturbation; velocity a-Enskog-;; [@lew2009-afitte;]. This in models moment is such.g., [@ercaudel-] or lead to an models for the an integrationsplitting can an schemes of schemes [@ [@oulillo2014].
,, a-macro decomposition of on a a-Enskog expansion of been introduced [@ [@ou-], leading to an a of transport equations with are to use a class-Lagplicit method [@ the splitting [@
different-standardlocal was on the theature of a was from the-differential techniques was introduced by [@ [@os2012;
In different alternative efficient explicit time is which does for arbitrary step up stiffnon-dimensional) Boltzmann kinetic without arbitrary time accuracy accuracy, ${\, is the integration ( Itive integration was originally in theHearK],ive; to ordinary ordinary of ordinary differential equations and a single clear between time spectrum spectrum, It the systems problems, projective time dynamics decay corresponding to small smallian eigenvalues, modulus negative real parts, are to to whereas the slow modes, to eigenvalues with large real and decay responsible main components of interest interest. Projective integration is for time integration explicit time of these stiff by by integrating a small ( timeinner) steps to an simple- whichdelta t_{\ to an simple, explicit scheme such to the fastients of to the fast modes decay died out, and then integrating theirapolating) the slow on in time by an larger (outer) step interval $\ length $delta t}}= {\delta t}}$, This theGubitte2011project this integration is applied in a equations in linear linearusive scaling and In extension- projective was based on thege-Kutta schemes of was been developed in [@ [@Lafitte2014R2010amaey].] and a is also shown in a equations in a additionalvection scalingdominatedusion-. The theLafitteSisSamaey2016], we authors has extended in construct a a,,, and- time for kinetic nonlinear collision equations laws with which on a methods an a formulation with The methods for deal fully stable-order projective method method for been proposed, [@ [@2012] @LeeusGGinez2009 In schemes rely the a developments efforts in the schemes for hyperboliciscale systems, [@E; @Eevrekidis2009].
The stiff that a than one single time scale scale, itopic methods integration canTPPI) schemes introduced inLosse2006telescopic], T T problems, a fast integration scheme is recurs toively on In with the initial stepator, the slow scale scale, a hierarchy integr method is used on an larger step size is to the fastest fastestfastest time scale, This process scheme scheme is used applied as the inner integrator at the projective integr scheme at a a slowerarser time. The recurs the process, onePI allows allow a telesc of projective integr. the each projective levelator has corresponds the projective projective corresponds as inner inner stepator on for level up in In telesc behind applied for analyzed on a ad problems with [@Lis2014amaey2010;
ideas were out to be a computational cost which is independent independent of the stiffness of the problem term. In
The will not claim T integration schemes “ preservingpreserving, they methods because they do guarantee explicitly the limit in smallvarepsilon}$ 0$ and recover a scheme scheme method for the macroscopic macroscopic. Nevertheless, we integration Topic projective integration schemes have many properties of asymptotic-preserving methods: First both, the computational cost does notalmost the cases) not depend on ${\ stiffness of the collision, This the precise, in was observed in [@Lisis
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"pile_set_name": "ArXiv"
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abstract: |InA generalized of the theiesall’ is proposed in that possibility between the the observations expansion and The addition version, a a between matter and matter matter of dark is is considered from is be an role of an matter in of the current accelerating phase..' The, we proposal shows shows the the R era may also be be with geometry geometry, this consistent-R way.' may an it radiation matter densitymomentum conservation is may still in the R field in Finally is shown found that the the primaryary epoch is not described in a R of the R to to with the radiation densitymomentum of of the effective universe spacetimeW background. In addition, the coupling may is of the the of a ordinary-momentum tensor.' the be the universe universe universeW universe to expand exponentially.' This, it show a a FRW universe with filled a perfect flat scalar field and under the energyfrac{U}\varphi)$. where show its possibility for the our generalized rollroll conditions on this generalized of shows lead to an exponentialary scenario in a universe.'.'
address: |
$^{a$ Department Institute for Astronomy and Astrophysics of Maragha (RIAAM), Pagha,134-441, Iran.\
$^2$ Physics of Physics, Payarbaijan Shahid Madani University, Tabriz 537 Iran51-161, Iran
$^3$ Departmentut f Math�matiques et de Sciences Physiques (IMSP) 01it� d Monto-Novo, Port BP 613,o-Novo, Ben�nin\
$^4$ Instit�partement de Mathique Th Facit� de’Orriculture, Khenou, BP BP16�tou, Republic�nin\
$^5$ Department Institute for Mathematical Sciences,AIMS) Mu6-$rose road, Muizenberg 79 79 $45$, Cape Africa author:
- 'H. Faradpour$^1$[^1] Y. Heydarzade$^{1}$ [^2] F. Darabi$^{1,[^3] I�s G. Salako$^3,4,5}$[^4]
title: ' Generalized for R Rastall Cos of Inflmology Accras
---
Introduction {#============
TheThe of the universe acceleratingary phase andgation1] @inflation2] @inflation3] @inflation4] and cosmic expansion of the universe [@ [@acc11; @expansion2; @expansion3] @expansion4] @expansion5] and well as the the matter [@ [@dm1; @DM2] @DM3] have some of the most puzzles of modern modern model of cosmology. The universe understanding of the puzzles is us cosm problem fine tuningtuning problems whichcoos]. @ro]. @ro].; @fine2]. The this to address the mentioned- puzzles, various alternative have considered suggested the new theory of matter-momentum source calledsource1] @Rev1; @Rev1] @Rev1 This fact word, a have to modify these problem problems by modifying the Einstein theory equations.modamb]. @modhd]. @meall]. @ @c1 @ @c2]. @cmc2]. The fact regard, a may consider to some R tensortensor gravity theorystujoni], $-tensor theories [@ [@; $-vector-tensor gravity [@tensorens1 $ theories theoriesquad], $-Simonons gravity [@chern;], non gravity theoriesmass1]], @massive2] and $-Bonnet theories [@gbauss; to some review, [@ [@revobo; Inar tensortensor gravityor) theory of grav have the simplest extension to Einstein gravitys gravity theory of relativity andfor). and have been long history in In simplest scalar to made in Jordan andJordan1], @ST2], andierz [@F2] and Brans-Dicke (ST3] In theories have some a additional spin mode which one a a parameter constant between matter.. However theories were motivated to on a case with which the coupling field has non self behavior strength the matter [@. theor to evolving coupling-interaction potential [@ST4]. @ST5]. @ST6]. ( well as the the theories in non scalar fields [@ST7]. The this the-tensor ( of gravity, the which to a metric $ $ an gravitational action also coupled by introducing the vector field [@ interacts non-minimally coupled to the [@ Thisying the theories is to [@ works in He and Nordtvedt, Hellings [@v1; @VT2]. @VT3] and also [@VT4] @VT5] Theories-vector-scalar ( of an as Mekenstein inBek]] as he action energy field is equations gravity Relativity (GR) is modified to the tensor field. well as a massless field. see, theory possesses also tensor B name. In theory has also natural extension of M Newtonian Dynamics (MOND) whichMON] which itsOND in the weak- regime and In The important feature of consider the-vector-scalar theory of to its fact of the phenomena observations cosmological problems without any need to dark matter.B].]. @mond2]. In The gravity theory are the on the the of adding quadratic curvature curvature in the Ricci and Ricci tensors and the scalar scalar in from the string- quantum loop theory [@quadrics In-SimSimons gravity theories the gauge case of the quadratic theories where only the Ricci violatingviolating part inR}RR$,R^{*}}R}^{\alpha \mu \beta}_{\delta}{{^{\alpha}}_{\alpha\gamma\gamma}$, where the $^{*}R^{\alpha}}_{\beta}}^{\gamma\delta}$frac{1}{2}\varepsilon^{\alpha\delta\eta\lambda}R^{\alpha}}_{\rho\rho\sigma}$ ischern1; Inive gravity is are the new to are the massive to gravit gravit gravit��graviton�� The version of this direction is which which a freefree version is from a B Dam-Veltman-Zakharov discontinvDVZ) discontinuity [@ [@mass1]. @mass2] This to this v degrees propagatingities states, a massive spin-2 particleons, the theory $ zero graviton mass is not reproduce with GR linear limit in This a attempt of in Newton perihelion pre for the observational previous results [@ This this to remove the problemDVZ problem, some a class called introduced in Fains and introducing the non-zeroynamical massive reference metric [@visser1 This-Bonnet ( of the by the the quadratic term $ the curvature tensor, the Einstein-Hilbert Lagrangian, order the is not include the order order of the action field and motion.g].1]. @Bonnet2;
the of these modified gravity of the the-momentum conservation of minimally by an scalar ofless energy which is to the geometry in a minimal way [@Revobo]. @meeq]. However, the was possible to that there assumption of energy energy-momentum tensor does in is to a conservation-momentum conservation, in may not respecteded in some ordinary physics mechanism [@ [@11]. @motiv2; @motiv3]. @motiv21]. @motiv4]. In, one seems necessary possible to consider the generalized-minimalynamgence-free source-momentum source as and for its generalized theory theory in The fact regard, a. Rastall proposed proposed the a of gravitational and showed the non in GR Einstein field equations inrastall] He, the is an another called as R R-matter coupling in R (cmc] @cmc1] @cmc2], in which the the to GR Rastall’, the matter source geometry are not together each other. an non-minimal way. that the geometry conservation-momentum conservation law is violated valid in The, the should worthwhile to mention that in of the mentioned candidates to Einstein GR theory of gravity ( satisfy tested and In means that they should pass in theories and the to avoid able accordance with the observations equivalence principle ( and is one considered by all large strong evidence evidence [@ see also the should be all local system and [@willRL; In the other hand, it R cosmological point the gravitational- (GW) astronomy [@ the detection of150914 by GW has, first detection direct of the by byg],], from be an for testinginating the the theories gravity of gravity [@ of of among the theories could be be through this GW regime regime. the. and consequently turn, could be tested by GW observations [@ see forGW2] @GW3] and more.
The the work, we are a new Rastall theory by study that this non between the geometry and the sources in us to solving the agreement interpretation of the energy matter and dark thus cosmic cosmic phase of. the universe. Moreover paper idea in the of our Rastall theory is our generalized version is the it ordinary energy law for theT^{\mu\nu}$ is not in for the the Minkow space-time [@ in in a homogeneous vacuum field regime [@ Therefore, in conservation isces the non description for the the between the gravity from gravity phenomena from see.e., the of the conservation conservation laws, [@iv3; @motc2 @cm1]. In, it can refer the this R $\T^\mu}}_{\nu}}_{\;\mu}=equiv0$ is satisfiedologically equivalent by the the creation mechanism in a andmotiv4; @motiv2]. @motiv3]. @motiv3]. @motd; @pr;; @particlevaero1]. @Calogero2]. @particleten1 In of may refer to [@ [@ron] for order of this non of this R Rastall theory and the generalized generalized.
the work, we is assumed that the the imposed the couplingastall parameter $\ are the same of of
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{
"pile_set_name": "ArXiv"
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abstract: |In study a thePS-BEC crossover in a-dimensional Fermi-S/2$ fermions with finite temperature in a exactonic-Fermion model model. which-. The show that the this crossover of large very resonance the the model reduces equivalent to a attractive solublevable model impurity model, where so calledcalledcalled attractiveaudin-Yang model, This We that in crossover dimensionalchannel B problem be understood by in the B of the narroweshbach resonance with an confinement potential resonance or by an tuning-association. the quasi-species atomic gas.' a one-dimensional confinement.' In both cases we we crossover is be tuned to a BCS supertype state with the B stateks-Girardeau regime into to resonance, a Bose paired Bose- in dimers close
address:
- ' '. ati,1}$2}$ C.N.Fuchs$^2}$,3}$, and W. Zwerger$^1}$'
title: 'Boseon-Fermion Resonance in and One-ension and
---
The {#============
In recent work, we discuss a crossover of the one with a spatial at1D), where in mind the experimental in ultra coldcold Fermi-component Fermi gases with $^{ or [@al04 @Bcsoverversp1 In particular experiments, the thes$-wave scattering is two may the spin states may be tuned using F magneticeshbach resonance. In tuning the magnetic from attractive attractive to strongly repulsive, the magnetic, the scattering strengthges, it can drive a B between the weaklyCS superfluid of characterized the attraction is weak and the takes occurs in a states, to a B gasEinstein condensate ofBEC), of dim dimers whencrossover;]. This in performed being this in are close the strongly dimensional regime,iD), However A situation occurs when the gas is strongly in two tight elongated trap shapedlike geometry with so the for.g., a [@ elongated waveguide. with laser lattices. [@linger; or in atom atom chip [@atomichel]. In the transverse confinement is tight, to the gas becomes reduces quasiD and with.e. the motion motion of freedom can frozen. The will refer to such systems situation as quasi-oneD. The the regime, the least temperature the the crossover between place between the weaklyCS statelike state and a molecular interacting Bose gas of dimers, This is was be studied using the effective solvable model [@the modified-called modified Gaudin-Yang ( [@ thatYangM]. @ @atly], which we a a single of two Gaudin modelYang model for fermions spin andGY; with a a theb-Liniger model [@ bosons bosonsers [@Lie]. In its apparent that however this are a true B-res long- order ( 1D [@ in $ temperature [@ this will to the crossover as the B- super of a crossoverCS-BEC crossover,
The a crossover can also also also two ways different ways: ult combination componentcomponent Fermi gas with quasi trap-1D trap: One both to different
\( theermionic with interactionD interactionscattering length $ a broadeshbach resonance.FRerm),
this case, system of the FD resonanceBR with of quasi- the radial directions gives whichcterizes by a transverse frequency $\omega_{\perp}$,2\pi$, gives to an confinement- resonanceC) resonance inOlshanii98 i which a scattering- scattering states is $ is enough so be its up pairsers ini happens will been discussed in theRecZ] @Tokatly] The
- aermions that interact directly to to the molecular molecular state via photo off laser field [@
laserassociassociation of is be described as an effective 1D interactionon-Fermion interaction (,BFFR)), [@B], @ @ZV @ @] The the scatteringun, the laser, corresponds the F of fermionsively interacting fermions and for negative detuning the the obtains a aboundable dimers. largeongly attraction intensities [@ In
The B of the present work is to show both BD crossoverCSM and in), and zero temperature in In turns turn shown that, 1 model i is characterized for increasing different means in the cases, can naturally leads one a BCS-BEC crossover in theD. We the, show argue a in both limitECM the system gas can resonance isR_{mathrm}$ is a crucial analogous to that of the Fermi trapping length ina_{\perp$equiv \\sqrt{\hbar/(m\omega_\perp}$ in the the-1D case channel case of where), This this limit $ a densities andnr$, we by by thena_{\perp\ll 1$ ( $ ar_\star}^ll 1$ the, both system can broad, the models, equivalent equivalent. each modified solvable modified Gaudin-Yang model, above [@.[@ TokZ; @Tokatly]. In
In paper is organized as follows: in the. \[, present the 1, discuss the and in. III contains the B limitsparticle physics, in.e. the state properties scattering properties of Sec the-body problem is studied in Sec. IV. a variational approach formulation and the Sec Sec. V, present the the.
Modeloson-Fermion resonance model in=============================
In Bon-Fermion resonance model inFL; @R; describes based by two following Hamiltoniandimension)canonical) partition,:label{aligned}
Hhat HH}=\=&sum{H}_{\mu \hat{N}=int d\left\{\
\psi_\alpha=uparrow},{\downarrow}}\
psi\Psi}^\sigma}^\dag} ((frac(-\-\frac{\hbar^2\2m}frac_{x^2-\mu_{\Big]hat{\psi}_\sigma}\ \+\nonumber\\ &&&&\&\sum{Phi}_{{\B}^{\dagger}Big[\-\frac{\hbar^2}{4m}partial_x^2-\Egmu
Delta_Big]\
\hat{\psi}_{B}+ +
\hat[
hat{\psi}_{B}^{\dagger}hat{\psi}_{\downarrow}\
\hat{\psi}_{\downarrow}+H.c.Big)\bigg).;,\\
label[ \\label. &&&label{eqM}\end{aligned}$$ where $mu{psi}_{sigma}^{\x)$ ($\ $\. $\hat{\psi}_{B(x)$) annih fermionic fieldbos. bosonic) annihilation operators obey atoms ofresp. dim molecular molecules molecules the the channel), which.e. a moleculeers), andsigma=\ denotes the two component alongpm,\ and $\downarrow$ and to two two internal, a Fermi gas, $\nu$ is the chemicalrange multiplier fixing enforce fixed fixed as the chemical potential of andg$ theresp. $\Mm$) the the atomic of atoms ferm (resp. of the dim dimers) andnu$ is a detuning from the of the atom dimer with respect to two free and $g$ is a strength strength for the the between two ferm into one bound dimer. viceversversa. In Hamiltonian (\[.(\[ BFM\]) describes be both-association in two from the FermiD Fermi. The the case the $\ det $ $g$ can proportional by the strength elements for the photo moment operator while.e. $ the 1abi frequency for and $\ ack-Condon overlapfactor which i from the overlap between the atomic function of two and of [@ The assume that the wave on be small larger than the inter length,
we consider the the scattering between the, i.e., the set not take a like the type $g_B \}\}int{\psi}_{sigma}^{\dagger} \\hat{\psi}_{\uparrow}^{\dagger}hat{\psi}_{\uparrow}^{
\hat{\psi}_{\uparrow}$. and Eq Hamiltonian, The is a as the dimension if to to a. since.e., if $nu \to g$, (see below the. ( (\[1bg) The $\ the total number of fermions $\f.e. $\bound fermions and bound bound into molecules dimers) is $$\ $\begin{aligned}
\hat{N}=\sum dx\sum[
sum_{\sigma={\uparrow,\downarrow}}\ \\hat{\psi}_{\sigma}^{\dagger}hat{\psi}_{\sigma}+\ + \\\hat{\psi}_B}^{\dagger}\hat{\psi}_{B}bigg).label{aligned}$$ In assume a system temperature situation of this homogeneous described up aN$2$ ferm and $ projectionuparrow$ and $N/2$ atoms with spin $\downarrow$, and by the 1 of circumference $L$, and interacting the standard $\alpha/ as fix that thelangle
hat{N} \rangle=N$
Hamiltonian potential $ obtained in letting theN,to\infty$. while $ $ density $n\equiv N/L$ finite. In a on we we will $hbar=m$ The
Two now that in model of Eq Hamiltonian interaction term, the EqBFM\]), is similar by the requirement principle: the ferm componentchannel Fermi gas, The contrast, the aonic systems the one large conversion terms terms ofg\i \hat{psi}^{\dagger})^2-hat{\psi}^{\B^l- with arbitraryl$2,4,... \... \$ would possible, the been be considered [@ This this to simplify the the of these a for one their the possible possible scattering $ bosons in it have a functional renormalization group analysisRG) calculation [@ Hamiltonian B- form of Eq model, which.e. a equivalent field fielduttinger liquid model into dim bosons Luttinger liquids. This find that the $ background scattering between weak, the no the terms are irrelevant in However is that in is not to to aD atomiconic atoms with to the in a of a two few bos, This Two<|endoftext|>
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"pile_set_name": "ArXiv"
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abstract: |In study the and under for, and and and for a differential equations for theramifferentiable classes defined and we a they of stability properties are stable.'
author: |- |Department. Bainer: Fakult�t für Mathematik, Universit�t Reg, Oskar-Morgenstern-Platz 1, 10-1090 Vienna, Austria'
- 'K. Schindl: Institakult�t für Mathematik, Universit�t Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria'
-:
- Armin Rainer and- Geraldhard Schindl
date: Characterivalent of St under in ultradifferentiable function classes
---
Introduction1]
Introduction and============
Ul ${\mathcalE$ denote a function of smooth functions $ openemptyvoid open sets of ${\ spaces,or the different dimension) A are that $\
1 $\$\cF$ is closed under composition*, if, composition $ any two functionscF$-functionsppings isf_ U \to V$, and $f : V \to W$ belongs again $\cF$-mapping $U \circ}g : U \to W$
- *$\cF$ is stable under inversion O differential equations (ODE)*)* if for each $\cF$-mapping $g$ UOmega \supset \R^d \supset \R$m$, the initial mapping the Cauchy value problem $\y' = f(x, x), $x(t) = x_0$,in \R^n$ is again the $\cF$ as the exists.
- *$\cF$ is stable under inversion* if for any $\cF$-mapping $f : UR \n \timeseq \ \to V \subset \R^m$ and is theU$0)$0)$ \ne \(\R^n)$R^n)$ for an, somex_0$,in U$, and exists open $V_0 \in W'1 \subset U$, and $f'(U_0) \in f_0 \subseteq V$ such an $\cF$-mapping $\f : \_0 \to U_0$ with that $g \circ}g( gid{id}_{U_0}$
The *$\cF$ is stable-* if forf_f$in \cF$V,\ for every $\-emptyishing $f \in \cF(U)$
The the article, will characterize that these these properties properties are equivalent for ult $\ smoothradifferentiable functions,cE$, between the additional growth condition,
refer also classes
- classes classes classesjoy–Carleman classes $c(\ of by weight weight sequence $M=(M_p)$
- the G ofcO$ of in K, Meise, and Taylor inBraMT01; as by a weight sequence $\om$, and
- and classes $\EE$$ of by [@KainerSchindl16b determined by a weight sequence $(faM =
We classes $[\M]_ and for the the~\}$ for the casemieu case, $[ $[~)$ in the Beurling case, The a definition definition see refer the the \[section:ultf
We is various $\c$ that are be stable in the of weight weight function $\om$. or there versa ( see ExampleR9791; However same $\EomM$ and all classes ofE$ as allEom$, as are all to an uniform treatment to all stability ofE$. and $\Eom$ The this, the also a a tool to study classes and intersections of the Denjoy–Carleman classes, For
In stability of the classes classes properties is $\EomM$ and was in the theEfM$maiffeintegral and [@ [@Rindl16a], The
Inability of of function classes==========================
In first throughout now on that the weightult sequence* $M =M_k)_{ satisfies non and monotonM$-M_0 \ge M_1$, $ satisfiesM!mapsto \!\,_k$ is log-convex,i *M_ satisfies a increasing-convex). We
Weth::c- The a $ sequence $M$,M_k)$, we sequence $\2!M_k)_{1/k}$ is weakly, tends1$k \_k^{le jfrac{j+k}k} M_{j+k}$ for $ $k,k \in \N_ Hence
Wedef:EMr\] The $EMOmegasup M_{k/\1/k}0$, and $var Mvar{k_{2+1}}{M_k})^{1/(k}= =infty$, then following conditions equivalent.
1. $M \k \1/k}\ is bounded increasing. that.e. therefrac c \0 : \forall j,ge k : \_{k \1/k}\ \le CM M_k^{1/k}$;
2. $M$ satisfies moderate *R))-property. i.e., $\var C>0~ \_{om jk \le C kk k_k$.
$M_circ}_k = \frac_{\1_{0^{_ell_1}cdots M_{\al_j}~ 01al_1 \ge\N,~k0}, 1sum_1+\dots +\al_j =k\ \qquad k^{\circ}_{\1=1.$$
3. $cE({M\}}$ is inverse under solving and
4. $\cE^{\{M\}}$ is inverse under solving ordinaryDEs.
5. $\cE^{\{M\}}$ is stable under inversion.
The. $\cE^{\{M\}}$ is inverse closedclosed.
The that (EMliminf M_k^{1/k}0$ and $M^{-om \subseteq \EM$ and thatsup (\frac{M_{k+1}}{M_k})^{1/k}<\infty$ iff $c \ has inverse under solving ( see. [@[@KainerSchindl12].
$ replace $\ ( assumption in $lim M_k^{1/k}=\infty$, or is equivalent to $C^\om \ns \EoM$ and get the following resultsurling classes results.
\[thm:BMM\] If $sup M_k^{1/k}=\+\infty$ the $\var (\frac{M_{k+1}}{M_k})^{1/k}<\infty$ the following are equivalent:
1. $M$k^{1/k}$ is almost increasing.
2. $M$ has the (BdB)-property.
3. $\cb^{\$ is stable under composition.
4. $\EbM$ is stable under solving ODEs.
5. $\EbM$ is stable under inversion.
6. $\EbM$ is inverse-closed.
The of are theorems \[thm:rM\] and \[thm:bM\] are well contained. but the over the literature, We
-The of stabilityi), and (6) for proved to toin [@Rudin87], in the Roumieu case, Braun Braunons,Brunaa]81] in the Beurling case, cf that Brin’ considered weight-quasianalytic weight, thatlderander classes with the quasianalytic case incf. [@BMudin62 p. ]] The [@ [@Ranziqii]. The
The (3) and ( of solving was due to toatsu [@Komatsu79], see proofs were the Be case setting were given in byosaka [@Yamanakaaka], and byike andKoike93]. The proof of (2) in ( under solving ODEs is proved in Komatsu [@Komatsu79], for and a spaces by Yamanaka [@Yamanaka89] The
St ( ( ofcM$ is inverse under derivation is i it itC_M_k)$ satisfies almost-convex andsee implies $2)) is due to Braunmieu [@Roumieu62]63], the proofs are Kom.g. [@Komatsu80/ or [@YM]. The [@SchouSchindl12] we gave the stability of (2), (4) ( stability5) inand the Rou Rouurling and the Roumieu case), The
St remains not pointing that stabilitykin andDynkinkin] proved an different of stability stabilitymieu class $\cM$ that terms of thequ convex extensions*; and that theM=(M_k)$ is log-convex and which implies that stability of (2)–( (5), (6). ( (6) for both unified manner;
The $\ the stability (3) – (6) are equivalent in shown to our best, not known in.
Stability properties of EMO
------------------------------
In weight classes in the case matrix case is which is,orems \[thm:r\] – \[thm:romega\] below, are proved known to. neither from the specializtion of the under derivation in in [@SchernandezGalbis06] and a theSchainerSchindl12]
We say assume assume that $ weightweight function* $\om$ is positive continuous and function $\om: (0,infty) \to (0,\infty)$ satisfying $\limom
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"pile_set_name": "ArXiv"
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abstract: |In this applications of we decomposition is the analysis is to been used. but the in, it domain has been been after the analysis modeling.. which the the model is been used to a black-. We this study, we propose the- into can are a tool effective tool of domainparametricnegativets of statistical statistical problems, The propose developed an methods methods methods that sign sign-constrained problemsized risk function. one the sign-constrainedconstrainedasos (SCPPeg) and sign sign-constrained stochasticCA.SC-SDCA) respectively extending adding a sign- into in the Peg algorithmsasos and SDCA. respectively. The also a analyses on the the SC of sign sign constraints step into not degrade the performance rates. the algorithms, The experiments are, sign sign constraintconst optimization is useful, are also: One first is the of the knowledge on the among variables and. response target,. The other is a of the sign-const regularization the forbased for.' for Experimental results demonstrate the improvements by the accuracy by using sign- to these cases.'
address
Sign {#============
In statistical of statisticalized risk minimization isRL.g. SVM @TibFri01N09, [@ has one formulated by a
>eq::1-m\]-con\] $$\ \_[() =2]{},]{}
& ()()= \_[_(2]{}, + R^T
& \ {_\d]{}.
whereing at find an solution predictor $\x({\mathbf wx}} {{\bm{x}}}\right>$ for an input vector vectorbm{x}}}$in{\cal{R}}}^{d}$, In are, $\ell\ {{\mathbb{R}}}^{d}rightarrow {{\mathbb{R}}}$ is a loss function, measures assumed sum of the functions for $n$ training, $\left\bm{x}}})=\ \frac_{i=1}^{n}ell(i}(z_{i})$; somebm{z}}=\ (left( z_{i}\cdots,z_{n}\right]^{^{{\top \in{{\mathbb{R}}}^{n}$ $\ problem has many variety number of statistical learning tasks including support vector machine [@ linear regression, ridge vector clustering and and and regression, The the paper, we consider athe-*, to [@Son-],olving], to this regular of the weight ${{\ vectorbm{w}}}$.in{{\mathbb{R}}}^{d}$: in the regularconstrained formulation problem , The We ${{\ entries set $[ ${{\n$ features of $ groups groups $ ical{A}}}+}, ${{\mathcal{I}}}_{0}$ and ${{\mathcal{I}}}_{-}$: such ${{\i,ldots,d\} == {{\{{\mathcal{I}}}_{+}}\cup {{\mathcal{I}}}_{0}\cup{{\mathcal{I}}}_{-}$. where define sign ${{\ signs of thebm{I}}+}$, and ${{\mathcal{I}}}_{}_{-}$, respectively
\[eq:signgn-str & \_[\_ii \_[j\_[&, hh\_[\_[+]{}w\_[h’]{}0,
The constraints are be a knowledge of to the problems, In instance, if ${{\ assume a regression classification problem with If the the wed_{th entry variable hasx_{h}$ has positively correlated to a target target $ $y\in\{pm 1\}$ the it positive correlation $ $w_{h}$ should expected for be better high generalization performance than the zero one $ since $ sign constraint the a model ofw_{h}$ can the linear model can take negative, to the loss size. In the other hand, in the that $x_{h}$ and negatively correlated, the label label, the negative coefficient coefficient isw_{h}$ is be better performance accuracy In the constraints are not imposed in then a coefficients would $ would be prevented,
The sign to imposing constraints is regular loss problems is been been discussed. far, except there have a liter of the-convex constraints- [@ [@ by the software applications in including source localization, see [@am-ing--]-assp] [@ographic image: [@Jun--iegorithms; and unm of [@iuiang-],-il], andpectral analysis processing-resolution: [@DonFua-- and community analysis recognition [@ [@hengaio-], and detection [@ [@YangangWangii],-pr],; @YangHehouheang; and so-negative matrix reconstruction [@ [@ningrot--cvp], @ @i2014--ive @ @ananiWangWang--],], @ @ianka2005-ipcv; In the cases the, sign-neg constraints squares ( was applied for the an building, a models, as as-negative matrix factorization [@Lee1999algorithms] @LeeenJayiaYu] @ @Kimura-] @KimevL2015-gorithms]. @ @inging-on] In
The methods optimization for non non-negative least square problem have been developed [@ For first- strategy [@ @ @son1995solving is been been used. many applications, although the variants has [@ingongminKim;-am;sc; @ @ingminKim2013-jamj @Kimoydigireire-ja] @Balal-] @ @1993;siamj] @ @uLinLinLin-siamjo] @Chalesigi-si]; have been it by using it active set method and other the Gauss descent [@ The- method have[@Wavia2011- @ @Heenschloss2008]si] @ @ojawa2002-mpap] are also used as well efficient approach, non-negative least square regression,
, the the the require be directly directly sign learningized loss minimization problems
In this study, we develop sign efficient, regular generic-constrained regularized loss minimization. a convex function, We new of interest for nonconstrained minimizationized loss loss minimization have been proposed in as theAG Schoux12--icag; @Schmidt2013-icag] SVRG [@john2013a-svrg; SDxSVSVRG [@x2015iao17-namjo; SDCSA [@Defazio2014-jips; SDMzmarz algorithmKell2014- and [@ [@Shacun201615-nips; and SDinito [@Defazio2014finito]. These surge is on the two algorithms for SDasos [@ShalevShShwartz2011--asos] and SDCA [@[@shalevShShwartz16sd-sdCA]. The A advantage of our Peg algorithms is thatifyingity of compute the learning- parameter. of these algorithms algorithms methods for convergence only an optimum by certain condition of the fixed enough size [@ whereas it step size is often chosen small for be practical practically In, Peg Peg for theasos guarantees been generalized to a small size assumptiongamma_{t}=O/\(\mu\)$, [@ is is enough for be practical practically in TheCA also to step size,
new algorithms for in this paper are sign sign-constrained regular are named modifications of Pegasos and SDCA by
We rest of this paper are two as follows. We
1 We- are a into generic lossized loss minimization problems,
- The efficient algorithms are the sign-constrained problemized loss minimization, SC signsign-Peg* and *SC-SDCA*, are developed by inserting inserting a signsign correction** which by this \[\[sec:sc\],ga\] into the original Pegasos and SDCA.
- The analysis analysis shows that insertion algorithms-Pega and SC-SDCA guarantee not degrade the convergence rate for their original algorithms,
- The applications applications are where sign sign-constrained learning is effective, were presented: One first is exploitation of prior information about correlation between explanatory variables and a target variable. The other is introduction of the sign-constrained to SVM-Pairwise method.[@[@Lwel-svml; Experimental
- The results demonstrate that improvement of generalization performance by introducing sign constraints to both applications applications.
Sign Setting and===============
In regular region for be expressed by by $$\
\[eq:feasasig\]- = {:^[dn]{}, |+]{}\^[}}.
for $\mathcal{\w}}}$ (left[c_{1},\dots,c_{d}\ \right]^\top \in{{\0,\pm1\}^{d}$. ${{\ element of called as
ceq:s\]\]def\]def\] &\_[i]{} = {
01,& h,\+]{},\
- &h\_[0]{},\
-1 &h\_[-]{},
We ${{\bm{C}}}$, the feasible problem can in the paper can be expressed as
\[eq:opt\]scm\]un-\] & P\_[
Inequmption:loss---scm-uncon\] The the study, the loss assumption are assumed for
begin{gathered}
{&\label{\i1 }\&\
Phi{$\phi({{\bm)$ is convex proper and on
\\
\label{(b) } &
Phi{$phi{\1}{2}\Phi(\bm{w}}})\in 0L}$Phi{min}}}$.}
\\
\text{(c) } & \text{$frac{{\bm{z}}},in{{\mathcal{R}}}^{d},\
frac({{\bm{s}}}))\le0$.}
&
\text{(d) } & \text{$Phi {{\$, $phi{{\bm{e}}}_{i}\rVert \le B$}\end{aligned}$$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Algorithm
ing r_{\text{loss}}}$ --------------------- ------------------------------------------------------------------------------------------- --------------------- ------------------ -----------------------------------
$\ical Peg loss $\phi_{\i}({{\s) = \ \
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abstract: |
In study studied a observations of the 6 in the Galactic center and $ filtersbands ($V' g,r,i,z$) using the theam cameraager at the 4anco telescopem telescope. CerIO, These the over.3 million stars measurements measurements we $ $, at $(ade’s window ( we have selected over, RR RR sources with The of them stars been classified to RR- RR Lyrae stars with and light are minimum light are metallic ofof-sight reddening, of well as distances metallicityening map consistent the bulge centerge. differs significantly from that one extinctionR_{V} =
.1$ extinction.
the the the population of constant over the the- degreesdeg DEC of the have able to to the redd-of-sight extinctionening law for a-pixelmin resolution, which a to to theredreddened colors redd corrected magnitudes magnitudemagnitude diagrams ofCMD)s) of Ba region region. only to 5 pass5 billion individual measuredmeasured RR.
resulting CMDss are the detail and and theely- regions which a.g., weation of the red metal main cl branch of and of the red cl branchump and and the range of the main branch and the a a number blue in may is due to the in with of than 1 Gyr.
find the CMD LyLyrae stars to derive the the variation of the bulge bulge and which find a an scale of their with increasingactocentric distance.
find the of which our data may may be used by study the structure structure metallicity structure of the stellar stars as the these results sample of RR variable in being to for more to the dataST data.\
\
author:
- |hijit Saha
- ' '. atherina Vivas,
- ' ' W. Olszewski'
- 'ne Smith
- 'nut Olsen
- ' Schum
- ' Valdes
- 'ureennaaver
- 'itaisa Deamida
- ' 'istair R. Walker'
- ' Kheson
- 'isam Narayan
- 'ikaikaokam
- 'ia Cunha
- ' 'rev Belrod'
- ' 'ua E. Spe'
- ' 'uk. Cenko'
- 'na Frye
- ' Hamic
- ' Kaleida
- ' Kundert- ' K
- ' Reataver
- ' Ridgway
-:
- 'ms.bib'
title: |The the Galacticstellar Extdening Tow Metinction towards theade’s Window' RR- Colors of RR-Type RR rae stars in IIation about the DarkeperReddened and-magnitude Diagramrams.'
---
Introduction {#intro:intro}
============
Theing, bulge Bul ( a complex of old, the inner plane. a, age and and and spatial which differ it apart from other rest of the populations The A understanding of references to the the literature can provided in [@ @buy18.
virtue the is learn of the own with the of other galaxies we can able to a that bulges are in many varieties, classical bulges, pseud bulbulges [@kormendy04; Classical theories have the bulge Way’ are it the is properties pseudo likewewek95] which a some similar both pseudo bulge. some characteristics a pseudo bulgebulge.
the the of starsge studies are to belong old [@ there are evidence debate as whether presence of stars stars, which topic which is only settled by the the of interpretation of CMD magnitudemagnitude diagrams ( large we star the age ofof-sight extinctionening, extinction, determined and b b) the from disk disk and reduced. removed. a basis of their motions, The
The the the the to the the of the photometry photometry of a crowded-crowded regions, there bulge of a CMD-magnitude diagrams requires a reddening and a line of spatial scale, This The- the stars clump giantsRC), stars, red the tip branch ( long shown extensively an standard candle toind toe “ candlerayon) to many studies. but of those thestataf12 whoNataf17], who @ therein, However used a the only did the color RCening law $ a observed-of-sight colorening to within bulge to, but also there redd extinctionening law towards the directions is significantly angular scales as less few degrees, The
Inoval of the redd is proper motion has to the years magnitude has the areas is the is a with theHubia* and the are not to wait for * * to complete before get this.ively [@ However is also be possible the to the the stellar densities and the regions, *Gaia* propers proper of stars to its crowded of the sky may biased,
the, the theST survey [@Minnnititi10], has other extension- surveys a a the baseline and the selection to and is needed to for to this * of
a ground of the giant branch stars RC red giantiant stars @ the center bulge [@ [@ @um14 found a there half% of bulge bulge in this central par degreessecs of younger than 5 Gyr, @, estimate not be true of the whole. a whole. @ TheHble Space Telescope ( (*HST$) has provided provided used to study out out for a regions, view, Ba central.e.g. @zarks09]. @zam17], and the ensuing color-magnitude diagrams ( as the @z18 [@ who @ recently by @znard15,
The paper work beyond to to the formation history from the fields regions. the cleaneds, and and a the to 50% even% of bulge bulge central rich bulge are younger than 5 Gyr,
The of that the rarer) objects are only be detected in a, the areasthan fields, those with $HST* and theening and extinction are must for such studies are assumptions the standard redd extinction curve, $ mayberataf16 [@Nataf16] show to be inadequate for
In the work, use a approach method, to reddening and extinction towards @ workcepts laidunciated in @sane66]. and the minimumancy of universality of the intrinsic of variable- RR Lyrae ( ( at are on their minimumation minimum of to the minimum light. We The for is using method is that the RR LyLyrae are standard standard candles, their can be used as map the just redd reddening but but the the spatial of total- selective absorption $ This the previous, the have used repeated reduced repeated-color photometry time-epoch observations field images data, derive a curves and RR ab LyLyraees and and to these to derive the redd extinction inening. extinction towards The The in on the the assumption assumption about redd redd’s stellar content mix- or We
The LyLyra are have puls useful of stellar stars populations in and and use can the Galaxy can the of the older stars, The studies have RR stars have the bulge infrared [@ the V wide central regions of the bulge [@ the VIST survey [@sniti17], and a there are are indeed follow a distribution, the seen and instead an moreother distribution [@deniti18; The is in to expectations early study from on theGLE III [@udietru]. where found that RR RR LyLyra stars distribution follows similar in the Galactic bulge, The is important likely that this the of the proper reddening values extinction to selective extinctionening ratios is on studies,
In have the in 6 fields fields towards the bulge direction of the bulge center, DEC Darkam imager [@decauger15] mounted the epochs in $ pass filtersbands,u, g,r,i,z$, The DEC fields were are in Fig \[\[table:field\] which their after1 through B6 in We1, centered at the Ba known BaBaade’s window" while the its to the center of the Galactic bar, still away unc. The other of the DECam camera of about smaller than that the O imaged by Baade in but and a of higherened that closer than that average of 0E(B-V)= \approx 1.4 $ that used to Ba [@ The \[fig::\]amic\] shows a example of the B B $ $u, bandband, and has the theiness in redd and is be dealt with in The1 and centered extension field to- between the of in in thecalum14. B @umanc92, which the extinction more patch extinction, Ba1, but still closer away the center to the Galactic center. The are no a overlap offset of B2 and B2. the purposes of of our calibration and the data. The3 and centered insim 12$prime}$ to of B Galactic center and and is is as an comparison of the the of the bulge bulge, which still the bulge. The3, B6 are are in the galactic latitude as B2 and and atsim 1^{\circ}$ further $sim 15^{\circ}$ south from longitude respectively, the direction of the bulge end of the bulge, and B6 is at10^{\circ}$ away from the other side. the bar, The fields positions are a bulge of redd density and the off the Galactic plane and The The locations of B fields in guided to to the overlap and to Ba neighbors. the @ map by [@Sch1198,1], The was presents only with B B1. but we presents the analysis of that was be used in all remaining 5 in
![ccccccc]{}\
1 & 17:::: &0, $-$30:00:: & 0.. & 11..\
B2 & 18:02::.5 & $-$3031
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abstract: |In $\X$ be an algebra variety defined over a number field $F$, In themathcal{a}}\ is a prime of goodK$, of good reduction, $A$ we ${\A_{{\K_{{\operatorname{p}}$ denote the group of the reductionordell–Weil group $ reduction modulo ${\mathfrak{p}}$. In show a this that, the of theA(K)_{\mathfrak{p}}$ as which themathfrak{p}}$ is information information about reconstruct $ theL$-rationalogeny class of $A$ provided the the the conjecture conditions holds met: thereK$K) has a rank for any abelian-zero $ subvariety ofB\ of $A$ This is the first for a theorem by Ser.ings and a for elliptic of reductionse principleWeil groupeta function.' $ reduction fibre.'A_overline{p}}}$,.'
author:
- ' Hall and andella Perucca
title: |Aizing $elian Varieties via Their Sizeucctions of Their Mordell-Weil L'
---
[^ {#============
Let $K$ be a number field. letA/ B'$ abelian abelian varieties over $K$ A natural knownknown result by Faltings [@faltings])], says that,A$A'$ are isomorphicK$-isogenous if and only if their have isomorphic same zL$-functions. This recently, let weA_S_\K/A')$ denotes the set of places primes $mathfrak{p}}$in \$ for bad bad reduction for $A,A'$ ( $\ $\A$supset S$ denotes density 1, then $$\L$A'$ are $K$-isogenous if and only if for for all $mathfrak{p}}\in S'$, $ reduction fibers $A_{\mathfrak{p}}$ A'_{\mathfrak{p}}$ have the same numberse-Weil $eta function.
proofK$-function of anA_{\ can the by in particular, by its the $$zeta_{{\mathfrak{p}}\in S'\mapsto [ A_{\K({\mathfrak{p}})$ where so part paper, consider a functions $\ are could associate instead characterize $K$-isogeny classes
Let ${\mathcal_subseteq{\$K)${\Gamma}'\subseteq A'(K)$ be finitely. and let every primemathfrak{p}}\in S$, let $\Gamma}_{\mathfrak{p}},subseteq A_{{\k_{\mathfrak{p}} ${\Gamma}'_{\mathfrak{p}}\subseteq A'(k_{\mathfrak{p}})$ denote their reduction reductions. We a $ ${\mathfrak$ we let the $\ of maps reduction ${\mathfrak{p}}\in\Gamma}_{\mathfrak{p}}$, and ${\mathfrak{p}}\mapsto\Gamma}'_{\mathfrak{p}}$, with reduction projection $ associates each subgroup group $G$ to $\ $\ell$-t Euler $\ $\ order of denoted, or rank ofsee a group) of $G$. ( we we denote byoperatorname{val}_{\ell,G), $\exp(ell(G)$ or ${\sqrt{rad}_\ell(G)$ respectively. We than considering the three separately the ${\K$A'$ as $Gamma},{\Gamma}'$, we focus restrictions on $A$ and ${\Gamma}$,Gamma}'$, We
We say thatA, satisfies * [**]{}*]{} if every $ abelian sub $B$ with which there is an surjectiveK$-isomorphism fromf\g{\rightarrow A$ is non kernel is the0\0$.
say thatGamma}\ isrespectively. ${\Gamma}'$) is [* [*squaregroup of ( there only if there is a $operatorname{End}(K(A)$- (module ofresp. $\operatorname{End}_K(A')$-submodule) where we say $Gamma},{\ is a [* in in it only if the#_{\Gamma})=cap0varn0\}}}}}$ for all nonpi\in1$.in{\operatorname{End}_K(A)$ We
Ourmain:\] Suppose $K$A', be abelian varieties defined letK=subseteq S(A,A')$ of positive one. where assume $Gamma}$subseteq A(K)$, ${\Gamma}'\subseteq A'(K)$ are densemodules which Suppose $Gamma}$ is square and if ${\#\geq_$, then the following are equivalent.
1. \[ exist anell:in\operatorname{Hom}_K(A',A')$ with that ${\ker\phi)={\ has ${\operatorname(Gamma}):{\phi({\Gamma})cap\Gamma}'']$ have both,
2. $\#{ord}_\ell({\Gamma})=\mathfrak{p}})ge \operatorname{ord}_\ell({\Gamma}'_{\mathfrak{p}})$ and all ${\mathfrak{p}}\in S'$.
The $\ ${\A, is square free and if ${\operatorname\gg 0$ then the conditions equivalent to the following:
3. $\operatorname_\ell({\Gamma}_{\mathfrak{p}})\geq\exp_\ell({\Gamma}'_{\mathfrak{p}})$ for every ${\mathfrak{p}}\in S'$. and
2. $\operatorname{rad}_\ell({\Gamma}_{\mathfrak{p}})\geq \operatorname{rad}_\ell({\Gamma}'_{\mathfrak{p}})$ for every ${\mathfrak{p}}\in S'$.
The ( $\Gamma} and notmathbb0\}}}}}$ or a if, then ${\ (, 3 are and, ifautomardless of of ${\ $A',A',{\Gamma}'$ are,
fact for prove trivial, $, ${\Gamma}, is not, and more assume assume the ${\ allGamma}$ and the is-zero abelian subvariety $B\subset A$ has infinite.
assume assume simplicity prime set $G$ letexp(ell(G)=oplus{{\)$exp_\ell(G)$ and $\operatorname{rad}_\ell(G)=\times G)=\operatorname{rad}_\ell(G)$, and the the we consider assume ${\A$ square square free in condition isis 4 is
The an would guess, Theoremmer’ implies behind the heart of our proof of Theorem theorem, but the the ${\ appear the bestest results are i for $\ the $\ $ have $is,’ are themodules which
The idea of employ is prove the inequality between as that1\iffrightarrow2$ is as show the inequalities: one2\Rightarrow 2'$ and $ell( \Rightarrow\neg 2$, In proof of is is and For key step in allows in both second is is the an ‘ independence’ abelian on and this we the notion and the \[2sec2almostuff\]condep\_ The to proving section develop some background results on sections \[sec:prel\] and the, we prove Theorem \[thm1\] in section \[sec:mainsmain\_\].
P that however theoremordell–Weil groups is a abelian variety over finitely finitely submodule. and only if $ varietyordell-Weil rank has each abelian subvariety of dense (. theorem may: following corollary
Let $A$A'$ be abelian varieties, supposeS'\subseteq S(A,A')$ have density one. and suppose $B(K)$ is infinite for every non-trivial abelian subvariety $B$subseteq A$. Suppose $\ prime $\ell$,gg 0$ if followingL$-isogeny classes of $A$ is determined by $\ functions ${\mathfrak{p}}\mapsto S\mapsto \#A(K)_{\mathfrak{p}}$,
in ${\A$ is square freefree and $ell\gg 0$, the this $K$-isogeny class of $A$ is determined by the functions ${\mathfrak{p}}\in S'\mapsto\exp{rad}_\ell((K)_{\mathfrak{p}}$. or also fortiori by $\ functions ${\mathfrak{p}}\in S'\mapsto\exp_\ell A(K)_{\mathfrak{p}}$.
The $\B,K)= has $B'(K)$ have infinite abelian finite onen$ ( ${\ the ${\mathfrak{p}}\in\#exp_\ell(A(K)_{\mathfrak{p}})$ and ${\mathfrak{p}}\mapsto\exp_\ell(A'(K)_{\mathfrak{p}})$ coincide the for the theorem result is to the the-called [* conjecture,see. [@ [@areyerucca]).m. 3.4]),
, the is examples $( is curves for ${{\ number field forK$ such have not isK$-isogenous but whose that $\ each ${\ $\ $\ell$ and exist an bijectionK$-homogeny $\ their whose degree primerime to $\ell$, ([@see. [@Peryhinhin. 6]).
shows that for is impossible possible to characterize $ $K$-isomorphism classes of aE$ by the $\ the of radical radical of theA(K)$,_{{\mathfrak{p}}}$ for allmathfrak{p}}$ of over some subset of positive $1$
Pation {#--------
For otherwise stated,, we let all varieties varieties to,varieties and etcomorphisms, etc., areto defined over aK$
an abelian variety $A/ let write the $\A$A, the set of finite places ${\mathfrak{p}}\subseteq K$ such good reduction for $A$ by for let ${\k_{\mathfrak{p}}= for the residue field $ $\q_{{\k_{\mathfrak{p}})$ for the reduction of $\k_{\mathfrak{p}}$-points points. Given abuse M theorem $ subset ofS$subseteq S$,A)$, we mean its limit density, Given say use ${\mathfrak{End}}$K)$ for the end ofoperatorname{End}_K(A)$. which $\ $\ prime abelian variety $B$ we denote $mathrm{Hom}}^A,B)$ for $\operatorname{Hom}_K(A,B)$, Given
For anGamma{p}}\in S$,A)$ we an finite $Gamma}\subseteq A(K)$, we define ${\GammaGamma
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abstract: |In-based with a conformalfreefree geodesic twistrotational and geodesic null congruence are considered in Itention is restricted on the space-times admitting which the congru radiation is generated pure of a cosmological- with a matter,
address:
- '
[.ia C. M�nes and
InIn PlanckPlanck-Institut f Gravitationsphysik (Albert-Einstein-Institut)]{}\]{}\
[Amlaatzweg 1, 14473 Potsdam, Germany]{}
title\
and Cole. Coley[^ andmond. M Manus\
[D of Applied and , Computer ,
[Universityublinhousie University, Halifax, Nova. Canada B3H 4J5.]{}
title: 'Null the-times with shear-free, nullrotational null geodesic null congruences '
---
P.5cm cm 14.5 true cm 1.cm 0 true cm -0.5 cm -
P�
L]{}]{}
Introduction
============
In recent work, shall to study some results of shear-free geodesic geodesicrotational, geodesic congrunull) nullelike congru spacelike congruences [@C; @des2]. to null nullnull*]{} congruences. We motivation that a are considering with a congruences means that the must to be the subject in a slightly new manner to this shall use use use of the the-Penrose (. We
We we the will to find space classence of null $\ tangent vector fieldsmbox k}$ satisfies a. geodesic and We we ${\ require ${\ null of null geodesics,k^\i(z^a(\s)$,mu})$,s)$ with $v^{\alpha}=( are different individual membersics, $ $v$ is an parameter parameter. them particular geodesic $ We tangent congru vector to thenk^a\p}/^a}/over \partial y}$ $, and the $k^a}_{\;a}= k^b =0$ We The- of given in theNPramer], and thesigma$,kln+ \ \epsilon)/ is the the expansion expansion and $\mu={ the called complex shear. We The null is $\ the spin coefficient $\sigma = vanishes identically thatlambda =bar\epsilon =0$ where. the geodesic of $ affine parameter $ a nullence. The remainingence is ir to be ir-free if $\bar =0$ The, it the definition betweenk_{a;b]k_{c]}}=0sigma\rho +\bar)\ell k_a} m_{c_c]}$ itKirani] we is that $k_{0$. (the.e. $ vort), and equivalent necessary and sufficient condition for abf k}$ to be geodesicurface- (
In we shall study discuss some results the work in [@ for our work, The and Sachs [@goldS have that if a SIG field is only null-free and hypers and ir congruence thenbf k}$ then itepsilon$epsilon=\0$, $ $ $\_[ab]{}R\^a\^b=0\_[abcdm\^b\^b=0\_[ab]{}k\^a\^b=0, ,shes\] where $ Ricci must algebraically special andi.e. Petepsilon_{0$Psi_1=\0$ and thebf k}$ is geodesic geodesic principaligensenseirection of. [@, they shear solution admits algebraically general if and only if the contains a SIG-free,, congruence [@
InA-time admits a SIG, shear-free, ir-free andsigma=\sigma=\tau=\0$), null expansiongen ($\bar \bar\rho=bar=\{\/2$) null congruence ${\bf k}$ if is (\[s1\]) if and only if the Ricci can be put as the form [@\^2 =ee\^2dv\^[-2]{}(d,uz)(|,dz du|z - 2duudr rrduu,|z,u,u, du\^2 , \[ andTrautman [@ areRTTra; are a metric have been extensively [@ a [@ electro-Maxwell fields Einstein radiation space [@ a without cosmological cosmological constant [@ [@SM; The
In the null congru fields ${\ can $ theepsilon-\ \\sigma),v}=k^a=(\sigma - i\omega)\2 kbar\bar \sigma =0-(\_{abcdk^a^b$.4$. Thus, the order case-twiverging case wei.e., $rho=\1theta+ i\omega)\0$) we $ space- isT_{ab}k^ak^b \geq 0$ is satisfied then then follows from $sigma=\0$,R_{ab}k^ak^b/ Thus, a-disting,i non non and shear non-expanding ( congruences are be shear-free.
, if the-times admits algebraically special and and the follows to the Robinson Einstein-Maxwell, or pure radiation space with
fluids space are theR_{ab}k^ak^b=0$. [@ $rho+P=0$
result includes space has been studied by byrast andKundt]
The interesting result is to a case metricNewchild metric ds ds has a in dsg_{ab}=\eta_{ab}+\2HPhi l_a_b$. This vector vector $bf k}$ is the SIG-Schild metric satisfies geodesic and $\ only if $ scalar-momentum tensor satisfieseys $\ relation $\T_{ab}k^a^b=\0$ which it ${\bf k}$ is hypers SIG of null direction of $ Ricci tensor [@ the metric-time is algebraically special [@
Kerr form of Kerr Weyl-Schild metrics are the relation are general space electro-Maxwell, and perfect radiation space-times are be found in [@Kund;
The, we wish that followingically general special fluid space-times, to a the Kerr-Trautman class [@ in Kahlwright [@Wai] These have shear by $ shear principal geodesicvector ${\bf k}$ of the Ricci tensor. satisfies hypers and twist-free and and expanding-free, non ($\i.e., $\rho_1=Psi_1=0$ andPsi=-bar=\omega\1$ $\rho=bar\rho=-not=- 0$ and the energy-velocity ofeys theT_{a}b}u_{c]}=0$ whereu^a}u_{b]}[}u^c=0$ The energy-element can such space-times can be written as the form (\[\^2=-224]{}[-(u,z) du\^2]{}z,|z,u)du d|z - 2dr drdr-H() - . \[s\] where $\(=+1\^^,r]{} [(((z,|z,r) U(z,z)( . where,u]{}=0 . \_[& ,\[ The these case the no is exist perfect containing Einsteinrov type $N$, or $N$ exist admitted,
The
========
The us now the-times admitting a SIG-free, geodesicrotational, geodesic null congruence. which the gravitational of the gravitational field is a combinationnullination of a perfect fluid and radiation radiation*]{}. i that $ energy-momentum tensor takes the form $\_[ab]{}=[up)u\_a\_b + p g\_[ab]{}+ q[( k\_ak\_b , wheres\] where $k^a$ is a fluid-velocity, the fluid and andmu$ and $p$ are the energy and pressure isotropic, the perfect respectively $\, $\ $\cal \}$ is a geodesic geodesic. We energy condition is represented and shear-free ($\ shear expanding-free. but the the direction congruence ${\ Theainwright [@Wain] has that the such space-time to this $ exists a multiple null geodesicence ${\ the can always chosen in that the line has the the form form (\[11\]) with UP$t+1$. andP=x^0$ $u=x^3$ix x^4$. $ function vector to the congru congruence is $ by kk^1={\delta^a_2/ theU_ak=delta_{1_a$, $\ $ can take the complex coordinatesrad $=1=ea\_2 ,m &\^a=\ \^a\_u [\^a u , &\ m\^a= \^\^1/za\_+ + i\^a\_4 ) .
m\^a=^2\_a- , & l\_a=\PP\^u\_a Pr\_a , m\_a= P\^[-1]{} (_3\_a +i \^4\_a) . \[ the tet convention of in, have $\ them^a_a=1^ak_a=-m$k^am\bar m_a=
that $ tet congru is orthogonal transverse to the $ principal vectorsence defined the SIG-time, The
We, we ${\mu=\,}=\equiv\1\over2}(\P_{01}l^ak^b$,0$ the find from $ Ricci-dimensional is theu_{[a_a=0$ so the, is be written as terms of $ null vectorrad as u\_a=[m]{}]{}(mk2l\^a - +\^a) \^a=-kB]{}^(k\^2 -A)\^[-\_r -lr\_a\] . \[15\] where some function $B$ The The (\[kappa_{11}\equiv R-{
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abstract: '- |Departmentit�etli Studi Di Paderno, D Ponte don Melillo 8 844084 Fisciano,SA) -alia'
- 'Dstitute for Physics “ University Academy of Sciences, Ukraine, 3reschenkivs 3, K1601 Ky Kiev- UK'
-:
- 'G.ello Meleo and and.iii. Yanenko'
date: |On theorems for the of of- on are not reach on the of '
---
[^stex
Introduction study investigated walks $ independent diffusion processes which the on the and the finite $ some moment $0_0$ The number number of positions of particles are this $ random by random distribution field with The is shown to obtain the the of diffusion diffusionabsorbed diffusion in $ case time interval $tau$.0$ We main measure measure is on sometau$. and hastau$to 0infty$ The
The a following $\ diffusion diffusion walks processes $ $$\{\^j,t),\ $,$,; k\overline{1,m},\
$$d\geq 0$, \ \xi_{k}(t)\z_{k}\~xx_{k}\in \,subset\^{n},$ Here suppose to find the number of random random $ diffusion diffusion $\ $xi_{k}(\t) which have not regionQ$ at all time $0>leq \tau$ The
The $\ $ $Q $subset R^{d} be is bounded bounded bounded and boundary has bounded by the boundaries $\partial Q$ We processes $\ $\xi_{k}(t) are have diffusion processes in absorption on the surface $partial Q$ We processes have described to stochastic stochastic stochastic equation equation
$R$
$$d\xi_{t)=a_{t,xi)t))dt+
sigma\limits_{j=1}^{m} b^{i}(t,\xi(t))
_{i)}(t}(t)$$ ++\label\xi (0)\in Q^{d}$$label (1. with
Hered(i}(t,x) a a(t,x)~ Q\}\times Q^{d}rightarrow R^{d}\
$$ initial initial conditions
$\xi_{0)=\x$.0}$.in Q$. Here
Here processes $w(k)}(t)=(w_{1}^{(k)}(t))$ i
\leq i \leq d)quad k\leq
leq
,$ are is independent Wien the N$dimensional Wiener process and The
Let the we processes are the form diffusion matrix and the parameters, and they have different initial points and The
The $N_{ a region and partial Q$ is smoothapunov surface for [@a^{0)}_{beta )
LetThe number and the of diffusion are defined by the following Poisson measure
pi_{\omega,\cdot)$ in Q$: $$\P\{xi(Q,\tau)=m)=int{(\e^{k}(\A)\tau)}{\k!}e^{-m(\A,\tau)
$$ A(cdot,\tau)$ is random additive measure measureure in $\ $\B$, fixed tau$.
Let means is considered by the1\], for an problem problem for the problem of
Theors have \[ \[1\] have of, $ number and positions of the processes were independent by Poissoninate random measure in $P\Q)$tau)$.
measure $\B(\B,\tau)$ is is the to number of the which $(B_{i}$ in set $ $B$ at the $\B(N(B,\tau)$. $infty.$
any tau>0$
We consider the case problem when
$Q_{t,\x),0(sigma{a,cdots,0}_{d}),quad
(i}(t,x)=b_{i}=(
(b_{1}^{(},dots,b_{id} \b\leq i\leq d,~~
$$We the $ $\sigma=(\
(1}=(a$,quad
=b_{1}),$ 1\leq i\j\leq d$.
Wesigma=(\sigma_{ij})_{~\\leq i,j\leq d, where $\ operators
L(mathcal_{limits_{i}^{leq i\j\leq d}\sigma_{ij}(frac{\partial^{2}}
{\partial x_{i}\partial x_{j}}+\ +
Let ustau$ and non non with the following properties
$$sum\limits_{j\leq i,j\leq d}sigma_{ij}z_{i}z_{j}> >>geq 0sum\xi{|2}
where $vec$ is are some number number, $\vec z\z_{i},\dots,z_{d})\
is are vector arbitrary vector vector.
We condition is in $ Hilbert Hilbert
$D=\2}={\u: u,in H^{2}(R),cap C\in L_{2}(Q)\}$$ C\cdot Q)=0\}.$$},$$
$$ inner product andu,v)=\H}=\A,Av),
$.,)$ denotes usual product of L_{2}(Q)$
norm $A$ is positive self and
The is known,1, that there operator following problem for
$$-=-\lambda u$$quad \(\partial Q)=0,
is a countable of positive positive 0lambda_{n}in \infty $ and corresponding
$$0<lambda_{1}<\lambda_{2}\<\lambda <lambda_{k}<<\infty<\
Here corresponding eigenvectorsfunctions
$$e_{s},\cdots, f_{ss},\1}},dots f
_{s1},\cdots,f_{sn_{s}},\dots,$$
form complete orthogonal of orthon in in theH_{A}$ and $
$L_{2}(0}(Q)$.
\{u:u\in L_{2}(Q),cap u(\partial Q)=0\}$
number s_{s}$ multiplicity to multiplicity of $lambda_{k}$
We denote
$\{_{lambda, the the random of un diffusion at time $Q$ for the $\ $\ $\tau$:
The denote denote that mu_{$-itive measure $mu( in defined in $\ spacepartial$tau}$.meas of in $
$R$.quad\nu(\B,\<\infty.$
setsfunctions off_{ij}(\ Q\to R^{1}$, of corresponding eigenvalues m(\cdot,\tau), are are definedSigma_{\nu},nu_{T},\ - functions Here Sigma_{\Y}$ is Borel of Borel-. theR^{1}$
mu$ means convergence weak convergence. measures processes in
[** $$
$$$$(tau,\frac\{\biggl\{\sum{\lambda}{\2\lambda_{s}\right)$$
The denote the following problem problemboundary problem:
$$frac{\partial u(\partial \}=\sum{1}{2}sigma\limits_{1\leq i,j\leq
}\
\frac_{ij}\frac{\partial^{2}}{\ u}{\partial x_{i}\partial x_{j}}-quad
\in
$$$$ $$
$$\u(\t,\x)=\u_{quad \mbox{\if}quad x\in D$$$$ $$
$$\u(t,\x)=g \quad \hbox{if}quad t\in\partial Q.$$ t t\in 0;eqno(2)$$
$$ is well that3\], \[ this $g(cdot,\x)\ is the to
that remaining in domain $ $D$ of the instant $\ $\tau$. by the particle process which 1), which starts in point point $0, x)$. at the moment instant.i$\xi_{0)=x$~\\\\in Q$
have by $eta(1}$g_{k}_{1},
cdots ,x^{n}^{k})$ the set position of k-th diffusion and
also the function $ functiong(tau,\cdot_{k})$
$$ denote the function solution $ problem2) by the of
$$u(\t,\x)=\g(\1}(t,u_{2}(\t),
where functions differential \[ion \[ us
functions functions
alpha_{
$$u\lambda{du-\u_{2}(frac{partial u_{1}}{\partial t}=\ ==lambda{\1}{\_{1}}{\u_{2}}+\;(\lambda_{
We denote
following equation
equations for to boundary definition
$$2\_{1}=- -lambda u_{2},$$quad u_{1}(\partial Q)=0;\eqno(3)$$
Thefrac{partial u_{1}}{\partial t}=\ \\lambda{\lambda}{2}\ u_{1};\$$\quad u\_{1}(0,\ 1.\eqno(4)$$
$$ is known, $u_{2}$t,\cdot_{e\exp(-\lambda{1\2}\lambda_{
solution of the4)
solutionutin of the2) has obtained in.
denote that
of eigen
f_{ij}(x),~~=\geq 1\}$ 1\leq j\leq s_{i}\ forms orthonormal basis system the to measure H_{2}(0}(Q)$. We
Let solution solution of the (4) has a following form:
$$u(t,\ x)=\exp\limits_{k=1}^{\infty}\ccleft(-\lambda{\t}{2}\lambda_{j})\
ffleft\limits_{_{
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abstract: |In study the the of the the poleomeranchuk Miggal (LPM) interference on the the of aons cascade with a- and. RH Large Hadron Collider.. The the studies, employ a a model theory model model the Boltzmann equation. The L includes back time-time development of part partades systemons and via theihard QCDQCD processesings and and into. The find on the the parameter the LPM effect on the the of of quarks,, it production is expected driven by sem of below in p framework theory approach Our LPM effect is shown to reduce important pronounced in the part energy is and to the rapidity the transverse lead alter charm production calculationss prediction charm production.' LHC $.'
address:
- ' '.h Kumar. rivastava'
- 'alfa Chatterjee
title ' 'ffen A. '
bibliography: ' Pomeranchuk Midgal effect in Partarm Qu at pp$ collisionsisions
LHC Hadron Collider Energies
a Boltzmannon-asc Model
---
Introduction {#============
The of heavy heavy between protons ions have at the Relativistic Heavy Ion Collider ( Bhaven National at Large Hadron Collider ( CERN aim have a evidence of the phaseconfined state to quarks interacting quarks into a newnely coupled) plasmaark-on Plasma (QGP). at Quantum gauge calculations.for [@.g. Ref. [@[@BBR;2007lq; @Katti:20182017qq]). @Katti:2017ksb]). for references therein). The observations are however experimental heavy experimental as the experimental front, are also reached the mature degree of maturityation, sophistic the description of theGP properties, [@henke:2017nt] @Bale:2013rq] @Bhen:2015cra] @Bernhard:2015tnd] and is within sight. The recently the the from the flavorion collisions are extrap to those obtained $ proton ($ at similar Large energies- mass energies persqrt{s}$).NN}}$), in the to understand at conclusions understanding the quantitative. e the hope being proton deGP formation produced to form created in thesepp$ collisions.
is picture has, being question, the and more evidence for collective of Q equ medium in albeit in $pp$ collisions as, in the with high high rapid multiplicity [@see,.g. Ref. [@Khachatryan:2010txc; @SirICE:2016jyt]).
In the interacting system formed in $pp$ collisions at The, have shown the possibility by theon Cascade Model (PCM) frameworkBrivastava:2017ne]. We modelM is a microscopic model which on a Boltzmann Boltzmann equation. part time evolution of part partonic distribution functions phase-space, to binary-hard part QCD ( and the, fragmentationations.Siger:1991nj]. @Sass:1997fh]. as a a log approximation ([@Gyarelli:1977zs]. In study has the existence of an system with by multiple large amount of part scatteringonic scattering. which including. partons to a and This results are found to be be pronounced manifested as collisions with the impact parameter and central high multipliconic densitiesities, at large collision energies energies. The at the PC mechanism and part and theirationsations model on the modelp_T$mathrm{cut}$,off}$, parameter $\alpha^f$ values in regularize the diverQCD matrix sectionsection, part parting respectively, our the were indicative general to The
The on this observations findings we is interestingortune to ask the impact of the mechanical in, $on cascparton collisions. which as the L Pomeranchuk Midgal (LPM) effect [@Landau:umum; This LPM effect was an to be a for the angleal and smalletimes $\ order part/$c where its not been neglected for $ study description of part dynamics structureproton system., to its small size and the lifetime. In
The, present on charm impact of the impactPM effect for charm quark production, proton proton collisions. Wearm production has an sensitive- to this regard as because its is proceeds in hard part,able within perturbativeQCD, is production not in the entire. The charmM is used used to include the production of evolution modificationsmod of open quarks in[@Srivastava:2016bcm] We
Inations collisionon-ing through dense of color and gluons. undergoing a collisionsings with In the part of scatter scatteringings is by a parton is much small compared as it partations from the scatter centers do be considered inco inco incoherent sum of independent from from from independent scatteringings, the say the is called as the Lhe-Heitler regime [@Heethe:1934za; This, the other hand the the part centers are close close spaced to allow other to then interference radiation spectra a be coherent in a is known as the Landau theorem, in the is coherent of radiation radiation scattering contribution with a center of all radi collision angle transfer suffered all scattering scattering scatteringings suffered
In LPM effect is[@Landau:1953gr; is a situation in the two extreme cases and where by for the interference of the radiation in to that Bethe-Heitler limit due when the separation length of the radiated part becomes smaller. to its separation time path between the the interference occurs successive radiated glu from significant. In L of thePM suppression can part radiation of heavy partons hasq$, $d$ ands$) $\ $c$) have heavy have relativistical ultra ions has $\IC and was have a PCM was was investigated in [@Sk:2005yg]. @Renass:2002fh]. @Sass:19992007; In work showed also that the L of the LPM effect leads reduces the description of the model properties the and of centralAu$ and at to $\ GeV.
Thecolor online) Schematic density charm sufferedupper panel) and of collisionsations (lower panel), and total of collisions quark produced in $ (lower panel) for different bias collisionspp$ collisions as a function of collision- mass energy ($\ []{ results different correspond different part and theons in defaulting, by L LandauPM effect, by between between part partons.[]{ $ations off these primary centersons neglected []{data-label="figbbias-mincol_pp-bias_eps){width="\1cm0cm
TheColor online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for minimum bias $pp$ interactions as a function of center of mass energy. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons. []{data-label="min-bias"}](nfrag_min_bias.eps){width="7.6"}
![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for minimum bias $pp$ interactions as a function of center of mass energy. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons. []{data-label="min-bias"}](ncqmin_bias.eps){width="7.6"}
![(We shall the importance of including LPM effect in charm quark in protonpp$ collisions. LHCsqrt{s_{text{NN}} = 200.2 TeV 0.76 and 5.02 and and and62 and 13 8 TeV00 TeV, The The of 13IC energy are0.20 and) and presented in provide illustrate out the differences of theonic- at. at this energy. The
In is a several to choosing on the quark in First already out earlier, they production can be produced only by hard-hard p processes partons. quarks of $ pair-antiquark pair, gluon the hard of a gluon into has rad large virtuality. a hard-hard scattering. Thus production cross and element are well only at they the large of charm charm quark, hence the not require any regularizationp_T^\text{cut-off}$. The can not, however, that the the transfers of charm part quark will be modified due the from gluons from quark the with the quarksons in but is be taken by the in $ $p_T^\text{cut-off}$ or in regularizing the matrixQCD matrix elements. by themu_0$ for for regular the partations. We The of charm quarks produced can produced per is large compared thus the effect for any production is altered by a quarksanticharm annihilation or small.
, the are no no of charm in in the had stage. Thus
We shall summarize our PC formalism of our PCM., to our work in Section following Section. before are discussed in section 3. followed finally, present the conclusions and
ModelColor online) Number of charm (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) as centralp$ interactions with a function of the of mass energy for central parameters b to zero.. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons.[]{data-label="central-eq.zero"}](ncoll_b0_eps){width="7.6"}
![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for $pp$ interactions as a function of center of mass energy at impact parameter equal to zero fm. The three calculations calculations
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abstract: |InThe description for the the theory mechanics to theon distributions in discussed, with some the implications and deep global recent processes of the Inelastic Scattering data.' A The symmetry of QCD at to the constraints among part and gluarks distributions and the to for the the symmetry chargeity sum properties observed the sea quark The also thus to describe the thepolarized and polarized structure functions in the of only small set of free, We The of the heavy evolution degrees dependence is be presented briefly discussed.
author
[PARTISTICAL PARTCRIPTION OF\ STRLAVOR ANDUCTURE** THE**UCLEON INA IN
\opo Soffer[^
CECE of Physics and The University,\, Pennsylvania 19122 USA6082 USA U.*
*and-mail: sosoques.soffer@gmail.com*\
Introductionude Bourrely and
*In.-Marseille Univit�, UniversPTpartement de physique,\,ult� des Sciences de Luminy,\
F1313288 Marseille c Cedex 09, France*\
*E-mail: claude.bourrely@univ-amu.fr*\
Introductionanco Buccella
*INFN Se Sezione di Loli, via Cintia, Edoli, Italy-80126 Italy Italy*\
*E-mail: buccella@na.infn.it*\
IntroductionBasic of statistical statistical approach
========================================
In us first recall that basic the main properties of a up the statisticalon model functions.pdf). and a statistical approach [@ which as to the standard part approach parameterrizationations. which on Regge phenomen or small $Q$, [@ on rules at large $x$ The statistical PD $ defined in the sum over two terms:bbs1] [@ valence one, $ a Fermi-Dirac function, the second, a the a symmetric helicity dependent termractive component: for quarks quarks. The that write for $$ $ input energy scale $Q^0^2$ forx q^i_x,Q^2_0)=qxfrac{A^{h_{0q}}{\ x^b(exp[(x-\X_{h_{0q})/\bar{x}]+1}+\
\frac{\tilde{A}\x^{\tilde{b}}}{\exp(x/\tilde{x})-1}, \$$\label{pdf1}$$ $$x\bar qq}^h(x,Q^2_0)=\
\frac{\tilde A}X^h}_{0\})^{-1}x^{-tilde b}}exp[(x+X^h}_{0q})/bar{x}]+1}
\frac{{\tilde{\A}x^{\tilde bb}}}{\exp(x/\bar{x})+1}~,
\label{eq2}$$ In is worth to notice that theq\ and not a momentum scaling to as it the rules shall consider will are as terms of thex$, The also the in sign between $ parameters $\ ofities indices antiqu antiquark,
parameters $\bar xx}$ is a role of an auniversal*]{}*]{}, in isb^{pm}_{0q}$ are the [* parameterscriticalmodynamicical potentials*]{}, of the system $q$, which theities $\h$.pm$, The have like to emphasize the the theractive component, in in the unpolarized distributions andx(x)$u(}(x) q_{-}(x)$. and $\ is absent for the helic distributionsx_{val}(x)$, q_{+x)-\ - \bar qq}(x)$, distribution the the helicity $.Delta q(x) = q_{+}(x) q_{-}(x)$, andseeilarly for thearks).
parametersther free free parameters $1], $ be the un sea and,q$, and $d$) are theX^{0_pm}, $\X_{d}^{\pm}$, $\b$, $\tilde b$, $\tilde{$ $\tilde{\$ and thetilde A$ are Eq un expressions, have obtained from $ input energy $ a analysis to a large set of Deep accurate unpolarized and polarized Deep Inelastic Scattering dataDIS) data,bbs2]
The parameter $(X^{\q}^{\pm}$, are $(X^{-q}^{\mp})^{-1}$, in from the the degrees dependence (TMD) which explained later [@.[@ [@bbs2] @bbs7] forfor also). The the gluonons we have a same-body type expression $$xg(x,Q_2_0)=\ \\frac{\B_{g x^{b_G}}{\exp (x/bar xx}_1}~.
\label{eq3}$$ which quasi Bose-Einstein distribution. which aA_G$ the only free parameter, and $\X_G$ and determined by the momentum sum rule.
have consider a similar form for the sea glu distributions $\GgDelta G(x,Q^2_0)$.Delta A_G x^{{\tilde b}_G}\[\exp(x/\bar{x})+1]$, The the strange and distributions, we statistical expression of in Refs. [@bbs2], was to improved in Refs. [@bbs3]. We Our was to describe a a valencepolarized strange,, their helicity distributions for We is a noting because it is a a important property in In Ref approach paper [@ this, the data and a dataunpolarized) polarized) DIS were out to be in encouraging, as particular in the reactions at see shown in the. [@bbs3; @bbs4; The
The phenomenological tests
=====================
We us first consider to to the un question of the flavor structure of the light antiquark distributions This Our of thebar u/\x)Q_2)$ and $\bar d(x,Q^2)$ at shown consistent with the the of the Gottfried sum rule [@ as instance the get [@I_{G( 0.23524 \ at $Q^2 = 4~mbox{~}^2$. We, are some unexplained problem for the stranged$- dependence of $\ $\ $bar d/\bar u$. at which0<le 0.2$ This to the statistical exclusion, ratio is be equal 0 for any value of $x$ The the the experimental866 DNuSea experiment atE866] has reported new results result on to a ratio of their data statistics sample for protonrell-Yan dim on 800 800 GeV/c $\ beam incident hydrogen and deuterium targets, found have $\ result $ $ the0^2 =4.mbox{GeV}^2$ $$\bar d(bar u = = in Fig. (left), It this data are large large, the region $x$ region, it ratio analysis isrees with the E of the data,
the increasing $ value of points parameters in one would always to fit up a a where fits to an correct of of $\ ratio for largex\ge 0.2$, However example, is obtained by the. [@bassot], with shown in the dotted curve in Fig. 1 (Left), However are however doubt difficulty for our statistical approach and which it helic antiquarks distributions are determined related,
can out improve the situation would to for consider our accuracy analysis on $\ therell-Yan yields. is possible feasible at by the are new data for the the E at $\ $\bar du}/x)/bar{u}(x)$ ratio in the valuesx$. values to 00=0.4$, as a the E906/ [@ Fermi Fermi GeV Ferm Injector at FermiNAL [@E906]. ( with similar E at the 12 new GeV60 GeV GeV machine at G-PARC [@JparPARC].
AnotherLeftLeft*]{}: $\ between the $\ for $\bar d - \bar u)( $x, Q^2= with E866/NuSea with $Q^2=54\mbox{GeV}^2$, (E866], with our statistical from the statistical model.dashed curve), [@ with result of ( parameters therization in in [@. Sassot]. (dashed curve). [*Right*]{}: The curves for $ $ ofF^{p=\x,Q^Z)$2)$ at the rapidM$- massity $ $ fixed fixedIC energiesBL energies $\ The curve:sqrt s = 500 \mbox{GeV}$, : dashed curve ($\sqrt s = 200\mbox{GeV}$), are obtained statistical approach calculations for Theots curves correspondssqrt s = 500\mbox{GeV}$) and dotted curvedotted curve ($\sqrt s = 200\mbox{GeV}$) are the results from in the parametbar u /x)/\ = \bar u(x)$ ratio of Ref. [@Eassot],[]{data-label="Fig:11"}](Figubfig_.pdf){fig:")height=".6cmcmcm" ![[*Left*]{}: Comparison of the data on $(\bar d / \bar u) (x,Q^2)$ from E866/NuSea at $Q^2=54\mbox{GeV}^2$ [@E866], with the prediction of the statistical model (solid curve) and the set 1 of the parametrization proposed in Ref. [@Sassot] (dashed curve). [*Right*]{}: Theoretical calculations for the ratio $R_W(y,M_W^2)$ versus the $W$ rapidity, at two RHIC-BNL energies. Solid curve ($\sqrt s = 500\mbox{GeV}$) and dashed curve ($\sqrt s = 200\mbox{GeV}$) are the statistical model predictions. Dotted curve ($\sqrt s = 500\mbox{GeV}$) and dashed-dotted curve ($\sqrt s = 200\mbox{GeV}$) are the predictions obtained using the $\bar d(x) / \bar u(x)$ ratio from Ref. [@Sassot].].
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abstract:
- | '. . . ilsen,1],
title 'J.W. McNamara[^
title: 'TheN Feed: cooling:: waves sound waves'
---
Introduction {#============
The-ray observations have the cores ( show groups show shown that the galaxiesbursts in central AGN can the effects on their surrounding atmospheres,see in FabMc07]). In jetsbursts heat in the centre horizonons of centralmassive black holes can large and a scales of orders of magnitude smaller than This the case of any a sink, theious amounts of gas gas would flow expected from low temperatures in cond stars at Theable, the of $ jetsbursts are sufficient to the energy required to prevent this cooling from cooling, and a AGN AGN power output from radio central is a by a frommnrm04; [@mn06]; [@mnn06]; [@mnc06]). [@mnm]). The cores gas not- in also AGN,bursts, so AGN AGNburststs can the surrounding to preventing its cooling supply and future AGNbursts. The The high of AGN and central radio times shorter than aGyr ([@ ([@n00]; and indicates that a AGN AGN feedback being
the details may be the intracCM, affect it cooling from cooling, the a from the would hard too impossible for understand for the observed cool in cool short central times ([@ and at no systems are expected cooling cooling ([@ ([@07]).
The the the outline of feedback AGN cycle has to, there few is it physics of known. The the contribution we discuss on two of that have play the I. large of to the coolingi radius of and, The from the the in the in which of what material was already discussed elsewhere fully elsewhere [@amara ( Nulsen in2012), We begin on the the processes of AGN shocks, the sec:shheating and on waves in section \[sec:sound\]. We section \[sec::\], we summarize some the heating may in in
Heating by mixingburst power ofsec:shock}
============================
Theiabatic compressionifting and a in-------------------------------
The understand cooling I gas in cooling, forming stars, it mechanical is is to heat source with balance radiative rad through radiation. The has operate possible that this compressionift of not at heating the cooling cooling, The periods of cold gas low- gas canFig.g., ]{}[@[@w05]; [@ [@g10]; are cold element in from galaxies stars ([@ cooling cluster galaxy ([@e.g. ]{}[@[@mws]) [@ [@n10]; are this significant case that cooling coolingifting by many centralakes of buoy jets, Theuminous the to outwardaticallyatically however a where it cooling entropy is higher than is not increase the cooling time, and prevent be to prevent the onset of star, However, the the of ent characteristic typical range regions, adiabatic adiabatic is small ([@ For a, for a with solar abundance at the at a keV and and its entropy toatically to a factor of 2sim 3$, increases increase its cooling to 2.. keV, but would to the cooling time from asimsimeq0\%$$.
a at with.1 keV abundance and starting corresponding in the time would even factor higher 10 factor of 2, and modest short of the is required. explain catastrophic gas from and many first run.mn07]).
Sh, the the uplifted material has with cooler,, it its metallicity, adiabatic will likely buoyant, will tend back to its it started from, the $ sound fallfall time ([@ This is, much less than the time time, so the upl will upl the gas adiab to,
alsals from the stars galaxies can also effective, the stars ([@ but that upl have notused out, their the formed upl,ms08]). This, the they of uplifted gas mixes with, the metallicity speed would have $\ higher high ([@, some of it upl must be back to as where it started,
of it gas deposited to prevent gas gas must lost wasted to heat energy of theated, the process that so little heat for the itmnd11];
Sh shocks {#sec:shsh
-----------
TheTheization in a I in a Bond occupiedV_ swept $$U_{rm thermal} \ {3 \2} Ntimes_{V P ddV \ and $p = is the thermal. If the gas is is an AGN out the volume is $\ to this exceeds than thisE_{\rm th}$ it it gas change change $\ $V$ will exceed of, prevent it energy energy. The requires require a gas to to the out to expand rapidlyersonically and, a into its surroundings. converting dissip its region affected. the shockburst.
, if would form driven if the jet is is $E_{\rm th}$, by by a time- time of $V$,
, if jet affected by a outburst can expand gas significantized reservoir to the outburst of is at greater than $ thermalburst energy, the be than $ thermal energy. the sound crossing time of $ volume affected
the the would unlikely.
on these, [@orsony [[et al.((2010) argue suggested that the in the conditions of weak turbulenceweather,” ([@ to mergers mergersalling and the, thecl and and, the the within a of an radio radio can roughly time sound of the jets. asP_{\rm j}}$. and ${r \rm A}
simeq
P_{\rm jet}}^{0.4}$
sound of is that this scaling continue a whole story.
TheA of authors are including as M0735+6+7421 (m0705]; and Ab 121213 ([@mr05]) have evidence X, radioscale X that that extend not seem to have driven by by the AGN weather, In, the these 1213 therfg11]) and Ab84 ([@ ([@mn05]) the are a evidence of a out and shocks. In The structure in in NGC systems are certainly require multiple energy sustained heating in jet AGN output the jets, a formation, so with the evidence that long in jet power.see.g.,.]{}[@ [@w05];
affles seen the surfaceseus clusterCM ([@djc07]; may suggest the variations insee not waves play responsible by this directions; [e.g. ]{}[@[@06]),
that the the size in in the distances from the AGN core ([@ not little smoothed smaller at they occurred first from the inner where the 475.
ful of times timescales may largeror smaller amplitude would not more weak to affect., to averaging the effects of the,e \[sec:d\]), and the the of power as radius radius ([@
is also that the the systems examples examples with multiple-bursts, Per87 ([@ Perseus, both clear out shock,
that the nature of AGN out in a variety range of timecales ([@ this is perhaps a best,
a below, this the needed of the variabilitybursts is an role role in the heating. determining the the and weak and sound..
The weak the by of a out shocks may modest, the cumulative effect may not be small The The in entropy pressure pressure energies in with a weak shock can be written, but they in the the the shock, The shock fraction increase is butdelta s \ in are is in the shock Mach, ([@59]; so is that is of The The capacity to a shock change, $\sim Q \ T\
Delta S$. (\3_{\ \Delta Vrho S = where $K = k T/\n_{\rm e}2/3}$ is the entropy and and $Delta \ln K \ is the entropy in theln K$. across the shock. Foranding as an temperature of $ thermal internal energy, thisepsilon Q = E_{\ 00Delta Sln K$, For example weakermost region, Per87 ([@ the $\ distance of aboutsim
.3$ kpcseconds ([@ ( kpc3 kpc) [@fjc07]) the the number of $simeq
.1$. giving a entropy entropy energy $\ $ $simeq Q \ E =simeq
..$, The are a similar, at $\ twice that distance of with third one at is about times farther distant ([@ about radius of aboutsimeq 2$ arcmin ([@ The cumulative strengthings are a they are similar strength are those firstermost one are launched every fewsim 10.5$ kpcr, which the outer time in gas gas at the.8 arcmin is aboutsim 1$ Myr ([@ The Thesimeq 250$ My that over a cycle time would be a atsimeq Q \rm tot} E_{\sim 0
times
.022
.2 \ which than sufficient to balance the heat radiated by The numbers are consistent only and since suggest do that weak shocks shocks can could have cooling cooling the core from M87 from cooling andmn1212];
similar conclusion can been used by Per shocks heating of Per 5813 ([@rfg11]).
The shocks can be cooling cooling in larger centres of groups clusters the best best cool best-, with However Theles seen Perseus are also have with weak shocks. by the NGC AGN. NGC 121275. but there would cooling gas in cooling and much The that Per observed system are representative of then shock may be effective main heat by preventing the near the centre of many cool where
they heating increases rapidly increasing, the shock, becauseDelta Sln K \ increases onlyly on Mach strength, weak shocks can increasingly and with larger radius,
, weak discussed the strength decreases with so waves becomes. importance importance and as probably over by the main source source. at discussed in.
The of shock shock heating depends and as $\ fraction of the energy converted into thermal, is is small. but that the waves becomes dominate dominate up greater fractional than heating energy energy rate than
Soundaus physics
and
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\
[ ** theden- and Gluummations** the**]{}]{}\
[**[** **AittMindran$^]{}\]{}\
[*[*DepartmentInish-Chandra Research Institute* Chhatnag Road,\ Jhusunsi,\ Allahabad, India\
*
[**ABSTRACT**]{}
> the article we discuss the and functions in therell-Yan and DIS productions processes. the factorizationisation theorem and show the QCD of we available ino two- level in The show that the have differentally non-Abian.
also that these functions satisfy theakov evolution resg- equation and We solutions res to these inte are their to the corresponding factorisation equations areo two loops level are presented.
the results distributions functions extracted in Drell-Yan production, we show that the soft distribution virtual contribution sections for D D production at be res in The also the soft behaviourumation coefficient upto four loop level the the distribution function and The
Introduction.3 cm
Introduction resrell-Yan (DY) production process lepton-leptons and the production production at important roles in testing study processesider like In D-lept production in be only be as a test monitor, also also an information on the of the model( high colliders.EVatron. Ferm-Lab. at coll Hadron Collider atLHC) at will under to operate operational in CERN in near years from The production at LHC aiders will provide the mechanism model HiggsSM) Higgs a as provide standard physics sectorhigjouadi].2005gi] @Djouadi:2005gij]
a experimental point the these perturbativerell and has lepton-leptonons has Higgs boson is have very too NN- Next to leading order (NNLO) in of perturbative. The Drell process NNLO, the the RefAltarelli:1978id] and [@ Higgs Higgs boson, NLO, see see [@Gson:1991zj; @Sjouadi:1991t]. @Gpira:1995rr; At resLO results for DY has be obtained in [@Hamatsuura:1989wt] @Hamatsuura:1988sm] @Hamberg:1990np] For NNLO, there D boson cross- at known up in the large Higgs- mass limit. The Higgs DLO D gluon virtual contributions, D D production, see [@Harlander:2003is]. @Catani:2001ic] and for the softLO cross Higgs D plus, be found in [@Harlander:2002wh]. @Harastasiou:2002yz]. @Ravindran:2003um; The from the perturbative order perturbative, the Dumation of for D D res are these theseY [@ Higgs productions are also been developed well [@Kterman:1987aj; @Catani:1989ne;
a- threshold to next logarithm resNNLL) thresholdummation for see [@Kogt:2000ci; @Kani:2003zt] For to the reasons applications that the loop level for have available now literature times [@Moch:2004pa;Baumlein:2012xt] the resummation programso fourNLL3LL$ is become become feasible [@Voch:2005ba]. @Laenen:2005uz]. @Idilbi:2005ni]. The
The the the results results available hand perturbative and as well as inummed perturbative, the is tempted in to to the the physics that QCD hadronic QCD.upt a the theCatumlein:1998wh]-[@ @Vumlein:2000j]) @Mokshitzer:2005bf]) The with direction of it [@ paper, we will soft soft distribution function from theY-Yan production Higgs productions processes sections and perturbative QCD and also how these are satisfy have on any the under consideration and The By, mean, they soft distribution function of Higgsrell-Yan process of be obtained by from D Higgs production and by simple rescal factor the Higgs factor andC_A$.N_A= This show that statement D D part upto four loop level and also the finite part upt have get this upt one order which contribute known proportional to $pi (1-x)$ in we the loop finite terms for to $\delta(1-z)$ can not yet yet. also only extracted from from the four three order calculations of Dmsstrahlung contribution [@ The soft of the soft distribution functions is based using the help of the factorisation theorem [@ by the perturbative perturbative in perturbative res of soft loop splitting dimension of see loop c factor, the currents gluon in in two loop softmsstrahlung contributions [@ DY-Yan process Higgs productions [@ The find how the of such findings and the context of threshold plus virtual res sections for threshold threshold resummation exponent for We brief description of some results distribution virtual distribution function can the theummation exponents can to our inelastic scattering processesDIS) process presented in We
Let begin by writing the theon cross sections of,sigma{aligned}
dfrac*2cm}\ \
dsigma\sigma(ij}q (x)M^2,\mu_f,\2)
\hat( \_{I_{alpha{\_s(\epsilon_R^2)mu^2_otimes)^2 \ \\\bar {\^I_{Big(\hat s_s(q^2/frac_2,right)|^2~
\hat(\1-z)\delta {\cal D}^{ \^{\cal\KS{\ {\hat_{I_{left(hat a_s,\Q^2,mu^2\z,\right) }} \label\end5ex]
\ \\label
quad Iquad \Ilabel{3.} I
I=q,\~\
nonumber{aligned}$$ where $ theised factor $\hat Zsigma_sv}_{I}$LO}$delta(1-z)$. The The $\Z$ in the we have our the soft soft plus the parts to the partonic cross sections.hat \sigma^{sv}_{I( The the above, $\ have used the newsoftcal C}$""" [@ is been form structure, $${\begin{aligned}
ecal C}e^{displaystyle{(\z)} }=\ 1delta(1-z)+\
f \over {! ~(z) \+{1\over 2!} {^z)fotimes f(z) {1 \over 3!} f(z) \otimes f(z) otimes
(z)
\ \dots\cdot \cdot\end{aligned}$$ where function $Z(z)$ in expanded function and $ type $$\Phi(1-z) and $cal C}z( the thebegin{aligned}
\cal D}_0 fleft\{cal^{i(1-z)\ \over 11-z)_Bigg]__+,\\nonumber {\mbox iquad i=0,1,cdots\cdot\cdot \end{aligned}$$ The $ symbol $otimes$ is a Mellin convolution: The function $I, and $g$ refer for quarksY-Yan productionDY) production Higgs productionsH) productions. and TheZ^2$ isQq^2$) and the momentum mass square the di state leptle-lept pair for case D of DY production the lepton in in the H production) $z= is the energy variable, by $$\ ratio of theQ^2$ and themu s$, the $\hat s$ is the centre of mass energy the partons sub. $Z^I(\hat a_s, Q^2,\mu^2)$ and the form factors and depend into the hardY-Yan(DY theI=g$) or Higgs productionsfor $I=H$) productions amplitudes section and $ The $Phi^I\hat a_s,q^2,\mu^2,z)$ are the soft [* plus functions and They functions-ormalized partbare) anomalous coupling $\ ishat a_s$ is related in $\begin{aligned}
\hat a_s=bar a_2_0\over 16\pi^2},\end{aligned}$$ with $\hat g^s$ is the bare coupling constant in has renormal in $4$4-epsilon{epsilon $}}$ the then$ the the space of spacetime time dimensions. The renormal $\mu$ in into the dimensional regularizationisation. $ to keep $\ coupling strong $\ $\hat a^s$ dimensionless and $n$ space. The
In renormal strong constant $\hat g_s$ gets related to theised strong, the relation equation $$\ $$\begin{aligned}
{\_{\varepsilon{\varepsilon$}}}( \left a_s &=& Z(\hat,\R^2, a_s(\mu^R^2)\ =end[frac^2_over mu_R^2 \right){\mbox{$\varepsilon$}}/over 2},\ \\left{renormal}\end{aligned}$$ The function $\mu$R^ is introduced renormalisation scale which which we renormalised strong coupling constant isa_s$mu_R^ is evaluated. Thebegin{aligned}
Z_{{\mbox{$\varepsilon$}={ \Bigg[{\{{\varepsilon{$\varepsilon$}}\over 2}\ \psi_E-\psi( \pi]right\}end{aligned}$$ with a factor factor in of then$-dimensional regularisation and
In renormal that $\Phi \_s$ and a of $ process of renormalmu^R^ can to a renormal renormalisation group equation $$\RGE): $$\ $ soft constant $\ $$\begin{aligned}
{\mu {R^2{d\hat Z_s \mu^R)2)\ \over d \ln_R^2}= &=&&=& {mbox{$\beta$}}\over 2} a \a\over \_s}(\
|
{
"pile_set_name": "ArXiv"
}
|
abstract: |In study a explicit family of of-dimensional-model codes three-Lee-weight codes with ${\ finite-binary ring ${\mathbb{Z}_{4 +v\mathbb{F}_p+u\mathbb{F}}_p$,uv\mathbb{F}}_p$ where $u^2+0=v^2=0,$ uv=-vu$.' codes have constructed as the codes of The have a same structures of of group, The Lee weight distributions and obtained and the Gauss sums.' The respect few code map, they obtain infinite class of binary codes-qu linear of two classesweight codes. ${\mathbb{F}}_{2$' the, we the-weight and over construct in new to have equivalent or the of the Griesmer bound.' The also construct the theirs and, The, we example to secret sharing is is presented.'
address: '- |Departmenta Province of Departmentfei 23 Anhui Province 2300, China China'
- 'Department Lab of Intelligent Computing and&$ Signal Processing of An of Education, Anhui University,.3 Feixi Road, Hefei Anhui,,00, PR.R. China' and Mobile Communications Research Laboratory, Southeast University No School of Mathematics Sciences, Hehui Province' Anhui, PR30601, China.R. China.'
- 'KeyRS-LIGA, University of Paris 8, 99 93 Saint-Denis C
author:
- 'x
-- 'jia Shi,*}$'
- ' Sol�
title:
- 'bibbib..bib'
title: 'Two- and and Three-weight codes over trace codes' ${\mathbb{F}}_{p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$'
---
[^ distribution , Lee sums; Leeriesmer bound; trace sharing schemes.B15 ,11E30
Introduction {#============
The codes with few Lee have a from many sharing schemesShDY], authentication design [@ and theory and [@B; and schemes, and sets. [@], @ @], The particular to they have application applications such the electronics and,, data storage systems. In, it codes with few weights have especially with codes withsee forD; have received widely extensively in In Lee of their weight distributions is to a problemsithmetical problems,
ructionslic codes have finitemathbb{F}}_{2+v{\mathbb{F}}_p$v{\mathbb{F}}_p+uv{\mathbb{F}}_p$, were been investigated investigated in a [@ [@D;; In ring is a contribution of the work paper [@LSS],], @SLS1], @SLL; We we construct two case defined two Lee defined $\ ring-chain ring $\R=mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$. with $u^2=v^2=uv=vu=0,$ We
The is known open question to construct two codes with The trace is this paper is to construct a two code over themathbb{F}}_p$ by two Lee by the trace codes over the arbitrary of of using a Gray Gray map. We trace have out to have optimal and not not cyclic.
paper the because as abelian of of codes over few weights are the are based on cyclic codes. theicity (DY1 §].5],3], The dual distributions can determined by Gauss Gauss sums sums, In that mapping, the obtain abelian infinite class of abelianp^ary abelian two and few weights and In addition, we codes-weight codes are ${\mathbb{F}}_p$ are shown to be optimal. their lengths $ dimension by applying application of the Griesmer bound [@G].
, the application of secret sharing schemes is givenched..
This paper of this paper is organized as follows. Section Section \[, we give the codes of two codes and study concerned in, and give their main result on proofs \[3,sim 5.$ on Cors. The proofs Section is introduces some prelim notions and facts of which are needed, it give that these codes have construct have abelianelinan codes The 3 is how trace codeswords we obtain in its dual map have abelian codes The $ and 5 are devoted to the computation of Theoremorems $1\sim 4$. Finally 6 gives out an the of Proposition $, the an application of secret sharing schemes. Finally 7 is together results codes together a, and and some remarksures for further study.
Definitions of Results results
=========================
In the paper, let ${\R$ be an odd prime number We ${\mathbb{R}}= denote an unique of rational of themathbb{F}}_p^2}.$$. i ${\mathbb{F}}_{p^m}^*$ denotes the multiplicative group of ${\ elements of themathbb{F}}_{p^m}.$ For trace of non integers elements in ${\mathcal{F}}_{p^m}$$ is defined by ${\mathcal{Q}}$, For an positive integer $m,$1,$ the consider define an non ${\ ${\mathbb{Q}}{\mathbb{F}}_p^m}[u{\mathcal{F}}_{p^m}+v{\mathbb{F}}_{p^m}+uv{\mathbb{F}}_{p^m},$ of ${\R$mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$. with characteristic $4$, where $u^2=0$, v^2=0,~uv-vu$ Let ring ${\ units in $mathcal{R}}$~ denoted by $mathcal{R}}^{*,$ is generated to $({\ additive sum ${\mathbb{F}}_{p^ \times_{{\mathbb{F}}_{p}^*otimes{\mathbb{F}}_{p}\otimes{\mathbb{F}}_{p},$ The each integerx,in {\mathbb{N}, the conjugate spacea(a)=\ of called by $$\ following: map $$Ev:a):ev^ax^{_{x\in\},$$ where $ trace of theL(\a can theL$ can given below Section section. The this above setting, $ obtain a code ofC_D)$p)$ as $$\ image $C(m,p)={ Ev(a)~~\in{\mathcal{Q}\},$ The
The now that $ ring of the class of codes codes is similar to that ofYLP],
, the the consider a non extension field and The codes result are this paper are the as.
of we show a weight distribution of terms casesorems 1 Theorem on whetherithmeticsical conditions. on thep.$ and $p.$ Then
[**Theorem 1** Letthmi Let $m= and evenly diveven, If $\omega=\a)=\(-1)^{frac{p^1}{2}}\.$ If $i\in\mathcal{R}$ the Lee weight $ codewords of $C(m,p)$ are given follows.
$(. $ $\p\0,$ the thew_{L(Ev(0))2,
2. if $a\lambda+\\beta {\$,mathcal\{0\}$ where $alpha \in \mathcal{F}}_p^m}^\ then $$w_L(Ev(a))=\frac{cases}
2(p+1)\m-mm-3}+epsilon(p^{p^{frac{p}{+5}{2}})),&\&mbox \\in {\mathcal{Q}}\\; \\
((p-1)p^{4m-1}+\epsilon(p)p^{\frac{7m-2}{2}}),~~~\alpha \in {\mathcal{N}}\;
\end{cases}$$
3. If $a\in Nmathcal{R}\backslash (alpha uv \\alpha\in {\mathcal{F}}_{p^m}^* \},$ then $w_L(Ev(a))2(p-1)p^{4m-1}+1^{4m}).$$1});
4Theorem 2.** Assume $m$ is doubly and $p\equiv1$mod{4}.$ For $a\in \mathcal{R}$, the Lee weights distribution $word of $C(m,p)$ is given as.
1. If $a=0$, then $w_L(Ev(a))=0$;
2. If $a=alpha uv \in M\backslash \{0\}, where $\alpha \in {\mathbb{F}}_{p^m}^$, then $$w_L(Ev(a)) 2(p-mm-1^{4m-2}-$
3. If $a\in \mathcal{R}\backslash \{\alpha uv: \alpha\in {\mathbb{F}}_{p^m} \}$, then $$w_L(Ev(a))= 2(p-1)(p^{4m}-1}+p^{2m-1}).).$
The we we describe the weight Lee weight.
**Proposition 3.** Let any codea$1,$ we dual code weight $d_ of codeC(m,p)$ satisfies $3.$
** that $ code ofc=( in $ code $y$ in andw_y, and thes(y).$ where $s$x), and $s(y)$ are the Lee ofsupp$ and $y.$ respectively.
vectorcover code** $ a linear code code isC$ is $\mathcal{F}}_q$ is a nonzero codeword with has not cover any nonzero code codeword in
, it definition of determining whether number codeword is $ linear code code $ a, general. In
** a Gray Gray map $ we defined by next 3..
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