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the broad mechanism for radiation emission from a hot tenuous plasma is simple . thermal kinetic energy of free electrons in the plasma is transferred by collisions to the internal energy of impurity ions , @xmath0 where @xmath1 denotes an excited state and @xmath2 the ground state of the impurity ion . this energy is then radiated as spectrum line photons which escape from the plasma volume @xmath3 where @xmath4 is the emitted photon energy and @xmath5 its frequency . similarly ions in general increase or decrease their charge state by collisions with electrons @xmath6 where @xmath7 denotes the next ionisation stage of impurity ion @xmath2 . the situation is often referred to as the _ coronal picture_. the coronal picture has been the basis for the description of impurities in fusion plasmas for many years . however , the progress towards ignition of fusion plasmas and to higher density plasmas requires a description beyond the coronal approximation . models of finite density plasmas which include some parts of the competition between radiative and collisional processes are loosely called _ collisional - radiative_. however , collisional - radiative theory in its origins ( bates , 1962 ) was designed for the description of dynamic plasmas and this aspect is essential for the present situations of divertors , heavy species , transport barriers and transient events . the present work is centred on _ generalised collisional radiative _ ( @xmath8 ) theory ( mcwhirter and summers , 1984 ) which is developed in the following sections . it is shown that consideration of relaxation time - scales , metastable states , secondary collisions etc . - aspects rigorously specified in collisional - radiative theory - allow an atomic description suitable for modelling the newer areas above . the detailed quantitative description is complicated because of the need to evaluate individually the many controlling collisional and radiative processes , a task which is compounded by the variety of atoms and ions which participate . the focus is restricted to plasmas which are optically thin and not influenced by external radiation fields , and for which ground and metastable populations of ions dominate other excited ion populations . the paper provides an overview of key methods used to expedite this for light elements and draws illustrative results from the ions of carbon , oxygen and neon . the paper is intended as the first of a series of papers on the application of collisional - radiative modelling in more advanced plasma scenarios and to specific important species . the practical implementation of the methods described here is part of the adas ( atomic data and analysis structure ) project ( summers , 1993 , 2004 ) . illustrations are drawn from adas codes and the adas fundamental and derived databases . the lifetimes of the various states of atoms , ions and electrons in a plasma to radiative or collisional processes vary enormously . of particular concern for spectroscopic studies of dynamic finite density plasmas are those of translational states of free electrons , atoms and ions and internal excited states ( including states of ionisation ) of atoms and ions . these lifetimes determine the relaxation times of the various populations , the rank order of which , together with their values relative to observation times and plasma development times determines the modelling approach . the key lifetimes divide into two groups . the first is the _ intrinsic _ group , comprising purely atomic parameters , and includes metastable radiative decay , @xmath9 , ordinary excited state radiative decay @xmath10 and auto - ionising state decay ( radiative and auger ) , @xmath11 . the intrinsic group for a particular ion is generally ordered as @xmath12 with typical values @xmath13 where @xmath14 is the ion charge . the second is the _ extrinsic _ group , which depends on plasma conditions - especially particle density . it includes free particle thermalisation ( including electron - electron @xmath15 , ion - ion @xmath16 and ion - electron @xmath17 ) , charge - state change ( ionisation @xmath18 and recombination @xmath19 ) and redistribution of population amongst excited ion states ( @xmath20 ) . the extrinsic group is ordered in general as @xmath21 with approximate expressions for the time constants given by @xmath22(1/(z+1)^2)(kt_e / i_h)^{1/2}(\rm{cm}^{-3}/n_e)~~s \nonumber\\ \tau_{ion } & \sim & [ 10 ^ 5 - 10 ^ 7](z+1)^4(i_h / kt_e)^{1/2}e^{\chi / kt_e}(\rm{cm}^{-3}/n_e)~~s \nonumber\\ \tau_{i - i } & \sim & [ 7.0\times10 ^ 7](m_i / m_p)^{1/2 } \nonumber\\ & & ( kt_e / i_h)^{3/2}(1/z^4)(\rm{cm}^{-3}/n_i)~~s \\ \tau_{i - e } & \sim & [ 1.4\times10 ^ 9](m_i / m_p)^{1/2 } \nonumber\\ & & ( ( kt_e / i_h)+5.4\times10^{-4}(kt_i / i_h)(m_p / m_i))^{3/2 } \nonumber\\ & & ( 1/z^2)(\rm{cm}^{-3}/n_i)~~s \nonumber\\ \tau_{e - e } & \sim & [ 1.6\times10 ^ 6](kt_e / i_h)^{3/2}(\rm{cm}^{-3}/n_i)~~s . \nonumber \end{aligned}\ ] ] the ion mass is @xmath23 , the proton mass @xmath24 , the ionisation potential @xmath25 , the ion density @xmath26 , the electron density @xmath27 , the ion temperature @xmath28 , the electron temperature @xmath29 and the ionisation energy of hydrogen is @xmath30 . @xmath20 may span across the inequalities of equation [ eqn : eqn6 ] and is discussed in a later paragraph . from a dynamic point of view , the intrinsic and extrinsic groups are to be compared with each other and with timescales , @xmath31 , representing plasma ion diffusion across temperature or density scale lengths , relaxation times of transient phenomena and observation times . for most plasmas in magnetic confinement fusion and astrophysics @xmath32 where @xmath33 represents the relaxation time of ground state populations of ions ( a composite of @xmath19 and @xmath18 ) and it is such plasmas which are addressed in this paper . these time - scales imply that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions . the dominant populations evolve on time - scales of the order of plasma diffusion time - scales and so should be modelled dynamically , that is in the time - dependent , spatially varying , particle number continuity equations , along with the momentum and energy equations of plasma transport theory . illustrative results are shown in figure [ fig : fig1]a . the excited populations of impurities and the free electrons on the other hand may be assumed relaxed with respect to the instantaneous dominant populations , that is they are in a _ quasi - equilibrium_. the quasi - equilibrium is determined by local conditions of electron temperature and electron density . so , the atomic modelling may be partially de - coupled from the impurity transport problem into local calculations which provide quasi - equilibrium excited ion populations and emissivities and then effective source coefficients ( collisional - radiative coefficients ) for dominant populations which must be entered into the plasma transport equations . the solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re - associated with the local emissivity calculations for matching to and analysis of observations . for excited populations , @xmath20 plays a special and complicated role due to the very large variation in collisional excitation / de - excitation reaction rates with the quantum numbers of the participating states . in the low density coronal picture @xmath34 and redistribution plays no part . critical densities occur for @xmath35 and for @xmath36 and allow division of the ( in principle ) infinite number of excited populations into categories including _ low levels _ , _ high singly excited levels _ and _ doubly excited levels _ for which important simplifications are possible . these are examined in section [ sec : sec2 ] . light element ions in fusion plasmas are generally in the singly excited state redistibutive case , approaching the doubly excited redistribution case at the higher densities . highly ionised ions of heavy species in fusion plasmas approach the coronal picture . illustrative results on critical densities are shown in figure [ fig : fig1]b . finally , because of the generally short @xmath15 compared with other timescales ( including those of free - free and free - bound emission ) , it is usually the case that the free electrons have close to a maxwellian distribution . this assumption is made throughout the present paper , but is relaxed in the next paper of the series ( bryans , 2005 ) . the basic model was established by bates ( 1962 ) . the ion in a plasma is viewed as composed of a complete set of levels indexed by @xmath37 and @xmath38 and a set of radiative and collisional couplings between them denoted by @xmath39 ( an element of the _ collisional - radiative matrix _ representing transition from @xmath38 to @xmath37 ) to which are added direct ionisations from each level of the ion to the next ionisation stage ( coefficient @xmath40 ) and direct recombinations to each level of the ion from the next ionisation stage ( coefficient @xmath41 ) . thus , for each level , there is a total loss rate coefficient for its population number density , @xmath26 , given by @xmath42 following the discussion in the introduction , it is noted that populated metastable states can exist and there is no real distinction between them and ground states . we use the term _ metastables _ to denote both ground and metastables states . metastables are the dominant populations and so only recombination events which start with a metastable as a collision partner matter . we condider the population structure of the @xmath14-times ionised ion , called the recombined or child ion . the @xmath43-times ionised ion is called the recombining or parent ion and the @xmath44-times ionised ion is called the grandchild . the metastables of the recombined ion are indexed by @xmath45 and @xmath46 , those of the recombining ion by @xmath47 and @xmath48 and those of the grandchild by @xmath49 and @xmath50 . therefore the ion of charge state @xmath14 has metastable populations @xmath51 , the recombining ion of charge @xmath43 has metastable populations @xmath52 and the grandchild ion of charge @xmath44 has metastable populations @xmath53 . we designate the remaining excited states of the @xmath14-times ionised ion , with the metastables separated , as _ ordinary _ levels for which we reserve the indices @xmath37 and @xmath38 and populations @xmath26 and @xmath54 . there are then , for example , direct recombination coefficients @xmath55 from each parent metastable into each child ordinary level and direct ionisation coefficients from each child ordinary level to each parent metastable @xmath56 such that @xmath57 . also there are direct ionisation coefficients @xmath58to the metastables of the child from the metastables of the grandchild . then the continuity equations for population number densities are @xmath59 = \left[\begin{array}{llll } \scriptnew{c}_{\mu \mu ' } & n_e~\scriptnew{r}_{\mu,\sigma } & 0 & 0 \\ n_e\scriptnew{s}_{\rho \mu ' } & c_{\rho \sigma } & c_{\rho j } & n_e~r_{\rho \nu ' } \\ 0 & c_{i \sigma } & c_{i j } & n_e~r_{i \nu ' } \\ 0 & n_es_{\nu \sigma } & n_es_{\nu j } & \scriptnew{c}_{\nu \nu ' } \end{array } \right ] \left[\begin{array}{l } n^-_{\mu ' } \\ n_{\sigma } \\ n_{j } \\ n^+_{\nu ' } \end{array } \right ] \nonumber \\\end{aligned}\ ] ] where the equations for the @xmath44-times and @xmath43-times ionised ions have been simplified by incorporating their ordinary population contributions in their metastable contributions ( shown as script capital symbols ) as the immediate focus is on the @xmath14-times ionised ion . this incorporation procedure is shown explicitly in the following equations for the @xmath14-times ionised ion through to equations [ eqn : eqn16 ] and may be done for each ionisation stage separately . note additionally the assumption ( made by omission of the ( 3,1 ) partition element , where 3 denotes the row and 1 the column ) that state - selective ionisation from the stage @xmath44 takes place only into the metastable manifold of the stage @xmath14 . from the quasi - static assumption , we set @xmath60 and then the matrix equation for the ordinary levels of the @xmath14-times ionised ion gives @xmath61 where we have used summation convention on repeated indices . substitution in equations [ eqn : eqn10 ] for the metastables of the @xmath14-times ionised ion gives @xmath62n^-_{\mu ' } \nonumber \\ & & + \left [ c_{\rho \sigma}-c_{\rho j}c^{-1}_{ji}c_{i \sigma}\right ] n_{\sigma } \nonumber \\ & & + n_e\left [ r_{\rho \nu'}- c_{\rho j}c^{-1}_{ji}r_{i \nu'}\right]n^+_{\nu'}. \end{aligned}\ ] ] the left - hand - side is interpreted as a total derivative with time - dependent and convective parts and the right - hand - side comprises the source terms . the terms in square brackets in equations [ eqn : eqn12 ] give the effective growth rates of each metastable population of the @xmath14-times ionised ion driven by excitation ( or de - excitation ) from other metastables of the @xmath14-times ionised ion , by ionisation to the @xmath43-times ionised ion and excitation to other metastables of the @xmath14-times ionised ion ( a negatively signed growth ) and by recombination from the metastables of the @xmath43-times ionised ion . these are called the @xmath8 coefficients . following burgess and summers ( 1969 ) , who used the name ` collisional - dielectronic ' for ` collisional - radiative ' when dielectronic recombination is active , we use the nomenclature @xmath63 for the @xmath8 recombination coefficients which become @xmath64 the @xmath8 metastable cross - coupling coefficients ( for @xmath65 ) are @xmath66 /n_e.\ ] ] note that the on - diagonal element @xmath67 /n_e$ ] with @xmath68 is a total loss rate coefficient from the metastable @xmath45 . substitution from equation [ eqn : eqn11 ] in the equations [ eqn : eqn10 ] for the metastables of the @xmath69-times ionised ion gives @xmath70 n_{\sigma } \nonumber \\ & & + \left [ c_{\nu \nu'}- n_e^2s_{\nu j}c^{-1}_{ji}r_{i \nu'}\right]n^+_{\nu'}. \end{aligned}\ ] ] the @xmath8 ionisation coefficients resolved by initial and final metastable state are @xmath71\ ] ] and note that there is contribution to cross - coupling between parents via recombination to excited states of the @xmath14-times ionised ion followed by re - ionisation to a different metastable , @xmath72.\ ] ] consider the sub - matrix comprising the ( 2,2 ) , ( 2,3 ) , ( 3,2 ) and ( 3,3 ) partitions of equations [ eqn : eqn10 ] . introduce the inverse of this sub - matrix as @xmath73 = \left[\begin{array}{ll } c_{\rho \sigma } & c_{\rho j } \\ c_{i \sigma } & c_{i j } \\ \end{array } \right]^{-1}\end{aligned}\ ] ] and note that the inverse of the ( 1,1 ) partition @xmath74^{-1 } \equiv \scriptnew{c}_{\rho \sigma}= \left [ c_{\rho \sigma}-c_{\rho j}c^{-1}_{ji}c_{i \sigma}\right].\ ] ] this compact representation illustrates , that the imposition of the quasi - static assumption leading to elimination of the ordinary level populations in favour of the metastable populations , may be viewed as a _ condensation _ in which the influence of the ordinary levels is _ projected _ onto the metastable levels . the metastables can be condensed in a similar manner onto the ground restoring the original ( ground states only ) collisional - radiative picture . the additive character of the direct metastable couplings @xmath75 means that these elements may be adjusted retrospectively after the main condensations . there are two kinds of derived coefficients associated with individual spectrum line emission in common use in fusion plasma diagnosis . these are _ photon emissivity coefficients _ ( @xmath76 ) and _ ionisation per photon ratio _ ( @xmath77 ) . the reciprocals of the latter are also known as _ photon efficiencies_. from equations [ eqn : eqn11 ] , the emissivity in the spectrum line @xmath78 may be written as @xmath79 this allows specification of the _ excitation _ photon emissivity coefficient @xmath80 and the _ recombination _ photon emissivity coefficient @xmath81 the ionisation per photon ratios are most meaningful for the excitation part of the emissivity and are @xmath82 each of these coefficients is associated with a particular metastable @xmath46 , @xmath83 or @xmath84 of the @xmath85 , @xmath86 or @xmath87 ions respectively . the radiated power in a similar manner separates into parts driven by excitation and by recombination as @xmath88 called the _ low - level line power coefficient _ and @xmath89 called the _ recombination - bremsstrahlung power coefficient _ where it is convenient to include bremsstrahlung with @xmath90 . note that in the generalised picture , additional power for the @xmath14-times ionised ions occurs in forbidden transitions between metastables as @xmath91 for the @xmath14-times ionised ion . in the fusion context , this is usually small . radiated power is the most relevant quantity for experimental detection . for modelling , it is the _ electron energy loss function _ which enters the fluid energy equation . the contribution to the total electron energy loss rate for the @xmath14-times ionised ion associated with ionisation and recombination from the @xmath43-times ionised ion is @xmath92 where the @xmath93 and @xmath94 are absolute energies of the metastables @xmath45 of the @xmath14-times ionised ion and of the metastables @xmath47 of the @xmath43-times ionised ion respectively . thus the electron energy loss is a derived quantity from the radiative power coefficients and the other generalised collisonal - radiative coefficients . note that cancellations in the summations cause the reduction to relative energies and that in ionisation equilibrium , the electron energy loss equals the radiative power loss . in the following sections , it is shown how the quasi - static assumption and population categorisation allow us to solve the infinite level population structure of each ion in a manageable and efficient way . such solution is necessary for low and medium density astrophysical and magnetic confinement fusion plasmas . this is unlike the simpler situation of very dense plasmas where heavy level truncation is used , because of continuum merging . the handling of metastables in a generalised collisonal - radiative framework requires a detailed specific classification of level structure compatible with both recombining and recombined ions . for light element ions , russell - saunders ( l - s ) coupling is appropriate and it is sufficient to consider only terms , since although fine structure energy separations may be required for high resolution spectroscopy , relative populations of levels of a term are close to statistical . so the parent ion metastables are of the form @xmath95 with @xmath96 the configuration and the recombined ion metastables are of the form @xmath97 with the configuration @xmath98 and the excited ( including highly excited ) terms are @xmath99 . @xmath100 and @xmath101 denote individual electron principal quantum number and orbital angular momentum , @xmath102 and @xmath103 denote total orbital and total spin angular momenta of the multi - electron ion respectively in the specification of an ion state in russell - saunders coupling . the configuration specifies the orbital occupancies of the ion state . for ions of heavy elements , relative populations of fine structure levels can differ markedly from statistical and it is necessary to work in intermediate coupling with parent metastables of the form @xmath104 and recombined metastables of the form @xmath105 with the configuration @xmath98 and the excited ( including highly excited ) levels @xmath106 . there is a problem . to cope with the very many principal quantum shells participating in the calculations of collisional - dielectronic coefficients at finite density necessitates a grosser viewpoint ( in which populations are _ bundled _ ) , whereas for modelling detailed spectral line emission , the finer viewpoint ( in which populations are fully _ resolved _ ) is required . in practice , each ion tends to have a limited set of low levels principally responsible for the dominant spectrum line power emission for which a bundled approach is too imprecise , that is , averaged energies , oscillator strengths and collision strengths do not provide a good representation . note also that key parent transitions for dielectronic recombination span a few ( generally the same ) low levels for which precise atomic data are necessary . in the recombined ion , parentage gives approximate quantum numbers , that is , levels of the same @xmath100 ( and @xmath101 ) divide into those based on different parents . lifetimes of levels of the same @xmath100 but different parents can vary strongly ( for example through secondary autoionisation ) . also the recombination population of such levels is generally from the parent with which they are classified . we therefore recognise three sets of non - exclusive levels of the recombined ion : * metastable levels - indexed by @xmath45 , @xmath46 . * low levels - indexed by @xmath37 , @xmath38 in a resolved coupling scheme , being the complete set of levels of a principal quantum shell range @xmath107 , including relevant metastables and spanning transitions contributing substantially to radiative power or of interest for specific observations . * bundled levels - segregated according to the parent metastable upon which they are built and possibly also by spin system - which can include _ bundle - nl _ and _ bundle - n_. viewed as a recombining ion , the set ( i ) must include relevant parents and set ( ii ) must span transitions which are dielectronic parent transitions . time dependence matters only for the populations of ( i ) , high precision matters only for groups ( i ) and ( ii ) and special very many level handling techniques matter only for group ( iii ) . to satisfy the various requirements and to allow linking of population sets at different resolutions , a series of manipulations on the collisional - radiative matrices are performed ( summers and hooper , 1983 ) . to illustrate this , suppose there is a single parent metastable state . consider the collisional - radiative matrix for the recombined ion and the right hand side in the bundle - n picture , and a partition of the populations as @xmath108 $ ] with @xmath107 and @xmath109 . elimination of the @xmath110 yields a set of equations for the @xmath111 . we call this a ` condensation ' of the whole set of populations onto the @xmath100 populations . the coefficients are the effective ionisation coefficients from the @xmath100 , the effective cross - coupling coefficients between the @xmath100 and the effective recombination coefficients into the @xmath100 , which now include direct parts and indirect parts through the levels @xmath112 . exclusion of the direct terms prior to the manipulations yields only the the indirect parts . call these @xmath113 and @xmath114 . we make the assumption that @xmath113 and @xmath114 may be expanded over the resolved low level set ( see section [ sec : sec2.2 ] ) to give the expanded indirect matrix @xmath115 and @xmath116 where @xmath37 and @xmath38 span the resolved low level set ( ij ) . these indirect couplings are then combined with higher precision direct couplings @xmath117 and @xmath118 so that @xmath119 and @xmath120 the procedure is shown schematically in figure [ fig : fig2 ] . the process may be continued , condensing the low level set onto the metastable set . the generalised collisional - dielectronic coefficients are the result . the time dependent and/or spatial non - equilibrium transport equations which describe the evolution of the ground and metastable populations of ions in a plasma use these generalised coefficients . following solution , the condensations can be reversed to recover the complete set of excited populations and hence any required spectral emission . the progressive condensation described above can be viewed as simply one of a number of possible paths which might be preferred because of special physical conditions or observations . for light element ions , four types of bundling and condensation are distinguished in this work : * ground parent , spin summed bundle - n @xmath121 lowest n - shell . * parent and spin separated bundle - n @xmath121 lowest spin system n - shell ( the _ bundle - ns population model _ ) . * low ls resolved @xmath121 metastable states ( the _ low - level population model _ ) . * parent and spin separated bundle - n @xmath121 low ls resolved @xmath121 metastable states . type ( a ) corresponds to the approach used in summers ( 1974 ) . type ( c ) corresponds to the usual population calculation for low levels in which ( consistent ) recombination and ionisation involving excited states are ignored . it establishes the dependence of each population on excitation for the various metastables only . type ( d ) , effectively the merging of ( b ) and ( c ) in the manner described earlier , is the principal procedure to be exploited in this work for first quality studies . details are in the following sub - sections . .[table : tab1]bundle-@xmath122 calculation pathways . the parent / spin system weight factor is defined in equation [ eqn : eqn32 ] and [ eqn : eqn33 ] . @xmath123 indicates the number of metastables of the recombined ion associated with the parent / spin system . the _ parent index _ , sequentially numbering the different parents shown in brackets , is used as the reference for tabulation of coefficients . [ cols="^,^,^,^,^,^ " , ] figure [ fig : fig5 ] illustrates the behaviour of low level populations . the graph is of the parameter @xmath124 from equation [ eqn : eqn20 ] . energy levels for high bundle-@xmath122 levels and their a - values , maxwell averaged collision strengths , radiative recombination coefficients , dielectronic recombination coefficients and ionisation coefficients are generated from a range of parametric formulae and approximations described in earlier works ( burgess and summers , 1976 ; summers , 1977 ; summers and hooper , 1983 ; burgess and summers , 1987 ; summers , 2004 ) . high quality specific data when available are acccessed from archives and substituted for the default values . this is a systematic procedure for dielectronic coefficients ( see sections [ sec : sec2.3.1 ] and [ sec : sec5 ] below ) . for low levels , a complete basis of intermediate coupling energy level , a - value and scaled born approximation collision data , spanning the principal quantum shell range @xmath125 is generated automatically using the cowan ( 1981 ) or autostructure ( badnell , 1986 ) procedures . this is called our _ baseline _ calculation . these data are merged with more restricted ( in level coverage ) but similarly organised highest quality data from archives where available ( e.g. r - matrix data such as ramsbottom ( 1995 ) ) . the data collection is compressed by appropriate summing and averaging to form a complete ls term basis and augmented with comprehensive high quality ls resolved dielectronic recombination , radiative recombination and collisional ionisation coefficients mapped from archives ( see sections [ sec : sec2.3.1 ] , [ sec : sec2.3.2 ] and [ sec : sec5 ] below ) . the radiative data has its origin in the work of burgess and summers ( 1987 ) . the final data collection is called a _ specific ion file _ ( adas data format _ adf04 _ ) . the detailed content is examined in section [ sec : sec3 ] . details of state selective dielectronic recombination and ionisation coefficients are given in the following sub - sections . state selective dielectronic recombination coefficients are required to all resolved low levels and to all bundle-@xmath122 shells for the various initial and intermediate state metastable parents @xmath47 for @xmath8 modelling . these are very extensive data and have been prepared for the present @xmath8 work through an associated international ` dr project ' summarised in badnell ( 2003 ) [ hereafter called _ dr - paper i _ ] and detailed in subsequent papers of the series . based on the independent particle , isolated resonance , distorted wave ( ipirdw ) approximation , the partial dielectronic recombination rate coefficient @xmath126 from an initial metastable state @xmath47 into a resolved final state @xmath37 of an ion @xmath127 is given by @xmath128 where @xmath129 is the statistical weight of the @xmath130-electron doubly - excited resonance state @xmath38 , @xmath131 is the statistical weight of the @xmath132-electron target state and the autoionization ( @xmath133 ) and radiative ( @xmath134 ) rates are in inverse seconds . the suffix @xmath135 is used here to denote a parent ion state . @xmath136 is the energy of the continuum electron ( with angular momentum @xmath101 ) , which is fixed by the position of the resonances . note that the parent states @xmath135 are excited , that is they exclude the metastable parents @xmath83 . the code autostructure ( badnell , 1986 ; badnell and pindzola , 1989 ; badnell , 1997 ) was used to calculate multi - configuration ls and intermediate coupling energy levels and rates within the ipirdw approximation . the code makes use both of non - relativistic and semi - relativistic wavefunctions ( pindzola and badnell , 1990 ) and is efficient and accurate for both the resolved low level and high-@xmath100 shell problems . lookup tables ( see section [ sec : sec5 ] ; _ adf09 _ ) are prepared comprising state selective recombination coefficients at a standard set of @xmath14-scaled temperatures , for each metastable parent , to all ls resolved terms of the recombined ion with valence electron up to @xmath100-shell @xmath137 ( normally @xmath138 ) and to bundle-@xmath122 levels of a representative set of @xmath100-shells ( usually spanning @xmath139 to 999 ) . these bundle-@xmath122 coefficients are simple sums over orbital states @xmath140 and so apply at zero density . this provides an extensive , but still economical , tabulation . in _ dr paper - i _ , we introduced an associated code , the ` burgess bethe general program ' bbgp . bbgp is used here as a support function in a model for the @xmath101-redistribution of doubly - excited states which provide a correction to the accurate , but unredistributed , dielectronic data . the redistributed data ( regenerated in the same _ adf09 _ format ) are normalised to ipirdw at zero density . the procedure is similar to that for singly excited systems . for @xmath141-averaged levels , the number densities expressed in terms of their deviations , @xmath142 from saha - boltzmann , and referred to the initial parent @xmath83 , are given by @xmath143^{3/2 } \frac{\omega_{p , nl}}{\omega_{\nu'}}e^{-e / kt_\rme}b_{p , nl}\ , . \label{eq22}\end{aligned}\ ] ] thus , in the bbgp zero - density limit , with only resonant capture from the @xmath47 parent balanced by auger breakup and radiative stabilization back to the same parent , we have @xmath144 in the extended bbgp model , we include resonant - capture from initial metastables other than the ground , dipole - allowed collisional redistribution between adjacent doubly - excited @xmath101-substates of the same @xmath100 by secondary ion- and electron - impact , and losses by ` alternate ' auger break - up and parent radiative transition pathways . the population equations for the @xmath101-substates of a doubly - excited @xmath100-shell become @xmath145 @xmath146 denotes resonance capture coefficients . @xmath147 denotes the number of parent metastables which are the starting point for resonance capture whereas @xmath148 denotes the number of true excited parent states . ion impact redistributive collisions are effective and are represented in the equations as @xmath149 ion contributions . the density corrected bundle-@xmath122 recombination coefficients are then @xmath150 details are given in _ dr paper - i_. figure [ fig : fig6 ] shows the effects of redistribution of the doubly excited @xmath142 factors in the recombination of @xmath151 . autoionisation rate calculations in ls coupling exclude breakup of non - ground metastable parent based ` singly excited ' systems by spin change . such breakup only occurs in intermediate coupling but can not be ignored for a correct population assessment even in light element ions . these rates are computed in separate intermediate coupling dielectronic calculations . these rates are main contributors to the parent metastable cross - coupling identified earlier ( cf . the @xmath152 $ ] intermediate states in figure [ fig : fig4]b leading back to the @xmath153 parent ) . direct ionisation coefficients for excited n - shells in bundle-@xmath122 population modelling are evaluated in the exchange - classical - impact parameter ( ecip ) approximation ( burgess and summers , 1976 ) . the method which merges a symmetrised classical binary encounter model with an impact parameter model for distant encounters is relatively simple and of moderate precision , but has a demonstrated consistency and reliability for excited n - shells in comparison with more eleborate methods . there is a higher precision requirement for state selective ionisation from metastable and low levels . such ionisation includes direct and excitation / autoionisation parts , but the latter only through true excited parents . stepwise ionisation is handled in the collisional - radiative population models . there are extensive ionisation cross - section measurements , but these are in general unresolved and so are of value principally for renormalising of resolved theoretical methods . the most powerful theoretical methods used for excitation , namely r - matrix with pseudostates ( rmps ) ( bartschat and bray , 1996 ; ballance , 2003 ) and convergent - close - coupling ( ccc ) ( bray and stelbovics , 1993 ) , have some capability for ionisation , but at this stage are limited to very few electron systems as is the time - dependent - close - coupling ( tdcc ) ( pindzola and robicheaux , 1996 ; colgan , 2003 ) ionisation method . for @xmath8 calculations , we have relied on the procedures of summers and hooper ( 1983 ) for initial and final state resolution of total ionisation rate coefficients and on the distorted wave approximation . the distorted wave method is our main method for extended @xmath8 studies for many elements and most stages of ionisation . rmps and tdcc studies are directed mostly at the neutral and near neutral systems where the distorted wave method is least reliable ( loch , 2005 ) . we use the configuration average distorted ( _ cadw _ ) wave approach of pindzola , griffin and bottcher ( 1986 ) . it has reasonable economy of computation while allowing access to complex , multi - electron ions , highly excited states , excitation / autoionisation and radiation damping . it expresses the configuration averaged excitation cross - section for @xmath154 as @xmath155 and the configuration averaged ionisation cross - section for @xmath156 as @xmath157 where @xmath158 are squared two - body coulomb matrix elements , @xmath159 denotes the average cross - section and @xmath160 , @xmath161 and @xmath162 denote average initial , final and ejected electron momenta respectively in the configuration average picture . the configuration average direct ionisation cross - sections and rate coefficients may be unbundled back to resolved form using angular factors obtained by sampson and zhang ( 1988 ) . note that the excitations described here are to auto - ionising levels and so resolved auger yields may be used which are the same as those in the dielectronic calculations of section [ sec : sec2.3.1 ] above . in fact the ionisation coefficient calculation results are structured and archived in adas ( format _ adf23 _ ) in a manner very similar to state selective dielectronic recombination . extensive studies have been carried out on light and heavy elements ( colgan , 2003 ; loch , 2003 ) . for bundle-@xmath122 modelling , the expected fundamental data precision is @xmath163 30% for excitation and ionising collisional rate coefficients , @xmath163 10% for a - values and state selective recombination coefficients and @xmath163 1% for energies . cccccccc ion & e & a & & @xmath164 & @xmath165 & @xmath103 + & . & & @xmath166 & @xmath167 & & & + @xmath168 & a & b & b & b , c & b & b & b + @xmath169 & a & b & b & b , c & b & b & b + @xmath170 & a & b & a , b & b , c & b & b & b + @xmath171 & a & a , b & b & b , c & b & b & b + @xmath172 & a & a & b & b , c & b & b & b + @xmath173 & a & a & b & b & b & b & b + @xmath174 & a & b , c & b & b & b & b & b + @xmath175 & a & b , c & b & b & b & b & b + @xmath176 & a & b , c & a , b & b & b & b & b + @xmath177 & a & b & a & a & b & b & b + @xmath178 & a & b & a & a & b & b & b + @xmath179 & a & b & b & b & b & b & b + @xmath180 & a & a & b & b , c & b & b & b + @xmath181 & a & a & b & b , c & b & b & b + @xmath182 & a & b , c & b & c & b & b & b + @xmath183 & a & c & b & b & b & b & b + @xmath184 & a & b & a & b & b & b & b + @xmath185 & a & b & a & b & b & b & b + @xmath186 & a & b & a & b & b & b & b + @xmath187 & a & b & a & b & b & b & b + @xmath188 & a & b & a & b & b & b & b + @xmath189 & a & b & b & b & b & b & b + @xmath190 & a & a & b & c & b & b & b + @xmath191 & a & a & b & c & b & b & b + for low level modelling many sources are used . a rating is given for the classes of fundamental data for the ions of carbon , oxygen and neon in table [ table : tab4 ] which is based on the following considerations . for energy levels , categories are _ a _ spectroscopic , _ b _ @xmath163 0.5% and _ c _ @xmath163 1.0% . c _ is anticipated from ab initio multi - configuration structure calculations , _ b _ from such calculations with extended optimising and _ a _ reflects direct inclusion of experimental energies from reference sources . for a - values , categories are _ a _ @xmath163 5% , _ b _ @xmath163 10% and _ c _ @xmath163 25% . c _ is anticipated from our baseline calculations , _ b _ from optimised multi - configuration structure calculations with extended optimising and _ a _ from specific studies in the literature . for electron impact maxwell averaged collision strengths , @xmath192 , the categories are _ a _ @xmath163 10% , _ b _ @xmath163 20% and _ c _ @xmath163 35% . c _ is from our baseline calculations , _ b _ from distorted wave calculations and _ a _ from specific r - matrix calculations , equivalent methods or experiment . for radiative recombination , the categories are _ a _ @xmath163 5% , _ b _ @xmath163 10% and _ c _ @xmath163 50% . c _ is from scaled methods using hydrogenic matrix elements , _ b _ from distorted wave one - electron wave functions in an optimised potential using spectroscopic energies and _ a _ from specific r - matrix calculations and experiment . b _ is the baseline in adas . for dielectronic recombination , the categories are _ a _ @xmath163 20% , _ b _ @xmath163 30% and _ c _ @xmath163 45% . c _ is from bbgp approximations , _ b _ from ls - coupled calculations using autostructure from the dr project and _ a _ from ic - coupled calculations using autostructure with parent and lowest resonance energy level adjustments from the dr project . b _ is the baseline in adas . it is to be noted that the theoretical relative precision which is consistent with the variation between the three categories is 15% better but dielectronic recombination comparisons with experiment still show unexplained discrepancies at the 20% level so the present categories are safe . for ionisation , the categories are _ a _ @xmath163 10% , _ b _ @xmath163 25% and _ c _ @xmath163 40% . c _ is from ecip approximations , _ b _ from configuration average with angular factor term resolution and _ a _ from rmps , tdcc calculations and experiment . b _ is the baseline in adas . as indicated in section [ sec : sec2.3 ] , these various data are assembled in an _ adf04 _ file which is sufficient to support the primary low - level population calculation . the tabulations are at a set of temperatures arising from a fixed set of z - scaled temperatures ( see section [ sec : sec2.1.1 ] ) which spans the full range to asymptotic regions of reaction data . collision data are converted to these ranges using _ c - plots _ ( burgess and tully ( 1992 ) and this procedure also flags data errors or queries . the precision of the specific ion file determines the achievable precision of all derived populations , emissivities and collisional - radiative coefficients ( whiteford , 2005 ; omullane , 2005 ) . the assembling of data in the _ adf04 _ is systematic and orderly . a comment section at the end of the file details the assembly steps , implementer , codes used and dates . this includes baseline and supplementation files , merging , ls compression , dielectronic recombination data inclusion etc . also there is extended detail of orginal data sources and a history of updates . a given _ adf04 _ file represents a ` snapshot ' of the state of available knowledge at the time . it is subject to periodic review and adas codes ( see section [ sec : sec5 ] ) are designed to enable easy reprocessing of all derived data following fundamental data update . the grading for each ion given in table [ table : tab4 ] is justified and supported by the comments from its _ adf04 _ file . a summary from @xmath193 is given in the following paragraphs in illustration . for @xmath188 , the low levels span 44 terms , including up to the @xmath194 shell built on the parent @xmath195 and @xmath196 shell built on the @xmath197 . only the @xmath195 parent is treated as a metastable from the @xmath8 point of view . the intermediate coupled baseline data set is _copmm@xmath19810-ic@xmath198ne6.dat_ with preferred supplementary energy a - value and @xmath192 data merged from _ copjl#be - ic@xmath198ne6.dat_. ionisation potentials and energy levels are from the nist standard reference database apart from @xmath199 ( kelly , 1987 ) and @xmath200 levels by ( ramsbottom , 1995 ) . the categorisation is _ a_. a - values were drawn from the opacity project ( opacity project team , 1996 ; tully , 1991 ) as justified ( for n iv and o v ) by wiese , fuhr and deters ( 1996 ) and supported / adjusted by @xmath163 3% by fleming ( 1996a , b ) , jnsson ( 1998 ) , froese fischer , godefeid and olsen ( 1997 ) , froese fischer , gaigalas and godefried(1997 ) , nussbaumer and storey ( 1979 ) , sampson , goett and clark ( 1984 ) and ramsbottom ( 1995 ) . b _ is safe . @xmath192s are taken from ramsbottom ( 1995 ) . these are for a 26 ls eigenstate multichannel r - matrix calculation . the categories assigned are _ a _ and _ b_. radiative and dielectronic recombination and ionisation all follow the baselines in adas , that is categories _ b _ although the summed and averaged ionisation rate coefficient ( over metastables ) is normalised to experiment . figure [ fig : fig7 ] shows @xmath201 coefficients for the c ii 858 spectrum line . the coefficients depend on both electron temperature and electron density in general . a common practice in spectral analysis is to observe principally the strongest resonance line of an ion . such emission is driven largely from the ground state ( figure [ fig : fig7]a ) and because of the large a - value , the density sensitivity occurs at relatively high density . thus such resonance line emission at moderate to low densities mostly reflects temperature and the distribution of the ionisation stages . comparison in near equilibrium ionisation balance plasmas of line ratios from the same ion , by contrast , is mostly directed at electron density and relies on the presence of metastables and spin changing collisional processes to confer the sensitivity . as shown earlier , the balance of the dominant ground and metastable populations is disturbed in dynamic plasmas and so density sensitivity may be modified by the dynamic state . the distinction of the metastable driven @xmath201 in @xmath8 modelling , as illustrated in figure [ fig : fig7]b , allows more complete diagnostic study and the possibility of separation of the two effects . in strongly recombining plasmas ( most commonly photo - ionised astrophysical plasmas ) the direct contribution of recombination to the emission may dominate the excitation part . the @xmath202 , as illustrated in figure [ fig : fig7]c , is then required . in the fusion context , multi - chordal spectral observations are important for the study of impurity transport especially near sources . visible and quartz uv observations are convenient and this places a requirement for @xmath201s from higher quantum shells . it is this primarily which defines the span of our _ low - levels _ for population modelling . emission from higher @xmath100-shells is significantly affected by cascading from yet higher levels . the full machinery of projection as described in section [ sec : sec2.1 ] is necessary for our global ambition of 20% precision for emissivity coefficients . figure [ fig : fig8 ] illustrates the @xmath8 recombination coefficients . at low electron density , radiative and dielectronic recombination dominate . for capture from metastables , alternate auger branching can largely suppress the dielectronic part of the surfaces . thus figure [ fig : fig8]a shows the characteristic exponential rise at the temperature for excitation of the main parent ( core ) transition of dielectronic recombination and then the subsequent fall - off ( as @xmath203 ) , in contrast with [ fig : fig8]c . at moderate densities , suppression of the high @xmath100-shell populations , principally populated by dielectronic recombination , through re - ionisation occurs and the coefficient falls in the dielectronic recombination region . at very high density , three - body recombination becomes effective , preferentially beginning at the lower electron temperatures . it is evident that recombination is less effective from the metastable at relevant ionisation balance temperatures . models which ignore the role of capture from the metastable parent ( which may be the dominant population ) can lead to substantial errors in recombination coefficients , while simply excluding all capture from the metastable can not deliver the precision sought for current modelling . for light elements in astrophysical plasmas , the zero - density coronal assumption for the recombination coefficient is still frequently made . this can not be justified even at solar coronal densities . figure [ fig : fig9 ] illustrates the @xmath8 ionisation coefficients . at low electron density , the coefficient is dominated by direct ionisation , including excitation / auto - ionisation , from the driving metastable . relatively high electron densities are required before the stepwise contribution begins . it is primarily the excitation to , and then further excitation and ionisation from , the first excited levels which controls this . the ground and metastable resolved coefficients both show the same broad behaviour as the usual ( stage to stage ) collisonal - radiative coefficient , tending to a finite limit at very high density . it is to be noted that the @xmath204 coefficients are required to be able to construct a meaningful stage to stage collisional - radiative ionisation coefficient from the generalised progenitors . it remains the case that most plasma modelling ( certainly in the fusion area ) is not adjusted to the use of the generalised coefficients as source terms . reconstruction of stage to stage source terms ( at the price of a reduction in modelling accuracy ) from the generalised coefficients is still is a requirement and is addressed more fully in section [ sec : sec5 ] . the generalised coefficients may be used to establish the equilibrium ionisation balance for an element in which the dominant ground and metastable populations are distinguished , that is the fractional abundances @xmath205}}{n^{[tot ] } } \right ) : \sigma=1,\cdots , m_z ; z=0 , \cdots , z_0\ ] ] where @xmath206 is the number of metastables for ionisation stage @xmath14 and @xmath207}=\sum_{z=0}^{z_0 } n^{[z]}=\sum_{z=0}^{z_0}\sum_{\sigma=0}^{m_z } n_{\sigma}^{[z]}\ ] ] in equilibrium . writing @xmath208}$ ] for the vector of populations @xmath209}$ ] , the equilibrium population fractions are obtained from solution of the matrix equations . @xmath210 } & n_e~\underline{\scriptnew{r}}^{[1 \rightarrow 0 ] } & 0 & . \\ n_e\underline{\scriptnew{s}}^{[0 \rightarrow 1 ] } & \underline{\scriptnew{c}}^{[1,1 ] } & n_e~\underline{\scriptnew{r}}^{[2 \rightarrow 1 ] } & . \\ 0 & n_e\underline{\scriptnew{s}}^{[1 \rightarrow 2 ] } & \underline{\scriptnew{c}}^{[2,2 ] } & . \\ 0 & 0 & n_e\underline{\scriptnew{s}}^{[2 \rightarrow 3 ] } & . \\ . & . \end{array } \right ] \left[\begin{array}{l } \underline{n}^{[0 ] } \\ \underline{n}^{[1 ] } \\ \underline{n}^{[2 ] } \\ \underline{n}^{[3 ] } \\ . \end{array } \right]_{equil}=0.\end{aligned}\ ] ] these in turn may be combined with the @xmath211 and @xmath90 to obtain the equilibrium radiated power loss function for the element as @xmath212 } & = & \sum_{z=0}^{z_0 } p^{[z]}\left ( \frac{n^{[z]}}{n^{[tot ] } } \right ) _ { equil } \nonumber \\ & = & \sum_{z=0}^{z_0}\sum_{\sigma=0}^{m_z } ( \scriptnew{plt}^{[z]}_{\sigma}+\scriptnew{prb}^{[z]}_{\sigma})\left ( \frac{n^{[z]}_{\sigma}}{n^{[tot]}}\right ) _ { equil}.\end{aligned}\ ] ] the equilibrium fractional abundances and equilibrium radiated power function are illustrated in figure [ fig : fig10 ] . it is useful at this point to draw attention to emission functions which combine emission coefficients with equilibrium fractional abundances . they are commonly used in differential emission measure analysis of the solar atmosphere ( lanzafame , 2002 ) where they are called @xmath213 functions . in solar astrophysics , it is assumed that the @xmath213 are functions of the single parameter @xmath29 ( either from a zero - density coronal approximation or by specification at fixed density or pressure ) and usually the abundance of hydrogen relative to electrons @xmath214 is incorporated in the definition . for finite density plasmas , in the generalised collisonal - radiative picture , we define @xmath215 functions , parameterised by @xmath29 and @xmath27 as @xmath216}_{j \rightarrow k } & = & \sum_{\sigma}\scriptnew{pec}^{[z](exc)}_{\sigma , j \rightarrow k}\left ( \frac{n_{\sigma}^{[z]}}{n^{[tot ] } } \right ) _ { equil } \nonumber \\ & & + \sum_{\nu'}\scriptnew{pec}^{[z](rec)}_{\nu',j \rightarrow k}\left ( \frac{n_{\nu'}^{[z+1]}}{n^{[tot ] } } \right ) _ { equil}. \end{aligned}\ ] ] these are strongly peaked functions in @xmath29 . the precision of the present modelling and data , including the full density dependence , is in principle sufficient to allow bivariate differential emission measure analysis ( judge , 1997 ) although this remains to be carried out . the general behaviour of a @xmath215 function is illustrated in figure [ fig : fig10]c . the organisation of the main calculations and data flow is shown in figure [ fig : fig11 ] . the calculations executed for the paper have been implemented in general purpose codes and attention has been given to the precise specification of all data sets and the machinery for accessing and manipulating them . this includes initial data , intermediate and driver data as well as the final @xmath8 products and follows adas project practice . population structure modelling requires an initial input dataset of energy levels , transition probabilities and collisional rates of adas data format _ adf04 _ which is complete for an appropriate designated set of low levels . for the light elements , best available data were assembled and verified as described in section [ sec : sec3 ] . these data are substitutes for more moderate quality , but complete , _ baseline _ data prepared automatically ( see _ gcr - paper iii _ ) . the _ adf04 _ files are required for ls - terms . in practice , we find it most suitable to prepare the data for lsj levels and then bundle back to terms . for @xmath8 modelling , state selective recombination and ionisation coefficients must be added to form the fully specified _ the ls - coupled dielectronic coefficients are mapped in from very large comprehensive tabulations of data format _ _ prepared as part of the _ dr project _ ( badnell , 2003 ) . the resulting _ adf04 _ files are the adas preferred data sets and are available for elements helium to neon . the two population codes , called adas204 and adas208 , work together . adas204 is the bundle-@xmath122 model . data on the metastable parent structure , quantum defects , auto - ionisation thresholds and autoionsation rates are required . these data may be extracted from the _ adf04 _ files and it has been helpful to make the driver dataset preparation automatic . high quality shell selective dielectronic data are essential and this is part of the provision in the _ adf09 _ files described in the previous paragraph . adas204 provides complete population solutions , but extracts from these solutions the condensed influence on the low n - shells , as _ projection matrices _ , for connection with the calculations of adas208 . it is to be noted that the main on - going development is refinement of atomic collision rates between the key low levels . the projection matrices are not subject to frequent change and so are suitable for long - term archiving ( _ adf17 _ ) . adas208 is the low - level resolved population model which delivers the final data for application . it draws its key data from the fully configured _ _ file and supplements these with projection data . the evaluation of the population structure takes place at an extended set of @xmath14-scaled electron temperatures and densities ( see section [ sec : sec2 ] ) and this means that the resulting @xmath8 coefficient data are suited to interpolation along iso - electronic sequences . thus the initial tabulation of @xmath8 coefficients is in iso - electronic datasets . it is convenient to implement the gathering and mapping from iso - electronic to iso - nuclear in a separate step , which also supports the merging back to the unresolved stage - to - stage picture if required . the production of @xmath217 and @xmath218 coefficients is directly to iso - nuclear oriented collections . the number of @xmath217s from the population calculations can , in principle , be very large . we restrict these by a threshold magnitude and to particular important spectral regions . it is straightforward to rerun adas208 to generate @xmath217 or @xmath218 coefficients alone in spectral intervals of one s choice . the separation of the adas204 and adas208 tasks and the ease of modifying data within an _ adf04 _ file means that ` what - if ' studies on the sensitivity of the derived data to fundamental data uncertainly can readily carried out . special adas codes enable detailed study of cumulative error and dominating sources of uncertainty such that error surfaces , for example for a @xmath217 as a function of electron temperature and density , may be generated . such error ( uncertainty ) analysis for theoretical derived coefficients and its utilisation in the confrontation with diagnostic experiments is the subject of a separate work ( see omullane , 2005 ) . the requirements for precise modelling of spectral emission and the relating of ionisation stages in thermal plasmas have been considered . collisional - radiative methodologies have been developed and extended to enable the full role of metastables to be realised , so that this generalised ( @xmath8 ) picture applies to most dynamically evolving plasmas occurring in magnetic confinement fusion and astrophysics . the procedures are valid up to high densities . the studies presented in the paper explore the density effects in detail within the @xmath8 picture and show that the density dependencies of excited ion populations and of effective rate coefficients can not be ignored . specific results are presented for light elements up to neon and the computations are carried out in an atomic basis of terms ( ls - coupled ) . such modelling will remain sufficient up to about the element argon , beyond which a level basis ( intermediate coupling ) becomes necessary . heavier elements will be examined in further papers of this series . considerable attention has been given to the generation and assembly of high quality fundamental data in support of the @xmath8 modelling . also datasets of fundamental and derived data have been specified precisely and codes have been organised following the principles of the adas project . the product of the study is the preferred adas data for the light element ions at this time . the atomic data and analysis structure , adas , was originally developed at jet joint undertaking . bates d r , kingston a e and mcwhirter r w p 1962 _ proc . soc . a _ * 267 * 297 .

the paper presents an integrated view of the population structure and its role in establishing the ionisation state of light elements in dynamic , finite density , laboratory and astrophysical plasmas . there are four main issues , the generalised collisional - radiative picture for metastables in dynamic plasmas with maxwellian free electrons and its particularising to light elements , the methods of bundling and projection for manipulating the population equations , the systematic production / use of state selective fundamental collision data in the metastable resolved picture to all levels for collisonal - radiative modelling and the delivery of appropriate derived coefficients for experiment analysis . the ions of carbon , oxygen and neon are used in illustration . the practical implementation of the methods described here is part of the adas project .

hii galaxies are local , dwarf starburst systems ( e.g. , * ? ? ? * ; * ? ? ? * ) , which show low metallicity [ 1/50 @xmath0 z / z@xmath1 @xmath0 1/3](e.g . * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? wolf - rayet ( wr ) signatures ( commonly a broad feature at @xmath2 4680 or blue bump ) , indicating the presence of wr stars , have been found in the spectra of some hii galaxies ( e.g. , * ? ? ? * ; * ? ? ? this is an important observational fact since according to recent stellar evolution models for single rotating / non - rotating massive stars , hardly any wrs are expected in metal - poor environments @xcite . studying the wr content in hii galaxies is crucial to test stellar evolutionary models at low metallicities . we have initiated a program to investigate hii galaxies with wr features using integral field spectroscopy ( ifs ; e.g. , @xcite ) . so far , we have observed 15 wr galaxies with the optical ifus : pmas at the 3.5 m telescope at caha and integral at the 4.2 m wht in orm . ifs has many benefits in a study of this kind , in comparison with long - slit spectroscopy . using ifs one can locate and find wrs where they were not detected before , not only because it samples a larger area of the galaxy , but also because ifs can increase the contrast of the wr bump emission against the galaxy continuum , thus minimizing the wr bump dilution . also , ifs is a powerful technique to probe issues related with aperture effects , and allows a more precise spatial correlation between massive stars and nebular properties ( e.g. , * ? ? ? * ; * ? ? ? we summarize here some recent results on mrk178 , one of _ the most metal - poor nearby wr galaxies _ ( see * ? ? ? * for more details ) : \1 ) the origin of high - ionization nebular lines ( e.g. heii@xmath34686 ) , apparently more frequent in high - z galaxies , is still an open question . one widely favored mechanism for he@xmath4-ionization involves hot wrs , but it has been shown that nebular heii@xmath34686 is not always accompanied by wr signatures , thus wrs do not explain he@xmath4-ionization at all times @xcite . in mrk178 , we find nebular heii@xmath34686 emission spatially extended reaching well beyond the location of the wr stars ( fig . [ fig ] , left - panel ) . the excitation source of he@xmath4 in mrk178 is still unknown . \2 ) from the sdss spectra , we have found a too high ew(wr bump ) value for mrk178 , which is the most deviant point among the metal - poor wr galaxies in fig . [ fig ] , right - panel . using our ifu data , we have demonstrated that this curious behaviour is caused by aperture effects , which actually affect , to some degree , the ew(wr bump ) measurements for all galaxies in fig.1 . we have also shown that using too large an aperture , the chance of detecting wr features decreases , and that wr signatures can escape detection depending on the distance of the object and on the aperture size . therefore , wr galaxy samples / catalogues constructed on single fiber / long - slit spectrum basis may be biased ! 4686 line ; the spaxels where we detect wr features are marked with green crosses . _ right panel _ : ew(wr blue bump ) vs ew(h@xmath5 ) . asterisks show values obtained from sdss dr7 for metal - poor wr galaxies ; the red one represents mrk178 . the three blue circles , from the smallest to the biggest one , represent the 5 , 7 and 10 arcsec - diameter apertures from our ifu data centered at the wr knot of mrk178 , at which the sdss fiber was centered too @xcite.,title="fig:",scaledwidth=54.0% ] 4686 line ; the spaxels where we detect wr features are marked with green crosses . _ right panel _ : ew(wr blue bump ) vs ew(h@xmath5 ) . asterisks show values obtained from sdss dr7 for metal - poor wr galaxies ; the red one represents mrk178 . the three blue circles , from the smallest to the biggest one , represent the 5 , 7 and 10 arcsec - diameter apertures from our ifu data centered at the wr knot of mrk178 , at which the sdss fiber was centered too @xcite.,title="fig:",scaledwidth=45.0% ] this work has been partially funded by research project aya2010 - 21887-c04 - 01 from the spanish pnaya .

wolf - rayet hii galaxies are local metal - poor star - forming galaxies , observed when the most massive stars are evolving from o stars to wr stars , making them template systems to study distant starbursts . we have been performing a program to investigate the interplay between massive stars and gas in wr hii galaxies using ifs . here , we highlight some results from the first 3d spectroscopic study of mrk 178 , _ the closest metal - poor wr hii galaxy _ , focusing on the origin of the nebular heii emission and the aperture effects on the detection of wr features .

with the discovery of the standard model ( sm ) higgs - like scalar at the lhc @xcite , the sm particle content seems complete . in particular , the mass and couplings of the neutral higgs boson seem to disfavor an additional chiral generation of quarks and leptons @xcite . however , additional vector - like fermions , in which an sm generation is paired with another one of opposite chirality , and with identical couplings , are less constrained , as there is no quadratic contribution to their masses . these states appear as natural extensions of the sm particle content , in theories with warped or universal extra dimensions , as kaluza - klein excitations of the bulk field @xcite , in non - minimal supersymmetric extensions of the sm @xcite , in composite higgs models @xcite , in little higgs model @xcite and in gauged flavor groups @xcite . vector fermions have identical left- and right - handed couplings and can have masses which are not related to their couplings to the higgs bosons @xcite . depending on the dominant decay mode , the limits on new vector - like fermions range from @xmath2 gev @xcite , rendering them observable at the lhc . vector quarks can modify both the production and decays of the higgs boson at the lhc , while vector leptons do not carry @xmath3 charge and can only modify the decay patterns of the higgs . study of the latter would be a sensitive probe for new physics . the lepton states contribute to self - energy diagrams for electroweak gauge boson masses and precision observables , and consistent limits on their masses and mixings have been obtained @xcite . vector leptons have been studied in the context of the sm @xcite , but less so for models beyond the sm , where they can also significantly alter the phenomenology of the model . in the sm , introducing heavy fermions provides a contribution of the same magnitude and sign to the one of the top quark and interferes destructively with the dominant @xmath4-contribution , reducing the @xmath5 rate with respect to its sm value . recent studies indicate that cancellations between scalar and fermionic contributions allow a wide range of yukawa and mass mixings among vector states @xcite . an investigation of vector leptons in the two higgs doublet model @xcite showed that the presence of additional higgs bosons alleviates electroweak precision constraints . introducing vector leptons into supersymmetry @xcite can improve vacuum stability and enhance the di - photon rate by as much as 50% , while keeping new particle masses above 100 gev and preserving vacuum stability conditions . in the present work , we investigate vector leptons in the context of the higgs triplet model ( htm ) . we do not deal with lhc phenomenology ( pair production and decay ) of the extra leptons , which has been discussed extensively in the literature @xcite , choosing instead to focus on signature features of this model . we have previously shown that in the higgs triplet model , enhancement of the @xmath6 rate is possible only for the case where the doublet and triplet neutral higgs fields mix considerably @xcite . we extend our analysis to include additional vector - like leptons in the model and investigate how these affect the higgs di - photon decay rate , with or without significant mixing in the neutral higgs sector . originally , both cms and atlas experiments at lhc observed an enhancement of the higgs di - photon rate , while the di - boson rates ( @xmath7 ) have been roughly consistent with sm expectations . at present cms observes @xmath8 times the sm rate , while atlas observes @xmath9 times the sm rate @xcite . given these numbers , it is possible that either the sm value will be proven correct , or a modest enhancement will persist . a further test of the sm is the correlation of the decay @xmath1 with one for @xmath10 . we also include the prediction for the vector lepton effect on branching ratio of @xmath1 and comment on the relationship with the di - photon decay . we have an additional reason to investigate the effects of vector leptons in the higgs triplet model . the model includes doubly - charged higgs bosons , predicted by most models to be light . being pair - produced , the doubly - charged higgs bosons are assumed to decay into a pair of leptons with the same electric charge , through majorana - type interactions @xcite . assuming a branching fraction of 100% decays into leptons , _ i.e. _ , neglecting possible decays into @xmath4-boson pairs , the doubly - charged higgs mass has been constrained to be larger than about 450 gev , or more , depending on the decay channel . however , if the vector leptons are light enough , which they can be , the doubly - charged higgs bosons can decay into them and thus evade the present collider bounds on their masses . we investigate this possibility in the second part of this work . our work is organized as follows . we introduce the higgs triplet model with vector leptons in section [ sec : model ] . in section [ sec : neutral ] we analyze the effect of the vector leptons on the decays of the neutral higgs bosons , and discuss the constraints on the parameter space coming from requiring agreement with present lhc data . we include both loop - dominated decays , @xmath0 in [ subsec : gamgam ] , and @xmath11 in [ subsec : zgam ] . in section [ sec : doublycharged ] we analyze the effect of the vector leptons on the production and decay mechanisms of the doubly - charged higgs at the lhc . we summarize our findings and conclude in section [ sec : conclusion ] . the electroweak gauge symmetry is broken by the vevs of the neutral components of the doublet and triplet higgs fields , @xmath41 where @xmath42 and @xmath43 are the doublet higgs field and the triplet higgs field , with @xmath44 ( 246 gev)@xmath45 . higgs masses and coupling constants in the presence of non - trivial mixing in the neutral sector have been obtained previously @xcite . one can invoke new symmetries to restrict the interaction of the vector leptons with each other , or with the ordinary leptons , or disallow the presence of bare mass terms in the lagrangian . for instance , 1 . if an additional @xmath46 symmetry under which the primed , double primed and the ordinary leptons have different charges , this would forbid explicit masses @xmath47 , @xmath48 , @xmath49 and @xmath50 in the lagrangian . vector leptons would get masses only through couplings to the higgs doublet fields @xcite . 2 . if one imposes a symmetry under which all the new @xmath51 singlet fields are odd , while the new @xmath51 doublets are even , this forces all yukawa couplings involving new leptons to vanish , @xmath52 , and the masses arise only from explicit terms in the lagrangian @xcite . 3 . finally one can impose a new parity symmetry which disallows mixing between the ordinary leptons and the new vector lepton fields , under which all the new fields are odd , while the ordinary leptons are even @xcite , i.e. , such that @xmath53 ; alternatively one might choose these couplings to be very small . in this analysis the focus will be on higgs decays . we investigate the model subjected to symmetry conditions 1 and/or 2 ; and we neglect mixing between the ordinary and the new vector leptons . when allowed , stringent constraints exist on the masses and couplings with ordinary leptons . new vector leptons are ruled out when they mediate flavor - changing neutral currents processes , generate sm neutrino masses or contribute to neutrinoless double beta decay . recent studies of models which allow mixing between the ordinary leptons and the new ones exist @xcite , but there restrictions from lepton - flavor violating decays force the new leptons to be very heavy @xmath54 tev , or reduces the branching ratio for @xmath55 and of @xmath0 decay to 30 - 40% of the sm prediction , neither desirable features for our purpose here . in the higgs triplet model , distinguishing signals would be provided by lighter vector leptons . since imposing no mixing between ordinary and new leptons allows new lepton masses to be as light as @xmath56 gev , perhaps without a reduction in the higgs di - photon branching ratio , we investigate this scenario here . in the charged sector , the @xmath57 mass matrix @xmath58 is defined as @xcite @xmath59 with @xmath60 and @xmath61 , with @xmath62 the vev of the neutral component of the higgs doublet . the mass matrix can be diagonalized as follows : @xmath63 the mass eigenvalues are @xmath64 , \label{eq : eigenvalues}\ ] ] with @xmath65 by convention , @xmath66 . for simplicity we assume lepton yukawa couplings @xmath67 and @xmath68 are real so that the transformations that diagonalize the mass matrix are real orthogonal matrices : @xmath69 the angles @xmath70 are given by @xmath71 the eigenstates of the vector lepton mixing matrix enter in the evaluation of @xmath0 and @xmath1 in the next section . the presence of the vector leptons affects the loop - dominated decays of the neutral higgs , @xmath0 and @xmath1 , and possible relationships between them . in the higgs triplet model , singly- and doubly - charged bosons also enter in the loops , creating a different dynamic than in the sm . we analyze these decays in turn , and look for possible correlations between them . recently , the triplet higgs model has received renewed interest recently because of attempts to reconcile the excess of events in @xmath0 observed at the lhc over those predicted by the sm . such an enhancement hints at the presence of additional particles , singlets under @xmath3 , but charged under @xmath72 which affect only the loop - dominated decay branching ratio , while leaving the production cross section and tree level decays largely unchanged . vector leptons are prime candidates for such particles , so we study their contribution to the higgs decay branching . the decay width @xmath0 is @xmath73_{htm } & = & \frac{g_f\alpha^2 m_{h}^3 } { 128\sqrt{2}\pi^3 } \bigg| \sum_{f } n^f_c q_f^2 g_{h ff } a_{1/2 } ( \tau^h_f ) + g_{h ww } a_1 ( \tau^h_w ) + \tilde{g}_{h h^\pm\,h^\mp } a_0(\tau^h_{h^{\pm } } ) \nonumber \\ & + & 4 \tilde{g}_{h h^{\pm\pm}h^{\mp\mp } } a_0(\tau^h_{h^{\pm\pm } } ) + \frac{\mu_{e_1}g_{hff}}{m_{e_1 } } a_{1/2}(\tau^h_{e_1 } ) + \frac{\mu_{e_2}g_{hff}}{m_{e_2 } } a_{1/2}(\tau^h_{e_2 } ) \bigg|^2 \ , , \label{partial_width_htm}\end{aligned}\ ] ] with @xmath74 given in eq . ( [ eq : eigenvalues ] ) , and where the loop functions for spin @xmath75 , spin @xmath76 and spin @xmath77 are given by : @xmath78\ , \tau^{-2 } \ , , \label{eq : ascalar}\\ a_{1/2}(\tau)&=&-\tau^{-1}\left[1+\left(1-\tau^{-1}\right)f\left(\tau^{-1}\right)\right ] , \label{eq : afermion}\\ a_1(\tau)&= & 1+\frac32 \tau^{-1}+4 \tau^{-1}\left(1-\frac12 \tau^{-1}\right)f\left(\tau^{-1}\right ) , \label{eq : avector } \end{aligned}\ ] ] and the function @xmath79 is given by : @xmath80 ^ 2 \hspace{0.5 cm } & \tau>1 \ , , \end{array } \right . \label{eq : ftau } \end{aligned}\ ] ] with @xmath81 , and @xmath82 the mass of the particle running in the loop @xcite . in eq . ( [ eq : thm - h2gaga ] ) the first contribution is from the top quark , the second from the @xmath4 boson , the third from the singly - charged higgs boson , the fourth from the doubly - charged higgs boson , and the last two from the vector leptons . we used the following expressions for the couplings of the higgs bosons with charged vector leptons : @xmath83 the couplings of @xmath84 to the vector bosons and fermions are as follows : @xmath85 with @xmath86 , and the scalar trilinear couplings are parametrized as follows @xmath87 \ , , \nonumber \\ & & \tilde{g}_{h h^+h^-}= \frac{m_w}{2 g m_{h^{\pm}}^2 } \bigg\{\left[4v_\delta(\lambda_2 + \lambda_3 ) \cos^2{\beta_{\pm}}+2v_\delta\lambda_4\sin^2{\beta_{\pm}}- \sqrt{2}\lambda_5v_\phi \cos{\beta_{\pm}}\sin{\beta_{\pm}}\right]\sin\alpha \nonumber \\ & & + \left[4\lambda_1\,v_\phi \sin{\beta_{\pm}}^2+{(2\lambda_{4}+\lambda_{5 } ) } v_\phi \cos^2{\beta_{\pm}}+ ( 4\mu-\sqrt{2}\lambda_5v_\delta)\cos{\beta_{\pm}}\sin{\beta_{\pm}}\right]\cos\alpha\bigg\ } \ , . \label{eq : redgcallittlehhp}\end{aligned}\ ] ] since the new leptons do not affect the higgs production channels , the effect on the di - photon search channel at the lhc is expressed by the ratio @xmath88_{htm}}{[\sigma ( gg \to \phi ) \times br(\phi \to \gamma \gamma ) ] _ { sm } } \nonumber \\ & = & \frac{[\sigma ( gg \to h ) \times \gamma(h \to \gamma \gamma)]_{htm}}{[\sigma ( gg \to\phi ) \times \gamma(\phi \to \gamma \gamma)]_{sm } } \times \frac{[\gamma ( \phi)]_{sm}}{[\gamma(h)]_{htm } } , \end{aligned}\ ] ] where @xmath42 is the sm neutral higgs boson and where the ratio of the cross sections by gluon fusion is @xmath89 here @xmath90 is the mixing angle in the cp - even neutral sector : @xmath91 with @xmath92 in @xcite we investigated the higgs boson decay branching ratio into @xmath93 with respect to the sm considering the lightest higgs boson is the @xmath94 signal excess observed at the lep at 98 gev , while the heavier higgs boson is the boson observed at the lhc at 125 gev , in a higgs triplet model without vector leptons , and found that this is the only scenario which allows for enhancement of the @xmath0 branching fraction . we thus set the values 125 gev and 98 gev for the @xmath84 and @xmath95 masses respectively , and adjust the parameters @xmath96 accordingly . vector lepton masses and mixing parameters depend on @xmath47 and @xmath48 , the explicit mass parameters in the lagrangian ; and @xmath97 , the vector leptons yukawa parameters . in the limit of vanishing dirac mass terms @xmath47 and @xmath48 , the pre - factors @xmath98 in ( 11 ) go to one . it then follows that there is destructive interference between the dominant @xmath4- boson contribution and the charged leptons loops @xcite . in this limit , despite possible enhancement from singly- and doubly - charged higgs bosons in the loop , we find a large suppression of the di - photon rate . we present the plots for the relative signal strength of @xmath99 , defined in eq . ( [ eq : rgamgam ] ) as a function of @xmath100 ( or equivalently @xmath101 ) , for various values of doubly charged higgs bosons mass , in the left - side panel of fig . [ fig : hgg ] , for @xmath102 . clearly , for this case ( no mixing ) the decay of the @xmath84 is suppressed significantly with respect to the value in sm over the whole region of the parameter space in @xmath103 . -0.3 in @xmath104{fig1leftpanel.pdf } & \hspace*{-1.5 cm } \includegraphics[width=3.1in , height=3.1in]{fig1middlepanel.pdf } & \hspace*{-2.1 cm } \includegraphics[width=3.1in , height=3.1in]{fig1rightpanel.pdf } \end{array}$ ] allowing mixing in the neutral higgs sector changes the relative contributions of the charged higgs to the di - photon decay . we show decay rates for @xmath0 as a function of @xmath105 for different values of doubly - charged higgs boson mass considering @xmath106 gev ( and 200 gev ) in fig . [ fig : hgg ] middle ( and right ) panels respectively . considerations for relative branching ratios are affected by the fact that the total width of higgs boson in the htm for @xmath107 is not the same as in the sm . the relative widths factor is @xmath108_{htm}}{[\gamma(\phi)]_{sm}}=\frac{[\gamma ( h \to \sum\limits_f f { \bar f})+ \gamma ( h \to ww^ * ) + \gamma ( h \to zz^*)+ \gamma ( h \to \nu \nu)]_{htm}}{[\gamma ( \phi \to \sum\limits_f f { \bar f})+ \gamma ( \phi \to ww^ * ) + \gamma ( \phi \to zz^*)]_{sm } } . \end{aligned}\ ] ] the plots in fig . [ fig : hgg ] correspond to symmetry condition 1 in section [ sec : model ] , that is , @xmath109 . however , if mixing with sm leptons is forbidden , but the vector leptons are still allowed to mix with each other , the pre - factors @xmath110 in eq . ( [ eq : thm - h2gaga ] ) are not one , and can modify the higgs di - photon decay . in the next plots we investigate the effect of non - zero mass parameters @xmath47 and @xmath48 , for fixed values of the yukawa couplings . in fig . [ fig : hemlcont1 ] we present the contour plots of constant @xmath111 for @xmath112 in the plane of the explicit mass terms @xmath113 and @xmath48 , for various values of doubly - charged higgs bosons mass and @xmath114 . the contours are labeled by the value of @xmath111 . the vector lepton masses are restricted to values for which @xcite @xmath115 , where @xmath116 are given in eq . ( [ eq : eigenvalues ] ) . the plots indicate that it is difficult to obtain any significant enhancement of the ratio @xmath111 for @xmath117 , and this does not depend on the chosen values for the doubly - charged higgs mass ; while for @xmath107 , enhancements are possible for various values of @xmath118 . @xmath119{fig2upperleftpanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig2uppermiddlepanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3 in , height=2.2in]{fig2upperrightpanel.pdf}\\ \hspace*{-0.9 cm } \includegraphics[width=2.3in , height=2.2in]{fig2lowerleftpanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig2lowermiddlepanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig2lowerrightpanel.pdf } \end{array}$ ] in fig . [ fig : hemlcont2 ] we investigate the dependence of @xmath111 on the yukawa couplings and vector lepton masses . we show contour plots for fixed @xmath99 in a @xmath120 plane , with @xmath101 and @xmath121 , for various values of @xmath105 and doubly - charged higgs boson masses . enhancements are possible here for all values of @xmath105 , but while for @xmath117 the decay @xmath0 is enhanced for large vector lepton masses and yukawa couplings , for @xmath107 we observe enhancements for light vector lepton masses and small yukawa couplings . @xmath119{fig3upperleftpanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig3uppermiddlepanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3 in , height=2.2in]{fig3upperrightpanel.pdf}\\ \hspace*{-0.9 cm } \includegraphics[width=2.3in , height=2.2in]{fig3lowerleftpanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig3lowermiddlepanel.pdf } & \hspace*{-0.4 cm } \includegraphics[width=2.3in , height=2.2in]{fig3lowerrightpanel.pdf } \end{array}$ ] if we wish to study the light vector leptons parameter space where @xmath0 is enhanced , @xmath107 is preferred . the enhancement is affected by mixing in the vector lepton sector , the various values for doubly - charged higgs bosons mass and values of @xmath105 . as the plots cover only a limited range of the parameter space , in the tables below we give the ranges for the values of the ratio @xmath122 for the various scenarios . in table [ tab : y1/6_me_ml ] , we fix the value of the yukawa coupling to @xmath123 , allow the vector lepton masses to vary in the ( 100 - 500 ) gev range , and show the values for @xmath99 for different @xmath105 and doubly - charged higgs masses . we note that the relative branching ratios are very sensitive to both doubly charged higgs mass values and values of @xmath105 . enhancements in the branching ratio of @xmath0 are possible for light @xmath124 gev , and are much more pronounced at large @xmath105 . note that for @xmath117 , the result is independent of @xmath118 , in agreement with the results obtained in @xcite . the reason is the following . in eq . ( [ eq : redgcallittlehhp ] ) , for @xmath117 , the coupling between neutral and doubly - charged higgs @xmath125 = \frac{m_w } { g m_{h^{\pm \pm}}^2}\bigg [ 2\frac { m^2_{h^{\pm \pm}}}{v_\phi^2 } v_\phi \bigg]=2 \frac{m_w } { g v_\phi } , \end{aligned}\ ] ] where we used the expression for @xmath126 from @xcite , is independent of @xmath118 . in table [ tab : y1/6_h_e ] we allow , in addition to mass variations , variations in the yukawa coupling @xmath127 . this means allowing both explicit ( dirac ) masses and additional contributions by electroweak symmetry breaking , @xmath128 . the dependence on the yukawa coupling @xmath67 is much weaker than on @xmath105 or on @xmath118 . however , one can see from the tables that modest enhancements of the ratio @xmath99 are possible for @xmath117 for large vector leptons yukawa couplings , unlike in the case of the triplet model without vector leptons . this would then be a clear distinguishing feature : enhancements of the decay @xmath0 in the absence of mixing in the neutral sector . the absence of mixing would manifest itself in observing tree - level decays ( @xmath129 and @xmath130 ) identical to those in the sm . there seems to be a minimum value of @xmath99 for @xmath131 , where the contribution from the doubly - charged higgs bosons is important for small doubly - charged masses and counters the contribution from the vector leptons . this is a suppression of the branching ratio for @xmath0 due to the fact that the vector lepton contribution interacts destructively with the dominant @xmath132 contribution . as a general feature , @xmath99 increases when we lower the doubly charged higgs mass and increase @xmath105 . this rules out part of the parameter space . for instance , for @xmath133 gev , the mixing can not be larger than @xmath134 , and for @xmath135 gev , mixings larger than @xmath136 are ruled out . if the value of @xmath137 is increased to @xmath138 gev , only modest enhancements are possible , and only for @xmath117 , for vector lepton explicit masses in the @xmath139 gev range and @xmath123 . increasing the vector leptons yukawa coupling increases the overall ratio @xmath99 . in most models , the @xmath140 and @xmath141 partial decay widths are correlated or anti - correlated , though usually the enhancement / suppression in the @xmath142 channel is much smaller compared to that in the @xmath143 channel . however , as in models with new loop - contributions to @xmath144 , sensitivity to both is expected , we study the correlation between the two here , in the presence of vector leptons . investigation of the branching ratio of @xmath1 is also further justified by the recent results from cms and atlas @xcite , which indicate branching fractions consistent with the sm expectation at 1@xmath145 in the higgs boson @xmath84 mass region at 95% c.l . . the decay width for @xmath1 is given by @xcite : @xmath146_{htm } & = & \frac{\alpha g_f^2 m_w^2 m_{h}^3 } { 64\pi^4 } \left ( 1-\frac{m_z^2}{m_h^2 } \right ) ^3 \bigg|\frac{1}{c_w } \sum_{f } 2 n^f_c q_f ( i_3^f-2q_f s_w^2 ) g_{h ff } a^h_{1/2 } ( \tau^h_f , \tau^z_f ) \nonumber \\ & + & \frac{(i_3^e-2q_e s_w^2)(2q_e)}{c_w } \bigg[\frac{\mu_{e_1}g_{hff}}{m_{e_1 } } a_{1/2}(\tau^h_{e_1 } , \tau^z_{e_1 } ) + \frac{\mu_{e_2}g_{hff}}{m_{e_2 } } a^h_{1/2}(\tau^h_{e_2 } , \tau^z_{e_2 } ) \bigg ] \nonumber \\ & + & c_w g_{h ww } a^h_1 ( \tau^h_w , \tau^z_w)- 2s_w \tilde{g}_{h h^\pm\,h^\mp}g_{zh^\pm h^\mp } a^h_0(\tau^h_{h^{\pm } } , \tau^z_{h^\pm } ) \nonumber \\ & - & 4s_w \tilde{g}_{h h^{\pm\pm}h^{\mp\mp } } { g}_{z h^{\pm\pm}h^{\mp\mp } } a^h_0(\tau^h_{h^{\pm\pm } } , \tau^z_{h^{\pm \pm } } ) \bigg|^2 \ , , \end{aligned}\ ] ] where @xmath147 @xmath148 ( with @xmath149 ) , and the loop - factors are given by @xmath150i_1(\tau^h,\tau^z).\nonumber\end{aligned}\ ] ] the functions @xmath151 and @xmath152 are given by @xmath153\nonumber \\ & + & \frac{\tau^{h\,2}\tau^z}{\left(\tau^h-\tau^z\right)^2 } \left[g\left(\tau^{h\,-1}\right)-g\left(\tau^{z\,-1}\right)\right],\nonumber\\ i_2(\tau^h,\tau^z ) & = & -\frac{\tau^h\tau^z}{2(\tau^h-\tau^z)}\left[f\left(\tau^{h\,-1}\right)-f\left(\tau^{z\,-1}\right)\right],\end{aligned}\ ] ] where the function @xmath79 is defined in eq . ( [ eq : ftau ] ) , and the function @xmath154 is defined as @xmath155 , \hspace{0.5 cm } & ( \tau\geq 1 ) \ , . \end{array } \right . \label{eq : gtau } \end{aligned}\ ] ] in eq . ( [ eq : hzg ] ) we list , in order , the ordinary leptons , vector leptons , @xmath4 boson , singly - charged higgs , and doubly - charged higgs contribution . the scalar couplings @xmath156 and @xmath157 are given in eq . ( [ littleh1tt ] ) , and the scalar trilinear couplings @xmath158 and @xmath159 are given in eq . ( [ eq : redgcallittlehhp ] ) . the remaining couplings in eq . ( [ eq : hzg ] ) are given by @xmath160 we proceed to perform a similar analysis as in sec . [ subsec : gamgam ] . we show first the variation of the branching ratio @xmath1 with the mass @xmath161 , for various values of the doubly - charged higgs masses , for the case of no mixing in the neutral sector , @xmath162 ( shown in fig . [ fig : hgzme ] , left - hand panel ) , and as a function of the mixing angle @xmath105 for @xmath163 gev , and @xmath164 gev , in the middle and right panels of fig . [ fig : hgzme ] , respectively . we have chosen the same parameter values as in fig . [ fig : hgg ] , for comparison . it is clear that the branching ratio into @xmath165 is fairly independent of both @xmath166 and @xmath118 , and always just below the sm expectations . note that the severe suppression seen in @xmath5 for @xmath117 ( fig . [ fig : hgg ] , left side panel ) does not occur here , and the results of the left side of fig . [ fig : hgzme ] are consistent with the data at lhc . but the variation with the mixing angle @xmath90 is pronounced , and the branching ratio can reach almost twice its sm value for @xmath167 . however , correlated with our predictions from sec . [ subsec : gamgam ] and lhc measurements for @xmath99 , the parameter space corresponding to an enhanced @xmath1 , for both @xmath168 gev and 200 gev , for doubly - charged higgs mass @xmath169 gev is ruled out . for all other values considered , the value for @xmath170 is close to , or below the sm expectations . this is general prediction of the model . -0.3 in @xmath171{fig4leftpanel.pdf } & \hspace*{-1.7 cm } \includegraphics[width=3.1in , height=3.0in]{fig4middlepanel.pdf } & \hspace*{-1.9 cm } \includegraphics[width=3.1 in , height=3.0in]{fig4rightpanel.pdf } \end{array}$ ] for a large range of parameter space , the decay @xmath1 can be suppressed significantly with respect to the sm . we plot decay rates for @xmath172 as a function of @xmath105 for different values of doubly charged higgs boson mass , considering @xmath173 gev and 200 gev in fig . [ fig : hgzme ] , middle and right panel respectively . again , considerations for relative branching ratios are affected by the fact that the total width of higgs boson in the htm is not the same as in the sm . the widths are the same as those in the sm for @xmath84 for @xmath117 , while for @xmath107 we take into account the relative widths factors , eq.([eq : width ] ) . in tables [ tab : yz1/6_me_ml ] and [ tab : yz1/6_h_e ] we present the explicit ranges of @xmath170 for varying @xmath174 and for a range of @xmath67 parameters . we choose a fixed value for @xmath123 in table [ tab : yz1/6_me_ml ] , as the preferred choice from other analyses @xcite , and for comparison with table [ tab : y1/6_me_ml ] . comparison of tables [ tab : y1/6_me_ml ] and [ tab : yz1/6_me_ml ] shows that the decay @xmath1 is far more stable against variations in masses and values for @xmath105 than @xmath0 , making it a less sensitive indicator for the presence of vector lepton states . in table [ tab : yz1/6_h_e ] we also allow variations in the yukawa coupling @xmath127 . as before this amounts to allowing both explicit and contributions by electroweak symmetry breaking , @xmath128 , to vector lepton masses . comparison of the tables [ tab : yz1/6_me_ml ] and [ tab : yz1/6_h_e ] indicates that the results are not very sensitive to variations in the yukawa coupling @xmath67 , or the vector lepton mass parameters @xmath174 . however , the relative branching ratios are very sensitive to values of @xmath105 . while the branching ratio into @xmath165 is almost always suppressed with respect to its sm value , there is a small region of the parameter space , light @xmath175 and @xmath176 where enhancement is possible ; but as discussed before , this region is ruled out by constraints from @xmath0 measurement , table [ tab : y1/6_h_e ] . note that for @xmath117 the branching ratio is as before , independent of the mass of @xmath175 and about the same as in the sm . the discovery of the doubly - charged higgs bosons would be one of the most striking signals of physics beyond sm , and a clear signature for the higgs triplet model . the decay modes of @xmath175 depend on the value of the vev of the neutral triplet higgs component , @xmath177 . when @xmath178 mev , the dominant decay mode of @xmath175 is into lepton pairs . if @xmath179 mev , the main decay modes of @xmath175 are into @xmath180 pairs are kinematically allowed . ] , and into @xmath181 , if kinematically allowed . searches for @xmath175 were performed at large electron positron collider ( lep ) @xcite , the hadron electron ring accelerator ( hera ) @xcite and the tevatron @xcite . the most up - to - date bounds have been more recently derived by atlas and cms collaborations at the lhc . assuming a drell - yan - like pair production , these collaborations have looked for long - lived doubly - charged states , and after analyzing 5 fb@xmath182 of lhc collisions at a center - of - mass energy @xmath183 tev and 18.8 fb@xmath182 of collisions at @xmath184 = 8 tev , they constrained the masses to lie above 685 gev @xcite . the assumption is that the doubly - charged higgs bosons decay 100% into a pair of leptons with the same electric charge through majorana - type interactions @xcite , thus neglecting possible decays into @xmath4-boson pairs , shown to alter the pattern of @xmath175 branching fractions @xcite . in this work , we allow both decays into @xmath185 bosons , and also include the effects of decays into vector leptons , which , if light enough , would modify the decays of the doubly - charged higgs bosons further . we take @xmath186 gev throughout our considerations is small enough to satisfy electroweak precision conditions , but large enough to allow decay into gauge boson and charged higgs @xcite . ] . the main production mode for @xmath175 is the pair production @xmath187 and the associated production @xmath188 . the production cross sections for both the vector boson fusion @xmath189 , and for weak boson associated production @xmath190 , are proportional to @xmath191 and much less significant for @xmath192 . at hadron colliders the partonic cross section for the leading order ( lo ) production cross section for doubly charged higgs boson pair is @xmath193,\ ] ] where we defined @xmath194 with @xmath195 the third component of the isospin for particle @xmath196 , @xmath197 the square of the partonic center of mass energy , @xmath198 , and @xmath90 the qed coupling constant evaluated at scale @xmath199 . the hadronic cross section is obtained by convolution with the partonic density functions of the proton @xmath200 where @xmath201 is the parton luminosity and @xmath202 ( @xmath203 is the total energy squared at the lhc ) . the cross section for pair - production , including nlo corrections , has been evaluated in @xcite . depending on mass parameters in the model , the doubly - charged higgs boson can decay into lepton pairs , including vector leptons , @xmath132 pairs , or @xmath204 states . in the higgs triplet model , the decay rate for @xmath175 into leptons is @xmath205 ^ 2 , \end{aligned}\ ] ] where @xmath82 is the mass of the @xmath206th lepton ( @xmath207 or @xmath208 ) and @xmath209 for @xmath210 , ( @xmath211 ) . similarly the decay rate of @xmath175 into fourth generation vector leptons is , if kinematically allowed @xmath212 \frac{m_{h^{\pm \pm}}}{4 \pi}\left ( 1-\frac{m_{e_i}^2}{m_{h^{\pm \pm}}^2}-\frac{m_{e_j}^2}{m_{h^{\pm \pm}}^2 } \right ) \left [ \lambda \left(\frac{m_{e_i}^2}{m_{h^{\pm \pm}}^2 } , \frac{m_{e_j}^2}{m_{h^{\pm \pm}}^2 } \right ) \right]^2 , \nonumber \\\end{aligned}\ ] ] with @xmath213 the mass eigenvalue from eq . ( [ eq : eigenvalues ] ) . in addition , we use the decay rates of @xmath175 into @xmath214 and @xmath215 : @xmath216^{3/2 } , \end{aligned}\ ] ] where @xmath217 is the mixing angle in the singly - charged higgs sector and @xmath218 we investigate the branching ratios of @xmath175 in two distinct parameter regions : * condition 1 : when @xmath219 is not kinematically allowed . then @xmath175 decays into leptons and @xmath132 pairs only : @xmath220 with the total width for condition 1 @xmath221 * condition 2 : when @xmath219 is kinematically allowed . then @xmath175 is able to decay into charged higgs and gauge bosons as well : @xmath222 with the total decay width for condition 2 @xmath223 we investigate the decay patterns of @xmath175 and present plots of the production cross section times the branching fractions under various conditions . to cover a wide range of parameter space , we distinguish two cases for each condition , depending on the vector lepton masses . we set the yukawa coupling of the vector leptons with the doublet higgs bosons to be @xmath224 for both cases . * case a corresponds to very light vector leptons : @xmath225 gev . for this case we obtain for the eigenvalues , @xmath226 gev , @xmath227 gev , the latter of which is close to the allowed minimum . * case b corresponds to intermediate mass vector leptons @xmath228 gev , @xmath229 gev . for this case we obtain for the eigenvalues , @xmath230 gev , @xmath231 gev . in fig . [ fig : condition1 ] we show the corresponding graphs for condition 1 ( when the decay @xmath219 is not kinematically allowed ) , case a on the top row and case b on the bottom row . we plot @xmath232 with @xmath233 as specified in the attached panels , as functions of the doubly - charged masses . on the left side of the figure the yukawa couplings of vector lepton with the triplet higgs boson are @xmath234 and on the right side of the figure , @xmath235 . we set @xmath236 throughout . for small vector lepton coupling @xmath237 , the decay into @xmath238 is dominant at low @xmath175 masses ( this is because we assumed the couplings @xmath32 with ordinary leptons to be all equal ; should we have chosen them diagonal , the branching ratio into @xmath239 would dominate ) . at high @xmath240 masses the branching ratio into @xmath214 dominates and can reach 40% , branching ratios into ordinary leptons reaching at most 40% , and those into vector leptons remaining at or below 20% level . if the triplet yukawa coupling to vector leptons is allowed to increase to @xmath241 , the decay into the two lightest vector leptons @xmath242 becomes dominant , and can reach 80 - 90% when kinematically allowed ( in the @xmath243 gev region for @xmath227 gev ) and overwhelms the other decay modes , which are now below 5% . the only difference between case a and case b in this figure are threshold effects . for case b , @xmath244 gev for decay into pairs of @xmath242 states , as @xmath231 gev ; otherwise the branching ratios are the same . -0.6 in @xmath245{fig5topleftcondition1case1yukawa001.pdf } & \hspace*{-0.8 cm } \includegraphics[width=3.5in , height=3.5in]{fig5toprightcondition1case1yukawa01.pdf}\\ \hspace*{-1.2 cm } \includegraphics[width=3.5 in , height=3.5in]{fig5bottomleftcondition1case2yukawa001.pdf } & \hspace*{-0.8 cm } \includegraphics[width=3.5in , height=3.5in]{fig5bottomrightcondition1case2yukawa01.pdf } \end{array}$ ] in fig . [ fig : condition2 ] we plot the same quantities for condition 2 ( when @xmath219 is kinematically allowed ) , also case a on the top row and case b on the bottom row , @xmath235 on the left hand side and @xmath241 on the right hand side . the decay pattern is very different here , and it is dominated by @xmath219 . for small vector lepton coupling @xmath246 the branching ratios for decays into vector and ordinary lepton pairs , and @xmath132 pairs are very small , and reach at most 1% . increasing the vector lepton coupling to @xmath241 , the decay into @xmath204 still dominates throughout the parameter space where it is kinematically allowed and can reach a branching fraction of over 90% , while the decay into vector leptons can have branching ratios of up to 25% . again , the decay rates into @xmath132 boson pairs and ordinary leptons are below 1% , and the only difference between case a and case b are , as in fig . [ fig : condition1 ] , threshold effects . -0.6 in @xmath247{fig6topleftcondition2case1yukawa001.pdf } & \hspace*{-0.6 cm } \includegraphics[width=3.5in , height=3.5in]{fig6toprightcondition2case1yukawa01.pdf}\\ \hspace*{-1.2 cm } \includegraphics[width=3.5 in , height=3.5in]{fig6bottomleftcondition2case2yukawa001.pdf } & \hspace*{-0.6 cm } \includegraphics[width=3.5in , height=3.5in]{fig6bottomrightcondition2case2yukawa01.pdf } \end{array}$ ] despite no new signals of physics beyond the sm at the lhc , the sm can not be the complete theory of particle interactions . an extension of the sm by additional vector - like leptons is not ruled out experimentally , and has been shown to provide a dark matter candidate . in models beyond the sm , the vector leptons can alter not only the phenomenology of the higgs , but also of other additional particle representations predicted by the models . we provide an example within the higgs triplet model , where previously we showed that , in the absence of triplet - doublet higgs mixing in the neutral sector ( @xmath117 ) , there is no enhancement of the rate of decay of @xmath0 in this model with respect to the sm expectation . introducing vector leptons does not affect any of the tree - level decays of the neutral higgs boson observed at the lhc , nor its production decay . however , loop decays into electroweak particles , such as @xmath1 and @xmath0 would be affected . we show that , for the no - mixing scenario ( @xmath117 ) the decays rates into @xmath93 and @xmath165 do not depend on the doubly charged higgs mass , and thus without the additional vector leptons , these decays would be unchanged from the sm expectations . with vector leptons , modest enhancements or suppressions are possible , more so for @xmath0 , where for large yukawa couplings , the rate of decays could even double . under the same circumstances , the decay width for @xmath1 remains practically unchanged from its sm value . the model thus presents a mechanism for enhancing one loop - decays and not the other , which seems to be consistent with the lhc data ( so far ) . if @xmath107 , the effect of the doubly - charged higgs bosons is felt for both @xmath248 and @xmath249 , most spectacularly so for very light @xmath169 gev , which is ruled out for @xmath250 . parameter space regions where light doubly - charged higgs masses ( 200 - 250 gev ) , _ and _ significant mixing in the neutral sector coexist , are disfavored . in general , there are many parameter combinations for which the decay @xmath248 is enhanced , but few for an enhanced @xmath251 , and these regions are ruled out by the branching ratio for @xmath0 . however , if the decay @xmath248 is ( modestly ) enhanced , while @xmath1 is the same as the sm to 1@xmath145 , small mixing angles and light doubly charged higgs bosons @xmath252 gev are preferred . the fact that there are no regions of the parameter space consistent with present measurements of the branching ratio for @xmath0 and an enhanced rate for @xmath1 is a feature of this model , valid over the whole explored range of the parameter space . other than that , there are no definite correlations or anti - correlations between there two loop - dominated decays . the intermediate mass doubly - charged higgs boson can decay into light vector leptons , which would alter its decay profile significantly . we explored this possibility and found that , if the singly charged higgs mass is such that the decay @xmath219 is not kinematically accessible , dominant branching ratios into vector leptons , if kinematically accessible , are expected for triplet yukawa couplings @xmath253 , whereas for small @xmath254 branching ratios into ordinary leptons , vector leptons and @xmath132 pairs are comparable for @xmath255 gev . if and where the decay @xmath219 is kinematically accessible , its corresponding branching ratio is the largest , while the branching fraction into vector leptons could reach 20 - 25% for @xmath253 . under both these circumstances , the decay patterns of the doubly charged higgs bosons are significantly altered , raising the hope that they can be found at masses around 300 - 500 gev . the work of s. b. and m.f . is supported in part by nserc under grant number sap105354 . g. aad _ et al . _ [ atlas collaboration ] , phys . b * 716 * ( 2012 ) 1 ; s. chatrchyan _ et al . _ [ cms collaboration ] , phys . b * 716 * ( 2012 ) 30 . t. aaltonen _ et al . _ [ cdf collaboration ] , phys . lett . * 104 * , 091801 ( 2010 ) ; a. lister [ cdf collaboration ] , arxiv:0810.3349 [ hep - ex ] . k. agashe , g. perez and a. soni , phys . d * 71 * , 016002 ( 2005 ) ; k. agashe , g. perez and a. soni , phys . d * 75 * , 015002 ( 2007 ) ; a. carmona and j. santiago , jhep * 1201 * , 100 ( 2012 ) ; g. -y . huang , k. kong and s. c. park , jhep * 1206 * , 099 ( 2012 ) ; c. biggio , f. feruglio , i. masina and m. perez - victoria , nucl . b * 677 * , 451 ( 2004 ) ; d. e. kaplan and t. m. p. tait , jhep * 0006 * ( 2000 ) 020 ; h. -c . cheng , phys . d * 60 * , 075015 ( 1999 ) . t. moroi and y. okada , mod . phys . lett . a * 7 * ( 1992 ) 187 ; t. moroi and y. okada , phys . b * 295 * ( 1992 ) 73 ; k. s. babu , i. gogoladze , m. u. rehman and q. shafi , phys . d * 78 * ( 2008 ) 055017 ; s. p. martin , phys . d * 81 * ( 2010 ) 035004 ; p. w. graham , a. ismail , s. rajendran and p. saraswat , phys . d * 81 * ( 2010 ) 055016 ; s. p. martin , phys . d * 82 * ( 2010 ) 055019 ; j. kang , p. langacker and b. d. nelson , phys . d * 77 * ( 2008 ) 035003 . b. a. dobrescu and c. t. hill , phys . * 81 * ( 1998 ) 2634 ; r. s. chivukula , b. a. dobrescu , h. georgi and c. t. hill , phys . rev . d * 59 * ( 1999 ) 075003 ; h. collins , a. k. grant and h. georgi , phys . rev . d * 61 * ( 2000 ) 055002 ; h. -j . he , c. t. hill and t. m. p. tait , phys . d * 65 * ( 2002 ) 055006 ; c. t. hill and e. h. simmons , phys . * 381 * ( 2003 ) 235 [ erratum - ibid . * 390 * ( 2004 ) 553 ] ; r. contino , l. da rold and a. pomarol , phys . d * 75 * ( 2007 ) 055014 ; phys . d * 79 * ( 2009 ) 075003 ; k. kong , m. mccaskey and g. w. wilson , jhep * 1204 * ( 2012 ) 079 ; a. carmona , m. chala and j. santiago , jhep * 1207 * ( 2012 ) 049 ; m. gillioz , r. grober , c. grojean , m. muhlleitner and e. salvioni , jhep * 1210 * ( 2012 ) 004 . n. arkani - hamed , a. g. cohen , e. katz and a. e. nelson , jhep * 0207 * ( 2002 ) 034 ; t. han , h. e. logan , b. mcelrath and l. -t . wang , phys . d * 67 * ( 2003 ) 095004 ; m. perelstein , m. e. peskin and a. pierce , phys . d * 69 * ( 2004 ) 075002 ; m. schmaltz and d. tucker - smith , ann . nucl . part . * 55 * ( 2005 ) 229 ; m. s. carena , j. hubisz , m. perelstein and p. verdier , phys . rev . d * 75 * ( 2007 ) 091701 : s. matsumoto , t. moroi and k. tobe , phys . d * 78 * ( 2008 ) 055018 . a. davidson and k. c. wali , phys . * 60 * ( 1988 ) 1813 ; k. s. babu and r. n. mohapatra , phys . d * 41 * ( 1990 ) 1286 ; b. grinstein , m. redi and g. villadoro , jhep * 1011 * ( 2010 ) 067 ; d. guadagnoli , r. n. mohapatra and i. sung , jhep * 1104 * ( 2011 ) 093 . a. azatov , m. toharia and l. zhu , phys . d * 80 * , 035016 ( 2009 ) . atlas collaboration atlas - conf-2013 - 019 ; s. chatrchyan et al . [ cms collaboration ] , phys . b * 718 * , 348 ( 2012 ) . f. del aguila , j. de blas and m. perez - victoria , phys . d * 78 * , 013010 ( 2008 ) . a. joglekar , p. schwaller and c. e. m. wagner , jhep * 1212 * , 064 ( 2012 ) . j. kearney , a. pierce and n. weiner , phys . d * 86 * , 113005 ( 2012 ) ; h. davoudiasl , h. -s . lee and w. j. marciano , phys . d * 86 * , 095009 ( 2012 ) ; k. j. bae , t. h. jung and h. d. kim , phys . rev . d * 87 * , 015014 ( 2013 ) ; m. b. voloshin , phys . d * 86 * , 093016 ( 2012 ) ; d. mckeen , m. pospelov and a. ritz , phys . d * 86 * , 113004 ( 2012 ) ; h. m. lee , m. park and w. -i . park , jhep * 1212 * , 037 ( 2012 ) ; c. arina , r. n. mohapatra and n. sahu , arxiv:1211.0435 [ hep - ph ] ; b. batell , s. jung and h. m. lee , jhep * 1301 * , 135 ( 2013 ) ; h. davoudiasl , i. lewis and e. ponton , arxiv:1211.3449 [ hep - ph ] . ; j. fan and m. reece , arxiv:1301.2597 [ hep - ph ] ; a. carmona and f. goertz , arxiv:1301.5856 [ hep - ph ] ; c. cheung , s. d. mcdermott and k. m. zurek , arxiv:1302.0314 [ hep - ph ] ; w. -z . feng and p. nath , arxiv:1303.0289 [ hep - ph ] ; c. englert and m. mccullough , arxiv:1303.1526 [ hep - ph ] . k. ishiwata and m. b. wise , phys . d * 84 * , 055025 ( 2011 ) . s. k. garg and c. s. kim , arxiv:1305.4712 [ hep - ph ] . a. joglekar , p. schwaller and c. e. m. wagner , arxiv:1303.2969 [ hep - ph ] . g. f. giudice and o. lebedev , phys . b * 665 * , 79 ( 2008 ) ; e. del nobile , r. franceschini , d. pappadopulo and a. strumia , nucl . b * 826 * , 217 ( 2010 ) ; s. p. martin , phys . rev . d * 81 * , 035004 ( 2010 ) . f. arbabifar , s. bahrami and m. frank , phys . d * 87 * , 015020 ( 2013 ) . [ cms collaboration ] , cms - pas - hig-13 - 001 and cms - pas - hig-13 - 005 ; g. aad et al . [ atlas collaboration ] , [ arxiv:1307.1427 [ hep - ex ] ] and atlas - conf 2013 - 029 . a. g. akeroyd , mayumi aoki , and hiroaki sugiyama . d * 77 * , 075010 ( 2008 ) ; a. g. akeroyd , mayumi aoki , and hiroaki sugiyama . d * 79 * , 113010 ( 2009 ) ; takeshi fukuyama , hiroaki sugiyama , and koji tsumura . jhep * 03 * , 044 ( 2010 ) ; s. t. petcov , h. sugiyama , and y. takanishi . d * 80 * , 015005 ( 2009 ) ; takeshi fukuyama , hiroaki sugiyama , and koji tsumura . d * 82 * , 036004 ( 2010 ) ; a.g . akeroyd , cheng - wei chiang , and naveen gaur , jhep * 1011 * , 005 ( 2010 ) ; a. arhrib , r. benbrik , m. chabab , g. moultaka and l. rahili , jhep * 1204 * , 136 ( 2012 ) ; a. arhrib , r. benbrik , m. chabab , g. moultaka and l. rahili , [ arxiv:1202.6621 [ hep - ph ] ] ; a. arhrib , r. benbrik , m. chabab , g. moultaka , m. c. peyranere , l. rahili and j. ramadan , phys . d * 84 * , 095005 ( 2011 ) ; e. j. chun , h. m. lee and p. sharma , jhep * 1211 * , 106 ( 2012 ) ; a. melfo , m. nemevek , f. nesti , g. senjanovi and y. zhang , phys . rev . d 85 , 055018 ( 2012 ) ; m. aoki , s. kanemura and k. yagyu , phys . d * 85 * , 055007 ( 2012 ) ; s. kanemura and k. yagyu , phys . rev . d * 85 * , 115009 ( 2012 ) ; m. aoki , s. kanemura , m. kikuchi and k. yagyu , [ arxiv:1211.6029 [ hep - ph ] ] . k. ishiwata and m. b. wise , arxiv:1307.1112 [ hep - ph ] . r. dermisek and a. raval , arxiv:1305.3522 [ hep - ph ] . s. chatrchyan _ et al . _ [ cms collaboration ] , arxiv:1307.5515 [ hep - ex ] ; [ atlas collaboration ] , atlas - conf-2013 - 009 . c. -s . chen , c. -q . geng , d. huang and l. -h . tsai , phys . d87 , * 075019 * ( 2013 ) ; c. -s . chen , c. -q . geng , d. huang and l. -h . tsai , phys . b * 723 * , 156 ( 2013 ) ; p. s. bhupal dev , d. k. ghosh , n. okada and i. saha , jhep * 1303 * , 150 ( 2013 ) [ erratum - ibid . * 1305 * , 049 ( 2013 ) ] . g. abbiendi _ et al . _ [ opal collaboration ] , phys . b * 526 * ( 2002 ) 221 ; g. abbiendi _ et al . _ [ opal collaboration ] , phys . b * 577 * ( 2003 ) 93 ; p. achard _ et al . _ [ l3 collaboration ] , phys . b * 576 * ( 2003 ) 18 ; j. abdallah _ et al . _ [ delphi collaboration ] , phys . b * 552 * ( 2003 ) 127 . j. sztuk - dambietz [ h1 and zeus collaboration ] , j. phys . * 110 * ( 2008 ) 072042 ; a. aktas _ et al . _ [ h1 collaboration ] , phys . b * 638 * ( 2006 ) 432 ; v. m. abazov _ et al . _ [ d0 collaboration ] , phys . * 101 * ( 2008 ) 071803 ; v. m. abazov _ et al . _ [ d0 collaboration ] , phys . rev . * 108 * ( 2012 ) 021801 ; d. e. acosta _ et al . _ [ cdf collaboration ] , phys . * 93 * ( 2004 ) 221802 ; t. aaltonen _ et al . _ [ the cdf collaboration ] , phys . * 101 * ( 2008 ) 121801 . s. chatrchyan _ [ cms collaboration ] , arxiv:1305.0491 [ hep - ex ] ; g. aad _ et al . _ [ atlas collaboration ] , phys . b 722 ( 2013 ) 305 ; s. chatrchyan _ et al . _ [ cms collaboration ] , eur . j. c * 72 * ( 2012 ) 2189 ; g. aad _ et al . _ [ atlas collaboration ] , eur . j. c * 72 * ( 2012 ) 2244 . c. -w . chiang , t. nomura and k. tsumura , phys . d * 85 * , 095023 ( 2012 ) ; s. kanemura , k. yagyu and h. yokoya , arxiv:1305.2383 [ hep - ph ] . m. muhlleitner and m. spira , phys . rev . d * 68 * , 117701 ( 2003 ) .

we analyze the phenomenological implications of introducing vector - like leptons on the higgs sector in the higgs triplet model . we impose only a parity symmetry which disallows mixing between the new states and the ordinary leptons . if the vector leptons are allowed to be relatively light , they enhance or suppress the decay rates of loop - dominated neutral higgs bosons decays @xmath0 and @xmath1 , and affect their correlation . an important consequence is that , for light vector leptons , the decay patterns of the the doubly - charged higgs boson will be altered , modifying the restrictions on their masses .

at temperatures below the critical value @xmath2 the neutron fermi liquid in the bulk matter of neutron stars is expected to develop a triplet superfluid condensate @xcite . considerable work has done with the most realistic nuclear potentials for determining the magnitude of the superfluid gap at different matter densities @xcite , while the low - energy collective excitations of such superfluid liquid are not well investigated as yet . in the meantime such excitations can play an important role in the evolution of neutron stars . for example , the decay of collective spin waves into neutrino pairs , occuring in a superfluid core of neutron star through neutral weak currents , presents a new mechanism of intensive cooling in some domain of low temperatures @xcite . a recent investigation @xcite has shown sound - like oscillations of @xmath3 superfluid neutron matter ( called superfluid phonons ) due to a very large mean - free - path influence heat conduction in a magnetized crust of neutron stars , where the motion of electrons is very anisotropic . analogous effects might be expected in the neutron star core , since spontaneous breaking of rotation invariance and the baryon number caused by the triplet condensation should lead to the appearance of several goldstone modes @xcite . generally speaking under @xmath4-equilibrium the superdense core of neutron stars is composed of neutrons with a small admixture of protons and electrons . a fraction of hyperons can also appear at higher densities . it is well known however that long - range electromagnetic interactions push out collective oscillations of the charged particles up to the plasma frequency which is large sufficiently for ( approximate ) decoupling of the plasma modes from the sound - like oscillations of neutral component @xcite . in this case the effect of short - range proton - neutron interactions is reduced mostly to renormalization of the effective mass of neutrons participating in the collective oscillations @xcite . therefore the problem can be simplified considering the sound - like excitations in a pure neutron superfluid liquid . previously the sound modes at finite temperatures have been investigated for isotropic singlet - spin superfluids @xcite and for the case of triplet @xmath5-wave pairing in superfluid liquid @xmath6 @xcite . although a qualitative picture of the sound - like waves in superfluids is very similar , the above theories can not be immediately applied to the case of superdense superfluid neutron matter . the well - developed theory of isotropic pairing can not be applied because the triplet condensate in superdense nuclear matter is expected to be anisotropic . the theory of sound - like collective excitations in an anisotropic phase of superfluid @xmath6 is designed only for extremely fast waves with a velocity that is very large as compared to the fermi velocity . since spin - orbit and tensor interactions between neutrons are known to dominate at high densities the neutron pairing involves a mixing of @xmath7 and @xmath8 channels @xcite . the sum of spin - orbit and tensor interactions can not be described with the aid of a sole coupling constant . this complicates the investigation of collective excitations in standard ways with making use of an explicit form of the pairing interaction , where the sole coupling constant drops out of the equations by virtue of the gap equation . however , when the wave - length of the perturbation is large as compared to the coherence length in the superfluid matter and the transferred energy is small in comparison with the gap amplitude , as is typical for sound - like excitations , the collective motion of the condensate can be described in terms of total variable phase which ( in the bcs approximation ) can be derived immediately from the current conservation condition . this approach , for the first time suggested in ref . @xcite , allows to avoid any explicit form of the interaction in the pairing channel . residual fermi - liquid interactions can be incorporated into the theory as a set of molecular fields @xcite . in application to polarization functions , this approach is well developed in ref . @xcite . this paper is organized as follows . section ii contains some preliminary notes on how the fermi - liquid interactions can be reduced to molecular fields . in sec . iii we derive , in the bcs approximation , the linear response of the triplet - correlated superfluid neutron liquid onto an effective vector field given by the sum of external and molecular fields . in sect . iv we express self - consistently the effective fields via external fields , thus obtaining the linear medium response by taking into account the fermi - liquid interactions . in sect . v we analyze the poles of the longitudinal response function in order to derive the dispersion of sound - like oscillations in the condensate . section vi contains a short summary of our findings and the conclusion . throughout this paper , we use the system of units @xmath9 , and the boltzmann constant @xmath10 . it is well known that the landau theory of a normal fermi - liquid is based on the fact that a large part of the interactions can be taken into account with the aid of renormalizations effects . an effective hamiltonian of the system contains the renormalized single - particle energy of quasiparticles with occupation numbers @xmath11 and a residual interaction between changes @xmath12 in the quasiparticle occupation at the fermi surface . as has been shown by leggett @xcite , the quasiparticle pairing does not change the net occupation for a given direction on the fermi surface , if approximate particle - hole symmetry is maintained . thus the fermi - liquid interactions remain unchanged upon pairing . in other words , the fermi - liquid interactions do not interfere with the pairing phenomenon . in our analysis we shall also assume that the anisotropy of the order parameter , which takes place in the case of triplet - spin pairing , plays no significant role in the fermi - liquid interactions . this assumption is clearly justified because the characteristic length associated with the fermi - liquid interactions is of the order of the inverse fermi momentum , @xmath13 , and hence is much smaller than any other characteristic length entering the problem . this allows us to disregard the spin - dependent part of fermi - liquid interactions in the vector channel we shall consider . since we are interested in values of the neutron momenta near the fermi surface , @xmath14 , the amplitudes of the fermi - liquid interactions @xmath15 can be expanded into legendre polynomials . in the vector channel these interactions are spin - independent and can be completely described in terms of the infinite set of landau parameters @xmath16 . in practice even for a saturated nuclear matter one does not know the landau parameters @xmath16 for @xmath17 , and in actual calculations they are frequently put equal zero . the remaining fermi - liquid interactions can be written in the form @xmath18 where @xmath19 is the density of states near the fermi surface in the normal state , and the effective mass of a neutron quasiparticle is defined as @xmath20 , where @xmath21 is the fermi velocity . this approach can be considered as a model of the fermi - liquid interactions . it is known , however , that the landau interactions with @xmath17 do not modify the longitudinal response functions of normal ( nonsuperfluid ) fermi liquid . for a one - component fermi liquid this was demonstrated , for example , in ref . the same result was obtained for a one - component @xmath3 superfluid fermi liquid in ref . @xcite where the effective vertices and the polarization functions ( at @xmath22 ) have been found to depend only on @xmath23 and @xmath24 . interactions ( [ ph2 ] ) renormalize the normal energy of a quasiparticle in the weak external vector field @xmath25 as@xmath26 we denote as @xmath27 the quasiparticle energy related to the fermi energy in the normal state ; @xmath28 is the coupling constant , which depends on the nature of the external field , and @xmath29 is the distribution function of neutron quasiparticles with momenta @xmath30 and spin @xmath31 . from eq . ( [ ep ] ) it is clearly seen that fermi - liquid interactions can be reduced to molecular fields @xcite , defined as @xmath32 and@xmath33 then eq . ( [ ep ] ) can be reduced to the form@xmath34 where the effective fields are given by the sum of external and molecular fields , @xmath35 the molecular fields depend on the charge perturbation and current density and should be calculated consistently . therefore first we shall perform the calculation of the medium response onto the effective field and next find the explicit form of the effective fields in a self - consistent way . the triplet order parameter in the neutron superfluid is a symmetric matrix in spin space which can be written as @xmath36 where @xmath37 are pauli spin matrices ; @xmath38 , with @xmath39 ; and @xmath40 is the @xmath41 unit matrix in spin space . in the ground state , the gap amplitude @xmath42 is a constant ( on the fermi surface ) , and @xmath43 is a real vector in spin space which we normalize by the condition @xmath44 . hereafter the angle brackets denote angle averages , @xmath45 . the angular dependence of the order parameter is represented by the unit vector @xmath46 which defines the polar angles @xmath47 on the fermi surface . in the components , @xmath48 . making use of the adopted graphical notation for the ordinary and anomalous propagators , @xmath49{gn.eps}}$ ] , @xmath50{gn.eps}}$ ] , @xmath51{f1.eps}}$ ] , and @xmath52{f2.eps}}$ ] , it is convenient to employ the matsubara calculation technique for the system in thermal equilibrium . then the analytic form of the propagators is as follows ( see , _ e.g. _ , ref . @xcite ) @xmath53 where the scalar green s functions are of the form @xmath54 and@xmath55 in the above , @xmath56 with @xmath57 is matsubara s fermion frequency , and we assume the `` unitary gap matrix '' in the ground state , @xmath58 , thus obtaining the energy of a one - particle excitation in the form @xmath59 , where the ( temperature - dependent ) energy gap , @xmath60 , is anisotropic . the following notation will be used below . we designate as @xmath61 the analytical continuation of the matsubara sums : @xmath62 where @xmath63 , and @xmath64 with @xmath65 . it is convenient to divide the integration over the momentum space into an integration over the solid angle @xmath66 and over the energy @xmath67 and operate with integrals@xmath68 these are functions of @xmath69 , @xmath70 and the direction of the quasiparticle momentum @xmath71 . in deriving eq . ( [ ixx ] ) integration over @xmath67 is extended to @xmath72 since the neutron matter is extremely degenerate . consider the medium response onto the effective vector field ( [ aef ] ) in the bcs approximation . in this case the ordinary three - point vector vertices of a quasiparticle and a hole are defined in accordance with eq . ( [ etilde ] ) : @xmath73 we use greek letters for dirac indices , @xmath74 . variation of the anomalous self - energy @xmath75 in the field @xmath76 , can be described with the aid of anomalous three - point vertices @xmath77 , defined as : @xmath78 the anomalous vertices are @xmath41 matrices in spin space which , near the fermi surface , depend on the transferred energy momentum @xmath79 and the direction @xmath80 of the quasiparticle velocity . the ward identity implies the following relations between the anomalous vertices and the order parameter in the system @xcite ( see also refs . @xcite ) : @xmath81 we now restrict our consideration to the case , when the wave - length of the perturbation is large as compared to the coherence length and the transferred energy is small in comparison with the gap amplitude , @xmath82 . the only possible collective motion of the condensate in this case is a variation of the total phase without a change of the order parameter structure . then the ward identity reveals that for a uniform medium the anomalous vertices can be written in the form @xmath83 where the unknown vector function @xmath84 satisfies the condition@xmath85 for further progress , let us consider the retarded bcs polarization tensor in the vector channel @xmath86 . the latter can be found using the fact that the current in the system , @xmath87 is connected to the linear correction @xmath88 to the green s function of a quasiparticle in the effective external field @xmath76 and can be obtained by analytic continuation of the following matsubara sums @xmath89@xmath90 where @xmath91 is the total number density of neutrons . the linear correction to the green s function of a quasiparticle caused by the external field @xmath76 is given by the diagrams shown in fig . [ fig1 ] , and can be written analytically as @xmath92 where the anomalous vertices are to be taken in the form ( [ t1 ] ) , and we use the notation @xmath93 , etc . , inserting eq . ( [ dg ] ) into eqs . ( [ ro ] ) and ( [ j ] ) we can derive the retarded polarization tensor @xmath94 with the aid of the standard relation @xmath95 . in this way after a little algebra we find@xmath96 where @xmath84 is still an unknown vector to be found . the functions @xmath97 and @xmath98 are given by @xmath99 and@xmath100 to obtain the second equality we made use of the identity @xmath101 which can be verified by a straightforward calculation . the function @xmath84 can be found from the requirement that the polarization tensor ( [ pi ] ) must satisfy current conservation conditions , @xmath102 , and @xmath103 , which can be written as the two coupled equations@xmath104@xmath105 we made use of relation ( [ qw ] ) in the first equation . from the explicit form ( [ lmn ] ) for @xmath106 one can easily find the following relation @xmath107 the first term on the right - hand side of this expression can be replaced as @xmath108 , in accordance with eq . ( [ lmnq ] ) . inserting the obtained result into eq . ( [ qlmn ] ) , we arrive at@xmath109 this formula can be simplified making use of the explicit form ( [ lm ] ) for @xmath110 . in the components we obtain the expressions : @xmath111@xmath112 valid for @xmath22 . from eqs . ( [ pi])([lm ] ) , ( [ q0 ] ) and ( [ qv ] ) one can obtain the complete bcs polarization tensor @xmath113 in the vector channel . due to conservation of the vector current the polarization tensor can be decomposed into the sum of longitudinal ( with respect to @xmath70 ) and transverse components , where the longitudinal and transverse polarization functions are defined as @xmath114 , @xmath115 , with @xmath116 . we now turn to the fermi - liquid effects . our aim is to express the effective fields ( [ aef ] ) via external fields . for this we can use the fact that the density current commutes with the bare interactions and does not need to be renormalized . the current connection with the external field @xmath117 is given by the well - known relation @xmath118 inserting this into eqs . ( [ a0mol ] ) and ( [ amol ] ) we obtain the time component of the effective field ( [ aef ] ) in the form @xmath119 using also the continuity equation , @xmath120 , we obtain the space components of the effective field : @xmath121@xmath122 insertion of the effective fields ( [ a0ef ] ) and ( [ atef ] ) into @xmath123 and @xmath124 after a little algebra results in the following : @xmath125@xmath126 in obtaining this we have used the landau formula relating the bare mass of a particle with a renormalized mass of a quasiparticle in the translation - invariant system @xmath127 . the complete polarization functions @xmath128 relate the density perturbation and the density current with the external field as @xmath129 , and @xmath130 , respectively . then from eqs . ( [ dr ] ) and ( [ jt ] ) we obtain the complete polarization functions:@xmath131@xmath132 in agreement with the results of gusakov @xcite . equations ( [ pl ] ) and ( [ pt ] ) completely describe the non - equilibrium behavior of superfluid fermi liquids in the vector - linear - response region for not too high @xmath69 and @xmath133 . explicit expressions can be written using the following notation : @xmath134@xmath135@xmath136 @xmath137@xmath138 where @xmath139 , @xmath140 and we assume that the angle @xmath141 between the quasiparticle momentum @xmath71 and the momentum transfer @xmath70 is fixed by the relation@xmath142 where the unit vector @xmath143 defines the direction of the momentum transfer , and the unit vector @xmath144 defines the polar angles @xmath145 on the fermi surface . then the longitudinal polarization function ( [ pl ] ) can be written as @xmath146 where @xmath147 the pole of the density fluctuation propagator ( [ pll ] ) at@xmath148 defines the dispersion , @xmath149 , of the `` collisionless '' collective mode , with @xmath22 . equation ( [ qsh ] ) with @xmath150 as given in eq . ( [ plbcsl ] ) generalizes previous results of refs . @xcite and @xcite to the case of pairing caused by spin - orbit and tensor interactions . therefore before proceeding to the detailed analysis of the sound propagation in the @xmath0 neutron superfluid , we examine the obtained equations for the particular cases of the triplet - spin condensate in superfluid @xmath6 . consider first _ the case of isotropic pairing_. if the energy gap is isotropic , the angle integrals in eqs . ( [ alpha])([khi ] ) can be performed by assuming the polar axis along the transferred momentum . we then obtain @xmath151 and the longitudinal polarization function reduces to eq . ( [ pll ] ) with @xmath152 as obtained by leggett @xcite for the case of isotropic @xmath153-wave pairing . consider now _ the case of anisotropic pairing at _ @xmath154 . in this case the quantities @xmath155 and @xmath156 are independent of @xmath153 and again isotropic , while @xmath157 . thus for @xmath154 we find @xmath158 in agreement with the result obtained by wlfe @xcite for an anisotropic phase of superfluid @xmath6 at zero temperature . notice that the same was obtained also by leggett @xcite for the case of isotropic @xmath153-wave pairing . the pole at @xmath149 corresponds to the first sound ( `` bogolyubov - anderson '' mode ) undamped at zero temperature . we turn now to _ the case of anisotropic _ @xmath5-_wave pairing in liquid _ @xmath6 . the experimentally observable sound velocity in liquid @xmath6 is large , therefore it is traditional to calculate the sound dispersion in the limit @xmath159 . expanding eq . ( [ plbcsl ] ) in powers of @xmath160 , one can obtain up to accuracy @xmath161 , @xmath162 where@xmath163 in obtaining eq . ( [ qser ] ) we used the identity @xmath164 inserting eq . ( [ qser ] ) into eq . ( [ qsh ] ) one can obtain the dispersion law for high - frequency sound in superfluid @xmath6 . assuming @xmath165 we find @xmath166 in agreement with the expression derived by wlfe @xcite . we focus now on _ the sound propagation in the _ @xmath0 _ superfluid neutron liquid _ which is expected to exist in neutron stars at supernuclear densities . first one has to specify the order parameter ( [ dn ] ) for the particular case of neutron pairing . it is conventional to represent the triplet order parameter of the system as a superposition of standard spin - angle functions @xmath167 of the total angular momentum @xmath168 with partial amplitudes @xmath169 : @xmath170 in our calculations we use vector notation which involves a set of mutually orthogonal complex vectors @xmath171 defined as @xmath172 and normalized by the condition @xmath173 . we will focus on the @xmath0 condensation into the state with @xmath1 which is conventionally considered as the preferable one in the bulk matter of neutron stars . in this case one has@xmath174 where @xmath175 and @xmath176 are partial contributions of the @xmath177 and @xmath178 states , respectively , @xmath179 . for @xmath180 the behavior of @xmath181 in the intermediate region of @xmath153 depends essentially on the temperature . according to eqs . ( [ alpha])-([khi ] ) the imaginary part of the functions arises from the pole of the integrand at @xmath182 . this is cherenkov s condition which can be satisfied only if @xmath183 . neglecting the narrow temperature domain where the imaginary part of polarization is exponentially small @xcite , one can conclude that the well - defined ( undamped ) waves correspond to @xmath184 . further we consider only undamped sound - like oscillations with @xmath184 . . the dimensionless velocity of zero sound , @xmath185 is shown vs the temperature parameter @xmath186 for different sets of landau parameters . case ( a ) : @xmath187 . case ( b ) : @xmath188 . we present the curves for the wave propagation along the symmetry axis ( @xmath189 ) and in the perpendicular direction ( @xmath190 ) . ] the density - dependent landau parameters entering the dispersion equation are not reliably known . therefore in fig . [ fig2 ] we present solutions to eq . ( [ qsh ] ) for several sets of the landau parameters . the curves show the zero - sound velocity as a function of temperature parameter @xmath191 for a pure @xmath192 pairing ( solid curves ) and for the case of pairing into the mixed @xmath193 state ( dashed curves ) . the plots are made for the sound propagating along the axis of the wave function of the condensate ( @xmath189 ) and in the orthogonal direction ( @xmath190 ) . for the case of mixed pairing we have chosen @xmath194 , in agreement with that found in realistic calculations by different authors ( see , _ e.g. _ , ref . as one can see , a small admixture of the @xmath8 state does not modify markedly the dispersion curves obtained for the pure @xmath5-wave superfluid . the sound waves are anisotropic . at fixed temperature the velocity of the sound grows along with deviation of the wave vector from the axis of the order parameter . the sound speed is maximal for orthogonal propagation . as regards the temperature dependence , immediately below the critical temperature @xmath2 the velocity of zero sound goes down , and the undamped wave disappears at some temperature @xmath195 when the sound velocity becomes smaller than the fermi velocity , @xmath196 ( although the mode with exponentially small damping can exist in some region below this temperature ) . thus the undamped collective excitation may or may not exist at some temperature , depending on the values of @xmath23 and @xmath24 . if it does , its velocity @xmath197 will always be greater than unity . let us summarize our results . we have studied the linear response of a superfluid neutron liquid to an external vector field in the limit @xmath22 . the calculation is made for the case of @xmath0 condensate which is expected to exist in the superdense core of neutron stars due to spin - orbit and tensor pairing interactions . by analyzing the poles of the longitudinal response we have found the low - energy spectrum of sound - like collecive excitations caused by density fluctuations in the condensate . previously the sound - like excitations were investigated for a triplet condensate caused by central pairing forces in @xmath6 @xcite . our dispersion equation ( [ qsh ] ) represents a generalization of the above results to the case of pairing caused by noncentral spin - orbit and tensor interactions and naturally recovers previous results obtained for the case of central forces . the sound - like spectrum of a fermi liquid substantially depends on the residual particle - hole interactions which are conventionally described by a set of landau parameters . we have limited our consideration to the first two terms of this expansion . this approach can be considered as a model of the fermi - liquid interactions , although there are indications that the higher - order landau interactions do not affect the longitudinal response functions @xcite . unfortunately , the density - dependent landau parameters entering the dispersion equation are not reliably known for an asymmetric nucleon matter ( although , in principle , these can be evaluated theoretically @xcite ) . therefore we have studied solutions to eq . ( [ qsh ] ) for several sets of the landau parameters . we found that the sound waves are anisotropic and a small admixture of the @xmath8 state does not modify markedly the dispersion curves obtained for the pure @xmath5-wave superfluid neutrons . at fixed temperature the velocity of the sound grows along with deviation of the wave vector from the axis of the order parameter . the sound speed is maximal for orthogonal propagation . immediately below the critical temperature @xmath2 the velocity of zero sound decreases when the temperature goes down , and the undamped wave disappears at some temperature @xmath198 when the sound velocity becomes smaller than fermi velocity . ( although the mode with exponentially small damping can exist in some region below this temperature ) . thus the undamped collective excitation may or may not exist at some temperature , depending on the values of @xmath23 and @xmath24 . if it does , its velocity will always be greater than fermi velocity . in our analysis , we have assumed that the axis of the order parameter is equally oriented everywhere . it is necessary to notice , however , that texture effects can orient different parts of the sample differently , and therefore give a range of frequency shifts which together appear as a broad line . the texture effects can be minimized by an external magnetic field , although the magnetic field has no consequence on the dispersion of the sound wave except that it serves to fix the relative orientation of the spin - orbital wave function associated with the order parameter , if the dipole interaction is taken into account . on the other hand the texture effects could play an important role in neutrino cooling of neutron star at the latest stage . indeed , the sound wave can emit a neutrino pair through neutral weak currents while crossing the border where the axis changes its direction . neutrino radiation is possible also due to collisions of sound waves @xcite . as already mentioned in the introduction , the sound waves are known to play an important role also in a superfluid heat conduction when the transverse electron motion is strongly suppressed by a magnetic field @xcite . various applications of the results obtained in this paper will be considered elsewhere . all the results of this paper depend on the fermi - liquid functions @xmath23 , @xmath24 which parametrize the normal fermi liquid . the only way to evaluate these functions for a superdense asymmetric nuclear matter is to estimate their values from first - principles calculations . although , in practice , such work is in progress @xcite , the complete information on the landau parameters for a neutron matter is not available at the moment . only the well - known conditions @xcite of the matter stability with respect to long - wave perturbations can be used in order to limit the landau parameters .

the linear response of a superfluid neutron liquid onto external vector field is studied for the case of @xmath0 pairing . in particular , we analyze the case of neutron condensation into the state with @xmath1 which is conventionally considered as the preferable one in the bulk matter of neutron stars . consideration is limited to the case when the wave - length of a perturbation is large as compared to the coherence length in the superfluid matter and the transferred energy is small in comparison with the gap amplitude . the obtained results are used to analyse collisionless sound - like excitations of the superfluid condensate . zero sound ( if it exists ) is found to be anisotropic and undergoes strong decrement below some temperature threshold depending substantially on the intensity of fermi - liquid interactions .

a challenge to the contemporary relativistic cosmology is provided by a set of observational data , indicated on late time acceleration expansion of the whole universe as well as the initial era , the inflationary epoch @xcite-@xcite . in the framework of the general relativity ( gr ) , inflationary model , quasi stable de sitter can not be realized as a possible physically acceptable model without any extra scalar matter field . it is needed to include some matter fields , like scalar field(s ) to resolve it . another approach which is totally revolutionary is to replace einstein gravity by some extended forms of classical geometrical objects . a reasonable candidate to explain this situation is modified gravity , in which we replace einstein - hilbert action , given by @xmath5 ( here @xmath6 is the boundary term ) by another set of geometrical objects like second order invariants @xmath7 or functions of ricci scalar @xmath8 , gauss - bonnet topological invariant @xmath9 and so on ( see @xcite- @xcite for reviews ) . historically , to be more precise , the simplest potentially reasonable candidate was @xmath3 gravity , a theory which it was proposed originally before recent activities @xcite ) and later motivated in light of the recent observational data , as a valid , physically reasonable , ghost - free and stable alternative theory instead of gr @xcite- @xcite . another type of modified gravity is the one in which geometry has been coupled to the matter fields non- minimally ( see @xcite for an updated review of such models ) . different types of non - minimally coupled models have been proposed like @xmath10 where @xmath11 stands for matter lagrangian and @xmath1 @xcite , where @xmath12 is the trace of the energy momentum tensor of matter fields , which is defined by @xmath13 or after a simple checking , it can be rewritten as the following ] : @xmath14 in this theory , @xmath15 . because of simplicity and beauty form , this theory attracted several activities in literature @xcite-@xcite . symmetry is an important issue to be addressed in any physical system under study . there are two classes of symmetries : global symmetries , in which the physical system ( dynamical system ) respects some types of transformations , which are defined by functions of coordinates . another is local , in which the conservation law gives us a `` local '' conserved quantity like charge . as we mentioned it before , @xmath3 gravity is a ghost free , and conformally equivalent to the scalar field theory in einstein frame . there are several interesting features in this theory to be useful for late time acceleration , dark energy and dark matter halo problems . also , inflation can be realized successfully using this simple and effective modified theory of gravity . recently to resolve dark matter problem , a new type of modified gravities has been proposed as titled mimetic model and modified versions of it @xcite-@xcite . basic hidden idea behind this new modification of gravity is to propose a conformally invariant , scalar theory of gravity in which scalar degree of freedom does not cause any problem with ghosts . the way is to reparametrize metric as a conformal transformation of an auxiliary metric , `` unphysical '' metric . the point is , the scalar field appeared as conformal function and plays the role of an internal degree of freedom . this scalar field is not ghost and it is constructed to have unit norm @xmath16 . this norm is defined on physical metric . it is remembering for us the role of velocity of a test particle with unit norm in the comoving frame of particle coordinates . an interesting feature is if we write flrw cosmological equations , an extra term proportional to @xmath17 , appeared . it mimics dark matter . so , it was adequate to name it as `` mimetic dark matter '' or briefly mimetic gravity . to unify @xmath0 gravity with this very interesting mimetic theory , it has been proposed mimetic @xmath0 gravity @xcite as a new class of modified gravities with the same inspiration as mimetic theory . because of its complexity and more physical solutions , this new mimetic @xmath0 deserves further physical investigations . very recently the dynamical behavior of mimetic f ( r ) has been investigated @xcite . our aim here is to address noether symmetry issue of such mimetic models . in literature noether symmetry has been investigated for different types of modified gravity like @xmath0 , @xmath18 , galileons and so on @xcite-@xcite . our aim in this paper is to explore noether symmetry for the mentioned above to modify gravities : mimetic @xmath19 and @xmath20 . the present paper is organized as follows : formal framework for @xmath1 theory of gravity is presented in section [ f(r , t ) ] . the formalism of f ( r ) mimetic theory is motivated in section [ f(r)mimetic ] . the noether symmetry approach is well understood briefly in section [ ns ] . noether symmetry is applied to @xmath1 in section [ ns1 ] and for @xmath0 mimetic model in section [ ns2 ] . we summarize in the last section . we adopted a `` god '' given system of units @xmath21 , where the gravitational coupling constant is given by @xmath22 . let us to start by the following simple extension of @xmath0 gravity in four dimensional spacetime : @xmath24 in the action , @xmath8 stands for the ricci scalar of the riemannian space - time , @xmath12 is trace of the energy , momentum tensor of matter lagrangian @xmath25 , is defined by the simple expression @xmath26 . trace is an important quantity in quantum theory of inflation and quantum gravity . the dynamical quantity is just metric @xmath27 . so , there is only a single field equation for it . if we calculate the variation of action we obtain : @xmath28 \sqrt{-g}d^{4}x\ , , \end{aligned}\ ] ] in this variation , we define : @xmath29 and @xmath30 , respectively . since @xmath8 is the only geometrical quantity in our theory , we recall standard variational expressions for it as the following : @xmath31 a simple check point is to evaluate it in a local flat coordinate frame , when @xmath32 . a variation of the connection termination , @xmath33 gives us : @xmath34 consequently , we can write the final simplified form of variation of @xmath8 as the following : @xmath35 by plugging these expressions in the total variation of action , @xmath36 , we find : @xmath37 \sqrt{-g}d^{4}x\ , . \label{var1}\end{aligned}\ ] ] to simplify more these functional , we need to redefine an auxiliary tensor field @xmath38 as a part of the variation of trace @xmath39 as the following : @xmath40 and in a similar form as we define @xmath12 , we are able to define trace of @xmath38 as the following : @xmath41 using these simplifications , we write the following form of the equation of motion of @xmath1 : @xmath42 it is remarkable that , when geometry is decoupled from the matter part , in the limit @xmath43 , we recover equation of motion of @xmath0 gravity from ( [ field ] ) . trace of ( [ field ] ) gives us the following equation : @xmath44 an alternative form of ( [ field ] ) is obtained as the following : @xmath45 conservation of the energy , momentum tensor in this type of modified gravity is checked by computing the covariant divergence of eq . ( [ field ] ) . using the following geometrical identity we obtain : @xmath46 \equiv 0\ , , \end{aligned}\ ] ] so , conservation of energy , momentum tensor , gives us the following vector current term : @xmath47\ , .\ ] ] for a simplicity , we consider the flat flrw metric in the following form : @xmath48 here @xmath49 stands for the scale factor . if we write down lagrangian of @xmath1 for this metric and if we assumed that the universe is filled with matter fields with effective pressure @xmath50 and energy density @xmath51 , we obtain @xmath52 . since this equation is a constraint , and because of @xmath53 , we introduce a pair of lagrange multipliers as the following @xmath54 . the point like lagrangian after an integration part by part is written as the following ( we set @xmath55 ) : @xmath56 where we suppose that @xmath57 . if @xmath58 , the lagrangian density is written : @xmath59 this is the form of lagrangian in @xmath0 theory . we suppose that ( [ l1 ] ) is an acceptable assumption for many cosmological applications , like matter dominant era or radiation . the associated equations of motion are given by a set of second order ordinary differential equations , euler - lagrange ( el ) equations , are given by the following : @xmath60 for @xmath61 they are obtained as the foloowing : @xmath62 to pass the case of @xmath58 , we obtain : @xmath63 our aim in this paper is to investigate noether symmetry issue of ( [ l1 ] ) . briefly , we are interested to know how noether symmetry is able to `` fix '' mathematical forms of @xmath64 . although @xmath1 provides a reasonable and good extension of @xmath0 theory , it does not respect conformal symmetry . also , extra degrees of freedom are possible . so , instabilities due to ghosts probably are happening . to resolve conformal symmetry and to be ghost - free , a model recently proposed as titled mimetic @xmath65 gravity @xcite . it is inspired from the mimetic theory @xcite , a model in which dark matter problem is resolved as an integration constant . also , it is self consistent with conformal symmetry . the basis of any type of mimetic theory is to parameterize of the riemannian metric tensor @xmath27 as the following conformally transformed formula @xcite @xmath66 here we introduced a pair of auxiliary objects : the first is an auxiliary metric ( unphysical and without dynamics ) @xmath67 and the second is a scalar field degree of freedom @xmath68 which has generally ghost , freedom . it is well known that using an orthogonality of metric , this scalar field satisfies the following constraint equation of motion : @xmath69 if we know the background metric @xmath70 , this equation fixes the form of the scalar field . if we interpret @xmath71 as the components of a four velocity @xmath72 , then normalized @xmath73 implies a possible normalization of the @xmath68 . this normalization can be understood as the first integral of the equation of motion for @xmath68 . we emphasize here that the auxiliary metric @xmath67 is an internal object of the space - time manifold following @xcite , we write the following action for mimetic @xmath0 gravity in metric formalism : @xmath74\ , , \ ] ] here @xmath75 , @xmath8 is the ricci scalar which is computed by the physical metric @xmath27 , and we also include the matter lagrangian by @xmath11 . what we need is to parametrize the physical metric according to ( [ metricreparametrization ] ) . we rewrite the action of theory in the following equivalent form : @xmath76\ , .\ ] ] thus , we perform variation with respect to auxiliary metric @xmath67 , we obtain : @xmath77 = 0\ , , \end{aligned}\ ] ] as a convention , here @xmath78 means @xmath79 , @xmath80 and @xmath81 are different derivative operators with respect to @xmath27 . also by using a similar `` dictionary '' as we used in @xmath1 , we define @xmath2 as the effective matter , energy - momentum tensor for @xmath11 is given by ( [ en1 ] ) . variation with respect to the scalar field @xmath68 gives us : @xmath82 \big\ } = 0\ , , \end{aligned}\ ] ] here like @xmath83 , we define trace of the energy - momentum tensor as @xmath84 . it has been proven that this new theory is conformally invariant and ghost free @xcite . so , it is remarkable to consider it as a valid extension of @xmath0 gravities . we consider the same flrw metric as ( [ metric ] ) . in this case , the constraint equation leads to @xmath85 where @xmath86 is cosmic time . it is not so hard task to write flrw equations for motion . actually , because our aim is to investigate noether symmetry , so what we need is just point like lagrangian of the mimetic @xmath0 scenario . to be more generally speaking , we slightly modify the original mimetic @xmath0 by including a potential term and by introducing a lagrange multiplier @xmath87 as the following : @xmath88\ , .\ ] ] the point like lagrangian for a fluid with pressure @xmath89 and by taking in to account that @xmath90 , is written in the following form : @xmath91 note that in general @xmath92 . equations of motion are written in the following forms : @xmath93 the second equation is just the standard definition of ricci scalar for flrw metric . the third one is reduced to the klein - gordon equation in the case of @xmath94 . the first equation becomes familiar as the equation of motion in @xmath0 gravity , if we set @xmath95 . clearly this equation posses de sitter solution as @xmath96 , @xmath97 . because of the importance of this model , we will study fixed points of the associated dynamical system , corresponding to this last case , when @xmath98 . we applied it before in the context of general relativity @xcite . here we review the basic concepts of an non - autonomous system . consider the following differential equation for an non - autonomous dynamical state vector @xmath99 : @xmath100 the equilibrium point , or fixed point is located at @xmath101 if and only if it solves the following algebraic equation for an instant of time , namely @xmath86 : @xmath102 we define the jacobian matrix ; @xmath103 $ ] must be bounded function of @xmath86 on a finite domain @xmath104 and furthermore it satisfies smoothly the lipschitz lemma , as the following : @xmath105 there is an important theorem about the asymptotic stability of the system in the vicinity of the equilibrium point : + ` theorem ` i : it is possible to linearize the system of equations in the vicinity of the fixed point in the following form : @xmath106 ` theorem ` ii : suppose that @xmath101 be the fixed point of the system @xmath107 , and it satisfies the following auxiliary conditions : @xmath108 it is adequate to define the time dependent function @xmath109 . we say that the system has an ` exponential stable equilibrium ` point of the linearized system ( [ sys2 ] ) , then this point is the exponential stable equilibrium point of the nonlinear system ( [ sys1 ] ) in this case the system of equations reduces to the following form : @xmath110 the first attempt is done by rewriting the system of equations in terms of a dimensionless `` time '' coordinate @xmath111 and a new set of dimensionless parameters as @xmath112 . equations read as the following non - autonomous system ( due to the potential term @xmath113 : @xmath114- \frac{x}{72h_0 ^ 2 f_{rr}},\\ & & \zeta'=-3-\frac{v(n)}{2h_0}\frac{e^{-\zeta}}{h}.\end{aligned}\ ] ] stationary ( fixed ) points are located at : @xmath115 because of the critical point is function of @xmath116 , it s moving when time is running . the jacobian of the linearized system is given by the following : @xmath117\end{aligned}\ ] ] where @xmath118 and : @xmath119-\frac{x}{72h_0 ^ 2 f_{rr}},\\ & & f_4=-3-\frac{v(n)}{2h_0}\frac{e^{-\zeta}}{h}.\end{aligned}\ ] ] the characteristic equation is given by : @xmath120 one eigenvalue is @xmath121 . another eigenvalues read as the following : @xmath122 @xmath123 @xmath124 in the above expressions : @xmath125 to have stable solution we must have @xmath126 . another eigenvalues @xmath127 must satisfy @xmath128 . it is remarkable to mention here that the stable manifold is defineb by the de sitter space - time . so , the model for time intervals @xmath129 has stable de sitter solution . symmetry is an important issue to be addressed in any physical theory . the basic property of a system with a defenite type of symmetry is the existence of an associated conserved quantity under this kind of symmetry . let us consider a typical dynamical system , is defined by a set of configurations coordinates set @xmath130 . generally speaking , the dimension of the system is defined as the number of independent coordinates of the system . we assume that the dynamics of the system are defined by the point like lagrangian is given by:@xmath131 . for each coordinate , it is possible to define a `` unique '' first order conjugate momentum : @xmath132 euler - lagrange equation of motion is given by the following set of n - ordinary second order diffrential equations : @xmath133 what we call it as _ noether symmetry approach_@xcite-@xcite is the existence of a `` unique '' vector field @xmath134 on tangent space @xmath135 ) : : @xmath136\,{,}\ ] ] if we can find `` generators '' coefficients @xmath137 , then we can strictly say that our dynamical system must satisfy the following geometrical constraint ( is called as _ lie derivative of the lagrangian _ : @xmath138 where explicitly we have : @xmath139\,{.}\ ] ] or equivalently we can write it as the following : @xmath140 it is an easy task to show that , existence of noether symmetry implies that the system has the following conserved ( local ) quantity : @xmath141 the equations are obtained setting to zero the coefficients of the terms @xmath142 in ( [ ns ] ) . they are several interesting applications of this symmetry approach in different cosmological models in different models @xcite-@xcite . in our paper we ll apply this approach to @xmath1 and mimetic @xmath0 theories of gravity . the plan in this section is to study the system of noether equations for lagrangian given by ( [ l1 ] ) . if we write down ( [ ns ] ) equation for ( [ l1 ] ) , @xmath143 and by putting the coefficients of @xmath144 and constant terms , we obtain the following system of partial differential equations for @xmath134 components @xmath145 : @xmath146 + it is a hard job to find all possible solutions of this nonlinear system of first order coupled partial differential equations ( pdes ) . inspired directly from the case of general relativity we ll limit ourselves to the following simple cases : * the case of einstein gravity with matter components @xmath147 : because any theory of modified gravity must be reduced to einstein - hilbert action at low curvature regime , we are interested to study a solution of @xmath1 in which the einstein - hilbert term is dominated as the leading term of theory . if we substitute @xmath147 , we obtain the following exact solution : @xmath148 we mention here that this is an special class of solutions founded before in literature @xcite ( for the case of purely @xmath149 see @xcite ) . the associated noether charge reads : @xmath150 the corresponding scale factor is obtained as follows : @xmath151^{2/3},\ \ h_0=\frac{3\sigma_0}{8c_1}.\end{aligned}\ ] ] this solution can be written in terms of q - exponential family @xcite : @xmath152 the pressure term deserves more investigations . the first term implies on the existence of the `` background pressure '' @xmath153 , which it can be realized by expectation vacuum energy of some quantum fields . the second term is a dark matter term , if we identify @xmath154 , as the dark matter density at present era , @xmath155 . in the case of perfect fluid with equation of state @xmath156 , we have @xmath157 . by a power - law expansion , @xmath158 , to be accelerated universe we must have @xmath159 , in our case the solution is not accelerating solution . * case with @xmath160 : if we would like to pass to modified gravity , we should try to solve systems of equations by assuming that @xmath161 , and by taking into the account the matter sector @xmath162 . this simple assumption gives us the following exact solutions for the model ( [ l1 ] ) under noether symmetry approach : @xmath163 here @xmath164 stands for a pair of arbitrary functions . the conserved noether charge associated with this model reads as follows : @xmath165 due to the leakage of more information of the form of @xmath166 , we can not integrate it . but it is possible to solve it for starobinsky inflationary model @xmath167 solution as the following : @xmath168 here @xmath169 and we define @xmath170 in this section we ll see how noether symmetry gives us useful information about mimetic @xmath3 theory . especially we would like to search for the possible forms of potential function @xmath171 in this model . thanks to the normalization condition @xmath172 , the potential is an implicit function of @xmath86 . so , generally speak in the dynamical system lagrangian is time dependent . let us to start by the same method as we used in the previous section . we write the following condition of ( [ ns ] ) for @xmath0 mimetic model : @xmath173 where for simplicity we assume that @xmath174 . using ( [ l2 ] ) , we can write the following system of differential equations , linear in @xmath145 : @xmath175 which are obtained setting to zero the coefficients of the different terms @xmath176 . we can distinguish some possible cases : motivated by recent observational data , indicates that we live in an accelerating universe , several forms of modified gravities have been proposed to resolve and explain this physical phenomena . one of the most popular and physically acceptable candidates is @xmath0 gravity and its extensions . in our work we established noether symmetry issue for two types of @xmath0 theories : a type of non - minimally coupled model is called as @xmath1 and mimetic @xmath0 . we started by reviewing the basic physical foundations of these theories . in @xmath1 model we have been written point - like lagrangian for flat flrw metric . we studied equations of motion and noether symmetry form for it . two important classes of solution for @xmath1 were found . in the first class , we show that the generalized q - exponential scale factor is an exact solution which it mimics the background with the background pressure . other solutions were found as general family of additive models , with an exact solution for scale factor in terms of elementary functions . this cosmological solution was obtained by considering the starobinsky model @xmath225 . in mimetic @xmath0 theory , we have been considered noether symmetries . we observed that there are two classes of solutions : the first is equivalent to the gr with dark matter and the calibrated cosmological constant . this case mimics a type of cosmological solutions with type iv future singularities , where higher derivatives of @xmath194 diverge . another case is modified gravity with two specified forms of potential functions : hybrid inflationary model , in which the scale factor evolves in the bouncing scenario . the second family is exponential form , in this case scale factor mimics the form of @xmath226 model perfectly . so , all cosmological models including @xmath226,bouncing and oscillatory solutions with future singularities are described perfectly by noether symmetrized @xmath1 and mimetic @xmath0 theories . we conclude that noether symmetry is able to provide a very excellent way to study cosmological implications of extended @xmath0 theories . we would like to thank the anonymous reviewer for enlightening comments related to this work . a. g. riess et al.,observational evidence from supernovae for an accelerating universe and a cosmological constant , astron . j. * 116 * , 1009 ( 1998 ) . s. perlmutter et al.,measurements of omega and lambda from 42 high redshift supernovae - supernova cosmology project , astrophys . j. * 517 * , 565 ( 1999 ) . p. de bernardis et al.,a flat universe from high resolution maps of the cosmic microwave background radiation , nature * 404 * , 955 ( 2000 ) . s. hanany et al.,maxima-1 : a measurement of the cosmic microwave background anisotropy on angular scales of 10 arcminutes to 5 degrees , astrophys . j. * 545 * , l5 ( 2000 ) . s. i . nojiri and s. d. odintsov , unified cosmic history in modified gravity : from f(r ) theory to lorentz non - invariant models , phys . rept . * 505 * , 59 ( 2011 ) [ arxiv:1011.0544 [ gr - qc ] ] . k. bamba , s. i . nojiri and s. d. odintsov , modified gravity : walk through accelerating cosmology , arxiv:1302.4831 [ gr - qc ] . k. bamba and s. d. odintsov , universe acceleration in modified gravities : @xmath227 and @xmath228 cases , arxiv:1402.7114 [ hep - th ] . s. i . nojiri and s. d. odintsov , modified gravity as realistic candidate for dark energy , inflation and dark matter , aip conf . * 1115 * , 212 ( 2009 ) [ arxiv:0810.1557 [ hep - th ] ] . s. capozziello and m. de laurentis , extended theories of gravity , phys . rept . * 509 * , 167 ( 2011 ) [ arxiv:1108.6266 [ gr - qc ] ] . s. capozziello and a. stabile , the weak field limit of fourth order gravity , in * frignanni , vincent r. ( ed . ) : classical and quantum gravity : theory and applications * chapter 2 ( nova press ) [ arxiv:1009.3441 [ gr - qc ] ] . s. capozziello and m. de laurentis , a review about invariance induced gravity : gravity and spin from local conformal - affine symmetry , found . * 40 * ( 2010 ) 867 [ arxiv:0910.2881 [ hep - th ] ] . s. capozziello and m. francaviglia , extended theories of gravity and their cosmological and astrophysical applications , gen . * 40 * ( 2008 ) 357 [ arxiv:0706.1146 [ astro - ph ] ] . s. nojiri and s. d. odintsov , introduction to modified gravity and gravitational alternative for dark energy , econf c * 0602061 * , 06 ( 2006 ) [ int . j. geom . phys . * 4 * , 115 ( 2007 ) ] [ hep - th/0601213 ] . h. a. buchdahl , non - linear lagrangians and cosmological theory , mon . not . 150 , 1 ( 1970 ) . t. harko , f. s. n. lobo , s. nojiri and s. d. odintsov , @xmath1 gravity , phys . d * 84 * , 024020 ( 2011 ) [ arxiv:1104.2669 [ gr - qc ] ] . m. jamil , d. momeni , m. raza and r. myrzakulov , reconstruction of some cosmological models in f(r , t ) gravity , eur . j. c * 72 * , 1999 ( 2012 ) [ arxiv:1107.5807 [ physics.gen-ph ] ] . f. g. alvarenga , a. de la cruz - dombriz , m. j. s. houndjo , m. e. rodrigues and d. sez - gmez , dynamics of scalar perturbations in f(r , t ) gravity , phys . d * 87 * , no . 10 , 103526 ( 2013 ) [ arxiv:1302.1866 [ gr - qc ] ] . s. d. odintsov and d. sez - gmez , @xmath229 gravity phenomenology and @xmath4cdm universe , phys . b * 725 * , 437 ( 2013 ) [ arxiv:1304.5411 [ gr - qc ] ] . t. harko , thermodynamic interpretation of the generalized gravity models with geometry - matter coupling , phys . d * 90 * , no . 4 , 044067 ( 2014 ) [ arxiv:1408.3465 [ gr - qc ] ] . a. f. santos , gdel solution in @xmath1 gravity , mod . a * 28 * , 1350141 ( 2013 ) [ arxiv:1308.3503 [ gr - qc ] ] . h. r. s. moraes , cosmology from induced matter model applied to 5d @xmath1 theory , astrophys . space sci . * 352 * , 273 ( 2014 ) . m. j. s. houndjo , d. momeni and r. myrzakulov , cylindrical solutions in modified f(t ) gravity , int . j. mod . d * 21 * , 1250093 ( 2012 ) [ arxiv:1206.3938 [ physics.gen-ph ] ] . e. h. baffou , a. v. kpadonou , m. e. rodrigues , m. j. s. houndjo and j. tossa , cosmological viable f(r , t ) dark energy model : dynamics and stability , astrophys . space sci . * 355 * , 2197 ( 2014 ) [ arxiv:1312.7311 [ gr - qc ] ] . h. shabani and m. farhoudi , cosmological and solar system consequences of f(r , t ) gravity models , phys . d * 90 * , 044031 ( 2014 ) [ arxiv:1407.6187 [ gr - qc ] ] . x. m. deng and y. xie , solar system s bounds on the extra acceleration of f(r , t ) gravity revisited , int . j. theor . ( 2014 ) . c. j. ferst and a. f. santos , gdel - type solution in @xmath1 modified gravity , arxiv:1411.1002 [ gr - qc ] . v. singh and c. p. singh , friedmann cosmology with particle creation in modified @xmath1 gravity , arxiv:1408.0633 [ gr - qc ] . s. capozziello and r. de ritis , noether s symmetries and exact solutions in flat nonminimally coupled cosmological models , _ class . _ * 11 * 107 ( 1994 ) . s. capozziello and r. de ritis , _ phys . lett . a _ * 195 * 48 ( 1994 ) . a. aslam , m. jamil , d. momeni and r. myrzakulov , noether gauge symmetry of modified teleparallel gravity minimally coupled with a canonical scalar field , can . j. phys . * 91 * , 93 ( 2013 ) [ arxiv:1212.6022 [ astro-ph.co ] ] . m. jamil , s. ali , d. momeni and r. myrzakulov , bianchi type i cosmology in generalized saez - ballester theory via noether gauge symmetry , eur . j. c * 72 * , 1998 ( 2012 ) [ arxiv:1201.0895 [ physics.gen-ph ] ] . m. jamil , f. m. mahomed and d. momeni , noether symmetry approach in f(r ) tachyon model , phys . b * 702 * ( 2011 ) 315 [ arxiv:1105.2610 [ physics.gen-ph ] ] . s. capozziello and a. de felice , f(r ) cosmology by noether s symmetry , jcap * 0808 * , 016 ( 2008 ) [ arxiv:0804.2163 [ gr - qc ] ] . s. nojiri and s. d. odintsov , inhomogeneous equation of state of the universe : phantom era , future singularity and crossing the phantom barrier , phys . d * 72 * , 023003 ( 2005 ) [ hep - th/0505215 ] . s. nojiri , s. d. odintsov and s. tsujikawa , properties of singularities in ( phantom ) dark energy universe , phys . d * 71 * , 063004 ( 2005 ) [ hep - th/0501025 ] . r. h. brandenberger , the matter bounce alternative to inflationary cosmology , arxiv:1206.4196 [ astro-ph.co ] . y. f. cai , exploring bouncing cosmologies with cosmological surveys , sci . china phys . astron . * 57 * , 1414 ( 2014 ) [ arxiv:1405.1369 [ hep - th ] ] .

extended @xmath0 theories of gravity have been investigated from the symmetry point of view . we briefly has been investigated noether symmetry of two types of extended @xmath0 theories : @xmath1 theory , in which curvature is coupled non minimally to the trace of energy momentum tensor @xmath2 and mimetic @xmath3 gravity , a theory with a scalar field degree of freedom , but ghost - free and with internal conformal symmetry . in both cases we write point -like lagrangian for flat friedmann - lemaitre - robertson - walker ( flrw ) cosmological background in the presence of ordinary matter . we have been shown that some classes of models existed with noether symmetry in these viable extensions of @xmath0 gravity . as a motivated idea , we have been investigating the stability of the solutions and the bouncing and @xmath4cdm models using the noether symmetries . we have been shown that in mimetic @xmath0 gravity bouncing and @xmath4cdm solutions are possible . also a class of solutions with future singularities has been investigated .

the standard qcd evolution equation @xmath2 describes response of the parton distribution function ( pdf ) @xmath3 to a change of the large energy scale @xmath4 . variable @xmath5 is identified as a fraction of the hadron momentum carried by parton of the type @xmath6 ( quark , gluon ) . evolution kernel @xmath7 is calculable within perturbative qcd . the above evolution equation ( [ eq : evolequ ] ) is an important ingredient in many qcd perturbative calculations . it can be solved using variety of the numerical methods , including monte carlo method . the knowledge of @xmath8 at certain initial @xmath9 , is required for solving evolution equation at other @xmath10 . the initial pdf is fitted to experimental data . it is important to provide formal proof of eq . ( [ eq : evolsolu ] ) , because it is a critical ingredient in several new monte carlo algorithms of the non - markovian type described in refs . @xcite and @xcite , and possibly in other future works . let us now explain in details notation used in eqs . ( [ eq : evolequ]-[eq : evolga ] ) . in function @xmath3 variable @xmath18 is the fraction of the hadron momentum carried by the parton of the type @xmath19 , i.e. gluon , quark or antiquark , at the high energy scale @xmath20 , conveniently translated into the `` evolution time '' variable @xmath21 . in qcd the pdf represents the wave function of the hadron close to the light - cone . see ref . @xcite for an expert discussion on the precise meaning of pdf in qcd , in a wide context of the so - called _ factorization theorems _ @xcite in the gauge quantum field theories . in this work we shall restrict ourselves to the most common qcd evolution equations of the dglap type @xcite , with the kernel splitting functions scheme were calculated in qcd at the two levels beyond the leading - logarithmic ( ll ) approximation . ] incorporating the qcd coupling constant ( for the sake of the simplicity of notation ) @xmath22 the qcd kernel functions are singular , with singularities of the type @xmath23 . we shall typically regularize them with the help of an explicit small infrared ( ir ) cutoff parameter can be @xmath24-dependent , without any loss of generality in the following treatment ] @xmath25 as follows : @xmath26 the important sudakov formfactor @xmath27 is directly related to the virtual part of the kernels : @xmath28 finally the ( bremsstrahlung - type ) auxiliary distribution @xmath29 e^{-\phi_k(t , t_n ) } \bigg[\prod_{i=1}^n { \euscript{p}}_{kk}^\theta ( t_i , z_i ) e^{-\phi_{k}(t_i , t_{i-1 } ) } \bigg ] \delta\big(x- \prod_{i=1}^n z_i \big ) \end{split}\ ] ] is an iterative solution of the flavour - diagonal evolution equation of eq . ( [ eq : evolga ] ) . see also fig . [ fig : bremstree ] for graphical representation of the above gluonstrahlung segment of the evolution . if our only aim was to prove the correctness of eq . ( [ eq : evolsolu ] ) as a solution of eq . ( [ eq : evolequ ] ) , then the simplest approach would be just to substitute it into this equation and check with a little bit of algebra that indeed it is the solution . our aims are however more general : ( i ) to derive eq . ( [ eq : evolsolu ] ) in a more systematic way , ( ii ) to understand better its relation to the other widely known and used iterative solutions of eq . ( [ eq : evolequ ] ) , ( iii ) to prove that its _ exclusive _ content , in terms of the fully differential distribution in all variables @xmath30 and @xmath31 , @xmath32 , for each @xmath33 , is exactly the same as in other iterative solutions , commonly used in the mc approaches . having all the above in mind , let us proceed methodically , first with deriving solution of the evolution of eq . ( [ eq : evolequ ] ) , in terms of a time - ordered exponential , widely used in the literature . next , we shall present first example of the derivation of eq . ( [ eq : evolsolu ] ) by means of reorganizing the evolution equation and solving it once again . then , we shall present second example of the derivation , in which the above time - ordered exponential is algebraically reorganized ( transformed ) into eq . ( [ eq : evolsolu ] ) . finally the third derivation of eq . ( [ eq : evolsolu ] ) based on straightforward reorganization of the multiple sums and integrals will be included in the appendix . the solution of eq . ( [ eq : evolequ ] ) can be established quickly and rigorously , for instance by means iteration , as a time - ordered exponential of the kernel operator @xmath34 in the vector ( linear ) space indexed by one continuous variable @xmath5 and one discrete @xmath6 . more precisely , eq . ( [ eq : evolequ ] ) in a more compact matrix notation reads @xmath35 and its solution in the same compact matrix notation is given by @xmath36 where we employ the following well known _ time - ordered exponential _ evolution operator . ] @xmath37 for @xmath38 . it is familiar to all readers from the textbooks of the quantum mechanics to be hermitian and @xmath39 to be unitary . ] . we define function @xmath40 to be equal 1 when @xmath41 and equal 0 otherwise . having defined the time ordered exponential evolution operator @xmath46 , let us quote its basic features and extend its definition for the latter use . the well known rule @xmath47 helps to manipulate products of the time - ordered exponents . we may also define the inverse operator for the `` backward evolution '' ( @xmath48 ) as follows @xmath49 where the inverse operator is constructed using , using eq . ( [ eq : torder ] ) , we leave to the reader . the matrix elements of @xmath50 can be non - positive and highly singular . ] with help of the above definition validity of eq . ( [ eq : urule ] ) can be extended to any @xmath52 . in the following we show how to resume singular @xmath53 terms by going back to the evolution equation , reorganizing it and solving it once again . we are going to show this standard trick in a detail , because , subsequently , we shall generalize it to the case of an arbitrary part of the kernel ( instead of the @xmath53 part ) . it is essentially a warm - up example . let us remind the reader , that in the above warm - up exercise we have resummed the relatively simple component @xmath63 of the evolution kernel , which was completely diagonal , both in @xmath6 and in @xmath45 . in this special case @xmath64 is trivially calculable , contrary to more general case of non - diagonal @xmath65 discussed in the following . let us now come back to our principal aim , proving eq . ( [ eq : evolsolu ] ) , where less trivial component of the kernel will be isolated / resummed . in order to prove eq . ( [ eq : evolsolu ] ) , we need to resum ( exponentiate ) the following part of the kernel @xmath66 which is diagonal in the flavour indices , but not in @xmath45 . this part of the kernel is always ir divergent and generates multiple gluon emission process , that is _ gluonstrahlung_. the remaining _ flavour - changing _ part of the full kernel is defined as @xmath67 . the original full evolution equation and its solution read @xmath68 at this point , the @xmath69-function of eq . ( [ eq : gabrems ] ) can be identified with the following operator @xmath70 where @xmath71 see also eq . ( [ eq : evolga ] ) . in order to derive eq . ( [ eq : evolsolu ] ) we proceed analogously as in the derivation of eq . ( [ eq : soliter ] ) ; we shall introduce @xmath72 in the evolution equation , similarly as we have introduced @xmath73 . the main complication will be in the non - commutative nature of @xmath72 . let us introduce in the evolution an equation auxiliary pdf @xmath74 getting after the differentiation @xmath75 after inserting eq . ( [ eq : ga2 ] ) we obtain @xmath76 the term proportional to @xmath77 gets eliminated @xmath78 and we return to the usual evolution equation @xmath79 with the usual solution @xmath80 the last step on the way to eq . ( [ eq : evolsolu ] ) is elimination of the operator @xmath81 being part of @xmath82 . the reason for that is that @xmath50 is not well suited for any numerical evaluation , especially of the mc type , due to alternating sign in the exponential expansion , hence it is better to eliminate it from the final result . it is done with the help of the following identity @xmath83 { \bf g}_b(t_{1},t_{0 } ) , \end{split}\ ] ] where @xmath84 . this identity is derived rather easily by inspecting each @xmath33-th term in the expansion of the time ordered exponent and applying the following relation-[eq : backevol ] ) and the accompanying discussion . ] ( analogous to eq . ( [ eq : trivial ] ) ) @xmath85 for each pair @xmath86 sandwiched between adjacent @xmath77 s . the final solution reads @xmath87 { \bf g}_b(t_{1},t_{0})\ ; { \bf d}(t_0 ) . \end{split}\ ] ] when translated into the integro - tensorial notation , the above formula turns out to be identical with our target eq . ( [ eq : evolsolu ] ) . in this way we have completed its derivation . it is now obvious why eq . ( [ eq : evolsolu ] ) we call a _ hierarchic _ solution of the evolution equation . it is because its components @xmath88 are solutions of another simpler evolution equation ( gluonstrahlung ) of its own . higher level evolution embeds lower level simpler evolution as a building block . a disadvantage of the derivation presented above is that it exploits the inverse evolution operator @xmath50 , which is in a general case difficult to define properly , while it drops out from the final result of eqs . ( [ eq : evolsolu ] ) or ( [ eq : solumatr2 ] ) anyway . the natural question is therefore whether we could derive eq . ( [ eq : evolsolu ] ) without introducing the operator @xmath50 in the intermediate stages of the proof . furthermore , going back to the modified evolution equation and solving it once again obscures the relation between variables @xmath89 in the non - hierarchic solution of eq . ( [ eq : soliter ] ) on one hand and the hierarchic one of eq . ( [ eq : evolsolu ] ) on the other hand . in the following we shall , therefore , present an alternative example of the derivation of eq . ( [ eq : solumatr ] ) without explicit use of the inverse evolution operator @xmath50 . in such a case , the relation between variables @xmath89 in eq . ( [ eq : soliter ] ) and eq . ( [ eq : evolsolu ] ) can be traced back ( recovered ) more easily . the following derivation will be strongly reminiscent to a derivation of identity @xmath90 by means of the taylor expansion and @xmath91 , but more transparent algebraically . ] with respect @xmath92 i.e. @xmath93 . let us introduce slightly modified evolution operator @xmath94 where @xmath39 was already defined as the time - ordered exponential in eq . ( [ eq : torder ] ) . the additional @xmath95-factor ensuring @xmath38 is will make the following algebra more compact . we define @xmath96 ( we shall set @xmath97 at the very end of calculation ) . the whole derivation relies on the following identity - dependence in @xmath98 . ] @xmath99 which can be derived using definition of eq . ( [ eq : torder ] ) , and reorganizing all integrations over @xmath30 s . the second derivative follows trivially : @xmath100 and the @xmath101th derivative is to verify this . ] @xmath102 where @xmath103 . now , let us use taylor expansion @xmath104 noticing that @xmath105 , we obtain @xmath106 we may set @xmath97 at this point . identifying @xmath107 , @xmath108 and @xmath109 we obtain more familiar identity . ] @xmath110 which leads immediately to eqs . ( [ eq : solumatr ] ) and ( [ eq : evolsolu ] ) . in this way we have completed the second _ proof of eq . ( [ eq : evolsolu ] ) this time without any reference to backward evolution operator @xmath50 . in addition to two elegant proofs of eq . ( [ eq : evolsolu ] ) presented in the previous sections , we include in appendix third proof , which relies on a rather straightforward method it starts from eq . ( [ eq : solumatr2 ] ) and through tedious reorganization of the sums over flavour indices ( change of the summation order ) and relabeling of the variables transforms it into eq . ( [ eq : evolsolu ] ) . the advantage of this third proof is that relation between integration and summation variables in both formulas is exposed in a manifest way . this might be useful in the construction of the exclusive mc model of the parton shower type . we are fully aware , of course , that all three derivations of eq . ( [ eq : evolsolu ] ) , shown in this work represent a well established mathematical formalism , very similar to that in use in the quantum mechanics , theory of markovian processes and renormalization group in the quantum field theory . we did not add much to the development of the corresponding area of mathematics . rather , our main aim was to customize this known formalism to the specific needs of solving the qcd evolution ( also numerically ) , such that solution of eq . ( [ eq : evolsolu ] ) and the other similar ones are obtained in an effortless and rigorous way . having all this in mind , let us comment on certain selected aspects of the presented formalism , on their possible refinements , extensions and applications . we shall concentrate mainly on two points : * extension to beyond - dglap evolutions in qcd , like ccfm and others . * possible application in the markovian mcs and the related question of the momentum sum rules and normalization of pdfs . in our definitions of the evolution of pdfs eqs . ( [ eq : evolequ]-[eq : evolsolu ] ) and the rest of the paper we have restricted ourself to dglap type @xcite evolution , leading - logarithmic ( ll ) version or its next - to - ll extensions . this restriction is however inessential and the validity of our derivations can be extended to a more general evolution equation @xmath111 in which the dependency of the generalized kernel @xmath112 is more general than only through the ratio @xmath113 . the above more general evolution equation is used for instance in the ccfm - type models of pdf @xcite . the dglap case of eq . ( [ eq : evolsolu ] ) is obviously covered by eq . ( [ eq : genevoleq ] ) , with the following identification @xmath114 the compact matrix notation used in the time - ordered exponentials can easily accommodate multiplications of the kernels @xmath112 such that all relevant algebra in the previous sections remains unchanged . let us only indicate how the product of two kernels gets redefined @xmath115 the reader can easily verify that the rest of the compact matrix algebra in our derivations remains unchanged . as already mentioned , results of this work were instrumental in the modelling qcd evolution using _ non - markovian _ type monte carlo techniques in refs . @xcite and @xcite . the corresponding mc programs simulate dglap and ccfm class evolutions . however , the solution eq . ( [ eq : evolsolu ] ) may be also used to construct an interesting example of the markovian mc in which single step in the markovian chain is a markovian process of its own . without going into fine details , let us indicate how this can be done . to this end we have to invoke momentum sum rule @xmath116 and reorganize slightly eq . ( [ eq : evolsolu ] ) . staying for simplicity with the dglap case ( ll and beyond ) the above sum rule determines virtual part of the kernel @xmath117 the above is used to set up properly markovian mc and in particular to split sudakov formfactor @xmath118 into bremsstrahlung part and the rest ( flavour changing part ) @xmath119 by means of pulling out @xmath120 and multiplying both sides of eq . ( [ eq : evolsolu ] ) by @xmath5 , we obtain the following formula suitable for a markovian mc @xmath121\ ; \int\limits_0 ^ 1 dz'_{n+1}\ ; \bigg [ \prod_{i=1}^{n } \int\limits_0 ^ 1 dz'_i\;dz_i\ ; \bigg]\ ; \\&\times { { \cal u}}_{kk}^b(t , t_n , z'_{n+1})\ ; \bigg [ \prod_{i=1}^n e^{-\phi^{a}_{k_{i-1}}(t_i , t_{i-1 } ) } z_i{\euscript{p}}_{k_i , k_{i-1}}^{a } ( t_i , z_i)\ ; { { \cal u}}_{k_{i-1}k_{i-1}}^b(t_i , t_{i-1},z'_i ) \bigg]\ ; \\&\times x_0 d_{k_0}(t_0,x_0 ) \delta\bigg(x- x_0 \prod_{i=1}^n z_i \prod_{i=1}^{n+1 } z'_i \bigg ) , \;\;\ ; k_n = k , \end{split}\ ] ] where @xmath122 obeys evolution equation of eq . ( [ eq : gabrems ] ) , with the substitution @xmath123 and with both side multiplied by @xmath45 . the evolution operator @xmath124 obeys nice `` unitarity '' rule @xmath125 for any @xmath24 , @xmath38 , in addition to the usual boundary condition @xmath126 . ( [ eq : evolsolu2 ] ) can be now used to define hierarchic ( nested ) markovian monte carlo algorithm . the normalized probability distribution of the forward markovian step in the flavour - changing upper level markovian process reads : @xmath127 for the lower level bremsstrahlung process one may use standard markovian mc technique of ref . @xcite . let us discuss selected details of the above scenario . here , all @xmath128 and @xmath30 , @xmath129 can be generated before generation of any @xmath45-variables in a separate markovian algorithm with the stopping rule being the usual condition @xmath130 . ( see ref . @xcite for more details on the the markovian class mc algorithms . ) variables @xmath31 of the flavour - changing kernels can also be generated at this early stage . the interesting question is : how and when do we generate @xmath131 according to gluonstrahlung operator @xmath132 ? if we have known an analytical ( even approximate ) representation of this function , or its precise value from the look - up tables , then we could readily generate them before entering into mc simulation of the bremsstrahlung subprocess . that would lead us to the use of the constrained mc of refs.@xcite for the bremsstrahlung segments . alternatively , @xmath131 may come out from a separate markovian mc module simulating gluonstrahlung sub - process starting at @xmath133 and stopping at @xmath30 , with the normalized probability distribution of single markovian step defined in ref . @xcite . in this latter case we would have simulated in the mc a hierarchic system of markovian processes , with the master flavour - changing markovian process and many markovian subprocesses , each of them implementing pure bremsstrahlung , flavour conserving , emissions . the above hierarchic markovian mc scheme , although quite interesting , seems to have no immediate practical importance . however , it may find applications in some future works . the basic aim of this paper is to provide solid technical foundation to other works in the area of the monte carlo simulation of the evolution of pdfs and parton shower in qcd . our basic result is the solution of the evolution equation of eq . ( [ eq : evolsolu ] ) , which was proved algebraically using three method . its primary application is construction of the constrained mc algorithms of ref . in addition , we also describe hypothetical application of such a solution in the markovian mc algorithm . although we are aware of many interesting relation of the discussed problems and solutions to other areas in physics , we did not attempt to elaborate on that too much , in order to keep the paper compact and transparent . let us mention also , that our solution can be used many times leading to a nested structure with several levels of the hierarchy . * acknowledgments * we would like to thank w. paczek for useful discussions and reading the manuscript . we thank for warm hospitality of the cern ph / th division were part of this work was done . we are going to show how to transform non - hierarchic solution in eq . ( [ eq : soliter ] ) ( with resummed virtual corrections ) into hierarchic solution in eq . ( [ eq : evolsolu ] ) ( with resummed gluonstrahlung ) using straightforward method of changing summation order and relabeling integration variables . the critical point in isolating two levels in the evolution , flavour - changing transitions and gluonstrahlung , will be the change of the summation order in eq ( [ eq : evolsolu ] ) , such that one is able to resum separately the pure bremsstrahlung segments obeying @xmath134 . these segments will form ( many ) functions @xmath135 , as defined in eq . ( [ eq : gabrems ] ) . the corresponding transformation of the summation order ( indexing ) looks schematically as follows @xmath136 where we have @xmath137 and the purpose of the upper index in this context is simply to show that the same index @xmath6 is repeated @xmath138 times . on the other hand variables @xmath139 and @xmath140 are truly different ( independent ) , with the upper index truly differentiating them . the aim is now to show that one can factorize out the functions @xmath135 and identify precisely the remaining functions and integrations . employing the above index transformation in the product of the @xmath34-functions we obtain @xmath141 where curly bracket embrace the diagonal elements @xmath142 , to be collected into the @xmath135-functions ; the remaining , nondiagonal ones , are now clearly isolated . each @xmath143 , @xmath12 in eq . ( [ eq : soliter ] ) is accompanied by an exponential factors . all of them ( including the first one which does not belong to any @xmath34 ) are now reorganized as follows : @xmath144 where all factors entering into products in @xmath135-functions are shown inside the curly brackets . together with the flavour - changing @xmath34 s , the above exponential form - factors look as follows : @xmath145 the other diagonal @xmath34 s will enter into @xmath135-functions . in this rather sketchy way we have shown that indeed eq . ( [ eq : soliter ] ) can be transformed into eq . ( [ eq : evolsolu ] ) by means of the straightforward reorganization of multiple sums and integrals . ( more systematic proof would require completing mathematical induction with respect to @xmath33 . ) k. golec - biernat , s. jadach , w. placzek , and m. skrzypek , _ acta phys . polon . _ * b37 * ( 2006 ) 17851832 , http://www.arxiv.org/abs/hep-ph/0603031[hep-ph/0603031 ] . s. jadach and m. skrzypek , _ acta phys . * b35 * ( 2004 ) 745756 , http://www.arxiv.org/abs/hep-ph/0312355[hep-ph/0312355 ] . j. c. collins , _ acta phys . polon . _ * b34 * ( 2003 ) 3103 , http://www.arxiv.org/abs/hep-ph/0304122[hep-ph/0304122 ] . j. c. collins , d. e. soper , and g. sterman , _ nucl . _ * b250 * ( 1985 ) 199 . j. c. collins and d. e. soper , _ nucl . * b193 * ( 1981 ) 381 . g. t. bodwin , _ phys . rev . _ * d31 * ( 1985 ) 2616 . lipatov , _ sov . * 20 * ( 1975 ) 95 ; + v.n . gribov and l.n . lipatov , _ sov . phys . _ * 15 * ( 1972 ) 438 ; + g. altarelli and g. parisi , _ nucl . * 126 * ( 1977 ) 298 ; + yu . l. dokshitzer , _ sov . * 46 * ( 1977 ) 64 . s. jadach and m. skrzypek , _ comput . _ * 175 * ( 2006 ) 511527 , http://www.arxiv.org/abs/hep-ph/0504263[hep-ph/0504263 ] . s. jadach and m. skrzypek , report cern - ph - th/2005 - 146 , ifjpan - v-05 - 09 , _ contribution to the hera lhc workshop , cern desy , 20042005 , http://www.desy.de/@xmath146heralhc/_ , http://www.arxiv.org/abs/hep-ph/0509178[hep-ph/0509178 ] . s. jadach and m. skrzypek , _ nucl . * 135 * ( 2004 ) 338341 .

the task of monte carlo simulation of the evolution of the parton distributions in qcd and of constructing new parton shower monte carlo algorithms requires new way of organizing solutions of the qcd evolution equations , in which quark@xmath0gluon transitions on one hand and quark@xmath0quark or gluon@xmath0gluon transitions ( pure gluonstrahlung ) on the other hand , are treated separately and differently . this requires certain reorganization of the iterative solutions of the qcd evolution equations and leads to what we refer to as a _ hierarchic iterative solutions _ of the evolution equations . we present three formal derivations of such a solution . results presented here are already used in the other recent works to formulate new mc algorithms for the parton - shower - like implementations of the qcd evolution equations . they are primarily of the non - markovian type . however , such a solution can be used for the markovian - type mcs as well . we also comment briefly on the relation of the presented formalism to similar methods used in other branches of physics . * ifjpan - v-04 - 09 * * hierarchically organized iterative solutions of the evolution equations in qcd@xmath1 * * s. jadach , m. skrzypek * _ and _ * z. was * + + _ to be submitted to acta physica polonica _ * ifjpan - v-04 - 09 + december 2006 * @xmath1this work is partly supported by the eu grant mtkd - ct-2004 - 510126 in partnership with the cern physics department and by the polish ministry of scientific research and information technology grant no 620/e-77/6.pr ue / die 188/2005 - 2008 .

the problem of detecting the symmetries of a curve has been studied extensively , mainly because of its applications in pattern recognition , computer graphics and computer vision . in pattern recognition , a common problem is how to choose , from a database of curves , the one which best suits a given object , represented by means of an equation @xcite . before a comparison can be carried out , one must bring the shape that needs to be identified into a canonical position . thus it becomes necessary to compute the symmetries of the studied curve . in this context , the computation of symmetries has been addressed using splines @xcite , by means of differential invariants @xcite , using a complex representation of the implicit equation of the curve @xcite , and using moments @xcite . in computer graphics , the detection of symmetries and similarities is important , both in the 2d and the 3d case , to gain understanding when analyzing pictures , and also in order to perform tasks like compression , shape editing or shape completion . many techniques involve statistical methods and , in particular , clustering ; see for example the papers @xcite , where the technique of transformation voting is used . other techniques are robust auto - alignment @xcite , spherical harmonic analysis @xcite , primitive fitting @xcite , and spectral analysis @xcite , to quote a few . in computer vision , symmetry is important for object detection and recognition . in this context , an analysis has been carried out using the extended gauss image @xcite and using feature points @xcite . in addition , there are algorithms for computing the symmetries of 2d and 3d discrete objects @xcite and for boundary - representation models @xcite . in the case of discrete objects ( polygons , polyhedra ) , the symmetries can be determined exactly @xcite . this can be generalized to the case of more complicated shapes whose geometry is described by a discrete object , as done in @xcite , where an efficient algorithm is provided . examples of this situation appear with bzier curves and tensor product surfaces , where the shape follows from the geometry of the control points . however , in almost all of the other above references , the goal is to find _ approximate _ symmetries of the shape . this is perfectly adequate in many applications , because the input is often a ` fuzzy ' shape , with missing or occluded parts in some cases . in fact , even if the input is exact , it is often an approximate , simplified model of a real object . here we shall consider a different perspective . we assume that our input is exact , and we want to deterministically detect the existence and nature of its symmetries , without converting to implicit form . more precisely , our input will be either a plane or a space curve @xmath0 defined by means of a rational parametrization with integer coefficients . our goal is to ( 1 ) determine whether @xmath0 has any symmetries , and ( 2 ) determine all symmetries in the affirmative case . notice that since we are dealing with a _ object , i.e. , the whole curve @xmath0 , we do not have a control polygon from which the geometry of the curve , and in particular its symmetries , can be derived . this could be the case if we were addressing a piece of @xmath0 , at least when @xmath0 admits a polynomial parametrization . in that situation , @xmath0 could be brought into bzier form , and then an algorithm like @xcite could be applied . in fact , in that case the algorithm of @xcite would be computationally more effective than ours , since essentially the analysis follows from a discrete object . however , this idea is no longer applicable when the whole curve is considered . additionally , an analysis of _ approximate _ symmetries of rational curves could be attempted by sampling points on the curve , and then applying algorithms like @xcite . in that case , the question is how to choose suitable zones for sampling , which amounts to collecting some information on the shape of the curve @xcite . a natural strategy is to look for notable points on the curve , like singularities , inflection points or vertices : since any symmetry maps notable points of a certain nature to the same kind of points or leaves them invariant , one might sample around these points . one thus obtains clusters of points that must be compared . there would be various possibilities for comparing these clusters , depending on the kind of symmetry one is looking for , all which should be explored . still , this approach only leads to an approximate estimate on the existence of symmetries , which is a different problem than the one considered in this paper . up to our knowledge , the deterministic problem for whole curves has only been solved in the case of implicit plane curves @xcite and in the case of polynomially parametrized plane curves @xcite . the case of space curves seems absent from the literature . in @xcite , the authors provide an elegant method to detect rotation symmetry of an implicitly defined algebraic curve , and efficiently find the exact rotation angle and rotation center . the method uses a complex representation @xmath1 of the curve . some cases not treated in @xcite are completed in @xcite , where similar ideas are applied to detect mirror symmetry . in contrast , our method applies directly to the parametrization , which is the most common representation in cagd , avoiding the conversion into implicit form . the approach in @xcite is similar to ours , although it should be noted that restricting to polynomial parametrizations yields an advantage for solving the problem fast and efficiently . the main ingredient in our method is the underlying relation between two parametrizations of a curve that are _ proper _ , i.e. , injective except perhaps for finitely many values of the parameter . essentially , whenever a symmetry is present , this symmetry induces an alternative parametrization of the curve . furthermore , if the starting parametrization is proper , this second parametrization is also proper . since two proper parametrizations of a same curve are related by means of a mbius transformation @xcite , we can reduce the problem to finding this transformation . thus , _ involutions _ , i.e. , symmetries with respect to a point , line or plane , can be detected and determined for plane and space curves . for rotations , we need one more ingredient : a formulation in terms of complex numbers for plane curves , or the pythagorean - hodograph assumption for space curves . in practice , our methods boil down to computing greatest common divisors and finding real roots of univariate polynomials , which are tasks that can be performed efficiently . throughout the paper we shall consider a rational curve @xmath2 , where @xmath3 or @xmath4 , neither a line nor a circle , defined by means of a proper rational parametrization @xmath5 where @xmath6,\qquad \gcd(p_i , q_i ) = 1,\qquad i=1,\ldots , n.\ ] ] here `` @xmath7 '' refers to the greatest common divisor . since @xmath0 is rational , it is irreducible . one can check whether a parametrization of a plane curve is proper , and every rational plane curve can be properly reparametrized without extending the ground field . see @xcite for a thorough study on properness and a proof of these claims , and see @xcite for similar results for rational space curves . we recall some facts from euclidean geometry @xcite . an _ isometry _ of @xmath8 is a map @xmath9 preserving euclidean distances . any isometry @xmath10 of @xmath8 is linear affine , taking the form @xmath11 with @xmath12 and @xmath13 an orthogonal matrix . in particular @xmath14 . the isometries of the plane and space form a group under composition that is generated by reflections , i.e. , symmetries with respect to a hyperplane , or _ mirror symmetries_. an isometry is called _ direct _ when it preserves the orientation , and _ opposite _ when it does not . in the former case @xmath15 , while in the latter case @xmath16 . the identity map @xmath17 of @xmath8 is called the _ trivial symmetry_. an isometry @xmath18 of @xmath8 is called an _ involution _ if @xmath19 , in which case @xmath20 is the identity matrix and @xmath21 . the nontrivial isometries of the euclidean plane are classified into reflections , rotations , translations , and glide reflections . the special case of _ central symmetries _ is of particular interest and corresponds to a rotation by an angle @xmath22 . central and mirror symmetries are involutions . the classification of the nontrivial isometries of euclidean space again includes reflections ( in a plane ) , rotations ( about an axis ) , and translations , and these combine in commutative pairs to form twists , glide reflections , and rotatory reflections . composing three reflections in mutually perpendicular planes through a point @xmath23 , yields a _ central inversion _ with center @xmath23 , i.e. , a symmetry with respect to the point @xmath23 . the special case of rotation by an angle @xmath22 is again of special interest , and it is called an _ axial symmetry_. central inversions , reflections , and axial symmetries are involutions . by bzout s theorem , an algebraic curve other than a line can not be invariant under a translation or glide reflection , and a space curve can , in addition , not be invariant under a twist . we shall refer to the remaining isometries as _ symmetries _ , and we shall say that a plane or space curve @xmath0 is _ symmetric _ , if it is invariant under a nontrivial symmetry . any algebraic curve in the plane , neither a line nor a circle , has finitely many symmetries @xcite . we need the following lemma to show the same result for _ nondegenerate _ space curves , i.e. , space curves not contained in a plane . [ lem : angles ] let @xmath24 be a nondegenerate irreducible space curve , invariant under a rotation with axis @xmath25 and angle @xmath26 . then @xmath27 , with @xmath28 an integer . for any plane @xmath29 normal to @xmath25 , a rotation about @xmath25 induces a rotation of the same angle on @xmath29 around the point @xmath30 . but then @xmath27 , with @xmath31 an integer that is at most the number of points in the intersection @xmath32 ; however , this is at most @xmath33 by definition of the degree of a nondegenerate irreducible curve . [ lem : parallelrotations ] let @xmath24 be an irreducible space curve , invariant under two rotations @xmath34 with axes @xmath35 . then @xmath35 can not be parallel . suppose that @xmath35 are parallel . let @xmath29 be a plane normal to @xmath35 that intersects @xmath0 in at least one point . the set @xmath32 is invariant under both rotations . but if a set of planar points exhibits rotation symmetry , then the rotation center must be the barycenter of the points , implying that @xmath36 . [ finite - involutions ] let @xmath24 be a space curve different from a line or a circle . then @xmath0 is invariant under at most : * one central inversion ; * finitely many rotation symmetries , whose axes are all concurrent ; * finitely many mirror symmetries , whose planes share a point . this result is known to hold when @xmath0 is degenerate @xcite , so assume that @xmath0 is nondegenerate . ( i ) : if @xmath0 is invariant under two central inversions with symmetry centers @xmath37 and @xmath38 , then it is invariant under their composition , which is a translation by @xmath39 along the direction @xmath40 @xcite . since @xmath0 is not a line , it can not be invariant under a nontrivial translation , implying that @xmath41 . ( ii ) : the composition of two rotations with axes @xmath35 is : ( a ) when the axes are parallel , a rotation with axis parallel to @xmath42 ; ( b ) when the axes intersect , a rotation with axis passing through @xmath43 ; ( c ) when the axes are skew , a twist . we can discard the cases ( a ) ( by lemma [ lem : parallelrotations ] ) and ( c ) . in the remaining case ( b ) , if @xmath0 is invariant under three rotations with axes intersecting pairwise in three distinct points forming a plane @xmath29 , then the composition of any two rotations with axes @xmath35 yields a rotation with axis @xmath25 intersecting @xmath29 transversally in a point away from the third axis @xmath44 . but then the axes @xmath25 and @xmath44 are skew , which is case ( c ) and can not happen . finally , suppose we have an infinite number of rotation axes @xmath45 meeting in a point @xmath23 . the set of lines through @xmath23 forms a real projective plane @xmath46 , which is compact . the points @xmath47 corresponding to the axes @xmath45 will therefore have a point of accumulation @xmath48 . any neighbourhood of @xmath48 will contain an infinite number of points in @xmath49 , corresponding to an infinite number of axes in @xmath45 . these axes meet in infinitely many distinct angles . the composition of two rotations with concurrent axes is another rotation , about an axis perpendicular to the concurrent rotation axes . if the rotations have rotation angles @xmath50 and their axes meet with an angle @xmath51 , then the composition is a rotation by an angle @xmath52 , where @xcite @xmath53 since there are finitely many @xmath54 by lemma [ lem : angles ] but infinitely many angles @xmath51 , we get infinitely many angles @xmath52 as well , therefore contradicting lemma [ lem : angles ] . we conclude that @xmath0 has at most finitely many rotation symmetries , whose axes are concurrent . ( iii ) : the composition of two mirror symmetries with planes @xmath55 is : ( a ) if the planes are parallel , a translation ; ( b ) if the planes intersect , a rotation about @xmath56 making twice the angle as between @xmath55 . case ( a ) can be discarded , so all mirror symmetries of @xmath0 have intersecting planes , and the statement follows in the case that @xmath0 has at most two mirror symmetries . suppose @xmath0 has at least three mirror symmetries @xmath57 with corresponding planes @xmath58 . if these three planes intersect in a point at infinity , then any two pairs , say @xmath59 and @xmath60 , intersect in parallel lines @xmath61 and @xmath62 in the finite plane , which can not happen by lemma [ lem : parallelrotations ] . it follows that the three planes intersect in a point @xmath63 in the finite plane . suppose there is a fourth mirror symmetry @xmath64 with plane @xmath65 not containing @xmath23 . then @xmath65 intersects @xmath66 in a certain line @xmath67 . if @xmath67 does not pass through @xmath23 , then @xmath0 has rotation symmetries about the two skew axes @xmath25 and @xmath67 . but then @xmath0 would be invariant under their composition , which is a twist , and this can not happen . so @xmath65 also contains @xmath23 and ( iii ) holds . [ invol - finite ] the number of symmetries of a plane or space curve , other than a line or a circle , is finite . the following theorem forms the foundation of our method . we need the definition of a _ mbius transformation _ ( on the affine real line ) , which is a rational function @xmath68 in particular the identity map is a mbius transformation , which we refer to as the _ trivial _ transformation . [ fundam - result ] the curve @xmath0 in is invariant under a nontrivial symmetry @xmath10 of the form if and only if there exists a nontrivial mbius transformation @xmath69 , with real coefficients @xmath70 , such that @xmath71 moreover , for any @xmath10 there is a unique mbius transformation @xmath69 satisfying . if there are two mbius transformations @xmath72 satisfying , then @xmath73 . since @xmath74 is proper , it follows that @xmath75 . for the first claim : `` @xmath76 '' : let @xmath77 . since @xmath74 is proper , @xmath78 is defined for all but finitely many values , and @xmath79 is a birational map @xmath80 from the real line to itself . any such map lifts to a birational automorphism of the complex projective line @xmath81 , which are known to be mbius transformations @xcite . it follows that @xmath69 takes the form and is nontrivial because @xmath10 is nontrivial . moreover , since @xmath69 maps the real line to itself , we can assume that the coefficients of @xmath82 are real . `` @xmath83 '' : if @xmath84 for some nontrivial isometry @xmath10 and nontrivial mbius transformation @xmath69 , we observe that @xmath77 is an alternative parametrization of @xmath0 , and therefore that @xmath0 is invariant under @xmath10 . equation relates the symmetries of @xmath0 to mbius transformations in the parameter domain . we obtain the following lemma . [ lem : order ] suppose an isometry @xmath10 and mbius transformation @xmath69 are related by . for any integer @xmath31 , the composition @xmath85 if and only if @xmath86 . because @xmath74 is proper , its inverse @xmath78 exists as a rational map , and @xmath87 . the result follows from @xmath88 . at this point one could in principle find the symmetries of @xmath0 by determining @xmath10 and @xmath69 satisfying the equation . however , the resulting polynomial system would involve too many variables in the coefficients of @xmath69 and @xmath10 to be solved efficiently . in the following section we propose an efficient method to determine the symmetries of @xmath0 . to consider plane and space curves in one go , we embed @xmath89 into @xmath90 as the plane of points with zero third component . the mappings on @xmath89 are lifted to mappings of @xmath90 leaving the third component invariant . for technical reasons , we assume that @xmath91 is well defined , and that @xmath92 are also well defined , nonzero , and not parallel . notice that this amounts to requiring that the _ curvature _ @xmath93 is well defined and nonzero at @xmath94 . since this is the case for almost all parameters @xmath95 , this condition holds after applying an appropriate , even random , linear affine change of the parameter @xmath95 . assume that the plane or space curve @xmath0 in is invariant under a nontrivial symmetry @xmath18 . by theorem [ fundam - result ] , there is a mbius transformation @xmath69 satisfying . our strategy will be to first , in sections [ sec : dzero][sec : spacerotations ] , express all unknown parameters in @xmath96 , and @xmath69 as rational functions of a single parameter @xmath97 of @xmath69 . substituting these rational functions into and clearing denominators , one obtains three polynomials in @xmath95 , whose coefficients are polynomials in @xmath97 . for to hold identically for all @xmath95 , each of these polynomial coefficients must be zero , which happens if and only if their greatest common divisor vanishes . removing from this polynomial all factors for which the mbius transformation or symmetry is not defined or not invertible , one obtains a polynomial @xmath98 in which every real root corresponds to a symmetry . let @xmath10 be a plane rotation or an involution in @xmath89 or @xmath90 . alternatively , let @xmath10 be any isometry and @xmath0 be a pythagorean - hodograph curve . we can now formulate the main theorem of the paper , which will be proved case - by - case in sections [ sec : dzero][sec : spacerotations ] . [ th - alg ] the curve @xmath0 has a nontrivial symmetry @xmath10 if and only if @xmath98 has a real root @xmath97 at which the parameters of @xmath96 , and @xmath69 are well defined . each real root of @xmath98 determines a mbius transformation , which corresponds uniquely to a symmetry of @xmath0 by theorem [ fundam - result ] . by corollary [ invol - finite ] , @xmath0 has at most finitely many symmetries , implying that @xmath98 can not be identically zero . finally , observe that one can directly find the symmetry type and its elements by analyzing the set of fixed points of the symmetry @xmath99 . in particular , @xmath100 is 1 for a symmetry with respect to a plane ; 2 for a symmetry with respect to a line ; 3 for a rotation symmetry or central inversion . if @xmath101 , equation becomes @xmath102 where @xmath103 and @xmath104 . applying the change of variables @xmath105 and writing @xmath106 , we obtain @xmath107 without loss of generality , we assume that @xmath108 is well defined at @xmath109 and that @xmath110 are well defined , nonzero , and not parallel . evaluating at @xmath94 yields @xmath111 while differentiating once and twice and evaluating at @xmath109 yields @xmath112 taking inner products and using that @xmath113 is orthogonal , we get @xmath114 from which we can write @xmath115 as a rational function of @xmath116 , @xmath117 a straightforward , but lengthy , calculation yields @xmath118 for any invertible matrix @xmath119 and vectors @xmath120 . taking the cross product in and using that @xmath113 is orthogonal , one obtains @xmath121 we analyze separately the cases @xmath122 and @xmath123 . for @xmath122 , implies that multiplying @xmath113 by the matrix @xmath124 $ ] gives the matrix @xmath125 $ ] so that @xmath126 . for @xmath123 , multiplying @xmath113 by the matrix @xmath127 $ ] gives the matrix @xmath128\ ] ] and @xmath129 . one sets @xmath15 to find the direct transformations and @xmath16 to find the opposite transformations . substituting @xmath130 , one expresses @xmath113 as a matrix - valued rational function of @xmath116 . finally from one expresses @xmath131 as a vector - valued rational function of @xmath116 . if @xmath133 , we may and will assume @xmath134 after scaling the coefficients of @xmath69 if necessary . differentiating twice , we get @xmath135 evaluating and at @xmath94 yields @xmath136 from , and using that @xmath113 is orthogonal , we can express @xmath137 solely in terms of @xmath97 . taking the cross product of , and using again and that @xmath113 is orthogonal , one obtains @xmath138 using and taking norms , one reaches @xmath139 which amounts to @xmath140 . if this equation does not have a real root , then we know that @xmath0 does not have any symmetry of any type . if it does , then we proceed to write @xmath113 in terms of @xmath97 . by computing the dot product of , and using that @xmath113 is orthogonal , we get @xmath141 next we consider separately involutions and plane rotations for which the corresponding mbius transformation has parameter @xmath134 . assume that @xmath18 is a nontrivial involution . then @xmath142 implying that @xmath143 . by lemma [ lem : order ] , @xmath144 , implying that @xmath145 , @xmath146 , and @xmath147 . if @xmath148 , then @xmath149 and @xmath150 , and therefore @xmath151 , which contradicts that @xmath152 is nontrivial . therefore @xmath153 . so , @xmath154 , and from we can write @xmath155 as a rational function of @xmath97 . by changing the parametrization if necessary , we can determine @xmath156 by assuming that the numerator and denominator of the above fraction have no real root in common . alternatively , we can take the gcd of the numerator and denominator and find the common real roots @xmath97 , determine the corresponding @xmath157 from by considering both signs separately , and @xmath156 from @xmath158 . moreover , the denominator of this expression vanishes iff @xmath159 which happens precisely when @xmath160 , with @xmath161 a nonzero constant . however , in that case @xmath162 is not defined , which contradicts one of our initial assumptions . hence , the above expression for @xmath156 is well defined . once the rational functions @xmath163 and @xmath164 are obtained , the matrix @xmath165 can again be determined from its action on @xmath166 , and @xmath167 , which is given by equations , , and . one finds @xmath168 from evaluating at @xmath94 . in order to detect rotation symmetries in the plane , we identify the euclidean plane with the complex plane as @xmath169 . thus the parametrization @xmath170 in yields a parametrization @xmath171 where we denoted the curve by the same symbol @xmath0 . writing @xmath172 for the rotation center and @xmath26 for the rotation angle , equation takes the form @xmath173 differentiating this expression we obtain @xmath174 without loss of generality , we assume that @xmath175 is well defined at @xmath94 and @xmath176 , so that evaluating and at @xmath94 gives @xmath177 expressing the symmetry in terms of the mbius transformation . differentiating , evaluating at @xmath109 , and solving for @xmath178 , we deduce @xmath179.\ ] ] since all coefficients of @xmath69 are real , the imaginary part of the above expression for @xmath178 must be zero , which yields rational expressions @xmath180 , and therefore also @xmath181 . the symmetry itself is determined from . some rotation symmetries of space curves are found by the previous algorithms . since axial symmetries are involutions , these rotations will be found directly by the method of section [ sec : involutions ] . by the cartan - dieudonn theorem , any rotation @xmath10 in @xmath90 is the composition of three reflections @xmath182 . however , if our curve @xmath0 has a rotation symmetry @xmath10 , then these reflections @xmath182 need not be mirror symmetries of our curve . taking compositions of the reflections found in section [ sec : involutions ] will therefore only yield some of the rotations of @xmath0 . in addition , the rotations whose corresponding mbius transformation have parameter @xmath183 will be found by the method of section [ sec : dzero ] . unfortunately it seems that the approach of the previous section can not be generalized to find all rotations of space curves . the complex numbers in the plane can be replaced by quaternions @xcite in space , which provides a convenient way to express rotations . for instance , one can give quaternion versions of and . the difficulty , however , comes from the fact that quaternions are not commutative , which makes it hard to eliminate the parameters defining the rotation in the resulting equations . because of this , we have not been able to prove a version of theorem [ th - alg ] for space rotations . in fact , we are uncertain whether it is possible in that case to write all parameters of the mbius transformation as rational functions of just one of them . while we might write these parameters in terms of two of them , which would yield a bivariate polynomial system , we feel that this solution is not satisfactory computationally . we therefore pose the question here as a pending problem . however , if the curve @xmath0 from is a _ pythagorean - hodograph curve _ , i.e. , if there exists a rational function @xmath184 such that @xmath185 then we do not run into the same obstacle . such curves form an important topic in computer aided geometric design , and they have been studied extensively both from the point of view of theory and of applications @xcite . in this case we can determine all rotation symmetries , since gives a rational function @xmath186 , from which we find a rational function @xmath156 by , and finally a rational function @xmath187 . after this we determine the symmetry as before . the sign of @xmath157 is not easily determined in advance , so it is necessary to carry out the algorithm for both cases . for all but the simplest examples , the computations quickly become too large to be carried out by hand , and a computer algebra system is needed . we have therefore implemented and tested the algorithms in @xcite . the resulting worksheet with implementations and examples can be downloaded from the website of the third author @xcite . 0.45 0.45 let @xmath0 be the _ deltoid _ from figure [ fig : deltoid ] , defined parametrically as the image of the map @xmath188 , @xmath189 we follow the recipe from section [ sec : planerotations ] to find its rotations . this parametrization is well defined at @xmath109 and satisfies @xmath190 . using that the imaginary part of is zero , we find @xmath191 , @xmath192 the symmetry is determined by , as @xmath193 @xmath194 substituting these expressions into yields a rational function in @xmath95 , whose coefficients are polynomials in @xmath97 . this rational function is identically zero if and only if the gcd @xmath195 of the coefficients in the numerator is zero . removing all factors for which either the mbius transformation or the symmetry is not defined or not invertible , we obtain a polynomial @xmath196 . substituting its three real zeros @xmath197 into , we find rotations about @xmath198 with angles @xmath199 . there are no additional symmetries for the case @xmath101 . m5.5em@*5ccccccc@ curve & deg . & parametrization @xmath200 & @xmath201 & # @xmath202 & # @xmath203 & @xmath204 & @xmath205 + cubic + & 3 & @xmath206 & @xmath207 & 1 & 1 & 0.09 & 0.09 + folium + & 3 & @xmath208 & @xmath209 & 0 & 1 & 0.23 & 0.29 + epitrochoid + & 4 & @xmath210 & @xmath211 & 0 & 1 & 0.12 & 0.14 + 3-leaf rose + & 4 & @xmath212 & @xmath213 & 2 & 3 & 0.53 & 0.82 + deltoid + & 4 & @xmath214 & @xmath209 & 2 & 3 & 0.17 & 0.47 + lemniscate + & 4 & @xmath215 & @xmath216 & 1 & 3 & 0.15 & 0.25 + astroid + & 6 & @xmath217 & @xmath218 & 3 & 5 & 0.75 & 1.65 + cardioid offset + & 8 & see worksheet @xcite & & 0 & 1 & 0.30 & 0.37 + m7em@*5ccccc@ curve & degree & parametrization @xmath74 & @xmath201 & # @xmath203 & @xmath205 + twisted cubic + & 3 & @xmath219 & @xmath220 & 1 & 0.26 + cusp + & 4 & @xmath221 & @xmath207 & 1 & 0.52 + axial sym . 1 + & 4 & @xmath222 & @xmath209 & 1 & 2.22 + crunode + & 4 & @xmath223 & @xmath216 & 3 & 39.6 + inversion 1 + & 7 & @xmath224 & @xmath225 & 1 & 6.8 + space rose + & 8 & @xmath226 & @xmath213 & 1 & 57.6 + inversion 2 + & 11 & @xmath227 & @xmath225 & 1 & 75.6 + let @xmath0 be the _ twisted cubic _ , defined parametrically as the image of the map @xmath228 from and using that @xmath154 we find @xmath229 @xmath230 one obtains @xmath231 from , , and and @xmath168 from evaluating at @xmath94 . substituting @xmath232 , and @xmath156 into , one finds that the coefficients of the powers of @xmath95 in the numerator have gcd @xmath233 for direct transformations and @xmath234 for opposite transformations . since the mbius transformation is invertible , @xmath157 is nonzero and there is only one relevant factor @xmath235 for the direct transformations . substituting @xmath236 into @xmath237 , and @xmath156 , one finds the mbius transformation @xmath238 with corresponding axial symmetry @xmath239 this symmetry is depicted in figure [ fig : twistedcubic ] by connecting corresponding points by lines . there are no additional symmetries for the case @xmath101 . we test the performance of the algorithms in section [ sec : determinesym ] for several classical curves on a dell xps 15 laptop , with 2.4 ghz i5 - 2430 m processor and 6 gb ram . additional technical details are provided in the worksheet @xcite . for each curve , tables [ tab : planesymmetries ] and [ tab : spaceinvolutions ] list the degree , a standard parametrization , a reparametrization @xmath201 that brings this curve into general position , the number @xmath240 and @xmath241 of ( nontrivial ) rotations and involutions found , and the average cpu times @xmath204 and @xmath205 ( seconds ) of the computations . in each case the algorithm for finding rotations performs better than the algorithm for finding involutions . the curves `` inversion 1 '' and `` inversion 2 '' are constructed to have precisely one central inversion . note that the algorithm for finding involutions of space curves performs significantly worse for the crunode in table [ tab : spaceinvolutions ] than for the other curves of degree four . the reason seems to be that , besides the degree of a parametrization , the sizes of the coefficients greatly influence the performance of the algorithms . even rational numbers with relatively small numerator and denominator get blown up by simple arithmetic operations . this is a common problem when computing in exact arithmetic . most of the computation time is spent by substituting the symmetry and mbius transformation in and finding the polynomial conditions on the parameter @xmath97 . we have provided effective methods for determining the involution symmetries of a plane or space curve defined by a rational parametrization , and the rotation symmetries of a rational plane or pythagorean - hodograph space curve . examples were given , and the algorithms have been implemented in . experiments show that we can generally quickly compute the rotation and involution symmetries of curves of relatively low degree . the current approach can not guarantee to find all the rotation symmetries of a space curve efficiently , except in the case of pythagorean - hodograph curves . for the general case , an alternative strategy seems to be required . calabi e. , olver p.j . , shakiban c. , tannenbaum a. , haker s. ( 1998 ) , _ differential and numerically invariant signature curves applied to object recognition _ , international journal of computer vision , 26(2 ) , pp . 107135 . muntingh g. , personal website , software + https://sites.google.com/site/georgmuntingh/academics/software stein , w.a . ( 2013 ) , _ sage mathematics software ( version 5.9 ) _ , the sage development team , http://www.sagemath.org . schnabel r. , wessel r. , wahl r. , klein r. ( 2008 ) , _ shape recognition in 3d - point clouds _ , the 16-th international conference in central europe on computer graphics , visualization and computer vision 08 . tarel j.p . , cooper d.b . ( 2000 ) , _ the complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition _ , ieee transactions on pattern analysis and machine intelligence , vol . 22 , no . 7 , pp . 663674 . taubin g. ( 1991 ) , _ estimation of planar curves , surfaces , and nonplanar space curves defined by implicit equations , with applications to edge and range image segmentation _ , ieee trans . patter analysis and machine intelligence , vol .

this paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization . we provide effective methods to compute the involution and rotation symmetries for the planar case . as for space curves , our method finds the involutions in all cases , and all the rotation symmetries in the particular case of pythagorean - hodograph curves . our algorithms solve these problems without converting to implicit form . instead , we make use of a relationship between two proper parametrizations of the same curve , which leads to algorithms that involve only univariate polynomials . these algorithms have been implemented and tested in the system .

the linear failure rate distribution with parameters @xmath1 and @xmath2 , ( @xmath3 ) which is denoted by @xmath4 , has the cumulative distribution function ( cdf ) @xmath5 and probability density function @xmath6 note that if @xmath7 and @xmath8 , then the lfr distribution is reduced to exponential distribution with parameter @xmath9 ( @xmath10 ) , and if @xmath11 and @xmath12 then we can obtain the rayleigh distribution with parameter @xmath13 ( @xmath14 ) . a basic structural properties of @xmath15 is that it is the distribution of minimum of two independent random variables @xmath16 and @xmath17 having @xmath18 and @xmath14 distributions , respectively ( sen and bhattachrayya , 1995 ) . if @xmath19 denotes the cdf of a random variable then a generalized class of distributions can be defined by @xmath20 for @xmath21 and @xmath22 , where @xmath23 is the incomplete beta function ratio and @xmath24 is the incomplete beta function . many authors considered various forms of @xmath19 and studied their properties : eugene et al . ( 2002 ) ( beta normal distribution ) , nadarajah and kotz ( 2004 ) ( beta gumbel distribution ) , nadarajah and gupta ( 2004 ) and barreto - souza et al . ( 2011 ) ( beta fr@xmath25chet distribution ) , famoye et al . ( 2005 ) , lee et al . ( 2007 ) and cordeiro et al . ( 2008 ) ( beta weibull distribution ) , nadarajah and kotz ( 2006 ) ( beta exponential distribution ) , akinsete et al . ( 2008 ) ( beta pareto distribution ) , silva et al . ( 2010 ) ( beta modified weibull distribution ) , mahmoudi ( 2011 ) ( beta generalized pareto distribution ) , cordeiro et al . ( 2011 ) ( beta - exponentiated weibull distribution ) , cordeiro et al . ( 2011 ) ( beta - weibull geometric distribution ) , singla et al . ( 2012 ) ( beta generalized weibull distribution ) , cordeiro et al . ( 2012 ) ( beta generalized gamma distribution ) and cordeiro et al . ( 2012 ) ( beta generalized normal distribution ) . in this article , we propose a new four parameters distribution , referred to as the blfr distribution , which contains as special sub - models : the beta exponential ( be ) , beta rayleigh ( br ) , generalized linear failure rate ( glfr ) and linear failure rate ( lfr ) distributions , among others . the main reasons for introducing blfr distribution are : ( i ) the additional parameters introduced by the beta generalization is sought as a means to furnish a more flexible distribution . ( ii ) some modeling phenomenon with non - monotone failure rates such as the bathtub - shaped and unimodal failure rates , which are common in reliability and biological studies , take a reasonable parametric fit with this distribution . ( iii ) the blfr distribution is expected to have immediate application in reliability and survival studies . ( iv ) blfr distribution shows better fitting , more flexible in shape and easier to perform and formula for modeling lifetime data . the reminder of the paper is organized as follows : in section 2 , we define the blfr distribution and outline some special cases of the distribution . we investigate some properties of the distribution in this section . some of these properties are the limit behavior and shapes of the pdf and hazard rate function of the blfr distribution . section 3 provides a general expansion for the moments of the blfr distribution . in section 4 , we discuss maximum likelihood estimation and calculate the elements of the observed information matrix . application of the blfr distribution is given in the section 5 . a simulation study is performed in section 6 . finally , section 7 concludes the paper . consider that @xmath26 is the density of the baseline distribution . then the probability density function corresponding to ( [ eq.fb ] ) can be written in the form @xmath27 we now introduce the blfr distribution by taking @xmath28 in ( [ eq.fb ] ) to be the cdf ( [ eq.flfr ] ) of the lfr distribution . hence , the blfr density function can be written as @xmath29 and we use the notation @xmath30 . the hazard rate function of blfr distribution is given by @xmath31 plots of pdf and hazard rate function of the blfr distribution for different values of it s parameters are given in fig . [ fig.den ] and fig . [ fig.hz ] , respectively . 1 . if @xmath32 , then we get the generalized linear failure rate distribution ( @xmath33 ) which is introduced by sarhan and kundu ( 2009 ) . 2 . if @xmath32 and @xmath7 , then we get the generalized exponential distribution ( ge ) ( gupta and kundu , 1999 ) . 3 . if @xmath32 and @xmath11 , then we get two - parameter burr x distribution which is introduced by surles and padgett ( 2005 ) and also is known as generalized rayleigh distribution ( gr ) ( kundu and raqab , 2005 ) . if @xmath34 , then ( 2.2 ) reduces to the linear failure rate distribution ( @xmath35 ) distribution . if @xmath7 , then we get the beta exponential distribution ( @xmath36 ) which is introduced by nadarajah and kotz ( 2006 ) . if @xmath11 , then we get the beta rayleigh distribution ( @xmath37 ) which is defined by akinsete and lowe ( 2009 ) and is a special case of beta weibull distribution ( famoye et al . , 2005 ) . if the random variable @xmath38 _ _ has blfr distribution , then the random variable @xmath39 satisfies the beta distribution with parameters @xmath40 and @xmath41 . therefore , @xmath42 satisfies the beta exponential distribution with parameters 1 , @xmath40 and @xmath41 ( @xmath43 ) . 8 . if @xmath44 and @xmath45 , where @xmath46 and @xmath47 are positive integer values , then the @xmath48 is the density function of @xmath46th order statistic of lfr distribution . the following result helps in simulating data from the blfr distribution : if @xmath49 follows beta distribution with parameters @xmath40 and @xmath41 , then @xmath50 follows blfr distribution with parameters @xmath51 , and @xmath41 . for checking the consistency of the simulating data set form blfr distribution , the histogram for a generated data set with size 100 and the exact blfr density with parameters @xmath52 , @xmath53 , @xmath54 , and @xmath55 , are displayed in fig [ fig.gd ] ( left ) . also , the empirical distribution function and the exact distribution function is given in fig [ fig.gd ] ( right ) . in this section , limiting behavior of pff and hazard rate function of the blfr distribution and their shapes are studied . * theorem 1 . * let @xmath48 be the pdf of the blfr distribution . the limiting behaviour of @xmath56 for different values of its parameters is given bellow : i. : : if @xmath57 then @xmath58 . ii . : : if @xmath59 then @xmath60 . iii . : : if @xmath61 then @xmath62 . : : @xmath63 . * proof : * the proof of parts ( i)-(iii ) are obvious . for part ( iv ) , we have @xmath64 it can be easily shown that @xmath65 and the proof is completed . * theorem 2 . * let @xmath48 be the density function of the blfr distribution . the mode of @xmath56 is given in the following cases : i. : : if @xmath57 and @xmath67 then @xmath48 has a unique mode in @xmath68 i. : : if @xmath57 and @xmath69 then @xmath48 has a unique mode in @xmath70 . ii . : : if @xmath59 then @xmath48 has at least one mode . * proof : * the proof is obvious and is omitted . * theorem 3 . * let @xmath71 be the hazard rate function of the blfr distribution . consider the following cases : i. : : if @xmath57 and @xmath72 then blfr distribution has an increasing hazard rate function . ii . : : if @xmath73 and @xmath72 then the hazard rate function of the blfr distribution is an increasing . : : if @xmath74 blfr distribution has a decreasing hazard rate function for @xmath75 , and @xmath71 is constant for @xmath76 . : : if @xmath77 and @xmath72 then @xmath71 is a bathtub - shaped . * proof : * \i . if @xmath57 then @xmath78 . therefore @xmath79 which is an increasing and linear function with respect to @xmath80 \ii . consider @xmath81 it implies that @xmath82 for @xmath83 and also , it is increasing with respect to @xmath84 . we have @xmath85 . now , rewriting the blfr density as function of @xmath86 , @xmath87 say , we obtain @xmath88 therefore , we have @xmath89 and we conclude that the hazard function of blfr distribution is increasing . \iii . if @xmath7 then @xmath90 and @xmath91 thus we have @xmath92 where @xmath93 , which implies the decreasing ( increasing ) hazard rate functions in this cases . \iv . it is difficult to determine analytically the regions corresponding to the upside - down bathtub shaped ( unimodal ) and bathtub - shaped hazard rate functions for the blfr distribution . however , by some graphical analysis we can shows : bathtub - shaped hazard rate function correspond to @xmath77 and @xmath72 . the proof is completed . here , we present some representations of cdf , pdf , and the survival function of blfr distribution . the mathematical relation given below will be useful in this section . if @xmath41 is a positive real non - integer and @xmath94 , then @xmath95 and if @xmath41 is a positive real integer , then the upper of the this summation stops at @xmath96 , where @xmath97 \1 . we can express ( [ eq.fb ] ) as a mixture of distribution function of generalized lfr distributions as follows : @xmath98 where @xmath99 and @xmath100 is distribution function of a random variable which has a generalized lfr distribution with parameters @xmath9 , @xmath13 , and @xmath101 . we can express ( [ eq.fblfr ] ) as a mixture of density function of generalized lfr distributions as follows : @xmath102 where @xmath103 is density function of a random variable which has a generalized lfr distribution with parameters @xmath9 , @xmath13 , and @xmath101 . the cdf can be expressed in terms of the hypergeometric function and the incomplete beta function ratio ( see , cordeiro and nadarajah , 2011 ) in the following way : @xmath104 where @xmath105 . the @xmath106th moment of blfr distribution can be expressed as a mixture of the @xmath106th moment of generalized lfr distributions as follows : @xmath107 where @xmath108 is density function of a random variable @xmath109 which has a generalized lfr distribution with parameters @xmath9 , @xmath13 , and @xmath101 . consider @xmath110 is a random sample from blfr distribution . the log - likelihood function for the vector of parameters @xmath111 can be written as @xmath112 where @xmath113 . the log - likelihood can be maximized either directly or by solving the nonlinear likelihood equations obtained by differentiating ( [ eq.like ] ) . the components of the score vector @xmath114 _ _ are given by @xmath115 where @xmath116 is the digamma function . for interval estimation and hypothesis tests on the model parameters , we require the observed information matrix . the @xmath117 unit observed information matrix @xmath118 is obtained as @xmath119.\ ] ] where the expressions for the elements of @xmath120 are @xmath121 @xmath122 where @xmath123 is the trigamma function . under conditions that are fulfilled for parameters in the interior of the parameter space but not on the boundary , asymptotically @xmath124 where @xmath125 _ _ is the expected information matrix . this asymptotic behavior is valid if @xmath125 is replaced by @xmath126 , i.e. , the observed information matrix evaluated at @xmath127 . for constructing tests of hypothesis and confidence region we can use from this result . an asymptotic confidence interval with confidence level @xmath128 for each parameter @xmath129 , @xmath130 , is given by @xmath131 where @xmath132 is the @xmath46th diagonal element of @xmath126 and @xmath133 is the upper @xmath134 point of standard normal distribution . in this section , we provide a data analysis to see how the new model works in practice . this data set is given by aarset ( 1987 ) and consists of times to first failure of fifty devices . the data is given by 0.1 , 0.2 , 1 , 1 , 1 , 1 , 1 , 2 , 3 , 6 , 7 , 11 , 12 , 18 , 18 , 18 , 18 , 18 , 21 , 32 , 36 , 40 , 45 , 46 , 47 , 50 , 55 , 60 , 63 , 63 , 67 , 67 , 67 , 67 , 72 , 75 , 79 , 82 , 82 , 83 , 84 , 84 , 84 , 85 , 85 , 85 , 85 , 85 , 86 , 86 . in this section we fit blfr , glfr , lfr , gr , ge , rayleigh and exponential models to the above data set . we use the maximum likelihood method to estimate the model parameters and calculate the standard errors of the mle s , respectively . the mles of the parameters ( with std . ) , the maximized log - likelihood , the kolmogorov - smirnov statistic with its respective _ p_-value , the aic ( akaike information criterion ) , aicc and bic ( bayesian information criterion ) for the blfr , glfr , lfr , gr , ge , rayleigh and exponential models are given in table [ table1 ] . we can perform formal goodness - of - fit tests in order to verify which distribution fits better to the first data . we apply the anderson - darling ( ad ) and cramrvon mises ( cm ) tests . in general , the smaller the values of ad and cm , the better the fit to the data . for this data set , the values of ad and cm statistics for fitted distributions are given in in table [ table1 ] . the empirical scaled ttt transform ( aarset , 1987 ) can be used to identify the shape of the hazard function . the scaled ttt transform is convex ( concave ) if the hazard rate is decreasing ( increasing ) , and for bathtub ( unimodal ) hazard rates , the scaled ttt transform is first convex ( concave ) and then concave ( convex ) . the ttt plot for this data in fig . [ fig.ex1 ] shows a bathtub - shaped hazard rate function and , therefore , indicates the appropriateness of the blfr distribution to fit this data . the empirical distribution versus the fitted cumulative distribution functions of blfr , glfr , lfr , gr , ge , rayleigh and exponential distributions are displayed in fig . [ fig.ex1 ] . the results for this data set show that the blfr distribution yields the best fit among the glfr , lfr , gr , ge , rayleigh and exponential distributions . for this data , the k - s test statistic takes the smallest value with the largest value of its respective _ p_-value for blfr distribution . also this conclusion is confirmed from the values of the aic , aicc and bic for the fitted models given in table [ table1 ] and the plots of the densities and cumulative distribution functions in fig . [ fig.ex1 ] . using the likelihood ratio ( lr ) test , we test the null hypothesis @xmath135 : glfr versus the alternative hypothesis h1 : blfr , or equivalently , @xmath135 : @xmath7 versus @xmath136 : @xmath137 . the value of the lr test statistic and the corresponding _ p_-value are 3.4 and 0.019 , respectively . therefore , the null hypothesis ( glfr model ) is rejected in favor of the alternative hypothesis ( blfr model ) for a significance level @xmath138 0.019 . for test the null hypothesis @xmath135 : lfr versus the alternative hypothesis @xmath136 : blfr , or equivalently , @xmath135 : @xmath139 versus @xmath136 : @xmath140 , the value of the lr test statistic is 15.3 ( _ p_-value = 0.00047 ) , which includes that the null hypothesis ( lfr model ) is rejected in favor of the alternative hypothesis ( blfr model ) for any significance level . we also test the null hypothesis @xmath135 : gr versus the alternative hypothesis @xmath136 : blfr , or equivalently , @xmath135 : @xmath141 versus @xmath136 : @xmath142 . the value of the lr test statistic is 8.3 ( _ p_-value = 0.0158 ) , which includes that the null hypothesis ( gr model ) is rejected in favor of the alternative hypothesis ( blfr model ) for a significance level @xmath143 . for test the null hypothesis @xmath135 : ge versus the alternative hypothesis @xmath136 : blfr , or equivalently , @xmath135 : @xmath144 versus @xmath136 : @xmath145 , the value of the lr test statistic is 19.2 ( _ p_-value = 6e-05 ) , which includes that the null hypothesis ( gr model ) is rejected in favor of the alternative hypothesis ( blfr model ) for any significance level . this section provides the results of simulation study . simulations have been performed in order to investigate the proposed estimator of @xmath146 @xmath41,@xmath9 and @xmath13 of the proposed mle method . we generated 10000 samples of size @xmath147 and @xmath148 from the blfr distribution for each one of the six set of values of @xmath149 we assess the accuracy of the approximation of the standard error of the mles determined though the fisher information matrix . the approximate values of @xmath150 , @xmath151 , @xmath152 and @xmath153 are computed . the results for the blfr distribution is shown in table [ table.2 ] , which indicate the following results : ( i ) convergence has been achieved in all cases and this emphasizes the numerical stability of the mle method . ( ii ) the differences between the average estimates and the true values are almost small . ( iii ) these results suggest that the mle estimates have performed consistently . ( iv ) the standard errors of the mles decrease when the sample size increases . we define a new model , called the blfr distributions , which generalizes the lfr and glfr distributions . the blfr distributions contain the glfr , lfr , gr , ge , rayleigh and exponential distributions as special cases . the blfr distribution present hazard functions with a very flexible behavior . we obtain closed form expressions for the moments . maximum likelihood estimation is discussed . finally , we fitted blfr distribution to a real data set to show the potential of the new proposed class . cordeiroa , g. m. , gomez , a. e. , de silva , c. q. , ortega , e. e. m. ( 2011 ) . the beta exponentiated weibull distribution , _ journal of statistical computation and simulation _ , doi : 10.1080/00949655.2011.615838 . cordeiroa , g. m. , nascimento , a. d. c. , cintra , l. c. , rago , l. c. ( 2012 ) . beta generalized normal distribution with an application for sar image processing , _ statistics _ , doi : 10.1080/02331888.2012.748776 .

we introduce in this paper a new four - parameter generalized version of the linear failure rate ( lfr ) distribution which is called beta - linear failure rate ( blfr ) distribution . the new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems . it can have a constant , decreasing , increasing , upside - down bathtub ( unimodal ) and bathtub - shaped failure rate function depending on its parameters . it includes some well - known lifetime distributions as special sub - models . we provide a comprehensive account of the mathematical properties of the new distributions . in particular , a closed - form expressions for the density , cumulative distribution and hazard rate function of the blfr is given . also , the @xmath0th order moment of this distribution is derived . we discuss maximum likelihood estimation of the unknown parameters of the new model for complete sample and obtain an expression for fishers information matrix . in the end , to show the flexibility of this distribution and illustrative purposes , an application using a real data set is presented . msc : 60e05 ; 62f10 ; 62p99 . keywords : beta distribution ; hazard function ; linear failure rate distribution ; maximum likelihood estimation ; moments ; simulation .

cataclysmic variable stars ( cvs ) consist of a white dwarf accreting from a roche lobe filling main - sequence , slightly evolved , or brown dwarf companion . in systems where the white dwarf only has a weak or no magnetic field , an accretion disc forms around the white dwarf . see @xcite for a detailed review . from the start of mass transfer , cvs evolve towards shorter orbital periods , due to loss of orbital angular momentum via magnetic braking and gravitational wave radiation @xcite . a period minimum occurs when the mass of the donor becomes too low to sustain hydrogen burning , and it is driven out of thermal equilibrium , becoming partially degenerate , and no longer shrinks in response to mass loss @xcite . as the donor continues to lose mass , the orbital period increases , and the system evolves back to longer periods , with decreasing mass transfer rates . the period minimum for normal hydrogen - rich cvs is predicted theoretically to occur at @xmath36570minutes @xcite , and is observed at @xmath380minutes @xcite . there are a small number of cvs and related systems that have orbital periods below this minimum , including three confirmed cvs that have evolved donors stars that have been stripped of most of their hydrogen by mass - transfer or prior to the onset of mass - transfer ( v485 cen , ; ei psc , @xcite ; css100603:112253@xmath4111037 , @xcite ) . the majority of the known ultracompact mass - transferring binaries belong to the small class known as the am canum venaticorum ( am cvn ) binaries . these consist of a white dwarf accreting from a hydrogen - deficient ( semi-)degenerate donor , allowing them to reach their short orbital periods ( 5 to @xmath365 minutes ; see @xcite for a recent review ) . see @xcite for further discussion of the evolution and population of systems below the cv period minimum . a subset of cvs , known as dwarf novae , shows outbursts , in which the system brightens by several magnitudes for a period of days to weeks . these outbursts are thought to be caused by thermal instabilities in the accretion disc , and have been the subject of considerable observational and theoretical study . some dwarf novae show superoutbursts in addition to the normal outbursts , these last longer and are generally brighter . during a superoutburst , tidal interactions between the disc and the donor star cause the disc to become asymmetric . this results in periodic modulations in the lightcurve , known as superhumps ( e.g. @xcite ) . this superhump period is typically a few percent longer than the orbital period , and is related to the mass ratio of the system ( e.g. @xcite ) . the origin of superhumps is discussed in detail by @xcite . here we present time resolved optical spectroscopy and photometry of the helium rich cv , sbss1108 + 574 ( sdssj111126.83 + 571238.6 ) . the system was identified via a survey designed to uncover am cvn binaries in the photometric database of the sloan digital sky survey @xcite . its spectrum shows unusually strong hei emission in addition to the balmer emission lines . it was discovered in outburst on 2012 april 22 by the catalina real - time transient survey ( crts ; @xcite ) , and reported as a new su uma dwarf nova by @xcite . @xcite presented photometry of the outburst , and identified a possible orbital period of 55.367 minutes , in addition to superhumps . we obtained time - resolved spectroscopy of sbss1108 + 574 on 2012 february 28 and 2012 april 20 with the gemini multi - object spectrograph ( gmos ; @xcite ) at the gemini - north telescope on mauna kea , hawaii . we used the b600 + grating with a 1 arcsec slit . gmos has three 2048@xmath54608 e2v deep depletion ccds , which were used in six amplifier mode . the resulting spectra cover the wavelength range 41206973 , with an average dispersion of 1.85 per pixel . the observations consist of 65 spectra in total , most of which have an exposure time of 173 seconds . the observations in february were cut short after only about one half of the binary orbit , and so are insufficient to determine the orbital period of the system . a second attempt was made to complete a 3 hour observing block on 2012 april 20 , when the system was caught in an unexpected outburst ( it had not been identified as a dwarf nova prior to this outburst ) . these spectra cover approximately three binary orbits . the log of our spectroscopic observations is given in table [ t : speclog ] . .summary of our spectroscopic observations of sbss1108 + 574 . [ cols="<,<,>,>",options="header " , ] there is considerable scatter around the radial velocity curve shown in fig . [ f:4686rv ] , which is likely caused by the unusual line structure during the outburst ( see fig . [ f : trail ] ) . calculating the radial velocities from the line wings using two narrower gaussians gives similar results , but the effect of the unusual structure causes greater scatter in the radial velocities , and greater errors , and so we prefer the single gaussian method . the uncertainty on the period was estimated by carrying out 10000 bootstrap selections of the radial velocity curve . for each subset , 54 radial velocities were selected from the full radial velocity curve , allowing for points to be selected more than once , and the periodogram calculated , taking the strongest peak as the period . the standard deviation of these computed periods , ignoring those that correspond to higher harmonics , is taken as a measure of the uncertainty in the derived orbital period . superhumps are caused by a resonant interaction between the accretion disc and the donor star , that causes the disc to become asymmetric . the increased viscous dissipation caused by this interaction between the donor and the distorted disc , leads to the brightness variations observed during dwarf nova superoutbursts . the observed superhump period is the beat period between the orbital period of the system and the precession period of the deformed disc @xcite . as the superhump phenomenon is due to resonance , and the precession rate for resonant orbits depends on the mass ratio of the system , there is a strong link between the superhump period and the mass ratio . the superhump period - excess , @xmath6 is found to increase with increasing mass ratio , @xmath2 . an empirical relation is derived from eclipsing dwarf novae , in which the mass ratio and superhump excess can be measured independently @xcite . of the three alternative @xmath7 @xmath2 relations , the @xcite form , @xmath8 and the @xcite form , both assume @xmath9 when @xmath10 . this is a reasonable assumption as we would expect a secondary with negligible mass to have a negligible tidal interaction with the disc . the third version of the relation , given by @xcite , does not use this assumption . the @xcite formulation is usually favoured for am cvn binaries @xcite , as observations of the only known eclipsing system , sdss j0926 + 3624 agree best with this form @xcite , and so we use that version here . the total time covered by our spectroscopic observations is not sufficiently long to reach the accuracy of the photometric periods , and so we take the weak additional signal in fig . [ f : pgram ] as the orbital period . as the superhump period varies during our observations , we conservatively take the separation of the two strongest peaks in fig . [ f : pgram ] as a measure of the uncertainty on the superhump period for calculation of the excess . this gives the superhump excess in sbss1108 + 574 , @xmath7=0.0176@xmath10.0032 , and the mass ratio , @xmath2=0.086@xmath10.014 . we note that this is very large compared to the mass ratio found for css1122@xmath41110 , @xmath2=0.017 @xcite , despite the similar orbital periods . our larger estimate for the superhump excess is consistent with the value found by @xcite for their stage b superhumps , @xmath7=0.0174@xmath10.0002 . using the smaller excess they measured for their stage c superhumps , and the @xcite form of the @xmath7 @xmath2 relation , they derive a mass ratio @xmath2=0.06 . we note that the values we derive for @xmath2 using the @xcite form of the @xmath7 @xmath2 relation , are consistent with the values calculated using the @xcite form . for our data the @xcite formulation gives a larger value of @xmath2 , however , this relation is calibrated using the shorter superhump periods that occur late in the superoutburst , and so likely overestimates @xmath2 for our longer periods from earlier in the superoutburst . the line profiles and trailed spectra for the strongest lines , folded on the orbital period , 55.3minutes ( 26.03cyclesd@xmath11 ) , are shown in fig . [ f : trail ] . it is clear that there is greater flux in the redshifted line peak than the blueshifted peak ; this unexplained asymmetry is often seen in the spectra of outbursting cvs . we also note the presence of underlying absorption due to the optically thick outbursting disc , this is particularly noticeable in the hei 5875 line , and makes the s - wave difficult to discern when it crosses the line centres . an s - wave is clear in the strongest line , heii 4686 , but becomes weaker with decreasing line strength . the presence of this s - wave can be seen in the other lines shown in fig . [ f : trail ] , however , the hei lines appear to show a brighter varying signal with higher velocities than the s - wave , that is almost in anti - phase . a second varying signal is also seen in the h@xmath0 trailed spectrum . this feature is likely responsible for the strength of the second harmonic in the periodograms of these lines . there is no coherent s - wave visible in the trailed spectra when folded on these second harmonic frequencies ; thus we are confident that we have identified the correct period of this system . the corresponding doppler tomograms @xcite give a similar picture of emission in the disc . a bright spot is clearly visible in the maps for heii 4686 , h@xmath0 and h@xmath12 . the bright spot is not usually seen in cvs during outburst , as the bright outbursting disc normally outshines it . since these were calculated using the same zero phase for all lines , assumed from the heii 4686 radial velocities , hjd@xmath13=2456037.7551 , the maps may be rotated due to the unknown phase shift between our zero phase and the true zero phase of the white dwarf . note that the bright spot appears at slightly different phases in each line . the extended bright spot seen in h@xmath0 , overlaps in phase with the bright spots seen in both heii 4686 and h@xmath12 . some evidence of the bright spot may also be seen in the doppler maps of the hei lines , however , they are dominated by the brighter varying signal causing the band of increased emission in the right - hand quadrants . we note that we do not detect the presence of spiral arms in our data . spiral arms have been observed in a number of dwarf novae during outburst ( e.g. @xcite ) , and are thought to be caused by the tidal affect of the donor on the large disc . as our observations may correspond to an early point in the evolution of the outburst , it is possible that spiral arms develop later in the outburst . the h@xmath12 line profile reveals a strong feature in the high velocity wing of the blueshifted peak , centred at about @xmath4800kms@xmath11 . the redshifted peak also extends beyond the range of the bright spot visible in the trailed spectrum . the origin of this feature is unclear , and it appears in the doppler map as a ring of emission at higher velocity than most of the disc emission . we detect no rotation between doppler maps created using only the first and only the third orbit , further verifying our identification of the orbital period . we plot the roche lobes and stream velocities for a @xmath2=0.086 binary together with the heii 4686 doppler map in fig . [ f : stream ] . = 0.086 binary . the map was rotated by applying a @xmath40.08 phase shift compared to the maps in fig . [ f : trail ] . [ f : stream],scaledwidth=48.0% ] the velocity positions of the accretor , donor and centre of mass are also shown . again , the map may be rotated about its origin due to the unknown phase shift between our assumed zero phase and the true zero phase . the trailed spectra ( fig . [ f : trail ] ) show a slight reduction in the line flux at a phase of @xmath31 . to examine this further we construct a lightcurve of the heii 4686 ew , shown in fig . [ f : ewlc ] . this reveals a clear dip , which we attribute to an eclipse of the outer disc . , scaledwidth=48.0% ] this allows us to constrain the inclination of the system , as it must be large enough that the outer edge of the disc can be eclipsed . since there is no eclipse of the inner disc or the accretor , we can also place an upper limit on its value . using our derived value for @xmath2 , the @xcite formula for the roche lobe radius of the secondary , and the approximate tidal limit for the maximum radius of the accretion disc ( e.g. @xcite ) , we find @xmath14 . the outbursting disc is expected to be larger than the quiescent disc , significantly increasing the likelihood of a grazing eclipse during outburst . it is therefore likely that there will be no eclipse detectable in quiescence . the strength of the helium emission lines compared to the hydrogen lines in the average spectrum ( fig . [ f : qavspec ] and table [ t : ew ] ) , highlights the unusual nature of this system . whilst a detailed abundance analysis can not be carried out with our current data , this is a strong indication of a much greater helium abundance than normally seen in cvs . models of accretion discs and donors in cvs and am cvns indicate that very little hydrogen is required to excite strong balmer lines , and the hydrogen abundance in sbss1108 + 574 may be significantly lower than 10 per cent . our derived period , 55.3@xmath10.8 minutes ( 26.03@xmath10.38 cycles d@xmath11 ) , is well below the cv period minimum , clearly indicating that the donor is significantly evolved , having been stripped of most of its hydrogen by mass - transfer or prior to the onset of mass - transfer . our spectroscopic period favours the weak 55.36 minute ( 26.01@xmath10.01 cycles d@xmath11 ) signal detected from the photometry and identified as the orbital period , over the 56.34 minute ( 25.56 cycles d@xmath11 ) signal identified as the superhump period . we are therefore confident that the candidate orbital period signal detected in our photometry and by @xcite , is the correct orbital period . this confirms sbss1108 + 574 as one of the shortest period cvs known , and places it well within the am cvn period range . there are three proposed formation channels for the am cvn binaries , defined by the type of donor . the donor can be ( 1 ) a second , lower mass white dwarf @xcite , ( 2 ) a semi - degenerate helium star , or ( 3 ) an evolved main - sequence star that has lost most of its hydrogen envelope . the latter is thought to form via the ` evolved cv ' channel @xcite , but this formation channel has generally been considered to be unimportant in comparison to the double white dwarf and helium star channels . @xcite , however , argue that the evolved cv channel could contribute a significant fraction of the total am cvn binary population . sbss1108 + 574 has all the characteristics of an am cvn progenitor in the evolved cv formation channel . it must be noted that not all cvs with evolved donors reaching periods below the normal period minimum , will form am cvn binaries as they are currently recognised ( yungelson et al . , in preparation ) . many systems will in fact reach their own period ` bounce ' , and evolve back towards longer orbital periods , without depleting their hydrogen sufficiently to appear as am cvn binaries . current models suggest that the fraction of evolved cv channel am cvn binaries may be lower than previously predicted ( @xcite ; yungelson et al . , in preparation ) . it is clear , however , that systems like sbss1108 + 574 and css1122@xmath41110 @xcite fall between the standard definitions of am cvn binaries and cvs . the large mass ratio we derive , @xmath2=0.086@xmath10.014 , indicates a different evolution to the standard am cvn binary population ; for gp com ( p@xmath15=46.6minutes ; @xcite ) , @xmath2=0.018 @xcite , and for v396 hya ( p@xmath15=65.1minutes ; @xcite ) , @xmath2=0.013 @xcite . the fact that the system shows outbursts indicates that the mass transfer rate is still relatively high , which is not expected for am cvn binaries that have passed @xmath16 and evolved back to long periods ( e.g. ) . compare our estimated accretion rate , @xmath17 ( for a @xmath18 accretor ) , to the value @xcite derived for gp com , @xmath19 , and the value estimated for v396 hya , @xmath20 , again presenting a strong contrast between sbss1108 + 574 and the long period am cvn binaries . this indicates that sbss1108 + 574 is still evolving towards shorter orbital periods , becoming increasingly helium - rich . we have presented time resolved spectroscopy of the helium - rich dwarf nova sbss1108 + 574 ( sdssj1111 + 5712 ) , confirming the period detected photometrically during the 2012 april outburst . the system shows unusually strong helium emission in both outburst and quiescence , suggesting a high helium abundance . we measure the superhump period from our photometry as 56.34@xmath10.18minutes , consistent with the result of @xcite . the spectroscopic period is found to be 55.3@xmath10.8minutes , significantly below the normal period minimum ( @xmath380minutes ) , confirming the system as an ultra - compact cv containing a highly evolved donor . the relatively high accretion rate , together with the large mass ratio , suggests that sbss1108 + 574 is still evolving towards its period minimum . we thank the anonymous referee for useful comments and suggestions . pjc acknowledges the support of a science and technology facilities council ( stfc ) studentship . ds , trm , btg and eb acknowledge support from the stfc grant no . st / f002599/1 . t. kupfer acknowledges support by the netherlands research school for astronomy ( nova ) . ghar acknowledges an nwo - rubicon grant . gn acknowledges an nwo - vidi grant . based on observations obtained under programme gn-2012a - q-54 at the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/.

we present time - resolved spectroscopy and photometry of the dwarf nova sbss1108 + 574 , obtained during the 2012 outburst . its quiescent spectrum is unusually rich in helium , showing broad , double - peaked emission lines from the accretion disc . we measure a line flux ratio hei 5875/h@xmath0=0.81@xmath10.04 , a much higher ratio than typically observed in cataclysmic variables ( cvs ) . the outburst spectrum shows hydrogen and helium in absorption , with weak emission of h@xmath0 and hei 6678 , as well as strong heii emission . from our photometry , we find the superhump period to be 56.34@xmath10.18minutes , in agreement with the previously published result . the spectroscopic period , derived from the radial velocities of the emission lines , is found to be 55.3@xmath10.8minutes , consistent with a previously identified photometric orbital period , and significantly below the normal cv period minimum . this indicates that the donor in sbss1108 + 574 is highly evolved . the superhump excess derived from our photometry implies a mass ratio of @xmath2=0.086@xmath10.014 . our spectroscopy reveals a grazing eclipse of the large outbursting disc . as the disc is significantly larger during outburst , it is unlikely that an eclipse will be detectable in quiescence . the relatively high accretion rate implied by the detection of outbursts , together with the large mass ratio , suggests that sbss1108 + 574 is still evolving towards its period minimum . [ firstpage ] accretion , accretion discs binaries : close stars : individual : sbss1108 + 574 novae , cataclysmic variables white dwarfs .

a theoretical foundation for understanding complex networks has developed rapidly over the course of the past few years @xcite . more recently , the subject of detecting network _ communities _ has gained an large amount of attention , for reviews see refs @xcite . community structure describes the property of many networks that nodes divide into modules with dense connections between the members of each module and sparser connections between modules . in spite of a tremendous research effort , the mathematical tools developed to describe the structure of large complex networks are continuously being refined and redefined . essential features related to network structure and topology are not necessarily captured by traditional global features such as the average degree , degree distribution , average path length , clustering coefficient , etc . in order to understand complex networks , we need to develop new measures that capture these structural properties . understanding community structures is an important step towards developing a range of tools that can provide a deeper and more systematic understanding of complex networks . one important reason is that modules in networks can show quite heterogenic behavior @xcite , that is , the link structure of modules can vary significantly from module to module . for such heterogenic systems , global measures can be directly misleading . also , in practical applications of network theory , knowledge of the community structure of a given network is important . access to the modular structure of the internet could help search engines supply more relevant responses to queries on terms that belong to several distinct communities . in biological networks , modules can correspond to functional units of some biological system @xcite . this section is devoted to an analysis of the modularity @xmath0 . identifying communities in a graph has a long history in mathematics and computer science @xcite . one obvious way to partition a graph into @xmath1 communities is distribute nodes into the communities , such that the number of links connecting the different modules of the network is minimized . the minimal number of connecting links is called the _ cut size _ @xmath2 of the network . consider an unweighted and undirected graph with @xmath3 nodes and @xmath4 links . this network can be represented by an adjacency matrix @xmath5 with elements @xmath6 this matrix is symmetric with @xmath7 entries . the degree @xmath8 of node @xmath9 is given by @xmath10 . let us express the cut - size in terms of @xmath5 ; we find that @xmath11,\ ] ] where @xmath12 is the community to which node @xmath9 belongs and @xmath13 if @xmath14 and @xmath15 if @xmath16 . minimizing @xmath2 is an integer programming problem that can be solved exactly in polynomial time @xcite . the leading order of the polynomial , however , is @xmath17 which very expensive for even very small networks . due to this fact , most graph partitioning has been based on spectral methods ( more below ) . newman has argued @xcite that @xmath2 is not the right quantity to minimize in the context of complex networks . there are several reasons for this : first of all , the notion of cut - size does not capture the essence of our ` definition ' of network as a tendency for nodes to divide into modules with dense connections between the members of module and sparser connections between modules . according to newman , a good division is not necessarily one , in which there are few edges between the modules , it is one where there are fewer edges than expected . there are other problems with @xmath2 : if we set the community sizes free , minimizing @xmath2 will tend to favor small communities , thus the use of @xmath2 forces us to decide on and set the sizes of the communities in advance . as a solution to these problems , girvan and newman propose the modularity @xmath0 of a network @xcite , defined as @xmath18\delta(c_i , c_j).\ ] ] the @xmath19 , here , are a null model , designed to encapsulate the ` more edges than expected ' part of the intuitive network definition . it denotes the probability that a link exists between node @xmath9 and @xmath20 . thus , if we know nothing about the graph , an obvious choice would be to set @xmath21 , where @xmath22 is some constant probability . however , we know that the degree distributions of real networks are often far from random , therefore the choice of @xmath23 is sensible ; this model implies that the probability of a link existing between two nodes is proportional to the degree of the two nodes in question . we will make exclusive use of this null model in the following ; the properly normalized version is @xmath24 . it is axiomatically demanded that that @xmath25 when all nodes are placed in one single community . this constrains the @xmath19 such that @xmath26 we also note that @xmath27 , which follows from the symmetry of @xmath5 . comparing eqs . and , we notice that there are two differences between @xmath0 and @xmath2 . the first is that @xmath0 implies that we _ maximize _ the number of intra - community links instead of minimizing the the number of inter - community links as is the case for @xmath2this is the difference between multiplying by @xmath28 and @xmath29 $ ] . the second difference lies in the the introduction of the @xmath19 in equation . the subtraction of @xmath19 serves to incorporate information about the inter - community links into the quantity we are optimizing . use of modularity to identify network communities is not , however , completely unproblematic . criticism has been raised by fortunato and barthlemy @xcite who point out that the @xmath0 measure has a resolution limit . this stems from the fact that the null model @xmath30 can be misleading . in a large network , the expected number of links between two small modules is small and thus , a single link between two such modules is enough to join them into a single community . a variation of the same criticism has been raised by rosvall and bergstrom @xcite . these authors point out that the normalization of @xmath19 by the total number of links @xmath4 has the effect that if one adds a distinct ( not connected to the remaining network ) module to the network being analyzed and partition the whole network again allowing for an additional module , the division of the original modules can shift substantially due to the increase of @xmath4 . in spite of these problems , the modularity is a highly interesting method for detecting communities in complex networks when we keep in mind the limitations pointed out above . what makes the modularity particularly interesting compared to other clustering methods is its ability to inform us of the optimal number of communities for a given network term in the eq . and is therefore directly linked to the conceptual problems with @xmath0 mentioned in the previous paragraph . ] . the question of finding the optimal @xmath0 is a discrete optimization problem . we can estimate the size of the space we must search to find the maximum . the number of ways to divide @xmath3 vertices into @xmath1 non - empty sets ( communities ) is given by the stirling number of the second kind @xmath31 @xcite . since we do not know the number of communities that will maximize @xmath0 before we begin dividing the network , we need to examine a total of @xmath32 community divisions @xcite . even for small networks , this is an enormous space , which renders exhaustive search out of the question . motivated by the success of spectral methods in graph partitioning , newman suggests a spectral optimization of @xmath0 @xcite . we define a matrix , called the modularity matrix @xmath33 and an @xmath34 _ community matrix _ @xmath35 . each column of @xmath35 corresponds to a community of the graph and each row corresponds to a node , such that the elements @xmath36 since each node can only belong to one community , the columns of @xmath35 are orthogonal and @xmath37 . the @xmath38-symbol in equation can be expressed as @xmath39 which allows us to express the modularity compactly as @xmath40 this is the quantity that we wish to maximize . the next step is the ` spectral relaxation ' , where we relax the discreteness constraints on @xmath35 , allowing elements of this matrix to possess real values . we do , however , constrain the length of the column vectors by @xmath41 , where @xmath42 is a @xmath43 matrix with the number of nodes in each community @xmath44 along the diagonal . in order to determine the maximum , we take @xmath45 + { \mathrm{tr}}[(\mathbf{s}^t \mathbf{s } - \mathbf{m})\tilde{\lambda } ] \right ) = 0,\ ] ] where @xmath46 is a @xmath43 diagonal matrix of lagrange multipliers . the maximum is given by @xmath47 where @xmath48 for cosmetical reasons . eq . is a standard matrix eigenvalue problem . optimizing in the relaxed representation , we substitute this solution into eq . , and see that in order to maximize @xmath0 , we must choose the @xmath1 largest eigenvalues of @xmath49 and their corresponding eigenvectors . since all rows and columns of @xmath49 sum to zero by definition , the vector @xmath50 is always an eigenvector of @xmath49 with the eigenvalue @xmath51 . in general the modularity matrix can have both positive and negative eigenvalues . it is clear from eq . that the eigenvectors corresponding to negative eigenvalues can never yield a positive contribution to the modularity thus , the number of positive eigenvalues presents an upper bound on the number of possible communities . however , we need to convert our problem back to a discrete one . this is a non - trivial task . there is no standard way to go from the @xmath3 continuous entries in each of the @xmath1 largest eigenvectors of the modularity matrix and back to discrete @xmath52 values of the community matrix @xmath35 . one simple way of circumventing this problem is to use repeated bisection of the network . this is the procedure that newman @xcite recommends . in newman s scheme , the only eigenvector utilized is the eigenvector corresponding to the largest eigenvalue @xmath53 of @xmath49 ( with highest contribution to @xmath0 ) . the @xmath52 vector most parallel to this continuous eigenvector , is one where the positive elements of the eigenvector are set to one and the negative elements zero . this is the first column of the community matrix @xmath35 . the second column must contain the remaining elements . we can increase the modularity iteratively by bisecting the network into smaller and smaller pieces . however , this repeated bisection of the network is problematic . there is no guarantee that that the best division into three groups can be arrived at by finding by first determine the best division into two and then dividing one of those two again . it is straight forward to construct examples where a sub - optimal division into communities is obtained when using bisection @xcite . spectral optimization is not perfect especially when only the eigenvector corresponding to @xmath53 is employed . therefore , newman suggests that it should only be used as a starting point . in order to improve the modularity , newman has devised an algorithm inspired by the classical kernighan - lin ( kl ) scheme @xcite . the procedure is as follows : after each bisection of the network we go through the nodes and find the one that yields the highest increase in the modularity of the entire network ( or smallest decrease if no increase is possible ) if moved to the other module . this node is now moved to the other module and becomes inactive . the next step is to go through the remaining @xmath54 nodes and perform the same action . we continue like this until all nodes have been moved . finally , we go through all the intermediate states and pick the one with the highest value of @xmath0 . this is the new starting division . we proceed iteratively from this configuration until no further improvement can be found . let us call this optimization the ` kln - algorithm ' . in the spectral optimization , the computational bottleneck is the calculation of the leading eigenvector(s ) of @xmath49 , which is non - sparse . naively , we would expect this to scale like @xmath55 . however , @xmath49 s structure allows for a faster calculation . we can write the product of @xmath49 and a vector @xmath56 @xcite as @xmath57 this way we have a divided the multiplication into ( i ) sparse matrix product with the adjacency matrix that takes @xmath58 , and ( ii ) the inner product @xmath59 that takes @xmath60 . thus the entire product @xmath61 scales like @xmath58 . the total running for a bisection determining the eigenvector(s ) is therefore @xmath62 rather than the naive guess of @xmath55 . using eq . during the kln - algorithm reduces the cost of this step to @xmath62 @xcite . simulated annealing was proposed by kirkpatrick _ et al_. @xcite who noted the conceptual similarity between global optimization and finding the ground state of a physical system . formally , simulated annealing maps the global optimization problem onto a physical system by identifying the cost function with the energy function and by considering this system to be in equilibrium with a heat bath of a given temperature @xmath63 . by annealing , i.e. , slowly lowering the temperature of the heat bath , the probability of the ground state of the physical system grows towards unity . this is contingent on whether or not the temperature can be decreased slowly enough such that the system stays in equilibrium , i.e. , that the probability is gibbsian @xmath64 here , @xmath65 is a constant ensuring proper normalization . kirkpatrick et al . realized the annealing process by monte carlo sampling . the representation of the constrained modularity optimization problem is equivalent to a @xmath1-state potts model . gibbs sampling for the potts model with the modularity @xmath0 as energy function has been investigated by reichardt and bornholdt , see e.g. , @xcite . mean field annealing is a deterministic alternative to monte carlo sampling for combinatorial optimization and has been pioneered by peterson et al . mean field annealing avoids extensive stochastic simulation and equilibration , which makes the method particularly well suited for optimization . there is a close connection between gibbs sampling and mf annealing . in gibbs sampling , every variable is updated by random draw of a potts state with a conditional distribution , @xmath66 where the sum runs over the @xmath1 values of the @xmath9th potts variable and @xmath67 denotes the set of potts variables excluding the @xmath9th node . as noted by @xcite , eq . ( [ eq : gibbs2 ] ) is local in the sense that the part of the energy function containing variables not connected with the @xmath9th cancels out in the fraction . the mean field approximation is obtained by computing the conditional mean of the set of variables coding for the @xmath9th potts variable using eq . ( [ eq : gibbs2 ] ) and approximating the potts variables in the conditional probability by their means @xcite . this leads to a simple self - consistent set of non - linear equations for the means , @xmath68 for symmetric connectivity matrices with @xmath69 , the set of mean field equations has the unique high - temperature solution @xmath70 . this solution becomes unstable at the mean field critical temperature , @xmath71 , determined by the maximal eigenvalue @xmath53 of @xmath49 . this mean field algorithm is fast . each synchronous iteration ( see section [ sec : experiments ] for details on implementation ) requires a multiplication of @xmath49 by the mean vector @xmath72 . as we have seen , this operation can be performed in @xmath58 time using the trick in eq . . in these experiments , we have used a fixed number of iterations of the order of @xmath60 , which gives us a total of @xmath62 similar to the case of by spectral optimization . ( a forthcoming paper discusses the relationship between gibbs sampling , mean field methods , and computational complexity . ) we will perform our numerical experiments on a simple model of networks with communities . this model network consists of @xmath1 communities with @xmath73 nodes in each , the total network has @xmath74 nodes . without loss of generality , we can arrange our nodes according to their community ; a sketch of this type of network is displayed in figure [ fig : networksketch ] . @xmath75[c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ p$ \vspace{.25 cm } \end{center}\end{minipage } } } & { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ q$ \vspace{.25 cm } \end{center}\end{minipage } } } & \cdots & { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ q$ \vspace{.25 cm } \end{center}\end{minipage } } } & $ $ \bigg\}n_c$$\\ { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ q$ \vspace{.25 cm } \end{center}\end{minipage } } } & { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ p$ \vspace{.25 cm } \end{center}\end{minipage } } } & & & \\ \vdots & & \ddots & & \\ { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ q$ \vspace{.25 cm } \end{center}\end{minipage } } } & & & { \framebox[1cm][c]{\begin{minipage}[c]{1 cm } \begin{center}\vspace{.25 cm } $ p$ \vspace{.25 cm } \end{center}\end{minipage}}}&\\ \multicolumn{4}{c}{\underbrace{\hspace{4cm}}_n } & \end{array } \renewcommand{\arraystretch}{1}\ ] ] communities are defined as standard random networks , where the probability of a link between two nodes is given by @xmath22 , with @xmath76 . between the communities the probability of a link between is given by @xmath77 . the networks are unweighted and undirected . let us calculate @xmath0 for this network in the case where @xmath78 and @xmath79 . in this case , we can calculate everything exactly . first , we note that all nodes have the same number of links , and that the degree of node @xmath9 , @xmath80 ( since a node does not link to itself ) . thus the total number of links @xmath81 in each sub - network is @xmath82 and since our network consists of @xmath1 identical communities the total number of links is @xmath83 . we can now write down the contribution @xmath84 from each sub - network to the total modularity @xmath85.\end{aligned}\ ] ] if we insert @xmath4 and use that @xmath86 , we find @xmath87 we see explicitly that when @xmath88 the modularity approaches unity . now , let us examine at the general case . since our network is connected at random , we can not calculate the number of links per node exactly , but we know that the network is well - behaved ( poisson link distribution ) , thus we can calculate the _ average _ number of links per node . we see that @xmath89 which is equal to the number of expected intra - community links plus the number of expected number of inter - community links . the number of links in the entire network is therefore given by @xmath90.\ ] ] we write down @xmath0 @xmath91\nonumber\\ & = & \frac{(n_c-1)p}{(n_c-1)p + n_c(c-1 ) q}-\frac{1}{c}.\end{aligned}\ ] ] when @xmath92 ( which is always the case ) , we have that @xmath93 when we write @xmath94 as some fraction @xmath95 of @xmath22 , that is @xmath96 , with @xmath97 , we find @xmath98 which is independent of @xmath22 . thus , for this simple network , the only two relevant parameters are the number of communities and the density of the inter - community links relative to the intra - community strength . we can also see that our result from eq . is valid even in the case @xmath99 , as long as the communities are connected and @xmath100 . . this figure displays @xmath0 as a function of @xmath95 ( the relative probability of a link between communities ) , with @xmath101 for the simple network defined in figure [ fig : networksketch ] . the blue line is given by eq . and the black dots with error - bars are mean values of @xmath102 in realizations of the simple network with @xmath103 and @xmath104 ; each data - point is the mean of @xmath105 realizations . the error bars are calculated as the standard deviation divided by square root of the number of runs.,title="fig : " ] + if we design an adjacency matrix according to figure [ fig : networksketch ] , we can calculate the value @xmath106 , where @xmath107 is a community - matrix that reflects the designed communities . values of @xmath108 should correspond to eq . . we see in figure [ fig : curve ] that this expectation is indeed fulfilled . the blue curve is @xmath0 as a function of @xmath95 with @xmath101 . the black dots with error - bars are mean values of @xmath102 in realizations of the simple network with @xmath103 and @xmath104 ; each data - point is the mean of @xmath105 realizations and the error bars are calculated as the standard deviation divided by square root of the number of runs . the correspondence between prediction and experiment is quite compelling . we should note , however , that the value of @xmath108 may be lower than the actual modularity found for the network by a good algorithm : we can imagine that fluctuations of the inter - community links could result in configurations that would yield higher values of @xmath0especially for high values of @xmath95 . we can quantify this quite precisely . reichardt and bornholdt @xcite have shown that demonstrated that random networks can display significantly larger values of @xmath0 due to fluctuations ; when @xmath109 , our simple network is precisely a random network ( see also related work by guimer _ @xcite ) . in the case of the network we are experimenting on , ( @xmath104 , @xmath103 ) , they predict @xmath110 . thus , we expect that the curve for @xmath111 with fixed @xmath1 will be deviate from the @xmath112 displayed in figure [ fig : curve ] ; especially for values of @xmath95 that are close to unity . the line will decrease monotonically from @xmath113 towards @xmath114 with the difference becoming maximal as @xmath115 . we know that the running time of mean field method scales like that of the spectral solution . in order to compare the precision of the mean field solutions to the solutions stemming from spectral optimization , we have created a number of test networks with adjacency matrices designed according to figure [ fig : networksketch ] . we have created @xmath105 test networks using parameters @xmath116 , @xmath117 , @xmath118 and @xmath119 $ ] . varying @xmath95 over this interval allows us to interpolate between a model with @xmath1 disjunct communities and a random network with no community structure . we applied the following three algorithms to our test networks 1 . spectral optimization , 2 . spectral optimization and the kln - algorithm , and 3 . mean field optimization . spectral optimization and the kln - algorithm were implemented as prescribed in @xcite . the @xmath120 non - linear mean field annealing equations were solved approximately using a @xmath121-step _ annealing schedule _ linear in @xmath122 starting at @xmath123 and ending in @xmath124 at which temperature the majority of the mean field variables are saturated . the mean field critical temperature @xmath125 is determined for each connectivity matrix . the synchronous update scheme defined as parallel update of all means at each of the @xmath126 temperatures @xmath127 can grow unstable at low temperatures . a slightly more effective and stable update scheme is obtained by selecting random fractions @xmath128 of the means for update in @xmath129 steps at each temperature . we use @xmath130 in the experiments reported below . a final @xmath131 iteration , equivalent to making a decision on the node community assignment , completes the procedure . we _ do not _ assume that actual the number of communities @xmath132 is known in advance . in these experiments we use @xmath133 . this number is determined after convergence by counting the number of non - empty communities the results of the numerical runs are displayed in figure [ fig : compare ] . this figure shows the point - wise differences between the value of @xmath134 found by the algorithm in question and @xmath135 plotted as a function of the inter - community noise @xmath95 . , @xmath117 , @xmath118 and @xmath119 $ ] . all data points display the point - wise differences between the value of @xmath134 found by the algorithm in question and @xmath135 . the error - bars are calculated as in figure [ fig : curve ] . the dash - dotted red line shows the results for the spectral method . the dashed blue line shows the results for the spectral optimization followed by kln post - processing . the solid black curve shows the results for the mean field optimization . the grey , horizontal line corresponds to the theoretical prediction ( eq . ) for the designed communities . , title="fig : " ] + the line of @xmath136 thus corresponds to the curve plotted in figure [ fig : curve ] . we see from figure [ fig : compare ] that the mean field approach uniformly out - performs both spectral optimization and spectral optimization with kln post - processing . we also ran a gibbs sampler @xcite for with a computational complexity equivalent to the mean field approach . this lead to communities with @xmath0 slightly lower than the mean field results , but still better than spectral optimization with kln post - processing . we note that the obtained @xmath137 for a random network ( @xmath138 ) is consistent with the prediction made by reichardt and bornholdt @xcite . we also see that the optimization algorithms can exploit random connections to find higher values of @xmath137 than expected for the designed communities @xmath108 . in the case of the mean field algorithm this effect is visible for values of @xmath95 as low as @xmath139 . figure [ fig : communities ] shows the median number of communities found by the various algorithms as a function of @xmath95 . . all optimization schemes consistently pick four or five communities for the highest values of @xmath95 . this finding is consistent with theoretical and experimental results by reichardt and bornholdt @xcite , title="fig : " ] + it is evident from figs . [ fig : compare ] and [ fig : communities ] that for this particular set of parameters the problem of detecting the designed community structure is especially difficult around @xmath140 . spectral clustering with and without the kln algorithm find values @xmath137 that are significantly lower than @xmath141 . the mean field algorithm manages to find a value of @xmath137 that is higher than the designed @xmath0 but does so by creating extra communities . as @xmath142 it becomes more and more difficult to recover the designed number of communities . we have introduced a deterministic mean field annealing approach to optimization of modularity @xmath0 . we have evaluated the performance of the new algorithm within a family of networks with variable levels of inter - community links , @xmath95 . even with a rather costly post - processing approach , the spectral clustering approach suggested by newman is consistently out - performed by the mean field approach for higher noise levels . spectral clustering without the kln post - processing finds much lower values of @xmath0 for all @xmath143 . speed is not the only benefit of the mean field approach . another advantage is that the implementation of mean field annealing is rather simple and similar to gibbs sampling . this method also avoids the inherent problems of repeated bisection . the deterministic annealing scheme is directed towards locating optimal configurations without wasting time at careful thermal equilibration at higher temperatures . as we have noted above , the modularity measure @xmath0 may need modification in specific non - generic networks . in that case , we note that the mean field method is quite general and can be generalized to many other measures . o. goldscmidt and d. s. hochbaum . polynomial algorithm for the @xmath144-cut problem . in _ proceedings of the 29th annual ieee symposium on the foundations of computer science _ , page 444 . institute of electrical and electronics engineers , 1988 .

we study community structure of networks . we have developed a scheme for maximizing the modularity @xmath0 @xcite based on mean field methods . further , we have defined a simple family of random networks with community structure ; we understand the behavior of these networks analytically . using these networks , we show how the mean field methods display better performance than previously known deterministic methods for optimization of @xmath0 .

the galactic neighborhood ( gan ) panel was charged to identify key scientific questions ( as well as a potential major discovery area ) , which could be effectively addressed in the upcoming decade , regarding galaxies and their surroundings out to redshifts @xmath0 . this local volume of the universe contains a diverse array of objects ( e.g. , galaxies of vastly different masses , morphologies , and star formation rates ) . but the basic constituents of the gan objects may be divided into three classes : 1 ) stars ( including their remnants ) ; 2 ) gaseous systems [ the interstellar medium ( ism ) , circumgalactic medium ( cgm ) , and intergalactic medium ( igm ) ] ; 3 ) dark components [ massive black holes ( mbhs ) , typically seen at centers of galaxies , and dark matter ] . the study of the interconnection among these constituents is a key part of the gan science . why is the gan science important in astronomy and astrophysics ? the gan is where astronomical phenomena can be examined in great detail . because of their proximity , objects can be observed with unparalleled sensitivities , on small physical scales , across the electromagnetic spectrum , and within a relatively well - determined galactic or intergalactic environment . for example , stellar populations can be resolved into individual stars only in the gan . such observations are often necessary in order to firmly identify underlying astrophysical processes and , occasionally , even new physics ( e.g. , the discovery of dark matter around galaxies ) . the understanding of astronomical phenomena and astrophysical processes in the gan thus represents the cornerstone for properly interpreting observations of distant universe . the gan further provides a test bed to check the validity of various assumptions , approximations , or recipes that are often needed in modeling / simulating the structure formation and evolution of the universe . locally calibrated empirical relations ( such as the peak to width relation of type ia sn light - curves and star formation laws ) are also very useful tools for studying distant galaxies . moreover , local measurements ( e.g. , star formation rates and total stellar masses of galaxies ) provide important anchors in determining how the universe has evolved . the panel strived to identify key science questions that the gan can potentially offer the most powerful and unique constraints to . in doing so , a broad range of questions were synthesized and ranked without regard for cost , current agency plan , or specific proposed instrumentation , although the panel was mindful of various limitations , both technical and financial . the four embraced questions , as well as the two identified potential discovery areas , exploit the use of the gan as a venue for studying interconnected astrophysical systems , for constraining complex physical processes , and for probing small scales . high signal - to - noise observations , which can often be obtained for gan objects , need to be interpreted with correspondingly accurate physical data . they are sometimes best obtained by experiment , and sometimes by theoretical calculation , jointly referred to as `` laboratory astrophysics '' . its importance is highlighted in the summary of the gan @xcite : `` the prospects for advances in the coming decade are especially exciting in these four areas , particularly if supported by a comprehensive program of theory and numerical calculation , together with laboratory astrophysical measurements or calculations . '' in the following , i briefly describe my interpretation of the laboratory astrophysics needs as well as the questions / discovery areas , particularly with consideration of the observing capabilities likely available from existing and upcoming nasa missions . * q1 : what are the flows of matter and energy in the circumgalactic medium ? * how a galaxy evolves depends largely on how it interacts with its environment . cosmological simulations have hinted that the accretion of matter onto a galaxy can have various different modes : hot " , cold " , or recycled wind " , depending primarily on galaxy mass . the biggest uncertainty in this emerging picture of galaxy formation and evolution is our poor understanding of galactic feedback . a number of feedback mechanisms have been proposed , ranging from pre - heating by the extragalactic uv background generated collectively by early star formation and agn activities , to _ in situ _ momentum- and/or energy - driven superwinds from starbursts and/or agns , and to long - lasting gentle outflows powered by type ia supernovae in galactic spheroids ( e.g. , @xcite and references therein ) . however , none of these mechanisms are well understood . while observations have shown strong evidence for infall ( accretion ) and outflow ( feedback ) of matter around galaxies , little is yet known about how mass , energy , and chemical elements actually circulate between galaxies and the environment . the cgm the galaxy / igm interface where this circulation occurs thus needs to be carefully studied in order to answer such fundamental questions as : where is the missing " baryon matter in galaxies ? how are they fed ? and how does galactic feedback work ? in the gan , it is possible to directly observe the working of the cgm , which may extend from a few kpc up to about 1 mpc around galaxies ( e.g. , fig . [ f12]a ) . two effective observational strategies : uv / x - ray absorption - line tomography and spectral imaging , have been demonstrated , primarily in the study of gas in and around the milky way ( for a review , see @xcite ) . to obtain transformative gains , however , these techniques need to be applied to more targets at better velocity resolution and over a broad temperature range . such observations at wavelengths most sensitive to the mass , energy , and key element flows will help to remove uncertainties in simulations of galaxy formation and evolution , which currently lack the resolution required for direct modeling of all physical processes . * q2 : what controls the mass - energy - chemical cycles within galaxies ? * to understand such cycles , we need to study the ism and its interplay with stars inside galaxies . the scale considered here ranges from kpc ( spiral structure ) , to parsecs ( giant molecular clouds , starbursts , and star clusters ) , and down to the sub - pc level where individual stars form . first , we need to measure ism conditions that control the molecular cloud formation , the rate of star formation , and the stellar initial mass function ( imf ) in various environments , ranging from massive starbursts to ultra - faint , low - mass , low - metallicity , dwarf galaxies , and from outer regions of spiral galaxies to the close proximity of the mbh at the center of the milky way . such measurements , which can be made with existing observing facilities such as herschel and with upcoming ones such as jwst , alma , and large single - dish millimeter / submillimeter - wave telescopes , will allow us to calibrate our understanding of how galaxies build up their stellar component over cosmic time . second , we need to examine the effects of stellar energy / chemical feedback on the structure and physical state of the ism and on gas - to - star conversion efficiency in the milky way and nearby star - forming galaxies . the ism spans a huge range of densities and temperatures and includes magnetic fields and cosmic rays , both of which can be dynamically important . the statistical properties of this complex system remain poorly determined , including even the geometry and topology of the density field , as well as the thermal phase distribution . the application of the uv / x - ray spectroscopy and imaging techniques similar to those described in q1 will greatly improve this situation , particularly for the understanding of the ism in the milky way . sensitive surveys in other wavelength bands will further allow us to characterize various components of the ism in galaxies out to @xmath1 , and hence their complex galactic ecosystem . * laboratory astrophysics needs for q1 and q2 : * these two questions , dealing with the so - called gastrophysics , have some common laboratory astrophysics needs , which may be met with reasonable efforts in the near future . particularly important are accurate physical data required to provide the diagnostics of heating and cooling processes in the ism and cgm , especially in ( sub)millimeter / infrared and x - ray , which upcoming observing facilities with high spectral resolution capabilities ( e.g. , alma , jwst and astro - h ) will be sensitive to . to make full use of the uv / x - ray spectroscopic data , for example , it is important to complete the measurements of radiative rates , electron - impact excitation rates , ionization rates , and dielectronic recombination rates as well as the listing of importtant emission and absorption lines of collisional plasma ( e.g. , fe xvii ) . at present , many lines ( even fe l - shell ) are still not identified . line energies from theories are typically only good to @xmath2 , which is not sufficient for accurate velocity measurements of starburst - driven outflows . for such measurements , we also need better identifications of satellite lines , which could otherwise lead to confusion with doppler broadening . moreover , it is essential to investigate processes involved in the interaction between plasma and cool gas , such as charge exchange , which may be responsible for much of diffuse soft x - ray emission observed in galaxies ( e.g. , fig . [ f12]b ) . in addition , data on cosmic - ray heating of the ism and cgm ( e.g. , the proton impact excitation cross sections ) need to be substantially improved . it is reasonable to suspect that pahs might account for the diffuse interstellar bands , but only careful measurement of pah absorption cross sections in the gas phase in the laboratory can confirm this . laboratory measurements are also needed of photoelectric yields from dust grains over a range of sizes , including pahs . various other examples for the required improvements in understanding the important `` microphysics '' ( e.g. , on mhd , plasma , and shock waves ) are given in the panel report . * q3 : what is the fossil record of galaxy assembly from first stars to present ? * one way to probe how galaxies actually come to be is to study the evolution of their properties by looking back in time . however , inevitably limited by the angular resolution and sensitivity of observing distant universe , this look - back approach relies on the measurements of globally - averaged properties ( such as luminosity and color of galaxies ) . the approach can be complemented by examining stellar fossil record in local galaxies . one can read in the color - magnitude diagrams ( cmd ) of resolved stellar populations to determine star formation histories , which can be associated with such galaxy properties as gas content , environment , and morphology . even internal patterns can be examined , such as the relationship of stellar populations with spiral arms or as a function of galactocentric distance . one can further spatially and kinematically probe substructures in the galactic halos of the milky way and other local galaxies , unraveling stellar streams and dwarf satellite galaxies . such studies can provide critical information about how gas collapses and forms stars down to these small physical scales and faint luminosities , as well as about the merger histories of the galaxies , illuminating the process of galaxy formation more generally . in addition , one can measure metallicity of individual stars , particularly interesting for tracing extremely metal - poor stars formed in earliest epochs . all these fossil record studies can only be done in the gan ! with the expected improvements in optical and near - ir imaging / spectroscopy capabilities over the next decade , substantial progress can be made in this field . one will be able to determine the star - formation histories of galaxies across the hubble sequence , to detect a significant sample of the smallest galaxies , and to measure precise abundances for elements from all the important nucleosynthetic processes that act in stars , from which much information can be obtained about the population of stars that produced the metals . with these measurements , one can potentially address such questions as : how old are the oldest stars in the milky way ? where are the lowest metallicity stars in the milky way and when did they form ? did the imf vary with metallicity and galactic conditions ? can chemical tagging of metal - poor stars be used to identify coeval populations , later dispersed around the galaxy ? the enormous potential of the fossil record to probe galaxy assembly from first stars to present will then be realized in the next decade or so . * q4 : what are the connections between dark and luminous matter ? * while the cold dark matter ( @xmath3cdm ) paradigm has passed many serious tests , there are still apparent conflicts between the predictions and observations on scales of kpc or smaller . potential resolution of these conflicts has been complicated by the uncertain interplay between dark and baryon matters . this confusion should be minimal in lower - mass galaxies , which appear to be increasingly dominated by dark matter . such galaxies , observable in the local universe , can readily be identified from future large - scale , deep , multi - color , photometric surveys of stars , together with follow - up spectroscopy . a substantially increased local inventory of small galaxies will enable us to confront the well - known `` missing satellites '' problem . the best place to look for the signature of weakly interacting dark matter ( via possible @xmath4-ray and/or x - ray radiation from self - annihilation or decay ) is the heart of ultra - faint dwarf galaxies , because of the high central densities and minimum astrophysical confusion from proportionally fewer stars . dark matter distributions on galaxy scales can also be explored in multiple ways , ranging from studying kinematics of stars and gas to mapping properties of diffuse hot gas in hydrostatic equilibrium . particularly interesting are the distribution and kinematics of dark matter within the milky way at the solar circle , providing constraints useful to the _ direct _ detection experiments . mbhs play an increasingly important position in astronomy and astrophysics . however , how mbhs form and evolve is still a question that remains to be answered ; we are stil very uncertain about their seeds ( @xmath5 stellar remnants , @xmath6 `` intermediate - mass '' bhs from pop iii stars , or @xmath7 mbhs from direct collapses of matter ) , about their merger history , and about the accretion process . the formation and evolution of mbhs are intimately related to their interplay with host galaxies . the most apparent manifestation of this interplay is the mass relation between mbhs and surrounding stellar spheroids . large uncertainties in the relation remain , however , especially at the low - mass end , where the presence of nuclear stellar clusters may play a central role . mbhs are also known to be an important source of galaxy feedback , although the efficiency remains very uncertain at low - accretion rates . the coupling between this feedback and surrounding matter is also poorly understood . the local universe offers the most promising avenues to advance our understanding of mbhs and their interplay with galaxies . advances in spatial resolution ( adaptive optics systems ) and sensitivity ( larger telescopes ) will enhance the most used techniques for measuring mbh masses and for studying stellar properties . the nuclei of the milky way galaxy and several other nearby galaxies provide our best opportunity to observe the interaction of mbhs with their immediate environments . many fascinating questions remain to be fully addressed , regarding the formation and dynamics of stars under extreme hostile condition . x - ray observations with large collecting areas and good spectral resolutions will be essential to the study of the accretion process . interesting new constraints on the process can also be obtained from the spins of mbhs , which can be measured over a wide mass range for local galaxies . furthermore , one may find potential seed black holes through dynamical studies of nearby systems and through the possibility of measuring gravitational waves of black - hole inspiral events . detection of gravitational waves will in general open up a new avenue for characterizing the demographics and merger rate of black holes . * laboratory needs for q3 and q4 : * the star formation history and metal abundance measurements clearly depend on our knowledge about stellar evolution , which in turn relies on the accuracy of laboratory astrophysics data on nucleosynthesis , opacity , etc . there remain significant rooms for improvements in the quality of such data ( e.g. , @xmath8c(@xmath9o reaction rate and @xmath10-decay lifetime for many r - process isotopes ) . a true comprehension of dark matter ultimately requires a direct detection in laboratory , while the modeling of the accretion and feedback of bhs demands a good understanding of important plasma astrophysical processes , particularly the magneto - rotational instability and magnetic reconnection . * discovery areas * the time - domain astronomy is identified as a major gan discovery area , chiefly because enormous swaths of parameter space remain to be explored . large - scale , multiple - epoch surveys , together with ever increasing computational capability and algorithm development , make the transient sky an area particularly ripe for discoveries of new objects and/or physical processes . the gan is particularly suited for such discoveries , because measurements of the distance , energetics , rates , and demographics of newly observed phenomena , as well as their associations with stellar populations and galactic structure , is the first essential step in understanding the underlying physics . astrometry is considered to be another area with exceptional discovery potential . a variety of powerful astrometric techniques [ radio , ( sub)mm vlbi , time - resolved large optical surveys ] are now reaching maturity to open a new window for the discovery of vast numbers of extrasolar planets , kuiper belt objects , asteroids , and comets ; to test the weak - field limit of general relativity with unprecedented precision ( for the mbh at the milky way center ) ; to measure the aberration of quasars from the centripetal acceleration of the sun by the galaxy ; to provide a complete inventory of stars near the sun ; to measure orbits of the globular clusters and satellite galaxies of the milky way and galaxies of the local group ; and to fix properties of the major stellar components of the milky way . while progress in addressing the four science questions and in the areas of discovery potential can be made with existing and upcoming facilities , reaching the full science goals will require powerful new observing capabilities : imaging / spectroscopy abilities in uv and x - ray will be essential to the understanding of the interconnected , multiphase nature of galaxies and their surroundings , while enhanced capability at longer wavelengths from ratio to optical will be particularly important to probing the processes that transform accreted gas into stars , to measuring the fossil record , and to finding the connections between dark and luminous matter . in many of these efforts , the laboratory astrophysics can play a significant or even essential role ! astro2010 frontier science galactic neighborhood panel consisted of leo blitz , julianne dalcanton , bruce draine , rob fesen , karl gebhardt , juna kollmeier , crystal martin , michael shull , jason tumlinson , q. daniel wang , dennis zaritsky , and steve zepf , plus astro2010 survey science liaison , scott tremaine . i thank mike shull , the chair of the panel , for various inputs and comments on this write - up . astro2010 panel reports new worlds , new horizons in astronomy and astrophysics , 2010 , http://sites.nationalacademies.org/bpa/bpa@xmath11048094 liu , j. , wang , q. d. , li , z .- y . , & peterson , j. r. 2010 , mnras , 404 , 1879 wang , q. d. , et al . 2001 , apjl , 555 , 99 wang , q. d. 2010 , pnas , 107 , 7168

the galactic neighborhood , extending from the milky way to redshifts of about 0.1 , is our unique local laboratory for detailed study of galaxies and their interplay with the environment . such study provides a foundation of knowledge for interpreting observations of more distant galaxies and their environment . the astro 2010 science frontier galactic neighborhood panel identified four key scientific questions : 1 ) what are the flows of matter and energy in the circumgalactic medium ? 2 ) what controls the mass - energy - chemical cycles within galaxies ? 3 ) what is the fossil record of galaxy assembly from first stars to present ? 4 ) what are the connections between dark and luminous matter ? these questions , essential to the understanding of galaxies as interconnected complexes , can be addressed most effectively and/or uniquely in the galactic neighborhood . the panel also highlighted the discovery potential of time - domain astronomy and astrometry with powerful new techniques and facilities to greatly advance our understanding of the precise connections among stars , galaxies , and newly discovered transient events . the relevant needs for laboratory astrophysics will be emphasized , especially in the context of supporting nasa missions .

the chemical composition of extremely metal - poor stars is expected to reflect the yields from a quite small number of nucleosynthesis processes . recent abundance analyses for extremely metal - poor stars have provided quite valuable information on the origin of the elements , ( in particular when combined with galactic chemical evolution studies , e.g. , * ? ? ? * ; * ? ? ? * ) , and the individual nucleosynthesis processes involved . the rapid neutron capture process ( r - process ) is known to be responsible for about half of the abundances of elements heavier than the iron - group in solar system material . although observational data have been rapidly increasing in the past decade , the astrophysical site of the r - process is still unclear . previous nucleosynthesis studies suggest several possibilities , e.g. , neutrino - driven winds @xcite or prompt explosions @xcite of core - collapse ( type ii / ibc ) supernovae , neutron star mergers @xcite , jets from gamma - ray burst accretion discs @xcite . all the scenarios proposed above involve , however , severe problems that remain to be solved , and no consensus has yet been achieved . models of the r - process nucleosynthesis are usually examined by comparison with the abundance pattern of the r - process component in solar - system material . recent measurements for abundances of neutron - capture elements in very metal - poor stars have been providing useful constraints on these models . @xcite have studied the chemical abundances of the extremely metal - poor star cs 22892052 , the first example of a small but growing class of metal - poor stars that exhibit very large excesses of r - process elements relative to iron ( [ r - process / fe ] @xmath0 ) . an important result of their work is that the relative abundance pattern of the neutron - capture elements from the 2nd to the 3rd peak ( 56 @xmath1 _ z _ @xmath1 76 ) in this star is identical , within observational errors , to that of the ( inferred ) solar system r - process component . this phenomenon is sometimes referred to as `` universality '' of the r - process , having a large impact on the studies of the nature of the r - process , and its astrophysical site . however , the abundance patterns of light neutron - capture elements ( 38 @xmath1 z @xmath1 48 ) in r - process enhanced stars exhibit clear deviations from that of the solar system r - process component @xcite . this suggests the existence of another r - process site which has contributed to the light r - process elements in solar - system material . this process is sometimes called as `` weak r - process '' , while the process that is responsible for heavy neutron - capture elements ( ba @xmath2 u ) is referred to `` main r - process '' . following the previous studies ( e.g. , * ? ? ? * ; * ? ? ? * ) , @xcite and @xcite have investigated the correlation of sr and ba abundances in very metal - poor stars ( [ fe / h ] @xmath3 ) in some detail . they showed that the dispersion of the sr abundances clearly decreases with increasing ba abundance . this correlation indicates the existence of ( at least ) two processes : one enriches both heavy and light neutron - capture elements , while the other yields light neutron - capture elements with only small production of heavy ones . @xcite discuss the contribution of a primary process which has produced light neutron - capture elements in our galaxy . the abundance patterns produced by the main r - process have been studied previously based on high resolution spectroscopy of metal - poor stars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) as mentioned above . by way of contrast , the abundance pattern of neutron - capture elements produced by the weak r - process is still unclear . since the abundances of heavy neutron - capture elements yielded by this process are very low , it is quite difficult to accurately measure their abundances in metal - poor stars . in addition , most spectral lines of light neutron - capture elements , in particular those with @xmath4 exist in the near - uv region , in which atmospheric extinction makes the observation with ground - based telescopes difficult . we have been investigating light neutron - capture elements in very metal - deficient stars from near - uv spectroscopy with the 8.2 m subaru telescope . ishimaru et al . ( 2006 , in preparation ) studied the abundances of pd and ag , which exist between the first and second abundance peaks produced by neutron - capture processes , for several metal - poor stars . one of the targets is hd 122563 , a well - studied bright metal - poor ( [ fe / h ] @xmath5 ) giant . this object has very low abundances of heavy neutron - capture elements ( e.g. [ ba / fe ] @xmath6 * ? ? ? * ) , while light neutron - capture elements show large excesses relative to heavy ones ( e.g. [ sr / ba ] @xmath7 ) . therefore , the abundance pattern of neutron - capture elements in this star might well represent the yields of the process producing light neutron - capture elements at low metallicity . in this paper we report the abundance pattern of light and heavy neutron - capture elements in hd 122563 . in 2 we present our near - uv spectroscopy with subaru / hds , while we describe in detail the abundance analysis of neutron - capture elements in 3 . our results are compared with the previous studies in 4 . we discuss the derived abundances of hd 122563 , and the source of neutron - capture elements in 5 . finally , our conclusions are presented in 6 . high dispersion spectroscopy of hd 122563 was carried out with the subaru telescope high dispersion spectrograph ( hds : * ? ? ? * ) in 30 april , 2004 . our spectrum covers the wavelength range from 3070 to 4780 with a resolving power of @xmath8 = 90,000 . the total exposure time is 5400 seconds ( one 600 s exposure plus four 1200 s ones ) . the reduction was carried out in a standard manner using the iraf echelle package . the signal to noise ratio ( s / n ) of the spectrum ( per 0.9 km s@xmath9 pixel ) estimated from the photon counts is 140 at 3100 ; 480 at 3500 ; 860 at 3900 ; 1080 at 4200 and 1340 at 4500 . in order to measure ba lines at 5853 and 6141 , a spectrum obtained during the hds commissioning covering 5090 6410 was used . the s / n ratio of this spectrum is 350 at 6000 . we adopted the line list used in previous studies for line identifications and analyses of neutron - capture elements ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition , the recent measurements of transition probabilities and hyperfine splitting reported for some elements ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) are incorporated . the line list used in the analysis is given in table [ tab : table1 - 4 ] . + equivalent widths of clean , isolated lines are measured by fitting gaussian profile . the results are given in table [ tab : table1 - 4 ] . for our quantitative abundance measurements , we used the analysis program sptool developed by y. takeda , based on kurucz s atlas9/width9 @xcite . sptool calculates synthetic spectra and equivalent widths of lines on the basis of the given model atmosphere , line data , and chemical composition , under the assumption of lte . we adopted the model atmosphere parameters ( effective temperature : @xmath10 , gravity : @xmath11 , micro - turbulent velocity : @xmath12 , and metallicity : [ fe / h ] ) derived by @xcite : @xmath10 = 4570 k , @xmath13 = 1.1 , @xmath12 = 2.2 km s@xmath9 , and [ fe / h ] @xmath14 . the estimated uncertainties are @xmath15 = 100 k , @xmath16 = 0.3 dex , @xmath17 = 0.5 km s@xmath9 , and @xmath18[fe / h ] = 0.19 dex ( honda et al . 2004 ) . the abundance analyses were attempted for 24 neutron - capture elements from sr to th . the effects of hyperfine and isotopic splitting are taken into account in the analysis of ba , la , eu , and yb ( see references in table 1 ) . the derived abundances are given in table [ tab : table2 ] . the abundances determined from individual lines are given in table [ tab : table1 - 4 ] . we used the solar system abundances obtained by @xcite to derive [ x / fe ] values . the size of the random errors are estimated from the standard deviation ( 1@xmath19 ) of the abundances derived from individual lines for elements that have three or more lines available for the abundance analysis . for the abundances of elements based only on one or two lines , we employ the mean of the random errors estimated from those elements with three or more lines available . the uncertainties of model atmosphere parameters result in systematic errors in the abundances of neutron - capture elements . the effects of these uncertainties on the abundance measurements are given in table [ tab : error ] . the behavior of the errors is slightly different for abundances derived from neutral species and those from ionized ones . this should be borne in mind in the discussion on the abundances of light neutron - capture elements , some of which were determined using lines of neutral species . by contrast , such differences do not significantly affect our discussion of the abundance ratios of heavy neutron - capture elements , which were all measured from ionized species . the details of the analysis for individual neutron - capture elements are presented below . for comparison purposes , the abundances of cu and zn were also determined . in addition to the cu 5105 and zn 4810 lines , the uv lines ( 3250 - 3350 ) are also used in the abundance analyses ( table [ tab : table1 - 4 ] ) . although the abundances of sr , y , and zr have been determined by a number of previous studies for hd 122563 , measurements for other light neutron - capture elements are quite limited so far . we have newly detected 5 elements ( nb , mo , ru , pd , and ag ) in our spectrum , and derived an upper limit for a sixth element ( rh ) . two strong lines of ( 4077 and 4215 ) and the weak line of ( 4607 ) were measured . the two resonance lines are so strong that the abundance derived from these lines is sensitive to the microturbulent velocity and treatment of damping . however , the agreement of the abundances derived from and ( see table [ tab : table1 - 4 ] ) suggests the reliability of our measurements . two ( @xmath21 ) lines are detected . the detection of the 3163 line is reported only for canopus by @xcite and for the sun . this line would be useful for abundance analyses because of its strength ( figure [ fig : spectra ] ) . we note , however , that the abundance determined from this line is slightly higher ( by 0.25 dex ) than that from the 3215 line , possibly suggesting a small contamination by other spectral features . however , here we simply adopt the mean of the abundances derived from the two lines . the ( @xmath22 ) 3864 line is detected in cs 22892 - 052 by @xcite . although a blend of a cn line is reported by these authors , that is not severe in hd 122563 , because the carbon abundance of this star is more than 1 dex lower than that of cs 22892052 . we have detected two lines of ( @xmath23 ) . the equivalent width of the 3498 line is measurable due to there being no ( apparent ) blend with other lines . since a strong line exists in the bluer region of the other line at 3728 , we applied the spectrum synthesis technique to the analysis of that line . the ( @xmath24 ) 3692 line exists in the wing of h i line , but is not detected in our spectrum . we determined only an upper limit on the abundance of this element . we have detected two lines of pd . the 3242 line blends with oh lines . the contamination was estimated using the oh line list of @xcite , adjusting the oxygen abundance to match the strengths of neighboring oh lines ( [ o / fe ] @xmath250.4 ) . the 3404 line is clearly detected with no severe blend ( figure [ fig : spectra ] ) . two lines have been measured in some metal - poor stars . the 3280 line is detected , while the 3383 line is not . since nh lines blend with the 3280 line @xcite , we included the nh lines adopted from @xcite in the spectrum synthesis , calibrating the n abundance to reproduce the nh features around the line ( [ n / fe]@xmath26 ) . we note that the n abundance adopted is much lower than those derived by previous works ( e.g.,[n / fe]@xmath7 * ? ? ? this discrepancy is partially due to the inaccurate gf - values used in our analysis . recent studies of n abundances based on nh lines applied corrections of gf - values to the kurucz s list ( e.g. , * ? ? ? in addition to the nh lines , some unidentified feature appears in the red part of this line ( figure [ fig : spectra ] ) . if such unknown lines also affect the line , the values derived from this line must be regarded as an upper limit on the ag abundance . the abundances of ba , la , eu , and yb were derived by a spectrum synthesis technique , taking into account hyperfine splitting . the abundances of other elements were determined by applying a single line approximation , which is justified by the fact that the absorption lines are quite weak in general . in addition to the two resonance lines at 4554 and 4934 , two ba lines in the red region were measured . the line data of @xcite were used for the analysis of ba lines , assuming the isotope ratios of r - process component in solar - system material . there is a small discrepancy between the ba abundances from the two resonance lines and others . a possible reason for this discrepancy is the non - lte effect ( e.g. , * ? ? ? however , we here simply adopt the average of the ba abundances from the four lines . we detected three lines at 3819 , 4129 and 4205 . however , the 4205 line blends with a line , while the 3819 one is affected by a wing component of a strong fe line at 3820 . since these eu lines are quite weak in the spectrum of hd 122563 , in contrast to those in r - process enhanced stars , the effect of the blending is significant . hence , we adopt the result from the 4129 . we note that the eu abundances derived from the excluded lines , within their relatively large errors , agree with the value from the 4129 line . the effect of isotope shift for gd lines was included in the analysis by @xcite for the s - process enhanced star cs 31062 - 050 . this effect is , however , neglected in the present analysis , because the gd lines of hd 122563 are very weak . we have detected three gd lines ( table [ tab : table1 - 4 ] ) . the gd abundance derived from the 3481 line is significantly higher than those from the other two , and an upper limit estimated from another line ( 3331 ) . we regard this as a result of contamination of the 3481 line by some unidentified lines , and exclude this line in the determination of the final gd abundance . yb is the heaviest element detected in our spectrum . we measured two yb lines at 3220 and 3694 . a spectrum synthesis technique was applied to the analysis including the effect of hyperfine splitting for these lines ( sneden et al . 2003 , private communication ) . agreement of the results from the two lines is fairly good . the upper limits of ir and th abundances are obtained from the analyses of 3800 and 4019 line . the line of 4019 is affected by contamination by other lines ( figure [ fig : th ] ) . we analyzed this line using the line list of @xcite for this wavelength region . we adopted @xmath28c/@xmath29c = 5 to estimated the blending of @xmath29ch lines . since the blends of fe and co lines are severe , no clear th feature is identified , although the quality of the spectrum is very high ( s / n = 950 at 4020 ) . we derived the upper limit of the th abundance ( log @xmath30 ( th ) @xmath31 ) from the fitting of synthetic spectra by eye . neutron - capture elements have been studied for this object by many authors . recently , @xcite , @xcite , @xcite , and @xcite performed detailed abundance analyses based on high resolution spectroscopy ( @xmath32 ) for this object . @xcite also determined the abundances of eight neutron - capture elements , though their resolving power ( @xmath33 ) is not as high as those in the above studies . recently , @xcite derived the abundances of ge , zr , la and eu from the uv region ( 2410 @xmath2 3070 ) of this object using hubble space telescope , and also uv - blue region ( 3150 @xmath2 4600 ) with keck i hires . figure [ fig : comp ] shows comparisons of the abundances of neutron - capture elements derived by the present analysis with those of previous work . our results are in good agreement with @xcite , @xcite and @xcite . however , the abundances of neutron - capture elements determined by the present analysis are systematically lower than those by @xcite and @xcite . here we inspect the discrepancy between the results of @xcite and ours in some detail , because their work determined abundances of a larger number of elements than @xcite , and they reported the equivalent widths used in the abundance analysis for some lines ( unfortunately , the equivalent widths were not given for the lines for which spectrum synthesis technique was applied ) . the atmospheric parameters adopted in our analysis are in good agreement with theirs : the differences ( our results minus those of @xcite ) of @xmath10 , @xmath13 , [ fe / h ] and @xmath12 are + 70 k , 0.2 dex , 0.07 dex and 0.3 km s@xmath9 , respectively . these differences results in differences of abundances smaller than 0.1 dex for heavy neutron - capture elements ( see table [ tab : error ] ) . therefore , the difference of the adopted atmospheric parameters is not a reason for the abundance discrepancy . we also found good agreements in the equivalent widths of the lines reported by both works . however , the @xmath34 value of the nd 4061 line used by @xcite is lower by 0.25 dex than ours . though the @xmath34 values of la adopted by the two works are quite similar , no information on the treatment of the hyperfine splitting was given by @xcite , while the effect is fully included in our analysis . we adopted recent line data for @xcite and @xcite , which should be more accurate than those used by @xcite . thus , the discrepancy between the results of the two work might be partially explained by the difference of the adopted line data . however , there seems to exist small ( @xmath20.2 dex ) systematic differences , for which no clear reason is identified . we investigated a high quality uv - blue spectrum of hd 122563 , and have detected 19 neutron - capture elements including nb , mo , ru , pd , ag , pr , and sm , which are detected for the first time by our study for this object , and upper - limits for 5 elements including th . the derived abundances are given in table [ tab : table2 ] . in this section , we discuss the abundance pattern of the neutron - capture elements , along with cu and zn , to investigate the origin of neutron - capture elements in this object . the abundance ratio between ba ( or la ) and eu is used as an indicator of the origin of neutron - capture elements . the value of [ ba / eu ] of the solar system r - process component is 0.81 , while that of the s - process component is + 1.45 @xcite . the value of the [ ba / eu ] is 0.50 in hd 122563 . in addition , the value of [ la / eu ] of the solar system r - process component is 0.59 , while that in hd 122563 is 0.50 . these results indicate that the heavy ( @xmath35 @xmath36 56 ) neutron - capture elements of this object are principally associated with the r - process , and the contribution of the ( main ) s - process is small if any . we recall that this object is very metal - poor ( [ fe / h]@xmath37 ) with no excess of carbon ( [ c / fe]=@xmath38 ) . figure [ fig : r ] shows the abundance pattern of hd 122563 , comparing with that of the solar - system r - process component . the solar system r - process abundance pattern is scaled to the eu abundance of hd 122563 . this figure clearly shows that the abundances of light neutron - capture elements ( 38 @xmath1 z @xmath1 47 ) are much higher than those of heavy ones in hd 122563 , compared to the solar - system r - process pattern . this behavior is very different from that found in r - process enhanced stars ( e.g. cs 22892 - 052 , cs 31082 - 001 : * ? ? ? * ; * ? ? ? although we indicated above that the heavy neutron - capture elements are principally associated with the r - process , figure [ fig : r ] indicates that even in the range 56 @xmath39 @xmath35 @xmath39 59 the abundances in hd 122563 exceed those of the scaled solar r - process , with only la falling on the eu - normalised curve . this suggests that the tendency for the light neutron - capture elements ( 38 @xmath1 @xmath35 @xmath1 47 ) to fall above the scaled solar r - process curve is part of a general , atomic - number dependent , trend . in order to demonstrate this behavior more clearly , figure [ fig : f7 ] shows the logarithmic difference of the abundances of this object from the solar system r - process pattern as a function of atomic number . we also show that of the r - process enhanced star cs 22892 - 052 @xcite in the same way . we find good agreement between the abundance pattern of cs 22892 - 052 and the solar - system r - process one , at least for elements with 56@xmath40 . by way of contrast , our new measurements for hd 122563 clearly demonstrate a quite different abundance pattern from that of the solar - system r - process component and of cs 22892052 . the excess of light neutron - capture elements ( e.g. sr ) with respect to the heavy ones ( e.g. ba ) was known for hd 122563 by previous studies ( see section 1 ) . however , the abundances of elements having intermediate mass ( e.g. mo , pd ) were measured in detail for the first time by the present work . the abundances of these elements are intermediate , and , hence , the abundances of neutron - capture elements gradually and continuously decrease with increasing atomic number from @xmath41 to @xmath42 . this trend is a key to investigating the nucleosynthesis process that is responsible for neutron - capture elements in this object . for comparison purposes , we also attempted to compare the abundances of hd 122563 with the solar system s - process distribution ( figure [ fig : s ] ) . in this case , ba , la , ce , and eu show large deviations from the solar system s - process curve , although the overall abundance pattern from sr to yb seems to be similar . since the s - process abundance pattern in the solar - system is a result of a combination of some individual processes ( at least the weak and main components of the s - process ) , we presume this apparent `` agreement '' is physically not important . we also have detected four neutron - capture elements heavier than eu ( gd , dy , er , yb : @xmath43 64 ) . the abundances of these elements also show a decreasing trend as a function of atomic number , with respect to the solar - system r - process pattern . this also has an impact on the understanding of the origin of neutron - capture elements in this object . th is a radioactive element and is synthesized only by the r - process . the upper limit of the th abundance determined by the present work indicated that this object does not show an excess of th with respect to eu ( and other heavy neutron - capture elements ) , compared to most r - process enhanced stars . the cu and zn abundances of hd 122563 are typical values found in very metal - poor stars ( e.g. , * ? ? ? * ; * ? ? ? that is , no clear difference is found in the abundances of these elements between the stars having high and low abundances of light neutron - capture elements . this result may give a hint for the origins of cu and zn , as well as of light neutron - capture elements , which are still unclear . here we consider the origin of neutron - capture elements in this object based on the abundance pattern determined by the present work . the abundance ratios between light and heavy neutron - capture elements ( e.g. , sr / ba ) in hd 122563 are clearly different from those in r - process enhanced stars , as already mentioned in 1 . therefore , the r - process responsible for the heavy neutron - capture elements in the solar system ( the so - called `` main '' r - process ) is not an important source of , at least , light neutron - capture elements in hd 122563 . the main s - process is also excluded from the possible source of light neutron - capture elements , because that yields even lower abundance ratios between light and heavy neutron - capture elements at low metallicity ( e.g. * ? ? ? * ; * ? ? ? the weak component of the s - process was introduced to interpret the light s - process nuclei in the solar system . however , the gradually decreasing trend of elemental abundances from sr to ba found in hd 122563 does not resemble that of the weak s - process , which predicts a rapid drop of abundances at @xmath44 ( i.e. y or zr ) . it should be noted in addition that the weak s - process is expected to be inefficient at low metallicity , because of the lack of the neutron source @xmath45ne ( e.g. * ? ? ? * ; * ? ? ? * ) in addition to the lack of iron seeds . thus , another process that has efficiently yielded light neutron - capture elements at low metallicity is required to explain the abundance pattern of neutron - capture elements in hd 122563 . the presence of such a process has been indicated by recent observational studies of metal - deficient stars with no excess of heavy neutron - capture elements ( * ? ? ? * and references therein ) . this process was called the `` light element primary process '' ( lepp ) by @xcite , who estimated its contribution to the solar abundances . @xcite have shown that such light r - process nuclei ( up to @xmath46 ) are produced in neutrino winds as a result of weak " ( or failed ) r - processing ( see also * ? ? ? it is likely , therefore , that these light neutron - capture elements originate from the core - collapse supernovae , although a contribution from other sources can not be excluded . the abundance pattern of light to heavy neutron - capture elements measured for hd 122563 provides a unique constraint on such model calculations . the abundance pattern of heavy neutron - capture elements of hd 122563 shows a significant departure from the r - process component in solar - system material , though the departure is not as large as that found for light neutron - capture elements ( figure [ fig : r ] ) . figure [ fig : r+s ] shows a comparison between the abundances of elements with @xmath47 measured for hd 122563 ( filled circles ) and the solar - system r - process abundance pattern @xcite , which is scaled to achieve the best fit to the observed data ( the solid line ) . ( recall that the scaling in figure [ fig : r ] was to eu . ) significant disagreements are found for ce and pr , as well as for dy , er and yb . given the fact that the abundance patterns of heavy elements measured for r - process enhanced stars show excellent agreement with that of the solar - system , the different abundance pattern of hd 122563 must be significant . some process that produces a different abundance pattern of heavy neutron - capture elements from that of the main r - process is at least required to explain the abundance pattern of this star . one possibility is to assume a small contribution of the ( main ) s - process . in order to examine this possibility , we adopted the r - process and s - process abundance patterns in the solar - system as determined by @xcite , and searched for the combination that gives the best fit to the abundance pattern of hd 122563 . the result is shown by the dashed line in figure [ fig : r+s ] . in this case , the contributions of the s - process to ba , la , and ce are significant ( @xmath2 70% ) , while those to elements heavier than nd are small ( @xmath48 10% ) . it should be noted that such a small contribution of the main s - process does not affect the abundance pattern of light neutron - capture elements discussed in the previous subsection . the root - mean - square ( rms ) of the logarithmic abundance difference between the observed and calculated abundance patterns is 0.29 dex for the case that a contribution from the s - process is introduced , compared with 0.34 dex if only the r - process is assumed . thus , the fit is slightly better if an s - process contribution is introduced . however , the improved fit comes at the cost of an additional free parameter ( the relative contribution of the s- and r - processes ) , and it is not clear that assuming an s - process contribution is justifiable from the point of view of the star s nucleosynthetic history . in particular , the discrepancy found in the abundance pattern of elements heavier than sm can not be explained by this approach , because the production of such heavy elements by the s - process is very small . this discrepancy leads us to consider an alternative possibility that the heavy neutron - capture elements of this star are also a result of the unidentified ( single ) process that is responsible for the large enhancement of the light neutron - capture elements discussed in 5.2.1 . @xcite have suggested that the small ( but non - negligible ) amount of heavy r - process nuclei ( @xmath49 ) with non solar r - process abundance pattern can be produced in neutrino winds if the entropy of the neutrino - heated matter is enough high ( but smaller than that required for the main r - process ) , in addition to the light r - process nuclei . if this is true , the abundance pattern of heavy neutron - capture elements in this object may give another hint for , or constraint on , the modeling of this process . the differences between the abundance patterns of heavy neutron - capture elements in hd 122563 and the solar - system r - process abundance pattern are not large . further observational data are clearly required to derive definitive conclusions . however , it would be very difficult to improve the measurement of abundances for hd 122563 , because the quality of the spectrum used in the present work is extremely high . one possibility is to reconsider the atomic line data to reduce the uncertainties in abundance measurements . another important observational study is to apply similar analysis to other metal - poor stars having high abundance ratios of light to heavy neutron - capture elements . in particular , measurements for stars with lower metallicity than hd 122563 are important , because no s - process contribution is expected for such stars . we obtained a high resolution , high s / n uv - blue spectrum of the very metal - poor star hd 122563 with subaru / hds . the abundances were measured for 19 neutron - capture elements , among which seven elements are detected for the first time in this star . our new measurements for hd 122563 clearly demonstrate that the elemental abundances gradually decrease relative to the scaled solar r - process abundances with increasing atomic number , at least from sr to yb . we have considered whether the higher abundances of lighter elements can be explained by contributions from the weak s- or main s - processes , and find these not to provide satisfactory explanations . the abundance pattern of elements with @xmath50 does not agree with any prediction of neutron - capture process models known . the abundance pattern of a wide range of neutron - capture elements determined for hd 122563 therefore provides new strong constraints on models of nucleosynthesis for very metal - poor stars , in particular those with excesses of light neutron - capture elements but without enhancement of heavy ones . this work was supported in part by a grant - in - aid for the japan - france integrated action program ( sakura ) , awarded by the japan society for the promotion of science , and scientific research ( 17740108 ) from the ministry of education , culture , sports , science , and technology of japan . most of the data reduction was carried out at the astronomical data analysis center ( adac ) of the national astronomical observatory of japan . aoki , w. , et al . 2002 , apj , 580 , 1149 aoki , w. , et al . 2005 , apj , 632 , 611 asplund , m. , grevesse , n. , & sauval , a. j. 2005 , in asp conf . 336 , cosmic abundances as records of stellar evolution and nucleosynthesis , ed . t. g. barnes iii & f. n. bash ( san francisco : asp ) , 25 asplund , m. 2005 , ara&a , 43 , 481 burris , d. l. , pilachowski , c. a. , armandroff , t. e. , sneden , c. , cowan , j. j. , & roe , h 2000 , apj , 544 , 302 busso , m. , gallino , r. , & wasserburg , g. j. 1999 , ara&a , 37 , 239 cameron , a. g. w. 2001 , apj , 562 , 456 cayrel , r , et al . 2004 , a&a , 416 , 1117 cowan , j. j. , et al . 2002 , apj , 572 , 861 cowan , j. j. , et al . 2005 , apj , 627 , 238 den hartog , e. a. , lawler , j. e. , sneden , c. & cowan , j. j. 2003 , apjs , 148 , 543 ecuvillon , a. , israelian , g. , santos , n. c. , mayor , m. , garcia lopez , r. j. , & randich , s. 2004 , a&a , 418 , 703 freiburghaus , c. , rosswog , s. , & thielemann , f. -k . 1999 , , 525 , l121 hill , v. , et al . 2002 , a&a , 387 , 560 honda , s. , et al . 2004 , apj , 607 , 474 ishimaru , y. & wanajo , s. 1999 , , 511 , l33 ishimaru , y. , wanajo , s. , aoki , w. , & ryan , s. g. 2004 , , 600 , l47 johnson , j. a. & bolte , m. 2001 , apj , 554 , 888 johnson , j. a. 2002 , apjs , 139 , 219 johnson , j. a. & bolte , m. 2002 , apj , 579 , 616 johnson , j. a. & bolte , m. 2004 , apj , 605 , 462 kurucz , r. l. , & bell . b 1995 , kurucz cd - rom , no.23 ( harvard - smithsonian center for astrophysics ) kurucz , r. l. 1993 , kurucz cd - rom , no.13 ( harvard - smithsonian center for astrophysics ) lawler , j. e. , sneden , c. , & cowan , j. j. , 2004 , apj , 604 , 850 lawler , j. e. , bonvallet , g. , & sneden , c. 2001a , apj , 556 , 452 lawler , j. e. , wickliffe , m. e. , den hartog , e. a. , & sneden , c. 2001b , apj , 563 , 1075 mcwilliam , a , preston , g. w. , sneden , c , & searle , l 1995 , aj , 109 , 27 mcwilliam , a 1998 , aj , 115 , 1640 noguchi , k.,et al . 2002 , pasj , 54 , 855 prantzos , n. , hashimoto , m. , & nomoto , k. 1990 , a&a , 234 , 211 sneden , c. , mcwilliam , a. , preston , g. w. , cowan , j. j. , burris , d. l. , & armosky , b. j. 1996 , apj , 467 , 819 sneden , c , cowan , j. j. , ivans , i. i. , fuller , g. m. , burles , s , beers , t. c. , & lawler , j. e. 2000 , apj , 533l , 139 sneden , c. , et al . 2003 , apj , 591 , 936 sumiyoshi , k. , et al . 2001 , apj , 562 , 880 surman , r. , mclaughlin , g. c. , & hix , w. r. 2005 , submitted to ( astro - ph/0509365 ) raiteri , c. m. , gallino , r. , busso , m. , neuberger , d. , & kappeler , f. 1993 , apj , 419 , 207 reynolds , s. e. , hearnshaw , j. b. , & cottrell , p. l. 1988 , mnras , 235 , 1423 rosswog , s. , liebendorfer , m. , thielemann , f .- k . , davies , m. b. , benz , w. , & piran , t. 1999 , a&a , 341 , 499 travaglio et al . 2004 , apj , 114 , 1293 truran , j. w. , cowan , j. j. , pilachowski , c. a. , & sneden , c. 2002 , pasp , 114 , 1293 wanajo , s. , kajino , t. , mathews , g. j. , & otsuki , k. 2001 , , 554 , 578 wanajo , s. , tamamura , m. , itoh , n. , nomoto , k. , ishimaru , y. , beers , t. c. , & nozawa , s. 2003 , apj , 593,968 wanajo , s. & ishimaru , i. 2005 , , in press westin , j. , sneden , c. , gustafsson , b. , & cowan , j. j. 2000 , apj , 530 , 783 woosley , s. e. , wilson , j. r. , mathews , g. j. , hoffman , r. d. , & meyer , b. s. 1994 , , 433 , 229 lcccccc wavelength & l.e.p.(ev ) & log@xmath34 & log@xmath30 & _ w_(m ) & ref + , @xmath51 & & & & & + 3247.53 & 0.000 & 0.060 & 1.04 & 115.1 * & 8 + 3273.95 & 0.000 & 0.360 & 1.24 & 112.3 & 8 + 5105.55 & 1.390 & 1.520 & 0.66 & 3.3 & 2 + , @xmath52 & & & & & + 3302.98 & 4.030 & 0.057 & 2.07 & 20.3 * & 1 + 3345.02 & 4.078 & 0.246 & 1.78 & 19.5 * & 1 + 4722.15 & 4.030 & 0.390 & 2.09 & 14.5 & 5 + 4810.54 & 4.080 & 0.170 & 2.10 & 20.5 & 5 + , @xmath41 & & & & & + 4607.33 & 0.000 & 0.280 & 0.14 & 2.4 & 7 + , @xmath41 & & & & & + 4077.71 & 0.000 & 0.170 & 0.18 & 163.3 & 6 + 4215.52 & 0.000 & 0.170 & 0.03 & 155.6 & 6 + , @xmath53 & & & & & + 3327.88 & 0.410 & 0.130 & 0.98 & 49.3 * & 8 + 3549.01 & 0.130 & 0.280 & 0.97 & 49.0 & 7 + 3584.52 & 0.100 & 0.410 & 0.98 & 44.1 * & 10 + 3600.74 & 0.180 & 0.280 & 1.06 & 68.2 & 7 + 3611.04 & 0.130 & 0.010 & 1.06 & 59.0 & 7 + 3628.70 & 0.130 & 0.710 & 0.90 & 31.8 & 10 + 3710.29 & 0.180 & 0.460 & 0.98 & 82.3 & 10 + 3747.55 & 0.100 & 0.910 & 0.98 & 22.8 & 7 + 3774.33 & 0.130 & 0.210 & 0.91 & 79.0 & 6 + 3788.70 & 0.100 & 0.070 & 0.95 & 66.6 & 6 + 3818.34 & 0.130 & 0.980 & 0.74 & 29.4 & 6 + 3950.36 & 0.100 & 0.490 & 0.94 & 47.4 & 6 + 4398.01 & 0.130 & 1.000 & 0.82 & 28.0 & 6 + 4883.69 & 1.080 & 0.070 & 0.82 & 25.0 & 6 + 5087.43 & 1.080 & 0.170 & 0.91 & 13.8 & 6 + , @xmath54 & & & & & + 3438.23 & 0.090 & 0.420 & 0.57 & 76.8 * & 7 + 3457.56 & 0.560 & 0.530 & 0.17 & 27.3 & 7 + 3479.02 & 0.530 & 0.690 & 0.34 & 16.7 & 7 + 3479.39 & 0.710 & 0.170 & 0.40 & 40.2 * & 7 + 3499.58 & 0.410 & 0.810 & 0.48 & 13.6 * & 7 + 3505.67 & 0.160 & 0.360 & 0.42 & 47.0 * & 7 + 3536.94 & 0.360 & 1.310 & 0.33 & 7.6 * & 7 + 3551.96 & 0.090 & 0.310 & 0.35 & 57.3 * & 8 + 3573.08 & 0.320 & 1.040 & 0.26 & 16.4 & 7 + 3578.23 & 1.220 & 0.610 & 0.29 & 3.8 * & 7 + 3630.02 & 0.360 & 1.110 & 0.28 & 12.8 & 7 + 3714.78 & 0.530 & 0.930 & 0.25 & 13.6 & 7 + 3836.77 & 0.560 & 0.060 & 0.31 & 47.3 & 6 + 3998.97 & 0.560 & 0.670 & 0.07 & 36.1 & 7 + 4050.33 & 0.710 & 1.000 & 0.09 & 11.0 & 7 + 4208.98 & 0.710 & 0.460 & 0.16 & 27.1 & 6 + 4317.32 & 0.710 & 1.380 & 0.08 & 5.3 & 6 + , @xmath21 & & & & & + 3163.40 & 0.376 & 0.260 & 1.30 & 24.2 * & 1 + 3215.59 & 0.440 & 0.190 & 1.65 & 4.5 * & 6 + , @xmath22 & & & & & + 3864.10 & 0.000 & 0.010 & 0.87 & 3.3 & 8 + , @xmath23 & & & & & + 3498.94 & 0.000 & 0.310 & 0.90 & 4.1 & 6 + 3728.03 & 0.000 & 0.270 & 0.82 & 4.7 * & 6 + , @xmath24 & & & & & + 3692.36 & 0.000 & 0.174 & @xmath48 - 1.20 & syn & 6 + , @xmath55 & & & & & + 3404.58 & 0.810 & 0.320 & 1.31 & 6.7 * & 6 + , @xmath56 & & & & & + 3280.68 & 0.000 & 0.050 & 1.88 & 6.5 * & 6 + , @xmath42 & & & & & + 4554.04 & 0.000 & 0.170 & 1.76 & 95.4 * & 6 + 4934.10 & 0.000 & 0.150 & 1.76 & 82.2 * & 6 + 5853.70 & 0.604 & 1.010 & 1.54 & 8.5 * & 6 + 6141.70 & 0.704 & 0.070 & 1.55 & 40.6 * & 6 + @xmath57 & & & & & + 3794.77 & 0.240 & 0.210 & 2.35 & 5.9 * & 7 + 3988.52 & 0.400 & 0.210 & 2.75 & 1.7 * & 4 + 3995.75 & 0.170 & 0.060 & 2.65 & 2.1 * & 4 + 4086.71 & 0.000 & 0.070 & 2.53 & 4.6 * & 4 + 4123.23 & 0.320 & 0.130 & 2.70 & 2.0 * & 4 + @xmath58 & & & & & + 4222.60 & 0.120 & 0.180 & 1.90 & 2.0 * & 6 + 4523.08 & 0.520 & 0.080 & 1.62 & 1.6 * & 10 + 4539.78 & 0.330 & 0.080 & 2.03 & 1.1 & 10 + 4562.37 & 0.480 & 0.190 & 2.01 & 1.4 & 10 + 4572.28 & 0.680 & 0.290 & 1.98 & 1.1 * & 8 + @xmath59 & & & & & + 4179.40 & 0.200 & 0.480 & 2.15 & 7.8 * & 7 + 4189.48 & 0.370 & 0.380 & @xmath48 - 2.15 & & 8 + @xmath60 & & & & & + 3784.25 & 0.380 & 0.150 & 2.14 & 2.0 & 9 + 3826.42 & 0.064 & 0.410 & 2.00 & 2.0 * & 9 + 4061.08 & 0.471 & 0.550 & 2.00 & 5.5 & 9 + 4232.38 & 0.064 & 0.470 & 1.89 & 2.3 * & 9 + @xmath61 & & & & & + 4318.94 & 0.280 & 0.270 & 2.16 & 1.8 * & 7 + 4642.23 & 0.380 & 0.520 & 2.16 & 0.8 & 7 + @xmath62 & & & & & + 3819.67 & 0.000 & 0.510 & 2.95 & syn & 5 + 4129.70 & 0.000 & 0.220 & 2.77 & 9.0 * & 5 + 4205.05 & 0.000 & 0.210 & 2.92 & syn & 5 + @xmath63 & & & & & + 3331.40 & 0.000 & 0.140 & @xmath48 - 2.15 & syn & 7 + 3481.80 & 0.490 & 0.230 & 1.79 & 4.0 * & 7 + 3549.37 & 0.240 & 0.260 & 2.40 & 2.1 & 7 + 3768.40 & 0.080 & 0.360 & 2.48 & 3.8 * & 7 + @xmath64 & & & & & + 3460.97 & 0.000 & 0.070 & 2.42 & 3.2 * & 7 + 3531.71 & 0.000 & 0.770 & 2.81 & 8.3 * & 8 + @xmath65 & & & & & + 3398.94 & 0.000 & 0.410 & @xmath48 - 2.00 & syn & 11 + @xmath66 & & & & & + 3499.10 & 0.060 & 0.136 & 2.75 & 2.7 & 10 + 3692.65 & 0.050 & 0.138 & 2.57 & 4.5 * & 6 + @xmath67 & & & & & + 3701.36 & 0.000 & 0.540 & @xmath483.00 & syn & 8 + @xmath68 & & & & & + 3289.37 & 0.000 & 0.020 & 2.70 & 27.5 * & 8 + 3694.19 & 0.000 & 0.300 & 2.86 & 13.3 * & 8 + @xmath69 & & & & & + 3220.76 & 0.350 & 0.510 & @xmath48 - 1.12 & syn & 7 + 3800.12 & 0.000 & 1.450 & @xmath48 - 1.60 & syn & 7 + @xmath70 & & & & & + 4019.12 & 0.000 & 0.270 & @xmath483.05 & syn & 7 + lccccc species & _ z _ & log@xmath30 & @xmath19 & [ x / fe ] & n + cu & 29 & 0.98 & 0.29 & 1.15 & 3 + zn & 30 & 2.01 & 0.16 & + 0.18 & 4 + sr & 38 & 0.11 & 0.08 & 0.26 & 3 + y & 39 & 0.93 & 0.09 & 0.37 & 15 + zr & 40 & 0.28 & 0.16 & 0.10 & 17 + nb & 41 & 1.48 & 0.17 & 0.13 & 2 + mo & 42 & 0.87 & 0.17 & 0.02 & 1 + ru & 44 & 0.86 & 0.17 & + 0.07 & 2 + rh & 45 & @xmath481.20 & & @xmath48 + 0.45 & 1 + pd & 46 & 1.36 & 0.17 & 0.28 & 2 + ag & 47 & 1.88 & 0.17 & 0.05 & 1 + ba & 56 & 1.62 & 0.12 & 1.02 & 4 + la & 57 & 2.66 & 0.10 & 1.02 & 5 + ce & 58 & 1.83 & 0.18 & 0.64 & 6 + pr & 59 & 2.15 & 0.17 & 0.09 & 1 + nd & 60 & 2.01 & 0.10 & 0.69 & 4 + sm & 62 & 2.16 & 0.17 & 0.40 & 2 + eu & 63 & 2.77 & 0.17 & 0.52 & 1 + gd & 64 & 2.44 & 0.17 & 0.76 & 2 + dy & 66 & 2.62 & 0.17 & 0.99 & 2 + ho & 67 & @xmath482.00 & & @xmath48 + 0.26 & 1 + er & 68 & 2.66 & 0.17 & 0.82 & 2 + tm & 69 & @xmath483.00 & & @xmath480.23 & 1 + yb & 70 & 2.78 & 0.17 & 1.09 & 2 + ir & 77 & @xmath481.60 & & @xmath480.21 & 2 + th & 90 & @xmath483.05 & & @xmath480.34 & 1 + lccccccccccc species & & & & & & & + & @xmath71 & @xmath72 & & @xmath73 & @xmath74 & & @xmath75 & @xmath76 & & @xmath76 & @xmath75 + & @xmath770.22 & @xmath780.17 & & @xmath780.06 & @xmath770.07 & & @xmath770.15 & @xmath780.06 & & @xmath780.23 & @xmath770.26 + & @xmath770.05 & @xmath780.04 & & @xmath770.05 & @xmath780.04 & & 0.00 & @xmath770.01 & & @xmath780.03 & @xmath770.02 + & @xmath770.13 & @xmath780.12 & & @xmath780.02 & @xmath770.03 & & @xmath770.04 & 0.00 & & @xmath780.01 & 0.00 + & @xmath770.12 & @xmath780.11 & & @xmath770.05 & @xmath780.03 & & @xmath770.03 & @xmath780.05 & & @xmath780.31 & @xmath770.39 + & @xmath770.08 & @xmath780.07 & & @xmath770.08 & @xmath780.08 & & @xmath780.02 & @xmath770.01 & & @xmath780.16 & @xmath770.09 + & @xmath770.08 & @xmath780.07 & & @xmath770.09 & @xmath780.08 & & @xmath780.03 & @xmath770.02 & & @xmath780.08 & @xmath770.05 + & @xmath770.09 & @xmath780.09 & & @xmath770.10 & @xmath780.10 & & @xmath780.04 & @xmath770.02 & & @xmath780.03 & @xmath770.02 + & @xmath770.19 & @xmath780.17 & & @xmath780.02 & @xmath770.01 & & @xmath770.03 & @xmath780.00 & & @xmath780.00 & 0.00 + & @xmath770.20 & @xmath780.17 & & @xmath780.01 & @xmath770.02 & & @xmath770.04 & @xmath780.00 & & @xmath780.01 & 0.00 + & @xmath770.20 & @xmath780.17 & & @xmath780.01 & @xmath770.03 & & @xmath770.04 & @xmath770.01 & & @xmath780.01 & @xmath770.01 + & @xmath770.21 & @xmath780.17 & & @xmath780.00 & @xmath770.01 & & @xmath770.04 & @xmath780.00 & & @xmath780.01 & @xmath770.01 + & @xmath770.09 & @xmath780.07 & & @xmath770.09 & @xmath780.07 & & @xmath780.01 & @xmath770.01 & & @xmath780.08 & @xmath770.06 + & @xmath770.08 & @xmath780.07 & & @xmath770.10 & @xmath780.09 & & @xmath780.03 & @xmath770.02 & & @xmath780.00 & @xmath770.00 + & @xmath770.08 & @xmath780.07 & & @xmath770.09 & @xmath780.09 & & @xmath780.02 & @xmath770.02 & & @xmath780.00 & 0.00 + & @xmath770.10 & @xmath780.08 & & @xmath770.10 & @xmath780.09 & & @xmath780.03 & @xmath770.02 & & @xmath780.01 & @xmath770.01 + & @xmath770.08 & @xmath780.08 & & @xmath770.09 & @xmath780.10 & & @xmath780.04 & @xmath770.02 & & @xmath780.00 & 0.00 + & @xmath770.09 & @xmath780.07 & & @xmath770.09 & @xmath780.08 & & @xmath780.03 & @xmath770.02 & & @xmath780.00 & @xmath780.00 + & @xmath770.09 & @xmath780.09 & & @xmath770.10 & @xmath780.10 & & @xmath780.04 & @xmath770.02 & & @xmath780.00 & @xmath770.00 + & @xmath770.10 & @xmath780.08 & & @xmath770.11 & @xmath780.09 & & @xmath780.03 & @xmath770.02 & & @xmath780.00 & 0.00 + & @xmath770.09 & @xmath780.10 & & @xmath770.10 & @xmath780.10 & & @xmath780.05 & @xmath770.02 & & @xmath780.01 & @xmath770.01 + & @xmath770.09 & @xmath780.09 & & @xmath770.10 & @xmath780.09 & & @xmath780.04 & @xmath770.03 & & @xmath780.00 & @xmath780.00 + & @xmath770.09 & @xmath780.08 & & @xmath770.10 & @xmath780.10 & & @xmath780.04 & @xmath770.02 & & @xmath780.03 & @xmath770.02 +

we obtained high resolution , high s / n spectroscopy for the very metal - poor star hd 122563 with the subaru telescope high dispersion spectrograph . previous studies have shown that this object has excesses of light neutron - capture elements , while its abundances of heavy ones are very low . in our spectrum covering 3070 4780 of this object , 19 neutron - capture elements have been detected , including seven for the first time in this star ( nb , mo , ru , pd , ag , pr , and sm ) . upper limits are given for five other elements including th . the abundance pattern shows a gradually decreasing trend , as a function of atomic number , from sr to yb , which is quite different from those in stars with excesses of r - process elements . this abundance pattern of neutron - capture elements provides new strong constraints on the models of nucleosynthesis responsible for the very metal - poor stars with excesses of light neutron - capture elements but without enhancement of heavy ones .

optical emission - line images of planetary nebulae ( pns ) reveal a fascinating range of morphologies ( e.g. , the _ hubble space telescope _ gallery of pns ) , indicating complex internal structures in the nebulae . among these pns , ngc 6543 , also known as the cat s eye nebula , has perhaps the most interesting morphology . as reported by @xcite , the h@xmath4 and [ ] @xmath55007 line images of ngc 6543 are similar , showing an inner shell surrounded by an envelope with multiple , interlocking , semi - circular features . the [ ] @xmath56584 line image , on the other hand , shows bright clumps strung along arcs that appear to wrap around the envelope . in addition , the [ ] image shows two small curly features along the major axis and two linear jet - like features at 18@xmath6 from the major axis . ( see figure 1 of @xcite for the [ ] and [ ] images of ngc 6543 . ) what is the formation mechanism that has produced the complex nebular morphology of ngc 6543 ? it has been suggested that pns are formed by the current fast stellar wind sweeping up the circumstellar material lost previously via the slow agb wind ( e.g. , * ? ? ? * ; * ? ? ? * ) . in this interacting - stellar - winds model , the physical structure of a pn is similar to that of a wind - blown bubble , as modeled by @xcite . the pn will comprise a central cavity filled with shocked fast wind at temperatures of 10@xmath7 k , a dense shell of swept - up agb wind at 10@xmath8 k , and an outer envelope of unperturbed expanding agb wind . this morphology is obviously too simple compared to the observed morphology of ngc 6543 . ngc 6543 is a known x - ray source and was marginally resolved by _ rosat _ observations @xcite . diffuse x - ray emission implies the existence of hot gas . it is thus important to resolve the diffuse x - ray emission and compare the distribution of hot gas to the location of the dense , cooler , nebular shell in order to understand the physical structure and formation mechanism of this nebula . we have obtained _ chandra _ observations of ngc 6543 . its diffuse x - ray emission is clearly resolved into several components and shows excellent correspondence with some of the optical features . in addition , a previously unknown point x - ray source is detected at the central star . we have extracted and modeled the spectra of the diffuse x - ray emission to derive the physical conditions of the hot gas . the results and their implications are reported in this letter . the analysis of the point source will be reported in a future paper . ngc 6543 was observed with the advanced ccd imaging spectrometer ( acis ) on board the _ chandra x - ray observatory _ on 2000 may 1011 for a total exposure time of 46.0 ks . the nebula was placed at the nominal aim point for the acis - s array on the back - illuminated ( bi ) ccd chip s3 . the bi chip has a moderately higher sensitivity than the front - illuminated ( fi ) chips at energies below 1 kev . furthermore , the bi chip is not affected by the inadvertent radiation damage ( occurred immediately after the deployment of _ chandra _ ) as were the fi chips . the point spread function of the acis observation has a half power radius ( the radius encircling 50% of the energy ) @xmath1 05 at @xmath9 1 kev . the energy resolution , @xmath10 , of the bi chip is @xmath11 at 0.5 kev and @xmath12 at 1.0 kev . the observations were carried out at a ccd operating temperature of @xmath13 c. the background count rate is consistent with the quiescent background @xcite ; therefore , no background flares " affected the observations and no time intervals needed to be removed . a total of 1,950@xmath1440 counts ( background - subtracted ) are detected from ngc 6543 . we received level 1 and level 2 processed data from the _ chandra _ data center . the data reduction and analysis were performed using the _ chandra _ x - ray center software ciao v1.1.5 and heasarc ftools and xspec v11.0.1 routines @xcite . the x - ray emission from ngc 6543 is clearly resolved into a point source and a diffuse component , as shown in figure 1 . to describe the distribution of the x - ray emission , it is convenient to use optical features as points of reference ; therefore , we plot the x - ray emission as contours over an archival _ hst _ h@xmath4 image . the initial overlay showed an offset of @xmath11@xmath15 between the point x - ray source and the central star . as this offset is within the range of combined pointing errors of _ hst _ and _ chandra _ , we have shifted the x - ray image to register the point source at the central star . the resultant x - ray contours overlaid on the h@xmath4 image are presented in figure 1d . the diffuse x - ray emission is well bounded by sharp , bright h@xmath4 filaments . this tight morphological correlation supports our choice of alignment between the x - ray and h@xmath4 images . the x - ray - bounding h@xmath4 filaments outline a 10@xmath16 elliptical shell with two @xmath17 outward extensions along the major axis ( figure 1c ) . the northern extension appears to be comprised of two closed lobes , while the southern extension appears to consist of an incomplete lobe that is open toward the east . the diffuse x - ray emission from the central elliptical shell is limb - brightened and follows closely the inner wall of the nebular shell in the h@xmath4 image . diffuse x - ray emission is also present in the extensions . in the northern extension , diffuse x - ray emission fills each nebular lobe . in the southern extension , diffuse x - ray emission appears to be bounded even though the nebular lobe appears incomplete . it is interesting to note that the western lobe in the northern extension and the eastern side of the southern extension appear to be radially aligned with the two linear , jet - like features best seen in reed et al.s ( 1999 ) [ ] image at 17@xmath18 from the central star . this alignment may be naively used to suggest a physical connection between the diffuse x - ray emission along this direction and the jet - like features ; however , as we show in 3.2 , there is no spectral evidence supporting this hypothesis . we have extracted spectra from the entire nebula and from three regions corresponding to the central elliptical shell , the northern extension , and the southern extension , respectively . the x - ray spectrum of the central elliptical shell was extracted with the central star excised . these spectra can be used to constrain the temperature , density , and abundances of the hot , x - ray - emitting gas , and to search for spatial variations in these physical conditions . the extracted spectra are plotted in figure 2 . in all cases the emission peaks at 0.55 kev , and then drops abruptly to a faint plateau between 0.7 and 0.9 kev . no significant emission is detected at energies greater than 1.0 kev . below 0.5 kev , the spectra of the central shell and the southern extension drop off to nearly zero at 0.3 kev , while the spectrum of the northern extension levels and stays high at 0.3 kev . the observed spectral shape depends on many physical parameters , including the temperature and chemical abundances of the x - ray - emitting gas , the intervening absorption , and the detector response . we adopt the thin plasma emission model of @xcite . it is expected that the x - ray - emitting gas should have a chemical composition consistent with either the fast stellar wind itself or a mixture of the fast wind and the nebular material . the nebular abundances of he , c , n , o , and ne have been reported to be 0.11 , 2.3@xmath210@xmath19 , 5.9@xmath210@xmath20 , 5.6@xmath210@xmath19 , and 1.4@xmath210@xmath19 relative to hydrogen by number , respectively @xcite . accordingly , we have adopted nebular abundances , relative to the solar values @xcite , of 1.13 , 0.63 , 0.53 , 0.66 , and 1.14 for he , c , n , o , and ne , respectively , and 1.0 for the other elements . for the abundances of the fast stellar wind , we have adopted he and n abundances 60 and 3 times the solar values , respectively , but kept the abundances of the other elements solar @xcite . for the intervening absorption , we have assumed solar abundances and adopted absorption cross - sections from @xcite . we first fit the observed spectrum of the diffuse x - rays from the entire nebula with models using the aforementioned nebular abundances , but with temperature , absorption column density , and the normalization factor as free parameters in the model . no models with nebular abundances can match the observed spectral shape . more specifically , the models produce either too strong lines at @xmath10.56 kev or too weak lines at @xmath10.43 kev . the spectral fits can be improved only if the n / o ratio is raised above the nebular value . the stellar wind has an enhanced n / o ratio , roughly three times the nebular value ; consequently , models using stellar wind abundances fit the observed spectral shape much more satisfactorily . the best - fit model with stellar wind abundances has a plasma temperature of @xmath21 = 1.7@xmath210@xmath3 k , an absorption column density of @xmath22 = 8@xmath210@xmath23 @xmath24 , and a normalization factor of 7@xmath210@xmath20 cm@xmath20 . this best - fit model is overplotted on the spectrum of the entire nebula in figure 2 . the reduced @xmath26 of the fits as a function of @xmath22 and @xmath27 is plotted in figure 3 . the 99% confidence contour spans @xmath22 = 5.512@xmath210@xmath23 @xmath24 and @xmath21 = 1.61.8@xmath210@xmath3 k ( or @xmath27 = 0.1350.155 kev ) . the observed ( absorbed ) x - ray flux is 8@xmath210@xmath28 ergs @xmath24 s@xmath0 ; the unabsorbed x - ray flux is 8@xmath210@xmath29 ergs @xmath24 s@xmath0 , and the x - ray luminosity is 1.0@xmath210@xmath30 ergs s@xmath0 for a distance of 1 kpc @xcite . we have also fit the spectra extracted from the central shell , and the northern and southern extensions . in these spectral fits we used the same stellar wind abundances as those adopted in the fits for the entire nebula . these three spectra have fewer counts , so the best - fit plasma temperature and absorption column are not as well constrained as those for the entire nebula . as indicated by the @xmath26 grid plots in figure 3 , the northern and southern extensions have very similar temperatures , but the northern extension has a smaller absorption column . the temperature and absorption column of the central shell are the least well constrained . its @xmath26 grid plot indicates that the gas in the central shell may be cooler , more absorbed , or both . the spectral fits suggest appreciable variation of absorption column across the 16@xmath15 extent of the x - ray emission region of ngc 6543 . ngc 6543 is at a high galactic latitude , @xmath31 , so the foreground interstellar absorption column is not likely to vary rapidly over the 16@xmath15 nebular extent . the absorption column density through the nebular envelope of ngc 6543 , on the other hand , is expected to increase from the north to the south because the major axis of the nebula is tilted with its north end toward us @xcite . furthermore , the nebular envelope is brighter and denser near the equatorial plane of the central shell , thus would produce higher absorption columns toward the central shell and the southern extension . therefore , the intervening absorption of the x - ray emission from ngc 6543 occurs mostly within its cool nebular shell . finally , we note that the temperature appears uniform , no variations greater than 50% are detected . we have extracted spatially - resolved spectra for the two lobes in the northern extension ( not shown here ) . as described in 3.1 , the western lobe is radially aligned with a jet - like [ ] feature , but both the eastern and western lobes display similar spectral shape , showing no evidence of additional dynamical heating in the western lobe . therefore , either the jet - like [ ] feature is not a physically energetic phenomenon or the alignment is fortuitous . the physical properties of the hot gas in ngc 6543 can be compared with those expected in the wind - blown bubble model of @xcite . in this model , the bubble interior is filled by the adiabatically shocked fast stellar wind ; however , the heat conduction and mass evaporation across the interface between the hot interior and the cool nebular shell raise the density , lower the temperature , and alter the abundances of the hot gas near the interface . the limb - brightened x - ray morphology of ngc 6543 indicates that the hot gas near the interface is responsible for the x - ray emission . furthermore , the temperature derived from x - ray spectral fits , @xmath11.7@xmath210@xmath3 k , is 100 times lower than the post - shock temperature of ngc 6543 s 1,750 km s@xmath0 wind @xcite , but may be consistent with that expected near the interface . the location and temperature of ngc 6543 s x - ray - emitting gas require that this hot gas contains a significant fraction of nebular material . in fact , if the mixing is adiabatic , the observed low temperature requires the x - ray - emitting gas to be dominated by nebular material . this is contrary to the results of our spectral fits , which indicate stellar wind abundances ! as an independent test to the origin of ngc 6543 s x - ray - emitting gas , we compare the mass of the hot gas to the amount of fast stellar wind supplied during the lifetime of the nebula . the electron density of the x - ray - emitting gas can be derived from the normalization factor of the spectral fit and distance . for a normalization factor of 7@xmath210@xmath20 cm@xmath20 , a distance of 1 kpc , and a he / h number ratio of 6 , the rms electron density is 50 @xmath32 @xmath33 , where @xmath34 is the volume filling factor . the emitting volume , including the central shell and the extensions and taking into account of the 35@xmath6 inclination of the major axis @xcite , is 2.5@xmath210@xmath35 @xmath34 @xmath36 . the total mass of the x - ray - emitting gas is thus 2.5@xmath210@xmath19 @xmath32 m@xmath37 . the dynamical age of the cat s eye s central shell is @xmath11,000 yr @xcite . the stellar wind mass loss rate of the cat s eye s central star is uncertain , and ranges from 4@xmath210@xmath38 m@xmath37 yr@xmath0 @xcite to 3.2@xmath210@xmath39 m@xmath37 yr@xmath0 @xcite . within the dynamical age of the central shell , the total mass supplied by the stellar wind is 4@xmath210@xmath20 m@xmath37 to 3.2@xmath210@xmath19 m@xmath37 . comparisons between these values to the value derived from x - ray observations indicate that the fast stellar wind can supply most or all of the hot gas only if the high mass loss rate is adopted . if the x - ray - emitting gas is comprised of largely shocked fast stellar wind , the low temperature of the x - ray - emitting gas is puzzling . how does the hot shocked wind cool from 1.4@xmath210@xmath40 k to 1.7@xmath210@xmath3 k within 1,000 yr ? it is possible that the fast stellar wind is not steady and its velocity was lower in the past , as suggested by @xcite for bd + 30@xmath63639 , another pn whose diffuse x - ray emission has been unambiguously resolved @xcite . future observations of more pns with diffuse x - ray emission may help us solve this puzzle . finally , it is interesting to note that the _ chandra _ image of ngc 6543 shows the hot , x - ray - emitting gas to be confined within the central shell and the two extensions along its major axis , but not associated with any of the intriguing optical features in the outer envelope . the thermal pressure of the hot interior is about twice as high as the pressure in the nebular shell , for a nebular electron density of 4,000 @xmath33 and temperature of 7,000 k @xcite . clearly , the hot gas drives the expansion of the central shell , and governs the ongoing evolution of ngc 6543 , but is not responsible for the complex optical features in the outer shell . we thank the referee , joel kastner , for useful and constructive suggestions to improve this paper . this work is supported by the _ chandra _ x - ray observatory center grant number go0 - 1004x . mag is supported partially by the dgesic of the spanish ministry of education and culture .

we have obtained _ chandra _ acis - s observations of ngc 6543 , the cat s eye nebula . the x - ray emission from ngc 6543 is clearly resolved into a point source at the central star and diffuse emission confined within the central elliptical shell and two extensions along the major axis . spectral analysis of the diffuse component shows that the abundances of the x - ray - emitting gas are similar to those of the fast ( 1,750 km s@xmath0 ) stellar wind but not those of the nebula . furthermore , the temperature of this gas is @xmath11.7@xmath210@xmath3 k , which is 100 times lower than the expected post - shock temperature of the fast stellar wind . the combination of low temperature and wind abundances is puzzling . the thermal pressure of this hot gas is about twice the pressure in the cool nebular shell ; thus , the hot gas plays an essential role in the ongoing evolution of the nebula .

the measurement of forward jets provides an important testing ground for qcd predictions of the standard model in the low - x region . the lhc ( large hadron collider ) can reach @xmath1 and @xmath2 values previously inaccessible to hera as displayed in figure [ fig : intro ] . to access the low - x region one must look at high rapidity . for such task the rapidity coverage of up to @xmath3 = 5.2 in cms @xcite has been used . the jet rapidity and transverse momenta is well described by the calculations at next - to- leading - order ( nlo ) in perturbative quantum chromodynamics ( qcd ) using the dokshitzer - gribov - lipatov - altarelli - parisi ( dglap ) @xcite approach and collinear factorization . the dijet cross - section is also well described @xcite . when the collision energy @xmath0 is considerably larger than the hard scattering scale given by the jet transverse momentum , @xmath4 , calculations in perturbative qcd require a resummation of large @xmath5 terms . this leads to the prediction of new dynamic effects , expected to be described by balitsky - fadin - kuraev - lipatov ( bfkl ) evolution @xcite and @xmath6 factorization @xcite . an effective theory has been developed which describes strong interactions in this kinematic domain @xcite . this description is particularly useful in events with several jets with large rapidity separation , which are not well described by dglap predictions . to extend the study of the parton evolution equations , the azimuthal angle differences were also measured . this observable has a sensitivity to bfkl effects when both jets are widely separated in rapidity ( eg : mueller - navelet jets ) . the inclusive forward jet cross - section was measured from an integrated luminosity of 3.14 @xmath7 @xcite . jets were reconstructed with the anti-@xmath6 clustering algorithm @xcite with a distance parameter r = @xmath8 = 0.5 . the energy depositions in the calorimeter cells were used as input for the clustering . assuming massless jets , a four momentum is associated with them by summing the energy of the cells above a given threshold . the forward region is defined as 3.2 @xmath9 4.7 . the jets are required to have a transverse momentum above @xmath4 = 35 gev . if more than one jet is present , the one with with highest @xmath4 is considered , as is illustrated in figure [ fig : feynman_inclusive_forward ] . the jets are corrected for the following systematic effects : @xmath10 and @xmath11dependent response of the calorimeters , overlap with other proton proton interactions and the migration of events across the @xmath4 bins due to jet energy resolution . in figure [ fig : inclusive_forward_unc ] the experimental systematic uncertainties are shown for the leading forward jet as function of @xmath10 . the jet energy scale is the dominant systematic uncertainty and the total uncertainty is around -25 + 30% . the inclusive forward jet production cross section corrected to hadron level is presented in figure [ fig : inclusive_forward ] . although all predictions describe the data within the uncertainty band , some of them do better . powheg @xcite + pythia 6 @xcite gives the best description . pythia 6 and pythia 8 @xcite describe the data reasonably well . cascade @xcite underestimates the cross - section while herwig 6 @xcite + jimmy @xcite tends to overestimate . nlojet++ overestimates the data but is still within the large theoretical and experimental uncertainties . the selection procedure for the simultaneous forward central dijet production is similar to the one for for the inclusive forward jet production . in addition , a central jet within @xmath12 2.8 with a transverse momentum above @xmath10 = 35 gev is required . a feynman diagram of the process is shown in figure [ fig : feymamn_forwardcentral ] . several mc predictions compared to the data cross - section is presented in figures [ fig : forwardcentral1 ] and [ fig : forwardcentral2 ] @xcite . forward jet cross - section is steeper than the central jet . the shape of the forward jet is poorly described when compared with the central jet . hej @xcite provides the best description being followed closely by herwig 6 and herwig + + @xcite . both pythia 6 , pythia 8 and the ccfm cascade have troubles describing the data for the central jets and for low @xmath10 forward jets . powheg + pythia 6 , which was the best prediction for inclusive forward jet production , yelds similar result as pythia 6 alone . the reconstruction and correction procedure is similar as for the inclusive forward jet production @xcite . mueller - navelet jets are the dijet pair with the highest rapidity separation . in this analysis only jets with @xmath10 above 35 gev and @xmath13 4.7 were considered . the azimuthal angle decorrelations of jets widely separated in rapidity is presented in figures [ fig : azimuthal_decorrelations1 ] and [ fig : azimuthal_decorrelations2 ] as function of rapidity separation . the first row of figure [ fig : azimuthal_decorrelations1 ] displays the azimuthal angle difference @xmath14 for jets with a rapidity separation @xmath15 less than 3 . pythia 6 and herwig + + describe the data within uncertainties , while pythia 8 and sherpa 1.4 @xcite with parton matrix elements matched show deviations at small and intermediate @xmath14 . the second row shows @xmath14 for a rapidity separation between 3 and 6 . herwig + + provides the best description , but all predictions show deviation beyond the experimental uncertainties . the last row shows the azimuthal angle difference for @xmath15 between 6 and 9 . the dijets are strongly decorrelated . herwig + + provides the best description while pythia 6 and pythia 8 fail for the lower @xmath14 region . the figure [ fig : azimuthal_decorrelations2 ] shows @xmath14 for mueller - navelet jets with different rapidity separations compared with with different pythia 6 predictions . the contributions of the angular ordering ( ao ) and multi parton interactions ( mpi ) are very similar . the intermediate @xmath15 region is better described without mpi . overall the data is better described with ao and mpi . using the same selection as in the previous section , the fourier coefficients of the average cosines have been measured @xcite and is presented in the figure [ fig : average_cosines ] . @xmath16 dglap contributions are expected to partly cancel in the @xmath17 ratio , which are described the by ll dglap based generators towards low @xmath15 . sherpa , pythia 8 and pythia 6 overestimate @xmath18 while herwig underestimate it . the ccfm based cascade predicts too small @xmath17 . at @xmath19 , a bfkl nll calculation describe @xmath18 within uncertainties . using jets with @xmath20 35 gev and @xmath13 4.7 the ratio of the inclusive to exclusive dijet production was measured as a function of @xmath15 @xcite . with increasing @xmath15 a larger phase space for radiation is opened . the inclusive dijet sample consists of events with at least 2 jets over the threshold and exclusive requires exactly two jets . the ratio of inclusive to exclusive dijet production is shown in the figure [ fig : ratio_dijets1 ] . pythia 6 and pythia 8 agree well with the data while herwig + + and hej + ariadne @xcite overestimate the data at higher @xmath15 . cascade is completly off . mpi gives only a small contribution . the ratio of inclusive to exclusive mueller - navelet dijets is presented in [ fig : ratio_dijets2 ] . at low @xmath15 the ratio of muller - navelet over exclusive is , by definition , smaller than inclusive over exclusive and at higher @xmath15 it is the same . the conclusions of the comparison between data and mc are the same as for the ratio inclusive over exclusive . inclusive measurements of forward and central forward jets , are reasonably well described by the mc predictions while more exclusive measurements are poorly described . a summary of the mc description is presented in table [ summary ] . the dglap based generators , pythia and herwig , seem to do a better job than the bfkl inspired cascade . the effort of description of the underlying events , development of the parton showers and tuning of pythia and herwig play an huge role into this result . .monte carlo description of the measurements [ cols="<,^,^,^,^,^",options="header " , ] to the cms collaboration for the oportunity to join this conference and to hannes jung for supervision in writing this proceeding . cms collaboration , measurement of the differential dijet production cross section in proton - proton collisions at @xmath21 7 tev , phys . b 700 ( 2011 ) 187 , arxiv:1104.1693 , doi:10.1016/j.physletb.2011.05.027 . cms collaboration , measurement of the inclusive production cross sections for forward jets and for dijet events with one forward and one central jet in pp collisions at @xmath22 7 tev " , jhep 1206 ( 2012 ) 036 , doi : 10.1007/jhep06(2012)036 , arxiv:1202.0704 cms collaboration , `` ratios of dijet production cross sections as a function of the absolute difference in rapidity between jets in proton - proton collisions at @xmath0 = 7 tev '' eur.phys.j.c72(2012)2216 , arxiv:1204.0696 .

the latest cms jet measurements in p - p collisions at @xmath0 = 7 tev , sensitive to small - x qcd physics , are discussed . these include inclusive forward jet and simultaneous forward - central jet production , as well as production ratios and azimuthal angle decorrelations of jets widely separated in rapidity .

the skyrme model @xcite is a nonlinear effective field theory of weakly coupled pions in which baryons emerge as localized finite energy soliton solutions . the stability of such solitons is guaranteed by the existence of a conserved topological charge interpreted as the quantum baryon number @xmath5 . more specifically , skyrmions consist in static pion field configurations which minimize the energy functional of the skyrme model in a given nontrivial topological sector . the model is partly motivated by the large-@xmath6 qcd analysis @xcite , as there are reasons to believe that once properly quantized , a refined version of the model could accurately depict nucleons as well as heavier atomic nuclei with mesonic degrees of freedom @xcite in the low energy limit . in the lowest nontrivial topological sector @xmath7 , the skyrmion is described by the spherically symmetric hedgehog ansatz which reproduces experimental data with an accuracy of @xmath8 or better @xcite . however , this relative success radically contrasts with the situation encountered in the @xmath9 sector , where the hedgehog ansatz is not the lowest energy configuration and would not give rise to bound state configurations @xcite . moreover , pioneering numerical investigations of verbaarschot @xcite clearly indicate that the @xmath0 skyrmion is not spherically symmetric , but rather possesses an axial symmetry reflected in its doughnut - like baryon density . further inspection of this numerical solution , in particular the profile function , suggests that the classical biskyrmion may be represented by an oblate field configuration . yet , most of the trial functions used to describe such a solution assume a decoupling from the angular degrees of freedom , i.e. @xmath10 , as this is the case for the instanton - inspired ansatz proposed in @xcite or in the early variational approach @xcite for example . on the other hand , some axially symmetric solutions were analysed in the @xmath7 sector , by @xcite and @xcite respectively , to include possible deformations due to centrifugal effects undergone by the rotating skyrmion and account for the quadrupole deformations of baryons . here , our aim is to extend the work on oblate skyrmions in @xcite to describe dibaryon states . the classical static oblate solution introduced in this manner will provide a quantitative estimate of the axial deformation , which is different from a uniform scaling in a given direction as performed in @xcite . it should also provide an adequate ground to perform the quantization of the @xmath0 soliton . there has been several attempts to decribed the angular dependence of the @xmath4 solutions which is much more complicated than the hedgehog form in @xmath11 fortunately , a few years ago , houghton et al . @xcite came up with an interesting ansatz based on rational maps . the rational map ansatz provides a simple alternative compared to the full numerical study of the angular dependence of the baryon density distribution of multiskyrmions . it also yields static energy predictions in good agreement with numerical solutions for several values of @xmath5 . the most interesting feature of this method remains without doubt that fundamental symmetries of multiskyrmions can easily be implemented in the ansatz solutions . this provides a clever way to identify the symmetries of the exact solutions , which are not always apparent , and in some cases , adequate initial solutions for lengthy numerical calculations . close as it may be , the rational map ansatz remains an approximation and in some cases , more accurate angular ansatz have been found . for example , houghton and krusch @xcite slightly improved the mass approximation of the biskyrmion by relaxing the requirement of holomorphicity imposed on rational maps . however , the profile function defined in this work still solely depends on @xmath12 . as may seems evident , some accuracy still may be gained by introducing a more appropriate parametrization of the soliton shape function . recently , ioannidou et al . @xcite obtained similar results by introducing an improved harmonic maps ansatz where the profile function depends on radial and polar degrees of freedom as well . however , they had to deal with a complicated second - order partial differential equation . from these considerations , we propose a @xmath0 oblate solution based on rational maps , which could be understood as the rational maps solution proposed in @xcite with the radial dependence @xmath13 replaced by an oblate form @xmath14 . consequently , the soliton undergoes a smoothly flattening along a given axis of symmetry . the parameter @xmath15 provides a measure of the scale at wich the deformation becomes important while the solution preserves the angular dependence given by the rational maps scheme for the @xmath0 case . this choice is obviously consistent with the toroidal baryon density of the @xmath0 skyrmion . implementing the oblate ansatz to the model , we first integrate analytically the angular degrees of freedom . this explains why other angular ansatz such as those in @xcite and @xcite were not chosen ; they led to complications . the second step involve solving the remaining nonlinear ordinary second - order differential equation resulting from the minimization of the static energy functional with respect to the profile function @xmath14 . thereby , the parameter @xmath15 is set as to minimize the static energy , i.e. the mass of the soliton . although the method applies to higher baryon numbers and other skyrme model extensions , the analysis is restricted here to @xmath0 skyrmion for @xmath1 skyrme model . in the next section , we present the axially symmetric oblate ansatz for the su(2 ) skyrme model introducing the oblate spheroidal coordinates . in section iii , we briefly describe rational maps and show how they can be used in the context of static oblate biskyrmions . a discussion of the numerical results follows in the last section , where we also draw concluding remarks about how the oblate - like solution could be a good starting point to perform the quantization of the non - rigid @xmath0 soliton as the deuteron . let us first introduce the oblate spheroidal coordinates @xmath16 which are related to cartesian coordinates through @xmath17 so a surface of constant @xmath18 correspond to a sphere of radius @xmath19 flattened in the @xmath20-direction by a factor of @xmath21 . for small @xmath18 , these surfaces are quite similar to that of pancakes of radius @xmath15 whereas when @xmath18 is large , they become spherical shells of radius given by @xmath22 . note that taking the double limit @xmath23 such that @xmath12 always remains finite , one recovers the usual spherical coordinates . thus , the choice of the parameter @xmath15 establishes the scale at which the oblateness becomes significant . finally , the element of volume reads @xmath24 neglecting the pion mass term , the chirally invariant lagrangian of the @xmath1 skyrme model just reads @xmath25^{2}%\ ] ] where @xmath26 with @xmath27 . here , @xmath28 is the pion decay constant and @xmath29 is sometimes referred to as the skyrme parameter . in order to implement an oblate solution , let us now replace the hedgehog ansatz @xmath30 by the static oblate solution defined as follow @xmath31 where the @xmath32 stand for the pauli matrices while @xmath33 is the standard unit vector @xmath34 . more explicitly , this unit vector is simply @xmath35 as will become apparent in the next section , we consider the case where @xmath36 and @xmath37 i.e. @xmath38 and @xmath39 depend only on the polar angle @xmath40 and the azimuthal angle @xmath41 respectively . furthermore , @xmath14 , which determines the global shape of the soliton , plays the role of the so - called profile function . in that respect , the oblate ansatz is clearly different from a scale transformation along one of the axis @xcite . as in its original hedgehog form , the field configuration @xmath42 constitutes a map from the physical space @xmath43 onto the lie group manifold of @xmath1 . finite energy solutions require that this @xmath1 valued field goes to the trivial vacuum for asymptotically large distances , that is @xmath44 . the expression for the static energy density is @xmath45^{2}\right)\ ] ] so , after substituting the oblate ansatz , the mass functional can be written as @xmath46=\int dv\left ( { \mathcal{m}}% _ { 2}+{\mathcal{m}}_{4}\right)\ ] ] with @xmath47 here , the notation is lighten by the use of the @xmath48 matrix defined as : @xmath49@xmath50 introducing an auxiliary variable for convenience , @xmath51 one easily deduces that @xmath52 and @xmath53 however , before minimizing the mass functional with respect to the chiral angle @xmath14 , in view to get the static configuration of the soliton , one must specify an angular dependence in @xmath54 and @xmath55 . this is the subjet of the next section , after a brief recall of some basic features related to the rational maps ansatz . formally , a rational map of order @xmath56 consists in a @xmath57 holomorphic map of the form @xmath58 where @xmath59 and @xmath60 appear as polynomials of degree at most @xmath56 . moreover , these maps are built in such a way that @xmath59 or @xmath60 is precisely of degree @xmath56 . it is also assumed that @xmath59 and @xmath60 do not share any common factor . any point @xmath20 on @xmath61 is identified via stereographic projection , defined through @xmath62 . thus , the image of a rational map @xmath63 applied on a point @xmath20 of a riemann sphere corresponds to the unit vector @xmath64 which also belongs to a riemann sphere . the link between static soliton chiral fields and rational maps @xcite follows from the ansatz @xmath65 inasmuch as @xmath13 acts as radial chiral angle function . to be well defined at the origin and at @xmath66 , the boundary conditions must be @xmath67 where @xmath68 is an integer and @xmath69 the baryon number is given by @xmath70 where @xmath71 is the degree of @xmath72 we consider only the case @xmath73 here , so @xmath74 . by analogy with the nonlinear theory of elasticity @xcite , manton has showed that the static energy of skyrmions could be understood as the local stretching induced by the map @xmath75 . in this real rubber - sheet geometry , the jacobian @xmath76 of the transformation provides a basic measure of the local distortion caused by the map @xmath42 . this enables us to build a symmetric positive definite strain tensor defined at every point of @xmath77 as @xmath78 this strain tensor @xmath79 , changing into @xmath80 under orthogonal transformations , comes with three invariants expressed in terms of its eigenvalues @xmath81 , @xmath82 and @xmath83 : @xmath84@xmath85@xmath86 since it is assumed that geometrical distorsion is unaffected by rotations of the coordinates frame in both space and isospace , the energy density should remain invariant and could be written as a function of the basic invariants as follows @xmath87 where @xmath88 and @xmath89 are parameter depending on @xmath28 and @xmath90 while the baryon density is associated with the quantity @xmath91 in this picture , radial strains are orthogonal to angular ones . moreover , owing to the conformal aspect of @xmath92 , angular strains are isotropic . thereby , it is customary to identify @xmath93 and @xmath94 thus , substituting these eigenvalues in ( [ energy ] ) and integrating over physical space yields @xmath95 with @xmath96@xmath97 and @xmath98 wherein @xmath99 corresponds to the usual area element on a 2-sphere , that is @xmath100 . at first glance , one sees that radial and angular contributions to the static energy are clearly singled out . now , focussing on the @xmath101 case , the most general rational map reads @xmath102 however , imposing the exact @xmath103torus symmetries ( axial symmetry and rotations of @xmath104 around cartesian axes ) , as expected from numerical analysis @xcite , restricts the general form above to this one @xmath105 it has been showed in @xcite that @xmath106 , and thus the mass functional alike , exhibits a minimum for @xmath107 . then , one must conclude that the most adequate choice of a rational map for the description of the biskyrmion solution boils down to @xmath108 . recasting this map in term of angular variables @xmath109 , we get @xmath110 it is easy to verify that for @xmath111 the rational map is simply @xmath112 or @xmath113 and @xmath114 , and one recovers the energy density in @xcite . we shall assume here that this angular function still holds in the oblate picture in ( [ m2]-[k2 ] ) . after analytical angular integrations are performed , the mass of the oblate biskyrmion can be cast in the form @xmath115 with @xmath116 and @xmath117 the explicit expressions of the density functions @xmath118 , which are reported in appendix a , follow from straightforward but tedious calculations . adopting the same conventions as in @xcite , for the sake of comparison , we also rescaled the deformation parameter as @xmath119 , with @xmath120 and @xmath121 . the values of @xmath122 mev and @xmath123 are chosen to coincide with those of @xcite . now , the chiral angle function @xmath14 can be determined from the minimization of the above functional , i.e. requiring that @xmath124=0 $ ] . thus static field configuration must obey the following nonlinear second - order ode : @xmath125 here , the primes merely denote derivatives with respect to @xmath18 . solving numerically for several values of @xmath126 , we obtain the set of chiral angle functions of figure 1 . when @xmath126 is small , we recover exactly the solution of the @xmath101 rational map ansatz . let us stress that increasing @xmath126 enforces a continuous displacement of the function @xmath14 which induces a smooth deformation of the soliton . . from left to right , we have @xmath127 0.5 , 0.2,0.1 , 0.05 , 0.01 , 0.005 and 0.001.,width=264,height=226 ] . the mass reaches its minimal value for @xmath128.,width=264,height=226 ] in figure 2 , we plot the mass of the oblate biskyrmion as a function of @xmath126 . the mass of the biskyrmion passes trough a minimum for a finite non - zero value of the parameter @xmath126 . this is a clear indication that the oblate solution is energically favored . note again that in the limit @xmath129 , we reproduce the mass value found in @xcite with the rational maps ansatz and profile function with radial dependence @xmath13 . numerical calculations carried out some years ago by verbaarschot @xcite and almost concurrently by kopeliovich and stern @xcite , establish that in the skyrme model the mass ratio @xmath130 is @xmath131 . the @xmath0 oblate solution also represents a bound state of two solitons since its mass is lower than twice the mass found in the @xmath7 sector @xcite . from our calculations , we get that the mass of the static oblate biskyrmion , being minimized for @xmath132 , is @xmath133 or @xmath134 mev . the parameter @xmath128 provides a measure of how the the @xmath0 solution is flattened . hence , the mass ratio of our flattened solution turns out to be @xmath135 . comparing to other ansatz for @xmath0 solutions , it is fairly smaller than that predicted by the familiar hedgehog ansatz with boundary conditions @xmath136 and @xmath137 , since then @xmath138 @xcite or and still better than the hedgehog - like solution with @xmath139 proposed as in @xcite , whose mass ratio is @xmath140 . note that none of these solutions are stable solutions since @xmath141 . let us mention that kurihara et al . @xcite achieved a mass ratio of @xmath142 using a different angular parametrization . however , there are no obvious physical grounds for their angular trial function and it remains that rational maps are far more superior when it comes to depict the symmetries of the @xmath4 solutions . our results still represent a slight improvement over those obtained in the original framework of rational maps , i.e. @xmath143 @xcite , where the chiral angle is strictly radial . hence the oblate solution which depends of a spheroidal oblate coordinate @xmath144 captures more exactly the profile shape of the biskyrmion than the original rational maps ansatz does although both rely on rational maps . the relatively small improvement also suggests that a better ansatz for classical @xmath0 static solution would require a different choice for the angular dependence . in that regard , @xcite and @xcite both achieved a mass ratio of @xmath145 by dropping the constraint on the rational maps to be holomorphic . these alternatives remain very difficult to implement for an oblate field configuration . it is worth emphasizing that the procedure presented here generalizes to any baryon number and any choice of angular ansatz consistent with multiskyrmions symmetries although analytical angular integration may become more cumbersome if not impossible in those cases . similarly , the approach can be generalized to other skyrme - like effective lagrangians . in this paper we only investigated the classical @xmath0 static solution but in principle , the full solution requires a quantization treatment to account for the quantum properties of dibaryons . the most standard procedure consists in a semiclassical quantization using collective variables . it only adds simple kinetic terms to the hamiltonian but these energy contributions should partially fill the energy gap between the @xmath146 deuteron mass ( @xmath147 mev ) and that of our deformed @xmath0 static skyrmion ( @xmath134 mev ) . so , even if our analytical oblate ansatz is not necessarily the lowest static energy solution for the @xmath0 skyrmion , the optimization of the oblateness parameter @xmath126 should prove adequate to take into account the soliton deformation due to centrifugal effects , as for the @xmath7 case @xcite . thus , following such a procedure , we can expect that the properly quantized biskyrmion solution would provide a good starting point for the description of the low energy phenomenology of the deuteron @xcite . the problem of quantization of the oblate biskyrmion solution is an important topic in itself and will be addressed elsewhere . j .- f . rivard is indebted to h. jirari for useful discussions on computational methods and numerical analysis . this work was supported in part by the natural sciences and engineering research council of canada . performing angular integrations in the oblate static energy functional , we get the following @xmath118 density functions : @xmath148@xmath149@xmath150@xmath151 where @xmath152 $ ] . t. krupovnickas , e. norvaisas and d.o . riska , lithuanian journal of physics 41 , 13 ( 2001 ) , arxiv : nucl - th/0011063 v1 17 nov 2000 ; a. acus , j. matuzas , e. norvaisas and d.o . riska , arxiv : nucl - th / nucl - th/0307010 v1 2 jul 2003 .

the numerical solution for the @xmath0 static soliton of the @xmath1 skyrme model shows a profile function dependence which is not exactly radial . we propose to quantify this with the introduction of an axially symmetric oblate ansatz parametrized by a scale factor @xmath2 we then obtain a relatively deformed bound soliton configuration with @xmath3 . this is the first step towards to description of @xmath4 quantized states such as the deuteron with a non - rigid oblate ansatz where deformations due to centrifugal effects are expected to be more important . [ [ section ] ]

apart from a recent work by doleschall and borbly@xcite , non - locality in the nn interaction has not been the object of dedicated studies for a long time . it is not absent however in models where it appears most often as a by - product of some prejudice in their construction . in the paris model@xcite , for instance , it was realized that an energy dependence could help in fitting nn scattering data . the transformation of this energy dependence into a @xmath4 dependence provides a non - local component . later on , the bonn group produced a model , field - theory motivated , taking into account the coupling of the @xmath5 channel to @xmath6 , n@xmath7 , @xmath8 channels@xcite . it contains both a spatial and a time - non - locality . moreover , the improvement consisting in introducing the dirac spinors to describe @xmath9-spin particles also provides non - locality . one could add other examples that have not been concretized in a high accuracy model . taking into account the substructure of nucleons and mesons in terms of quarks most often leads to a non - locality@xcite , which is better expressed in configuration space than in momentum one , contrarily to the above sources of non - locality . a double question may be raised about this non - locality . does it help in explaining nn scattering data and how this could be evidenced ? on the other hand , taking into account that a non - locality can be transformed away by a unitary transformation ( wave by wave ) , one can wonder whether the different models on the market are independent of each other ? concerning the first question , some answer is obtained by examining models such as the versions nij1 ( non - local ) and nij2 ( local ) of the nijmegen group@xcite . they equally fit the scattering phase shifts ( @xmath10 ) , but in the first case , this is achieved with 41 parameters while 47 are required in the other one . the slightly smaller number in the former case perhaps provides indication that the introduction of some non - locality is beneficial . the second question is the main object of the present paper . the plan will be as follows . in the second section , we present the different non - local terms which we are interested in . how they can be removed by a unitary transformation at the first order is given . the third section is devoted to a few selected results concerning the deuteron : static properties , form factors , structure function ( @xmath0 ) and the tensor polarization ( @xmath1 ) . it involves a comparison of these quantities obtained with different models when the effect of the non - locality is taken into account . section four contains the conclusion and a discussion . due to a lack of space , we concentrate here on the essential points . details and extended results could be found in refs . the interaction of interest here may be written : @xmath11 . \label{def}\ ] ] where the non - local terms take the form of an anticommutator or a commutator . another term of same order could be considered but those retained here are the only ones appearing in the pion - exchange contribution when this one is expanded up to order @xmath12 . due to its long range , it a priori provides larger contributions . moreover , they are theoretically well identified while shorter range contributions are likely to have some effective character . the last term in the above equation has been studied at length in refs . though it has a different origin , it has a strong similarity with a term arising from the difference of pseudo - scalar and pseudo - vector @xmath13 couplings , considered in an earlier work by friar@xcite . as for the anticommutator terms , only rough estimates were made in the past . they are considered more completely here . in principle , if two models are unitary equivalent , the corresponding hamiltonians , @xmath14 and @xmath15 , should fulfill the following relation : @xmath16 at the first order in the interaction , the quantity , @xmath17 , appearing in the unitary transformation , @xmath18 , can be determined by requiring : @xmath19= \delta\,v^0 \ ; , \label{req}\ ] ] where @xmath20 has to be local . this equation is fulfilled as follows : @xmath21 with @xmath22 it is noticed that the difference of @xmath23 and @xmath24 in eq . ( [ unit ] ) involves two - body but also many - body terms . similarly , if a one - body current is introduced in one representation , the other one contains two , ... body currents to ensure the unitary equivalence . the role of the two - body part is examined in the following section . applications involve the interaction models : nij2@xcite , argonne v18@xcite , reid93@xcite which are local ones , nij1@xcite , nij93@xcite , paris@xcite which have a linearly @xmath4 dependence , and the bonn - qb , bonn - cd ones@xcite . concerning static properties , a quantity of interest is the deuteron d - state probability , often referred to characterize different models . the difference between the paris and bonn - qb model is 0.78% . as the two models have quite close values for the mixing parameters , @xmath25 , their comparison is largely free of bias with this respect . taking into account the effect of the non - local term , @xmath26 , explains 0.74% ( 0.60% and 0.14% for the commutator and anti - commutator parts respectively ) . notice that the change in the deuteron d - state probability just reflects the fact that this quantity is not an observable one . similar results were reached by von geramb et al.@xcite , using the inverse scattering problem methods . another interesting quantity is the ratio @xmath27 , which , contrary to the above one , is observable . the difference is 0.23 @xmath28 while @xmath26 explains 0.20 @xmath28 . it is also instructive to look at quantities that only depend on the scalar anticommutator part of @xmath26 . in this order , we compare the squared charge radius for the models nij1 and nij2 . the models differ by an amount of 0.004 @xmath29 while the effect of the non - locality is estimated to be around 0.001 @xmath29 . the apparent failure to explain the discrepancy in this case actually points to the difference of the models for the asymptotic normalization @xmath2 , which as is well known , governs the size of the charge radius . this factor is not affected by the present analysis of non - local effects . ratios of form factors and related quantities calculated in different representations of the interaction should go to 1 when the non - locality effect is accounted for and provided that the unitary equivalence is ensured . it is not so in practice because the unitary transformation is treated at first order . in each case , there are thus two sets of results , depending on whether corrections are added to predictions made with one model or removed from the other ones . ratios of predictions made from the bonn - qb and paris models for the @xmath0 and @xmath1 observables are shown in fig . as it can be seen , the ratio of bare predictions ( continuous line ) tends to 1 when the effect of the anticommutator ( dashed and dotted lines ) and commutator terms ( small - dash and dashed - dotted lines ) is considered . a large part of the effect is due to the tensor part of @xmath26 . notice that a slight departure from 1 appears for @xmath1 around @xmath30 . the effect of the scalar part of the anticommutator , which is seen in fig . [ fig2 ] ( left part ) , shows features similar to the previous ones . it involves the s - wave function close to the origin , which is generally suppressed in local models compared to non - local ones . the right part of the same figure emphasizes a case where an agreement between two models turns into a disagreement . in fact , this one is consistent with what is expected from the comparison of the asymptotic normalization , unaffected by the present analysis . in fig . [ fig3 ] , all predictions corrected for the effect of a non - local term are compared to the paris ones . at low @xmath31 , it is seen that some discrepancy is still present while the initial motivation of the work was rather to explain it by non - locality effects . actually , for both @xmath0 and @xmath1 , the discrepancy reflects a sensitivity to the @xmath32 ratio in the first case and @xmath2 in the second one . around @xmath33 , the ratio becomes very close to 1 , while in absence of corrections departures up to 10% and 20% respectively could be observed . the decrease of the uncertainty has motivated schiavella and sick in using the quadrupole form factor to derive the neutron charge form factor@xcite . effects of non - local terms in nn interaction models have been considered . roughly , they explain a large part of the differences that the comparison of various model predictions for electromagnetic observables evidences . the part with a tensor character has been found to be the dominant one . it is also the best determined . similar conclusions hold in some cases for the scalar part but the effect is often masked by other effects related to the fact that models correspond to different values of the asymptotic normalization , @xmath2 , which is an observable and remains unchanged in the analysis performed here . while the original goal of present studies was rather to relate discrepancies between models to some non - locality , it appears that this is not always possible , especially at low @xmath31 . interestingly however , accounting for this effect tends to restore some hierarchy of the results , as expected from simple models . thus , in the above range , the structure function , @xmath0 , and the quadrupole form factor , @xmath34 , evidence a direct sensitivity respectively to the asymptotic normalizations , @xmath2 and @xmath3 , which are unaffected by the present analysis of non - local effects . the result of the analysis presented here was not a priori guaranteed . the fact that it points to a unique family of phase - equivalent models indicates that the sensitivty of the models to different parametrizations of the radial part for instance or to a different fit to experimental data is rather small . thus , the availability of various models is not without interest . the remaining sensitivity , as for @xmath2 or equivalently the scattering length , @xmath35 , strongly calls for a more accurate determination of these quantities . throughout the present study , we compare together predictions of models for electromagnetic observables . ultimately , a comparison to experiment should be done . in this respect , a model to be prefered , most probably non - local , is that one based on degrees of freedom of which effectice character , unavoidable in any case , is as low as possible . 9 p. doleschall and i. borbly : phys . rev . * c62 * , 054004 ( 2000 ) . m. lacombe et al . : phys . rev . * c21 * , 861 ( 1980 ) . r. machleidt , k. holinde and ch . elster : phys . rep . * 149 * , 1 ( 1987 ) ; r. machleidt : adv . in nucl . phys . * 19 * , 189 ( 1989 ) ; r. machleidt : phys . rev . * c63 * , 024001 ( 2001 ) . v. kukulin ( plenary talk , this conference ) . stoks et al . : phys . rev . * c49 * , 2950 ( 1994 ) . b. desplanques and a. amghar : z. phys . * 344 * , 19 ( 1992 ) . a. amghar and b. desplanques : nucl . * a585 * , 657 ( 1995 ) . a. amghar and b. desplanques : nucl - th/0209058 . j. forest : phys . rev . * c61 * , 034007 ( 2000 ) . friar : phys . * c22 * , 796 ( 1980 ) . wiringa , v.g.j . stoks and r. schiavella : phys . rev . * c51 * , 38 ( 1995 ) . von geramb , lectures notes in physics : vol . * 427 * , ( 1994 ) ; b.f . gibson , h. kohlhoff and h.v . von geramb : phys . rev . * c51 * , r465 ( 1995 ) . r. schiavella and i. sick : phys . rev . * c64 * , 041002(r ) ( 2001 ) .

the effect of non - locality in the nn interaction models is examined . it is shown that this feature can explain differences in predictions made from models evidencing a difference with this respect . this is done for both static and dynamical observables , taking into account that a non - local term can be transformed away by performing a unitary transformation . some results for the deuteron form factors , the @xmath0 structure function and the @xmath1 tensor polarization are given as an example . a few cases where discrepancies can not be explained are also considered . they point to differences in the models as for the deuteron asymptotic normalizations , @xmath2 and @xmath3 , which are not affected by the present analysis .

type ia supernovae ( sne ia ) are the most important standardized candles for cosmology , and have been used to discover dark energy and the accelerating universe @xcite . this was facilitated by the realization that supernovae with broader lightcurves are intrinsically brighter , while those with narrow lightcurves are dimmer @xcite . various schemes exist to correct sn ia luminosities based on their lightcurve shape here we use the `` stretch '' method , in which the time axis of a template lightcurve is multiplied by a scale factor @xmath6 to fit the data @xcite . some properties of sne ia have been found to correlate with environment brighter supernovae with broader lightcurves ( high @xmath6 ) tend to occur in late - type spiral galaxies @xcite , while dimmer , fast declining ( low @xmath6 ) supernovae are preferentially located in an older stellar population , leading to the conclusion that the age of the progenitor system is a key variable affecting sn ia properties @xcite . the fact that supernovae occur at a much higher rate in late type galaxies , and that the sn ia rate is proportional to the core - collapse rate is another indication that age plays an essential role . following this previous work , @xcite model sne ia as consisting of two populations a `` prompt '' component whose rate is proportional to the star formation rate of the host galaxy , and a second `` delayed '' component whose rate is proportional to the stellar mass of the galaxy . * hereafter s06 ) tie all of these results together using data from the supernova legacy survey ( snls ) , finding that slow declining , brighter sne ia come from a young population and have a rate proportional to star formation on a 0.5 gyr timescale , while dimmer , faster declining supernovae come from a much older population with a rate proportional to the mass of the host galaxy . s06 and @xcite predict that the sne ia whose rates are proportional to star formation will start to dominate the total sample of sne as cosmic star formation increases with redshift . since these sne are intrinsically brighter , the mean luminosity of the population should increase with redshift . here we use the two - component sn ia model of @xcite and the stretch distribution for each component from snls data ( s06 ) to quantify the expected magnitude of this effect . we then compare the predicted evolution in lightcurve stretch to sn distributions from the snls and the higher - z sn search @xcite . @xcite parameterize the supernova rate in a galaxy as @xmath7 where @xmath8 is the total stellar mass in the galaxy , @xmath9 is the star formation rate , and a and b are constants . these authors use the supernova rates in galaxies of different morphologies and colors to derive values for @xmath10 and @xmath11 . s06 use an alternate method they fit galaxy models to snls @xmath12 host galaxy photometry to derive masses and star formation rates . then , using the cosmic star formation history of @xcite , s06 predict the rate of sne from each component versus redshift in their fig . 10 . note that here we adopt the same definition of @xmath13 as s06 it is the mass turned into stars and does not include mass loss from supernovae . the prompt and delayed sn ia components have different stretch distributions ( s06 ) . to determine the stretch of each sn , here we use the sne from s06 , though we fit a new lightcurve template to the data using the sifto method @xcite . because stretches are always defined relative to the @xmath14 template lightcurve , stretch values should only compared within a publication . however , the stretches derived here are approximately 4% larger than those in , largely due to the use of a narrower @xmath14 template . all sne from passive galaxies ( i.e. those with no measurable star formation rate ) were assigned to the @xmath10 component . star forming galaxies have sne ia from both components , so the @xmath10 distribution from passive galaxies was scaled by mass and subtracted from the distribution of sne ia from star forming galaxies , leaving the @xmath11 distribution ( as in s06 ) . the resulting distributions , and gaussian fits are shown in figure [ abdist ] . note that to conserve the total number of sne one should add the sne subtracted from the @xmath11 distribution back to the @xmath10 distribution this is unnecessary for our purposes because the relative heights of the gaussians are normalized as a function of redshift in the next step . to estimate the expected stretch evolution with redshift , we take the observed @xmath10 and @xmath11 distributions and scale them to the predicted relative values with redshift from fig . 10 of s06 . increasing cosmic star formation with redshift produces a larger fraction of sne from the prompt component . stellar mass as a function of redshift is determined by integrating the star formation history from the earliest times , so the total stellar mass , and the number of sne from the a component , decreases with increasing redshift . the net result is that in the @xmath15 model the mean stretch increases from 0.98 at @xmath16 to 1.04 at @xmath5 . one caveat is that in the @xmath15 model there is no time dependence for the @xmath10 component . sne ia from 10 gyr old progenitors are just as likely as sne ia from 3 gyr old progenitors . if 10 gyr - old sne ia are actually more rare , the @xmath15 model will overpredict the number of @xmath10-component sne at @xmath16 , as they result from stars formed during the high star formation rate in the early universe ( see discussion in s06 ) . as an alternative to the @xmath15 model we tested the two component sn ia delay time distribution from @xcite , which has an exponential decrease in supernova probability from the delayed component with time . the drawback of this model is that the probability distribution is somewhat arbitrary . also , rather than the 50 - 50 split between prompt and delayed sne chosen by @xcite , here we scale each component by the @xmath10 and @xmath11 values measured by s06 . this gives similar results to the @xmath15 model , predicting a shift in mean stretch from 0.98 to 1.02 from @xmath0 . in figure [ gaussreal ] we compare the predicted stretch distributions from the @xmath15 model to the observed stretch distributions in three redshift bins from the low redshift data used by , the snls data in s06 , and the data of the higher - z supernova search @xcite . all lightcurves have been refit here using the same method . we also tested the data against the modified @xcite model , but we did not find it to be a better predictor of sn evolution with redshift ( table [ table ] ) . ccccc 0 - 0.1 & 0.81 & 0.63 & 15% & 39% + 0.1 - 0.75 & 0.64 & 0.83 & 30% & 28% + 0.75 - 1.5 & 0.60 & 0.84 & 52% & 35% + each survey has different selection effects the most serious for the current study is malmquist bias , the tendency to discover only the brightest members of a group near the detection limit of a magnitude - limited survey . to minimize the effect , for each of the high redshift searches we only consider supernovae from a reduced volume so that none of the supernovae used are near the magnitude limit . the snls regularly discovers sne ia out to @xmath17 , but here we use only the subset with @xmath18 , where malmquist bias is minimal . similarly , we only use @xcite sne with @xmath19 , where the authors report their sample is complete @xcite . lowering the redshift cuttoff to @xmath20 does not change the average stretch for the highest - z sample , but it reduces the sample size from 20 to 13 , and thus decreases the significance of the results . as an additional protection against selection bias , we only consider sne with @xmath21 . sne ia with @xmath22 are both dim and spectoscopically peculiar , like sn 1991bg @xcite , and have not yet been detected at @xmath23 , probably because of a combination of malmquist bias and spectroscopic selection bias @xcite as redshift increases , and the angular size of the host galaxy decreases , and it becomes more and more difficult to spectroscopically identify such faint supernovae when blended with their bright , often elliptical , hosts . this cut removes 3 sne ia from the low - z sample [ other low - z sne are already removed because we only consider hubble - flow sne ia , with @xmath24 , to be consistent with ] . in all cases we use only sne ia with at least 4 lightcurve points , and at least one detection before 10 rest - frame days after maximum light in the @xmath11-band , so that stretch is accurately measured . figure [ gaussreal ] shows that the average observed stretch increases with redshift , from @xmath25 at a median redshift of @xmath26 , to @xmath27 at @xmath28 , and @xmath29 at @xmath30 . simultaneously the percentage of sne ia with @xmath31 decreases from 24% to 15% to 1.4% . the ks test gives a 2% probability that the lowest and highest redshift sample are drawn from the same distribution . the predicted distributions from the @xmath15 model are overplotted . the observed trends match the predictions of the empirically - based models with increasing redshift fewer low - stretch sne ia are observed , and the mean sn ia stretch increases . we find the same result when this analysis is repeated with the salt and salt2 lightcurve fitters . these results are also consistent with the findings of , that the low - z sample had an average stretch 97% that of snls sne . estimate distances from sne ia using @xmath32 where @xmath33 is the distance modulus , @xmath34 is the peak @xmath11-band magnitude , @xmath35 is a color , and @xmath36 , @xmath37 and @xmath38 are parameters fit by minimizing residuals on the hubble diagram . found @xmath39 , so a drift in average stretch of @xmath40 from @xmath26 to @xmath30 results in a 12% drift in average intrinsic sn ia luminosity over this redshift range . evolution in the sn population will not necessarily bias cosmological studies , since sne are only used in this way after correction for lightcurve shape . however , we can no longer assume that any deficiencies in lightcurve width correction schemes will average out under the assumption the distribution of sne is similar over all redshifts . if there is a systematic residual between low stretch and high stretch sne when they are stretch corrected , this could cause a bias in the determination of cosmological parameters as the population evolves . to test an extreme case of evolution , we fit the equation of state parameter for dark energy , @xmath3 and the matter density , @xmath41 using the data from , ( dashed lines in fig . [ lowhighs ] ) and compared it to a fit using the same data , but retaining only @xmath42 sne ia at @xmath43 and only @xmath44 sne ia at @xmath45 ( solid lines ) . the strongly evolving subset gives estimates for @xmath3 and @xmath41 consistent with the full set , although the errors are larger because there are fewer sne in the subset . one worry with an evolving population is that stretch - magnitude or color - magnitude relations may evolve , i.e. @xmath37 or @xmath38 derived at one redshift may not be appropriate at another . the values derived here for the strong evolution subset and the full set are consistent , but again a strong test awaits a larger data set . measuring changes in @xmath3 with time will require much stricter control of potential evolutionary effects . possible biases depend critically on the exact nature of the evolution , the experimental design , the cosmology , and how the time variable component of @xmath3 is parameterized e.g. @xmath46 , @xmath47 , or @xmath48 @xcite . however , a good rule of thumb is that to keep systematic errors significantly below statistical errors for a mission such as snap , the corrected magnitudes of sne ia should not drift by more than 0.02 mag up to @xmath4 @xcite . unfortunately there are not enough well measured sne ia in the literature to determine whether or not stretch correction works to this level of precision . after stretch correction , the rms scatter of sne ia around the hubble line is @xmath49 mag in the best cases . therefore @xmath50 sne ia are required in each of several stretch or redshift bins to determine whether biases remain at the 0.02 level after correction . such precise tests will soon be possible by combining data from large surveys underway at low and high redshift . we have shown that there is some evolution in the average lightcurve width , and thus intrinsic luminosity of sne ia from @xmath16 to @xmath5 , although significant evolution is found only over a large redshift baseline . this evolution is consistent with our predictions from the @xmath15 model as star formation increases with redshift , the broader - lightcurve sne ia associated with a young stellar population make up an increasingly larger fraction of sne ia . though we have taken steps to minimize the effects of malmquist bias , it is possible that residual effects play some role in increasing average stretch with redshift . however , the net effect on cosmology studies is the same no matter the underlying cause . in either case , there is increased pressure on the light curve shape calibration method to correct for the evolution in sn ia properties with redshift . @xcite found that sne ia still give evidence for an accelerating universe even if sne ia are not corrected for stretch . this was possible because the difference in a universe with dark energy and one with @xmath51 is large 0.25 magnitudes at @xmath52 , whereas the population evolution seen here implies that average sn ia magnitudes should increase by 0.07 mag over the same redshift range . however , discriminating between dark energy models requires much more precise control of sn ia magnitudes over a larger redshift baseline . most theoretical studies addressing possible sn ia evolution have focused on metallicity . although theorists have proposed various mechanisms that could conceivably alter the properties of sne ia as the average metallicity changes with cosmic time @xcite there is no consensus regarding which effects are important or even the sign of these effects . observational studies have found no evidence that metallicity affects the properties of sne ia @xcite . instead , it is more likely that age differences between the two populations ( and thus almost certainly the mass of the secondary star ) play a role in the evolution of the observed stretch distribution with redshift @xcite . prompt sne ia are thought to be brighter because they produce more @xmath53ni . if the chandrasekhar - mass model describes most sne ia , they must then produce less intermediate mass elements , assuming that the amount of unburned material is negligible in normal sne ia . we therefore predict that high redshift sne ia will have less ca and si . this is confirmed by the most intensive study of high redshift sn ia spectra @xcite . one concern raised by these findings is that pathological sne ia such as sn 2001ay @xcite , sn 2002cx @xcite , sn 2002ic @xcite , and snls-03d3bb @xcite , which do not obey typical lightcurve shape correction schemes , are associated with star formation . since star formation density increases by a factor of ten from @xmath16 to @xmath5 @xcite , at high redshift these pathological supernovae will be an order of magnitude more common . thus the conventional wisdom that all high redshift supernovae will have counterparts at low redshift @xcite only holds if sample sizes are much larger than those currently used for cosmology . only 20 sne ia have published lightcurves at @xmath17 , and only @xmath5450 sne ia at @xmath55 have sufficient data to be cosmologically useful . fortunately , thus far it has been possible to identify these outliers so that they do not affect cosmological analyses , but future studies requiring increased precision must be vigilant of the effects of an evolving sn ia population .

recent studies indicate that type ia supernovae ( sne ia ) consist of two groups a `` prompt '' component whose rates are proportional to the host galaxy star formation rate , whose members have broader lightcurves and are intrinsically more luminous , and a `` delayed '' component whose members take several gyr to explode , have narrower lightcurves , and are intrinsically fainter . as cosmic star formation density increases with redshift , the prompt component should begin to dominate . we use a two - component model to predict that the average lightcurve width should increase by 6% from @xmath0 . using data from various searches we find an 8.1%@xmath12.7% increase in average lightcurve width for non - subluminous sne ia from @xmath2 , corresponding to an increase in the average intrinsic luminosity of 12% . to test whether there is any bias after supernovae are corrected for lightcurve shape we use published data to mimic the effect of population evolution and find no significant difference in the measured dark energy equation of state parameter , @xmath3 . however , future measurements of time - variable @xmath3 will require standardization of sn ia magnitudes to 2% up to @xmath4 , and it is not yet possible to assess whether lightcurve shape correction works at this level of precision . another concern at @xmath5 is the expected order of magnitude increase in the number of sne ia that can not be calibrated by current methods .

images from the hubble space telescope ( hst ) advanced camera for surveys ( acs ) suffer from strong geometric distortion : the square pixels of its detectors project to trapezoids of varying area across the field of view . the tilted focal surface with respect to the chief ray is the primary source of distortion of all three acs detectors . in addition , the hst optical telescope assembly induces distortion as does the acs m2 and im2 mirrors ( which are designed to remove hst s spherical aberration ) . the sbc s optics include a photo - cathode and micro - channel plate which also induce distortion . here we describe our method of calibrating the geometric distortion using dithered observations of star clusters . the distortion solutions we derived are given in the idc tables delivered in nov 2002 , and currently implemented in the stsci calacs pipeline . this paper is a more up to date summary of our results than that presented at the workshop . an expanded description of our procedures is given by meurer ( 2002 ) . * observations*. the acs smov geometric distortion campaign consisted of two hst observing programs : 9028 which targeted the core of 47 tucanae ( ngc 104 ) with the wfc and hrc , and 9027 which consisted of sbc observations of ngc 6681 . additional observations from programs 9011 , 9018 , 9019 , 9024 and 9443 were used as additional sources of data , to check the results , and to constrain the absolute pointing of the telescope . the ccd exposures of 47 tucanae were designed to well detect stars on the main sequence turn - off at @xmath0 in each frame . this allows for a high density of stars with relatively short exposures . the f475w filter ( sloan g ) was used for the ccd observations so as to minimize the number of saturated red giant branch stars in the field . for the hrc two 60s exposures were taken at each pointing , while for the wfc which has a larger time overhead , only one such exposure was obtained per pointing . simulated images made prior to launch , as well as archival wfpc2 images from gilliland et al . ( 2000 ) were used to check that crowding would not be an issue . for calibrating the distortion in the sbc we used exposures of ngc 6681 ( 300s - 450s ) which was chosen for the relatively high density of uv emitters ( hot horizontal branch stars ) . the pointing center was dithered around each star field . for the wfc and hrc pointings , the dither pattern was designed so that the offsets between all pairs of images adequately , and non - redundantly , samples all spatial scales from about 5 pixels to 3/4 the detector size . for the sbc pointings , a more regular pattern of offsets is used augmented by a series of 5 pixel offsets . * distortion model*. the heart of the distortion model relates pixel position ( @xmath1 ) to sky position using a polynomial transformation ( hack & cox , 2000 ) given by : @xmath2 here @xmath3 is the order of the fit , @xmath4 is the reference pixel , taken to be the center of each detector , or wfc chip , and @xmath5 are undistorted image coordinates . the coefficients to the fits , @xmath6 and @xmath7 , are free parameters . for the wfc , an offset is applied to get the two ccd chips on the same coordinate system : @xmath8 @xmath9 are 0,0 for wfc s chip 1 ( as indicated by the fits ccdchip keyword ) and correspond to the separation between chips 1 and 2 for chip 2 . the chip 2 offsets are free parameters in our fit . @xmath10 correspond to tangential plane positions in arcseconds which we tie to the hst @xmath11 coordinate system . next the positions are corrected for velocity aberration : @xmath12 , @xmath13 , where @xmath14 here * u * is the unit vector towards the target and * v * is the velocity vector of the telescope ( heliocentric plus orbital ) . neglect of the velocity aberration correction can result in misalignments on order of a pixel for wfc images taken six months apart for targets near the ecliptic . finally , we must transform all frames to the same coordinate grid on the sky @xmath15 : @xmath16 where the free parameters @xmath17 are the position and rotation offsets of frame @xmath18 . * calibration algorithm*. we use the positions of stars observed multiple times in the dithered star fields to iteratively solve for the free parameters in the distortion solution : fit coefficients @xmath19 ; chip 2 offsets @xmath20 ( wfc only ) ; frame offsets @xmath21 ; and tangential plane position @xmath22 of each star used in the fit . the stars are selected by finding local maxima above a selected threshold . the centroid in a @xmath23 box about the local maximum is compared to gaussian fits to the @xmath24 profiles , if the two estimates of position differ by more than 0.25 pixels , the measurement is rejected as likely being effected by a cosmic ray hit or crowding . further details of the fit algorithm can be found in meurer et al . ( 2002 ) . * low order terms*. originally only smov images taken with a single roll angle were used to define the distortion solutions . the solution using only these data is degenerate in the zeroth ( absolute pointing ) and linear terms ( scale , skewness ) . so we used the largest commanded offsets with a given guide star pair to set the linear terms . however , comparison of corrected coordinates to astrometric positions showed that residual skewness in the solution remained . hence , as of november 2002 , the idc tables for wfc and sbc are based on data from multiple roll angles . the overall plate scale is set by the largest commanded offset . for the hrc , the linear scale is set by matching hrc and wfc coordinates , since the same field was used in the smov observations . the zeroth order terms ( position of the acs apertures in the hst @xmath25 frame ) was determined from observations of an astrometric field . .summary of fit results [ cols="<,^,>,>,^,>,^,^,^ " , ] the distortion in all acs detectors is highly non - linear as illustrated in fig . [ f : nonlin ] . we find that a quartic fit ( @xmath26 ) is adequate for characterizing the distortion to an accuracy much better than our requirement of 0.2 pixels over the entire field of view . table [ t : res ] summarizes the rms of the fits to the various datasets . the wfc and hrc fits were all to f475w data as noted above . to check the wavelength dependence of the distortion we used data obtained with f775w ( wfc and hrc ) and f220w ( hrc ) from programs 9018 and 9019 . we held the coefficients fixed and only fit the offsets in order to check whether a single distortion solution is sufficient for each detector . table 2 shows that there is a marginal increase in the rms for the red data of the wfc , little or no increase in the fit rms for the red hrc data , but a significant increase in the rms using the uv data . an examination of the hrc f220w images reveals the most likely cause : the stellar psf is elongated by 0.1 " . a similar elongation can also be seen in sbc psfs . we attribute this to aberration in the optics of either the acs m1 or m2 mirrors or the hst ota ( hartig , et al . , the aberration amounts to 0.1 waves at 1600 , but is negligible relative to optical wavelengths , hence it is not apparent in optical hrc images . while it was expected that the same distortion solution would be applicable to all filters except the polarizers , recent work ( by tom brown , stsci , and our team ) has shown that at least one other optical filter ( f814w ) induces a significant plate scale change ( factor of @xmath27 ) . in the long term , the idc tables will be selected by filter in the stsci calacs pipeline . while a quartic solution is adequate for most purposes , binned residual maps ( fig . [ f : resid ] ) show that there are significant coherent residuals in the wfc and hrc solutions . these have amplitudes up to @xmath28 pixels . the small - scale geometric distortion is the subject of the anderson & king contribution to this proceedings . hack , w. , & cox , c. 2000 , isr acs 2000 - 11 , stsci . hartig , g. et al . 2002 , in `` future euv and uv visible space astrophysics missions and instrumentation '' , eds . blades & o.h . siegmund , proc . spie , vol . 4854 , in press [ 4854 - 30 ] . gilliland , r.l . 2000 , apj , 545 , l47 . meurer , g.r . 2002 , in `` future euv and uv visible space astrophysics missions and instrumentation '' , eds . blades & o.h . siegmund , proc . spie , vol . 4854 , in press [ 4854 - 30 ] .

the off - axis location of the advanced camera for surveys ( acs ) is the chief ( but not sole ) cause of strong geometric distortion in all detectors : the wide field camera ( wfc ) , high resolution camera ( hrc ) , and solar blind camera ( sbc ) . dithered observations of rich star cluster fields are used to calibrate the distortion . we describe the observations obtained , the algorithms used to perform the calibrations and the accuracy achieved .

azobenzene - functionalized self - assembled monolayers ( sams ) on metal surfaces represent a viable and efficient way to obtain ordered architectures of photo - switching molecules . @xcite however , it has been observed that in such closely packed systems photo - isomerization can be drastically hindered by steric effects , @xcite and even by excitonic coupling between the chromophores . @xcite in order to overcome these limitations and obtain sams with efficient switching rates , a number of strategies have been developed , such as modifying the morphology of the substrate , @xcite introducing organic spacers , @xcite and functionalizing azobenzene with end groups.@xcite to tune these complex systems in view of optimized performance , a deep knowledge of their chemical composition and structure - property relationship is required . x - ray absorption spectroscopy ( xas ) represents a powerful technique for this purpose , and a synergistic interplay with theory can provide an insightful interpretation of the experimental data . first - principles methods represent the most suitable tool . density - functional theory ( dft ) , both in the _ core - hole _ approximation @xcite and the @xmath1-self - consistent - field ( @xmath1scf ) approach , @xcite is routinely applied to simulate xas in a wide range of materials , from gas - phase molecules to solid - state systems . @xcite recently , also time - dependent dft has become popular to compute core - level excitations in molecular compounds . @xcite while these approaches can provide qualitative agreement with experiments , explicit many - body treatment has turned out superior to such approaches . in small molecules , coupled - cluster methods have been successfully applied to compute xas from the carbon and nitrogen k - edge . @xcite for solid - state materials , many - body perturbation theory ( mbpt ) represents the state - of - the - art formalism to describe neutral excitations.@xcite the electron - hole ( _ e - h _ ) interaction , effectively described by the bethe - salpeter equation ( bse ) , plays a crucial role not only in conventional semiconductors , @xcite but also in organic crystals @xcite and even in isolated molecules . @xcite a number of studies on core - level excitations , from different edges and in several materials , @xcite has demonstrated that bse can accurately reproduce xas . in this paper , we present an _ ab initio _ study of x - ray absorption spectra of azobenzene - functionalized sams . we consider excitations from the nitrogen ( n ) @xmath0 edge , i.e. , involving transitions from 1@xmath2 electrons to the conduction bands . in this manner , we obtain exciton binding energies and determine the character of the core - level excitations . going from the isolated molecule to a closely - packed sam , we analyze the xas at increasing density of azobenzene molecules , and we compare our results with experimental data . in order to understand whether and how functional groups affect the nature of the excitons and their binding energy , we consider molecules that are either h - terminated ( h - az ) or functionalized with trifluoromethyl ( -az ) . a sketch of an azobenzene - functionalized sam of alkanethiols on gold is presented in fig . [ figure1_h - az]a . the chromophores are covalently bonded to the alkyl chains , which are attached to the gold substrate through a thiol group . as suggested by scanning tunneling microscopy ( stm ) measurements , @xcite the sam has an orthorhombic supercell , with lattice vectors @xmath3=6.05 and @xmath4=7.80 , hosting two inequivalent azobenzene molecules . in our calculations , we neglect the alkyl chains and the gold surface , since they are expected not to play a role in the xas from the n @xmath0 edge . therefore , we consider only the azobenzene molecule , with a methoxy group added to one end ( see fig . [ figure1_h - az]b ) , in order to reproduce the chemical environment of the covalent bond to the alkyl chain . the reciprocal distance and orientation of the molecules in the unit cell is set according to the stm data . @xcite although the first experiments on these systems predicted a herringbone structure of the chromophores in the sams , @xcite a consensus about the orientation of azobenzenes is still missing . we consider the two inequivalent molecules in the unit cell being oriented parallel to each other , since we expect deep core levels to be hardly affected by the reciprocal orientation of the molecules . in this configuration ( see fig . [ figure1_h - az]d ) the azobenzenes are separated by about 2 in the lateral direction , and by @xmath5 3.8 in the direction perpendicular to the plane of the phenyl rings . we incorporate @xmath5 14 of vacuum in the vertical direction , to effectively simulate a two - dimensional system . in order to understand the effects of packing in the xas , we consider an additional structure , including only one molecule in the same unit cell . we refer to this system , shown in fig . [ figure1_h - az]c , as _ diluted _ sam ( d - sam ) , to distinguish it from the _ packed _ sam ( p - sam , fig . [ figure1_h - az]d ) . for comparison , we investigate an isolated azobenzene molecule in an orthorhombic supercell , with @xmath5 6 of vacuum in each direction . we also consider sams of -functionalized azobenzene , which have been recently synthesized . @xcite and other functional groups are used in experiments as markers , to identify the orientation of the molecules with respect to the surface , @xcite and/or to tune the switching properties of the sams by decreasing the steric hindrance due to intermolecular interactions . @xcite also for -az , we investigate p- and d - sams , as well as an isolated molecule for comparison . we adopt the same structures shown for h - az in fig . [ figure1_h - az ] . x - ray absorption spectra are computed from first principles by solving the bse , which is an effective equation of motion for the electron - hole two - particle green s function . @xcite by considering only transitions from core ( @xmath6 ) to unoccupied ( @xmath7 ) states , the bse in matrix form reads : @xmath8 in case of n @xmath0 edge , the n 1@xmath2 is the only initial state . the bse hamiltonian in eq . [ eq : bse ] can be written as the sum of three terms : @xmath9 the _ diagonal _ term @xmath10 accounts for single - particle transitions . including only this term corresponds to the independent - particle approximation ( ipa ) . the _ exchange _ ( @xmath11 ) and _ direct _ ( @xmath12 ) terms in eq . [ eq : h_bse ] incorporate the repulsive bare and the attractive screened coulomb interaction , respectively . the coefficients @xmath13 and @xmath14 in eq . [ eq : h_bse ] enable to select the _ spin - singlet _ ( @xmath13 = @xmath14 = 1 ) and _ spin - triplet _ ( @xmath13 = 0 , @xmath14 = 1 ) channels . in the latter case , the exchange interaction is not present . in eq . [ eq : bse ] , the eigenvalues @xmath15 represent excitation energies . exciton binding energies ( @xmath16 ) are defined , for each excitation , as the difference between excitation energies @xmath15 computed from ipa and bse . the eigenvectors @xmath17 carry information about the character and composition of the excitons . through the transition coefficients @xmath18 @xmath17 enter the expression of the imaginary part of the macroscopic dielectric function ( @xmath19 ) : @xmath20 all calculations are performed with the ` exciting ` code , @xcite a computer package implementing dft and mbpt . @xcite ` exciting ` is based on the all - electron full - potential augmented planewave method , which ensures an explicit and accurate description of core electrons . the calculation of xas via the solution of the bse in an all - electron framework has been successfully applied to different absorption edges in a number of bulk materials , @xcite including , very recently , small molecules . @xcite the kohn - sham ( ks ) electronic structure , used here as starting point for the bse , is computed within the local - density approximation ( perdew - wang functional ) . @xcite quasiparticle energies are approximated by ks single - particle energies , and a scissors operator is applied to match the experimental absorption onset for the p - sam , according to the available data for h - az @xcite and -az . @xcite the same correction is applied also to the respective isolated molecule and d - sam , since we do not have experimental data available for these systems . a @xmath21-point mesh of 6@xmath224@xmath221 ( 3@xmath222@xmath221 ) is used to sample the brilloiun zone for the p - sam ( d - sam ) , in both ground - state and bse calculations . for the basis functions , a planewave cutoff @xmath23=5 bohr is applied to the molecules ; for the sams it is reduced to 4.625 bohr . muffin - tin spheres of radii @xmath24=0.8 bohr are considered for hydrogen , @xmath24=1.1 bohr for nitrogen and fluorine , and @xmath24=1.2 bohr for carbon and oxygen . the atomic positions of each structure are optimized by minimizing the hellmann - feynman forces within a threshold of 0.025 ev / . for the calculation of the @xmath25-@xmath26 interaction term @xmath27 in the bse ( eq . [ eq : h_bse ] ) , the screening is evaluated within the random - phase approximation , including the n 1@xmath2 core states , all valence states and 100 conduction bands . local - field effects are included , with at least 400 @xmath28 vectors for the sams and about 2000 for the molecules . these parameters ensure convergence of the xas within 0.25 ev . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] finally , we investigate the x - ray absorption spectra of azobenzene molecules and sams terminated with a trifluoro - methyl group ( -az , see fig . [ figure4_cf3-az_xas ] ) . since no additional n atoms are introduced in the system , we expect to observe the same features as in the xas of the h - terminated counterparts . with this analysis we aim at understanding the influence of functionalization on binding energies and exciton character . we again consider an isolated -functionalized azobenzene molecule , as well as d- and p - sams , in order to inspect the role of packing density . the calculated xas are shown in fig . [ figure4_cf3-az_xas ] , and the ( binding ) energies of the bright excitons are summarized in table [ table2 ] . the spectra appear strikingly similar to those presented in fig . [ figure2_h - az_xas ] for h - terminated azobenzene , and so the main features analyzed previously . the xas are considerably blue - shifted from the molecule to the p - sam , i.e. , upon increasing intermolecular interactions . the intense peaks correspond to transitions to unoccupied states , having the same @xmath29 character as in the h - az systems . by inspecting carefully tables [ table1 ] and [ table2 ] , we notice that exciton binding energies slightly increase in presence of termination . this functional group has an electron - withdrawing character and introduces a sizable dipole moment in the molecule , of the order of 5 debye . this slightly enhances the @xmath25-@xmath26 attraction , thus strengthening the exciton binding energy of the main peaks in fig . [ figure4_cf3-az_xas ] , of about 0.2 ev on average . in the p - sam the binding energy of a is 0.4 ev larger than in the h - az system . on the contrary , in the d - sam the value of @xmath16 decreases by 0.2 ev for the lowest - energy resonance a , compared to its h - terminated counterpart . in a similar fashion , the optical absorption onset computed for polycyclic aromatic hydrocarbons is red - shifted by about 0.30.5 ev , in presence of edge - functional groups.@xcite also for -az , we observe singlet - triplet splittings ( @xmath550 mev ) , which are two orders of magnitude smaller than the binding energies . the comparison with the experimental data from ref . indicates good agreement with our results . in both theoretical and experimental spectra the intense low - energy peak a , as well as the weaker resonances b and c , present the same energy separation and relative intensity . since @xmath30 states are not included in our bse calculation , the bump above 405 ev is not reproduced in our spectrum . we have investigated n 1@xmath2 x - ray absorption spectra of azobenzene - functionalized sams , determining the nature of the excitations and discussing the role of many - body effects . our results , obtained from _ ab initio _ calculations , in the framework of many - body perturbation theory , reveal a clear excitonic character of the main peaks in the xas . binding energies for core - edge excitons , computed from the solution of the bethe - salpeter equation , decrease from 6 ev in the molecule to 4 ev in packed sams . this is a many - body effect assigned to an interplay between screening and exciton delocalization . based on this finding , we expect exciton coupling between different molecules to be even more pronounced in the optical range , where @xmath25-@xmath26 pairs are typically more delocalized . this could give insight into the loss of switching capability , as observed for densely packed sams . @xcite core - level excitations in these systems are ruled by the attractive @xmath25-@xmath26 correlation . the exchange interaction plays a negligible role , as testified by singlet - triplet splittings , which are two orders of magnitude smaller than exciton binding energies . functionalization with a group does not affect the overall spectral features , but only induces a slight increase in the exciton binding energies of @xmath50.2 ev on average . good agreement is observed with available experimental data . in conclusion , our work confirms the predictive power of many - body perturbation theory in determining the character of the resonances and in disclosing the microscopic mechanisms ruling the core excitation process . this confirms the indispensable role of theory not only in interpreting the experimental data , but also in gaining further insight into the physics of core - level spectroscopy . this work was funded by the german research foundation ( dfg ) , through the collaborative research center sfb-658 . c.c . acknowledges support from the _ berliner chancengleichheitsprogramm _ ( bcp ) . our theoretical results are compared with angle - resolved measurements . for a quantitative comparison , we refer here to data obtained for the so - called _ magic angle _ , for which the spectrum is independent of the light polarization . intensities are normalized to the height of the core - edge peak . the different intensity of these features in theory and experiment can be mainly ascribed to details in the structure , defects , or coupling to vibrations , which are not taken into account in our theoretical approach .

we study x - ray absorption spectra of azobenzene - functionalized self - assembled monolayers ( sams ) , investigating excitations from the nitrogen @xmath0 edge . azobenzene with h - termination and functionalized with groups is considered . the bethe - salpeter equation is employed to compute the spectra , including excitonic effects , and to determine the character of the near - edge resonances . our results indicate that core - edge excitations are intense and strongly bound : their binding energies range from about 6 to 4 ev , going from isolated molecules to densely - packed sams . electron - hole correlation rules these excitations , while the exchange interaction plays a negligible role .

adiabaticity is an interesting concept in physics both for theoretical studies and experimental practices @xcite . according to the adiabatic theorem @xcite , if the parameters of the system vary with time much more slowly than the intrinsic motion of the system , the system will undergo the adiabatic evolution . for a classical system , the adiabatic evolution means that the action of the trajectory keeps invariant . for a quantum system , an initial nondegenerate eigenstate remains to be an instantaneous eigenstate when the hamiltonian changes slowly compared to the level spacings @xcite . hence , the adiabatic evolution has been employed as an important method of preparation and control of quantum states @xcite . however , a problem may arise when the eigenstates become accident degenerate at a critical point , i.e. , when the level spacing tends to zero at a critical point . for a classical system it corresponds to that the frequency of the fixed point is zero at the critical point . the adiabatic condition is not satisfied at the critical point because the typical time of the intrinsic motion of the system becomes infinite . can adiabatic evolution still hold up when the adiabatic condition breaks down at the critical point ? our motivation , derives from practical applications in current pursuits of adiabatic control of bose einstein condensates ( becs ) @xcite , which can often be accurately described by the nonlinear schrdinger equation . here the nonlinearity is from a mean field treatment of the interactions between atoms . difficulties arise not only from the lack of unitarity in the evolution of the states but also from the absence of the superposition principle . this was recently addressed for becs in some specific cases band , kivshar . but then , however , for such systems , only finite number of levels are concerned . the nonlinear schrdinger equation of the system with finite number of levels can be translated into a mathematically equivalent classical hamiltonian system . the evolution of an eigenstate just corresponds to the evolution of a fixed point of the classical hamiltonian system . then , the accident degeneracy of eigenstates is just translated into accident collision of the fixed points . the latter one is quite well - known subject and has been studied widely at least as a purely mathematical problem @xcite . hence , our concern here is only focused on the adiabatic evolution of the fixed points of classical hamiltonian systems . in this paper , we present a supplemental condition of the adiabatic evolution for the fixed points of classical hamiltonian systems when the adiabatic condition breaks down at some critical points in the terms of topology . as an example , we investigate the adiabatic evolution of the fixed points of a classical hamiltonian system which has a number of practical interests . we show that the adiabatic condition will break down at bifurcation points of the fixed points . but the adiabatic evolution is destroyed only for the limit point . for the branch process , the adiabatic evolution will hold , and the corrections to the adiabatic approximation tend to zero with a power law of the sweeping rate . for clarity and simplicity , we consider a one - freedom classical hamiltonian @xmath0 with canonically conjugate coordinates @xmath1where @xmath2 is a parameter of this system . the equations of motion are : @xmath3 we can find two kinds of trajectories in the phase space for the system : fixed points and closed orbits . the fixed points are the solutions of eqs . ( [ fpp ] ) when the right hands of them are zero . for a hamiltonian system there are only two kinds of the fixed points : elliptic points ( stable fixed points ) , hyperbolic points ( unstable fixed points ) . the closed orbits are around each of the elliptic points . we denote the fixed points by @xmath4 where @xmath5 is the total number of the fixed points . the action of a trajectory is defined as @xmath6 where the integral is along the closed orbit . obviously , the action of a fixed point is zero . the action is invariant when system undergos adiabatic evolution . according to the adiabatic theorem @xcite , the adiabatic condition can be expressed as @xmath7 where @xmath8 is the frequency of the fixed point . if this condition holds , the system will undergo adiabatic evolution , and keep the action not varying . if @xmath9 the condition can always be satisfied . we can obtain the frequencies of the fixed points by linearized the equations of motion . let us define the jacobian matrix as @xmath10 it is well - known that when @xmath11 the fixed point is a stable fixed point ( elliptic point ) ; when @xmath12 the fixed point @xmath13 is a unstable fixed point ( hyperbolic point ) . the point with @xmath14 is a degenerate point at which the stability of the system is not determined . for a stable fixed point @xmath15 the frequency of this fixed point is @xmath16 obviously , @xmath17 depends on the parameter @xmath18 supposing at a critical point , namely @xmath19 we have @xmath20 therefore , the condition ( [ cond ] ) will break down at the point . we want to know what will happen when the adiabatic condition fails ( will the adiabatic evolution of the fixed point be destroyed when the adiabatic condition does not hold ? ) . in fact , if @xmath21 the point @xmath13 is a bifurcation point at which the fixed point will collide with the other fixed points @xcite . hence , the breakdown of adiabatic condition is equivalent to collision of the fixed points ( equivalent to accident degeneracy of eigenstates of a corresponding quantum system ) . in the collision process , fixed points may annihilate or merge into a stable fixed point . the collision of the fixed points can be described clearly in the terminology of topology @xcite . the equations of motion just define a tangent vector field @xmath22 on the phase space . obviously , the fixed points @xmath23 are the zero points of the vector field , i.e. , @xmath24 . we know that the sum of the topological indices of the zero points of the tangent vector field is the euler number of the phase space which is a topological invariant @xcite . for a hamiltonian system , the topological index for a stable fixed point is @xmath25 and for a unstable fixed point is @xmath26 indeed , if the fixed point is a regular point ( not a degenerated point ) , i.e. , @xmath27 , the topological index of the fixed point can be determined by determinant of the jacobian matrix defined by eq . ( [ jaco ] ) @xcite . if @xmath28 , @xmath13 is a stable fixed point and the topological index is @xmath25 ; if @xmath29 it is a unstable fixed point and the index is @xmath26 if @xmath30 i.e. if @xmath13 is a bifurcation point , the topological index of this point seems to be not determined . as we have shown before , the point is just the critical point of adiabatic evolution , corresponding to collision of the fixed points . however , because the sum of the topological indices is a topological invariant , the topological index is conserved in a collision process of the fixed points . therefore , the topological index of the bifurcation point can be determined by the sum of the indices of the fixed points involved in collision . so , if the topological index of the bifurcation point is not zero , it is still a fixed point after collision . but if the topological index of the bifurcation point is zero , the bifurcation point will not be a fixed point after collision . now , let us imagine what will happen when a fixed point is destroyed by a collision process . because there are only two kinds of trajectories for a classical hamiltonian system : fixed points and closed orbits around each of the stable fixed point , so when a fixed point is destroyed , it will form a closed orbit around the nearest stable fixed point . the action of the new orbit must be proportional to the distance between the critical point and the nearest stable fixed point . this sudden change of action ( from zero to finite ) is so - called `` adiabatic tunneling probability '' which has been studied in refs . @xcite . on the other hand , if the topological index of the bifurcation point is @xmath31 , i.e. , it is a unstable fixed point after the collision , we can not expect the adiabatic evolution can keep on after collision . but if the topological index of the bifurcation point is @xmath25 , i.e. , it is still a stable fixed point after the collision , or in other word , the stable fixed point survive after collision . for such case , the adiabatic evolution will not be destroyed . from above discussion , it is clear that when the adiabatic condition given by eq . ( [ cond ] ) does not hold at a critical point with @xmath32 the system will still undergo the adiabatic evolution if the topological index of the fixed point @xmath33 is @xmath25 . on the contrary , if the topological index of the point @xmath34 is zero or @xmath31 the adiabatic evolution will be destroyed . hence , we get a supplemental condition of the adiabatic evolution of the fixed points for a classical hamiltonian system when the adiabatic condition breaks down at a critical point . when the adiabatic condition is not satisfied at a critical point , the topological property of the bifurcation point plays an important role to judge whether the system will undergo adiabatic evolution over this critical point : if the index of the degenerated point @xmath34 is @xmath25 the adiabatic evolution will hold . if the index of the point @xmath34 is zero or @xmath35 , the adiabatic evolution will not hold . as a paradigmatic example , we consider the following system @xmath36 in which @xmath37 are canonically conjugate coordinates , and @xmath38 , @xmath39 are two parameters . the equations of motion are : @xmath40 , these yield @xmath41 @xmath42 this classical system can be obtained from a quantum nonlinear two - level system , which may arise in a mean - field treatment of a many - body system where the particles predominantly occupy two energy levels . for example , this model arises in the study of the motion of a small polaron @xcite , a bose - einstein condensate in a double - well potential liufu7,liufu8,liufu9 or in an optical lattice @xcite , or for two coupled bose - einstein condensates @xcite , or for a small capacitance josephon junction where the charging energy may be important . this quantum nonlinear two - level model has also been used to investigate the spin tunneling phenomena recently @xcite . the fixed points of the classical hamiltonian system are given by the following equations @xmath43 the number of the fixed points depends on the nonlinear parameter @xmath18 for weak nonlinearity , @xmath44 there exist only two fixed points , corresponding to the local extreme points of the classical hamiltonian . they are elliptic points located on lines @xmath45 @xmath46 and @xmath47 respectively , each being surrounded by closed orbits . for strong nonlinearity , @xmath48 two more fixed points appear on the line @xmath49 @xmath46 in the windows @xmath50 one is elliptic and the other one is hyperbolic as a saddle point of the classical hamiltonian , where @xmath51 in the following , we only consider the cases in the region @xmath52 . we can obtain the frequencies of the fixed points by linearized the eqs . ( [ eq1 ] ) and ( [ eq2 ] ) . for the elliptic fixed points on line @xmath49 @xmath53 the frequencies are equal , they are @xmath54 obviously , if @xmath55 the frequencies will be zero , i.e. , @xmath56 . from eq . ( [ fix ] ) , we can obtain , when @xmath57 @xmath58 one of the elliptic fixed point will be @xmath59 at this point the adiabatic condition will break down . hence , if we start from this elliptic fixed point on line @xmath45 @xmath46 at @xmath60 , and @xmath38 changes with time as @xmath61 ( keeping @xmath39 invariant in the window @xmath62)@xmath63 the adiabatic condition ( [ cond ] ) will break down at the point @xmath64 when @xmath38 reaches @xmath65 because @xmath66 . we want to know what will happen when the adiabatic condition is not satisfied ( will the adiabatic evolution be destroyed when the adiabatic condition does not hold ? ) . there are two different cases for discussing : @xmath67 and @xmath68 case 1 @xmath69 : for the convenience , we choose @xmath70 and @xmath71 . we start at the elliptic fixed point @xmath72 and @xmath38 varies with very small @xmath73 at the beginning , the system follows the @xmath74 $ ] adiabatically . but when @xmath38 reaches @xmath75 @xmath76 the adiabatic evolution is destroyed with a jump of action ( the action changes to a finite value from zero suddenly ) at the point @xmath77 . 1(a ) shows this process . obviously , the breakdown of adiabatical condition leads to the destroy of the adiabatic evolution . case 2 @xmath78 : from eq . ( [ ham ] ) and ( [ fix ] ) , we can have two elliptic fixed points on line @xmath45 @xmath46 for @xmath79 , @xmath80and for @xmath44 there is only one fixed point , @xmath81obviously , for @xmath82 and @xmath83 , @xmath84 so adiabatic condition can not be satisfied . we integrate the classical equations of the hamiltonian system ( [ ham ] ) , with the initial condition @xmath85 @xmath86 , and @xmath87 fig . 1(b ) shows the time evolution of this fixed point for a very small sweeping rate @xmath88 . the final state is a very small oscillation around the fixed point @xmath89 . in fig . 2 , we plot the dependency of the small oscillation amplitude @xmath90 on the sweeping rate @xmath88 . from this figure , it is clear to see that the amplitude of the small oscillation will tend to zero with the sweeping rate decreasing as a power law : @xmath91 . therefore , for this case , the system will evolve adiabatically and keep the action not changing for all the time if the sweeping rate is small enough , even when @xmath38 crosses the critical point @xmath92 at which @xmath93 , i.e. , though the adiabatic condition is not satisfied when @xmath38 crosses the point @xmath94 , the system is still undergoing adiabatic evolution . in fact , if we make the series expansion of the hamiltonian ( [ ham ] ) around the critical point , the system can be approximated to a double well system @xcite . therefore , the phenomenon of case 2 can be illustrated by the standard double well model . considering a particle in a double well , the system is described by the hamiltonian @xmath95 for @xmath96 it has two stable fixed points @xmath97 and @xmath98 and an unstable fixed point @xmath99 for @xmath100 it has a single stable fixed point @xmath101 . at the critical point @xmath102 , three fixed points merge into a stable fixed point . as the parameter @xmath103 varies from @xmath25 to @xmath31 the system goes from a double well to a single well . the stable fixed points are just the bottom of the wells , and the unstable point is just the saddle point of double well . if the particle is at the fixed point @xmath104 at the beginning , i.e. , the particle stays at the bottom of one well . then , let @xmath103 vary very slowly . at the critical point @xmath105 the two wells merge into a single well . at this time , the bifurcation point is @xmath106 which is the bottom of the single well . so if @xmath103 varies very slowly , one can imagine that the particle will stay at the bottom of the well all the time , even when the system goes from a double well to a single one ( at this time the adiabatic condition does not hold but the bifurcation point is still a stable fixed point , because the bifurcation point still corresponds to the bottom of the well ) . as we have discussed in sect . ii , the breakdown of the adiabatic condition @xmath107 corresponds to the trajectory bifurcation , i.e. , the points @xmath64 is just a bifurcation point of the fixed points . the properties of the fixed points are determined by the following jacobian @xmath108 where @xmath109 . if the jacobian @xmath110 the zero point ( fixed point ) is a regular point . but when @xmath111 , the zero point is a bifurcation point . there are two kinds of bifurcation points : limit points and branch points . the limit point satisfies that @xmath111 but @xmath112 @xmath113 which corresponds to generation and annihilation of the fixed points . if @xmath114 , the point @xmath115 is a branch point . the branch point corresponds to branch process of the fixed points . the directions of all branch curves are determined by the equations @xcite @xmath116 or @xmath117 where @xmath118 @xmath119 and @xmath120 are three constants . @xmath121 corresponds to @xmath2 or @xmath122 respectively . different solutions of the above equations correspond to different branch processes . for the zero point @xmath123 , i.e. , the fixed point on line @xmath124 we can obtain @xmath125 obviously , when @xmath66 , @xmath126 the critical point @xmath127 is a bifurcation point , at which the adiabatic condition fails . for the case 1 : we can find at the point @xmath128 the jacobian @xmath129 but @xmath130 this point is a limit point which corresponds to annihilating of zero points . at this point , the elliptic point annihilates simultaneously with a hyperbolic point . in fig . 1(a ) , the dashed lines is the trajectory of the hyperbolic point . apparently , the elliptic point evolves adiabatically until it annihilates with the hyperbolic point at @xmath131 after this annihilation the elliptic point turns to an ordinary closed orbit with a nonzero action , so the adiabatic evolution is destroyed . the annihilation process of the fixed points of the system ( [ ham ] ) has also been discussed in ref . @xcite in detail . for the case 2 : at the point @xmath132 the jacobian determinant @xmath133 and @xmath134 . this is a branch process of the fixed points . we can prove that for this case @xmath135 , so the solutions of equations ( [ d1 ] ) and ( [ d2 ] ) give two directions : @xmath136 and @xmath137 the branch process corresponds to merging process . at this branch point , three fixed points , two elliptic points and one hyperbolic point , merge together . one can see this point in fig . 1(b ) , in which the dotted line is the trajectory of the hyperbolic point , and the dashed line corresponds to the trajectory of another elliptic point . since the total topological index is invariant , the three fixed points merge to one point with index @xmath138 i.e. , merge to an elliptic point . the elliptic point evolves adiabatically until it reaches the critical point @xmath128 at which three fixed points merges to one elliptic point . therefore , after the branch process , the elliptic point turns to a new elliptic point , the action keeps zero and the adiabatic evolution still holds . from above discussion , we see that the adiabaticity breaks down at bifurcation points of the fixed points , but only for the limit point the adiabatic evolution is destroyed ( case 1 ) , while for this case the two fixed points annihilate . for case 2 , three fixed points merge to one , because the critical point @xmath127 is still a stable fixed point , the adiabatic evolution keeps with action zero . the phenomena discussed above can occur for the adiabatic change of @xmath139 with @xmath140 fixed . on the other hand , the hamiltonian ( [ ham ] ) is invariant under the transformations @xmath141 @xmath142 and @xmath143 hence , the phenomena can also be found under such transformations . in summary , at some critical points , the adiabatic condition fails , but the adiabatic evolution may not always be broken . we find that the topological property of the critical point plays an important role for adiabatic evolution of the fixed points when the adiabatic condition does not hold . if the topological index of the critical point is @xmath25 the adiabatic evolution of fixed point will not be destroyed . on the contrary , if the index of the critical point is zero or @xmath31 , the adiabatic evolution will be destroyed . as a paradigmatic example , we investigated the adiabatic evolution of a classical hamiltonian system which has a number of practical interests . for this system , the adiabaticity breaks down at bifurcation points of the fixed points . but only for the limit point the adiabatic evolution is destroyed . for the branch process , the adiabatic evolution will hold , and the corrections to the adiabatic approximation tend to zero with a power law of the sweeping rate . in general , the corrections to the adiabatic approximation are exponentially small in the adiabaticity parameter , both for quantum system and classical system @xcite . it is particularly interesting that the corrections of the adiabatic approximation may be a power law ( e.g. , for the case 2 ) . the power law corrections to daidabatic approximation have also been found in the nonlinear landau - zener tunneling @xcite . in ref . @xcite , the authors found that when the nonlinear parameter is smaller than a critical value , the adiabatic corrections are exponentially small in the adiabatic parameter , but when the nonlinear parameter equals to the critical value , the adiabatic corrections are a power law of the adiabatic parameter . furthermore , if the nonlinear parameter is larger than the critical value , the so - called non - zero adiabatic tunneling will occur wuniu , add . indeed , the cases , for which the corrects to the adiabatic approximation are not exponential law with the adiabatic parameter , correspond to the collision of fixed points this work was supported by the 973 project of china and national nature science foundation of china ( 10474008,10445005 ) . lb fu is indebted to dr . chaohong lee and alexey ponomarev for reading this paper , and acknowledges funding by the alexander von humboldt stiftung . s. das , _ et al _ phys . a bf 65 , 062310 ( 2002 ) ; n.f . bell , _ et al _ phys . a * 65 * , 042328 ( 2002 ) ; a.m. childs , _ et al _ phys . rev . a 65 , 012322 ( 2002 ) ; r.g . unanyan , _ et al _ phys . lett . * 87 * , 137902 ( 2001 ) . f. dalfovo _ phys . * 71 * , 463 ( 1999 ) ; a.j . leggett , rev . phys . * 73 * , 307 ( 2001 ) ; r. dum _ et al . _ , . lett . * 80 * , 2972 ( 1998 ) ; z.p . karkuszewski , k. sacha , and j. zakrzewski , phys . rev . a**63 * * , 061601(r ) ( 2001 ) ; t. l. gustavson,_et al _ , phys . 88 , 020401 ( 2002 ) ; j. williams , _ et al _ phys . a 61 , 033612 ( 2000 ) ; matt mackie , _ et al _ phys . 84 , 3803 ( 2000 ) ; roberto b. diener , biao wu , mark g. raizen , and qian niu , phys . lett . * 89 * , 070401 ( 2002 ) . c. lee , w. hai , l. shi , and k. gao , phys . a * 69 * , 033611 ( 2004 ) ; c. lee , w. hai , x. luo , l. shi , and k. gao . ibid . * 68 * , 053614 ( 2003 ) ; w. hai , c. lee , g. chong , and l. shi , phys . e * 66 * , 026202 ( 2002 ) ; c. lee , et al . , phys a * 64 * , 053604 ( 2001 ) .

for classical hamiltonian systems , the adiabatic condition may fail at some critical points . however , the breakdown of the adiabatic condition does not always make the adiabatic evolution be destroyed . in this paper , we suggest a supplemental condition of the adiabatic evolution for the fixed points of classical hamiltonian systems when the adiabatic condition breaks down at the critical points . as an example , we investigate the adiabatic evolution of the fixed points of a classical hamiltonian system which has a number of applications .

the new generation of cosmic microwave background ( cmb ) experiments ( see e.g. @xcite , and the future missions planck @xcite , and map @xcite ) promises to estimate the cosmological parameters within a precision of 1% . the current dataset already allows in some cases an uncertainty below 10% on such parameters as the baryon density or the primordial spectral slope . such a precision allows and demands a clear assessment of the theoretical assumptions . so far , all the estimations of cosmological parameters based on the cmb data assumed a gaussian distribution of the primordial fluctuations ( the only exception i know of is ref . @xcite in which the primordial slope @xmath2 was estimated in presence of skewness ) . such an assumption is based on the conventional models of inflation and has the obvious and enormous advantage of being simple and uniquely determined . however , the gaussianity of the primordial temperature fluctuations is still to be fully tested @xcite and there exist several theoretical models which actually predict its violation @xcite . therefore , it is necessary to quantify how the cosmological parameters derived from cmb experiments depend on the statistical properties of the fluctuation field . in this paper i derive the dependence of four cosmological parameters on the skewness of the fluctuations assuming a flat space . the parameters are the primordial slope @xmath2 , the baryon and cold dark matter rescaled density parameters @xmath3 ( @xmath4 is the hubble constant in units of 100 km / sec / mpc ) , and the cosmological constant density parameter @xmath5 . the flat space constraint reduces to the relation @xmath6 it is clear that removing the hypothesis of gaussianity leaves room for an infinity of different possible assumptions concerning the fluctuation distribution . i adopt here the edgeworth expansion ( ee ) , for three reasons : a ) it can be seen as a perturbation of a gaussian function ; b ) it is easy to manipulate analytically and c ) it is the distribution followed by any random variable that is a linear combination of @xmath7 random variables in the limit of large @xmath7 ( for @xmath8 the edgeworth distribution reduces to a gaussian ) . the latter property might be useful to describe fluctuations that arise due to several independent sources . the ee has been previously used in cosmology to model small deviations from gaussianity @xcite . the main drawback of the ee is that it is not positive definite . however , when the deviation from gaussianity is small , this problem is pushed many standard deviations away from the peak and does not affect the parameter estimation . this paper is meant to exemplify the effects that a non - zero skewness introduces on the likelihood estimation . for generality , i will not confine myself to any specific mechanism for generating the non - gaussianity . moreover , for simplicity , i will skip over several additional complications like bin cross - correlations , calibration , pointing and beam errors that an accurate analysis should take into account . the likelihood function usually adopted in cmb studies ( e.g. @xcite ) is an offset log - normal function . this function is an approximation to the exact likelihood that holds for gaussian data in presence of gaussian noise @xcite . the offset depends on quantities that are not yet publicly available ; since the offset can be neglected in the limit of small noise , we assume as starting point a simple log - normal that , neglecting factors independent of the variables , can be written as @xmath9 where @xmath10 , the subscripts @xmath11 and @xmath12 refer to the theoretical quantity and to the real data , @xmath13 are the spectra binned over some interval of multipoles centered on @xmath14 , @xmath15 are the experimental errors on @xmath16 , and the parameters are denoted collectively as @xmath17 . we neglect also the residual correlation between multipole bins , which should be anyway very small for the latest data . an overall amplitude parameter @xmath18 can be integrated out analytically adopting a logarithmic measure @xmath19 in the likelihood . writing @xmath20 it follows @xmath21 so that , neglecting the factors independent of the variables and putting @xmath22 , we obtain@xmath23 ^{2}}{2\sigma _ { \ell } ^{2}}\right ] dw\propto e^{-\frac{1}{2}\left ( \gamma -\frac{\beta ^{2}}{\alpha } \right ) } , \ ] ] where @xmath24 let us now introduce the edgeworth expansion . denoting with @xmath25 the normal variable in the gaussian function , the edgeworth expansion is @xcite@xmath26 \left [ \right . 1+\frac{k_{3,\ell } } { 6}h_{3}(x_{\ell } ) + & & \nonumber \\ \frac{k_{4,\ell } } { 24}h_{4}(x_{\ell } ) + \frac{k_{3,\ell } ^{2}}{72}h_{6}(x_{\ell } ) & + ... , \left . \right ] , & \end{aligned}\ ] ] where @xmath27 is the @xmath2-th cumulant of @xmath28 and @xmath29 is the hermite polynomial of @xmath2-th order . notice that the ee has the same norm , mean and variance as the gaussian , but different mode ( the peak of the distribution ) . here , as a first step , we limit ourselves to the first non - gaussian term containing the skewness @xmath30 . assuming that @xmath28 is distributed according to the edgeworth expansion , we can build the truncated edgeworth likelihood function @xcite to first order in @xmath30 : @xmath31 , \ ] ] with @xmath32 . now , integrating over @xmath33 we obtain@xmath34 dw= & & \nonumber \\ \sqrt{\frac{2\pi } { \alpha } } e^{-\frac{1}{2}\left ( \gamma -\frac{\beta ^{2}}{\alpha } \right ) } \left [ 1+\frac{1}{6}g(k_{3,\ell } , \delta _ { \ell } , \sigma _ { \ell } ) \right ] , & & \label{edgelik } \end{aligned}\ ] ] where@xmath35 this is the likelihood function that we study below . the effect of the extra terms is to shift the peak ( or mode ) of the distribution of each @xmath0 while leaving the mean unperturbed . since the shift depends on @xmath36 , @xmath15 and @xmath30 , the resulting _ mode spectrum _ will be distorted with respect to the _ mean spectrum_. therefore , the likelihood maximization will produce in general results that depend on @xmath30 . in fig . 1 we show the peak shift introduced in the simplified case in which the skewness is independent of @xmath1 : if @xmath37 is _ negative _ , the spectrum is shifted _ upward _ by a larger amount at the very small and very large multipoles , and by a smaller amount around @xmath38 , where the relative errors are the smallest ; if @xmath37 is positive the shift is downward . as a consequence of the distortion , we expect that a constant negative skewness favours spectra which are tilted downward with respect to the gaussian case , and the contrary for a positive skewness . in general , the cosmological parameters will depend on the multipole dependence of @xmath30 . for small @xmath39 , the shift can be approximated by@xmath40 clearly , if the peak shift introduced by the ee were independent of @xmath1 , the integration over the amplitude @xmath33 would erase the non - gaussian effect on the likelihood . that is , putting @xmath15 and @xmath36 equal to a constant independent of @xmath1 we obtain @xmath41 to evaluate the likelihood , a library of cmb spectra is generated using cmbfast @xcite . following @xcite i adopt the following uniform priors : @xmath42 @xmath43 @xmath44 . as extra priors , the value of @xmath4 is confined in the range @xmath45 and the universe age is limited to @xmath46 gyr . the remaining input parameters requested by the cmbfast code are set as follows : @xmath47 @xmath48 in the analysis of @xcite @xmath49 , the optical depth to thomson scattering , was also included in the general likelihood and , in the flat case , was found to be compatible with zero at slightly more than 1@xmath50 . therefore here , to reduce the parameter space , i assume @xmath49 to vanish . the theoretical spectra are compared to the data from cobe @xcite and boomerang @xcite . to specify the skewness @xmath30 three simplified cases are studied : in the first one ( `` constant skewness '' ) , @xmath51 is assumed independent of the multipole @xmath1 ; in the second ( `` gaussian skewness '' ) , the skewness is assumed to be generated by some process only in a particular range of multipoles:@xmath52 where , in the numerical examples below , i put @xmath53 and @xmath54 . in the third case ( `` hierarchical skewness '' ) , the `` hierarchical '' ansatz is assumed @xcite , in which the skewness of the temperature field is proportional to the square of its variance . at the first order , we can assume that the skewness of the @xmath0 distribution is proportional to the skewness of the fluctuation field , so i put@xmath55 where , for instance , @xmath56 . in all three cases @xmath57 is left as a free parameter . these three choices are of course purely an illustration of what a real physical mechanism might possibly produce . fig . 2 shows the one - dimensional edgeworth likelihood functions marginalized in turn over the other three parameters . for the `` constant skewness '' , @xmath58 varies from -1.6 to 1.2 ( light to dark curves ) : below and above these values the likelihood begins to show pronounced negative wings , which signals that the first order edgeworth expansion is no longer acceptable . while the likelihood for @xmath59 is almost independent of @xmath37 , it turns out that the other likelihoods move toward higher values for higher skewness . as anticipated , this can be explained by observing that a higher skewness implies smaller @xmath0 at small multipoles : a tilt toward higher @xmath2 and higher @xmath60 gives therefore a better fit . the effect is of the order of 10% for @xmath61 . in the `` gaussian skewness '' case the trend is qualitatively the opposite , as can be seen in fig . 3 , where @xmath58 ranges from -4 to 4 ( light to dark ) . here the cosmological parameters decrease for an increasing skewness . the reason is that now the effect is concentrated around the intermediate multipoles @xmath62 : a positive skewness induces smaller @xmath0 at these multipoles , and therefore a smaller @xmath2 and @xmath60 helps the fit . the third case , the `` hierarchical skewness '' , is not shown because is qualitatively similar to the previous case : the region around @xmath38 is in fact also the region where @xmath0 is larger and therefore @xmath30 given by eq . ( [ hier ] ) is larger . 4 summarizes the results : the trend of the estimated parameters ( mean and standard deviation ) versus @xmath57 in the `` constant skewness '' case . the constant plateau that is reached for @xmath63 depends on the fact that for large and negative skewness the peak shift is independent of @xmath37 . the cosmological parameters can be well fitted by the following expressions : @xmath64 for @xmath4 the fit is @xmath65 . notice that the trend for @xmath4 is stronger than for the other variables : @xmath4 goes from 0.65 to 0.85 when @xmath37 increases from -1.6 to 1.2 . similar relations can be found for the other cases as well . this paper illustrates quantitatively a basic and obvious fact about cosmological parameter estimation , namely the dependence on the underlying statistics . although the gaussianity of the cmb data is still to be proved , almost all the previous works estimated the cosmological parameters assuming vanishing higher order cumulants . here it has been shown that a non - zero skewness distorts the mode spectrum with respect to the mean spectrum , inducing a considerable variation to the best fit cosmological parameters . the edgeworth expansion we used in this paper is convenient for analytical purposes but its use is limited to relatively small deviations from gaussianity . in fact , the peak shift displayed in fig.1 is always smaller than the errobars , and as a result the parameters , although varying with @xmath37 , remain always within one sigma from the zero - skewness case . this , however , does not mean that the dependence on the higher order moments can be neglected , first because it is a systematic effect , and second because more general probability distributions which are not small deviations from gaussianity might introduce much larger shifts . we have shown that , to first order , the peak shift @xmath66 is proportional to @xmath67 . the error @xmath15 includes cosmic variance and experimental errors . in the future , the main source of error will be cosmic variance , at least below @xmath68 or so . a skewness of order unity will therefore introduce an additional `` skewness bias '' that will limit the knowledge of the cosmological parameters by an amount similar to the cosmic variance itself . at this point it will become necessary to estimate @xmath30 along with the other parameters . the first order ee is however inadequate , since it is linear in @xmath30 , and it will be necessary to extend the expansion to higher orders @xcite , or to adopt a non - perturbative non - gaussian distribution . 10 c. b. netterfield _ et al . _ , astro - ph/0104460 . n. w. halverson _ et al . _ , astro - ph/0104489 . a. t. lee _ et al . _ , astro - ph/0104459 . g. de zotti , et al . , proc . of the conference : " 3 k cosmology " , roma , italy , 5 - 10 october 1998 , aip conference proc , in press , astro - ph/9902103 l. page , proc iau symposium 201 eds a. lasenby & a. wilkinson astro - ph/0012214 c.r . contaldi , p.g . ferreira , j. magueijo , k.m . gorski , ap.j . , ( 2000 ) 534 , 25 a. kogut , banday a.j . , bennett c.l . , gorski k , hinshaw g. smoot g.f . , wright e.l . ( 1996 ) apj , 464 l29 p. ferreira , magueijo j. , gorski k.m . , ( 1998 ) apj , 503 , 1 a.j . banday , zaroubi s. , gorski k.m . , ( 1999 ) apj , 533 , 575;pando j. , valls - gabaud d. , fang l. , ( 1998 ) phys . lett . , 81 , 4568 ; t. falk , rangarajan r. , srednicki m. , ( 1993 ) apj , 403 l1 ; p.s . corasaniti , l. amendola & f. occhionero , mnras ( 2001 ) , 323 , 677 ; avelino p.p . , shellard e.p.s . , wu j.h.p . , allen b. ( 1998 ) apj , 507 l101;gangui a. , mollerach s. , phys.rev . d54 ( 1996 ) 4750;gangui a. , martin j. , mnras 313 , 323 ( 2000);a . linde and v. mukhanov , phys . d56 , 535 ( 1997 ) e. gaztanaga , p. fosalba , e. elizalde , mnras 295 ( 1998 ) 35 ; r. s. kim , m. a. strauss , ap . j. ( 1998 ) 493 , 39 l. amendola , apj , ( 1994 ) 430 , l9 l. amendola , mnras ( 1996 ) , 283 , 983 l. amendola , astro . lett . and communications , ( 1996 ) 33 , 63 r. juszkiewicz , weinberg d.h . , amsterdamski p. , chodorowski m. , bouchet f. , ( 1995 ) apj , 442 , 39 j.r . bond , a.h . jaffe and l. knox , ap . j. , 533 , 19 ( 2000 ) m. kendall , stuart a. , ord j.k . , kendall s advanced statistics , 1987 , oxford university press , new york u. seljak and m. zaldarriaga , ap.j . , 469 , 437 ( 1996 ) p.j.e . peebles , 1980 , the large - scale structure of the universe , princeton university press .

the estimation of cosmological parameters from cosmic microwave experiments has almost always been performed assuming gaussian data . in this paper the sensitivity of the parameter estimation to different assumptions on the probability distribution of the fluctuations is tested . specifically , adopting the edgeworth expansion , i show how the cosmological parameters depend on the skewness of the @xmath0 spectrum . in the particular case of skewness independent of @xmath1 i find that the primordial slope , the baryon density and the cosmological constant increase with the skewness .

electronic wave functions in a disordered lattice exhibit an exponentially localized envelope in space a phenomenon , commonly known as the anderson localization @xcite . the problem has kept itself alive and kicking over all these years in condensed matter physics , and has given quantum transport properties of disordered systems intriguing twists and turns . the recent development of fabrication and lithographic techniques has taken the phenomenon of anderson localization beyond the electronic systems , substantiated by remarkable experiments incorporating localization of light @xcite , ultrasound in three dimensional elastic networks @xcite , or even plasmonic @xcite and polaritonic @xcite lattices . direct observation of the localization of matter waves @xcite in recent times has made the decades old phenomenon even more exciting . the key point in anderson localization is the dimensionality . within the tight binding approximation , the electronic wave functions are localized for dimensions @xmath0 ( the band center in the off diagonal disorder case is an exception ) . for @xmath1 with strong disorder , the wave function decays exponentially @xcite . extensive analyses of the localization length @xcite , density of states @xcite , and multi - fractality of the single particles states @xcite have consolidated the fundamental ideas of disorder induced localization . intricacies of the single parameter scaling hypothesis its validity @xcite , variance @xcite , or even violation @xcite in low dimensional systems provided the finer details of the localization phenomenon that have subsequently been supported by experimental measurements of conductance distribution in quasi - one dimensional gold wires @xcite . however , in low dimensions , or more specifically , in one dimensional disordered lattices even a complete delocalization of electronic states can be seen . this path breaking result was initially put forward by dunlap et al . @xcite in connection with a sudden enhancement of conductance of a class of polyanilenes on protonation . known as the _ random dimer model _ ( rdm ) the phenomenon is attributed to certain special kinds of positional correlation in the potential profiles . the investigation of delocalization of eigenstates in correlated disordered models was taken up further over the years and interesting results such as the relation of localization length with the density of states @xcite were put forward . the work extended to quasi - one dimensional systems as well for which the landauer resistance and its relation to the localization length was examined in details @xcite for a two - leg ladder model , an extensive extension of which was later done by sedrakyan et al . @xcite . controlled disorder induced localization and delocalization of eigenfunctions took a considerable volume in contemporary literature , exploring solid non - trivial results involving electron or phonon eigenstates @xcite . extended eigenfunctions in all such works mostly appear at special discrete set of energy eigenvalues . eventually , the possibility of a controlled engineering of spectral continuum populated by extended single particle states and even a metal - insulator transition in one , or quasi - one dimensional discrete systems have also been discussed in the literature @xcite . but , on the whole , the general exponentially localized character of the eigenfunctions prevails , and the possibility of having a mixed spectrum of localized and extended states in a disordered system ( under some special positional correlations ) is now well established . can one generate , going beyond the rdm , a full band of only extended eigenfunctions in a disordered system with @xmath0 ? if yes , what would be the minimal models capable of showing such unusual spectra ? this is the question that we address ourselves in the present communication . we put forward examples of a class of essentially one dimensional disordered and quasiperiodic lattices where a _ complete delocalization _ of electronic states can be engineered , and _ absolutely continuous _ bands can be formed in the energy spectrum . this is shown to be possible when an infinite disordered or quasiperiodic array of two kinds of ` bonds ' is side coupled to a single or a cluster of quantum dots ( qd ) from one side at a special set of vertices . minimal requirements are discussed in details . in some of the examples cited here , the attachment of the dots form local loops which can be pierced by a constant magnetic field , breaking the time reversal symmetry of electron - hopping only locally , along the edges of such closed loops . the engineering of bands of extended states is shown to be the result of a definite numerical correlation in the values of the electron hopping amplitude along the chain ( backbone ) and the coupling of the linear backbone with the side coupled dots , the strength of the magnetic field or both . it should be mentioned that an early report of a rdm - kind of correlation leading to extended eigenfunctions in a fibonacci superlattice was put forward by kumar and ananthakrishna @xcite . the insight into the phenomenon was immediately provided by xie and das sarma @xcite . however , the fact that , certain specific _ numerical relationship _ among a subset of parameters of the hamiltonian is capable of producing , absolutely continuous _ bands _ of extended eigenfunctions is uncommon , and to the best of our knowledge , has not been addressed until very recently @xcite . we consider two bonds @xmath2 and @xmath3 arranged along a line forming an infinite linear chain . the sequence of the bonds may be random or quasiperiodic @xcite , offering either a pure point spectrum or a singular continuous one . the bonds connect identical atomic sites , an infinite subset of which is coupled to similar atoms ( mimicing single level quantum dots ( qd ) ) from one side giving the system a quasi one dimensional flavor . the disorder ( or , quasiperiodic order ) thus has a topological character . in addition to the basic interest of going beyond the rdm , two other facts motivate us in undertaking such a work . first , the fano - anderson effect @xcite caused by the insertion of a bound state into a continuum is an exciting field , and has been investigated recently in nanoscale systems @xcite . in this context , our study provides examples where one can observe at least one effect of inserting multiple bound states , in fact , an infinity of them in a _ singular continuum _ , or a pure point spectrum . second , the present advanced stage of growth techniques has motivated in depth studies of quasiperiodic nanoparticle arrays in the context of ferromagnetic dipolar modes @xcite or plasmon modes @xcite . also , the use of a scanning tunnel microscope ( stm ) tip to fabricate structures atom by atom , viz . , xe on ni substrates @xcite , or nanometer size gold particles on metals @xcite , or , putting individual atoms of si substrate @xcite has stimulated a lot of work in this field @xcite . our results can motivate future experiments in this direction . in section [ model ] we describe the lattice models . in section [ method ] , within subsections [ method]a and [ method]b the local , non - local and the mixed cases introduced in section [ model ] are discussed , with explicit remarks on the density of states profiles in each case . subsection [ method]c specially deals with the special case of a fibonacci quasiperiodic chain , using a real space renormalization group ( rsrg ) scheme . section [ transmission ] describes the two terminal transmission coefficient , while section [ univer ] provides a critical discussion on the evolution of the parameter space under the rsrg scheme and its relation with the extendedness of the wave function . in section [ othergeometries ] we briefly point out a triplet of other geometries which are less restrictive compared to the ones discussed here , and in section [ conclu ] we draw our conclusion . we refer the reader to fig . [ cells ] ( double line ) and @xmath3 ( red single line ) , such that a @xmath3-bond is always flanked by two @xmath2-bonds on either side . the atomic sites on the backbone are marked as @xmath4 , @xmath5 and @xmath6 as described in the picture . the hooping integrals are appropriately described by @xmath7 and @xmath8 . ( a ) a qd ( @xmath9 ) is locally connected to the @xmath4-site . this @xmath9-@xmath4 cluster is renormalized " into an effective site ( yellow circle surrounded by red dotted lines ) . ( b)a qd ( @xmath9 ) is non - locally coupled to the @xmath5-@xmath6 pair . the @xmath9-@xmath5-@xmath6 cluster is then renormalized into the immediate lower geometry , pointed by the arrow . ( c)the qds @xmath10 and @xmath11 exhibit a mixed connection to @xmath5-@xmath6 pair . the block @xmath5-@xmath10-@xmath11-@xmath6 is renormalized to the diatomic molecule shown by the arrowhead . in every case , the linear chain ( disordered or quasiperiodic ) is formed by arranging the cluster linked by the bent cyan double arrowheads in the desired order.,width=321 ] where the basic structural units are displayed . the backbone in each case is an infinite array of a single ( red ) bond @xmath3 and a double bond @xmath2 . we shall restrict ourselves to a geometry where the single ` @xmath3 ' bonds do not come pairwise . thus we have a kind of ` anti - rdm ' here . this is not always needed though , as will be discussed in the concluding section . three cases are separately discussed . the simplest one is that of a local connection ( lc ) , where a single qd ( marked as @xmath9 in fig . [ cells](a ) is tunnel - coupled to a site @xmath4 flanked by two @xmath2-bonds . the second case discusses a non - local connection ( nlc ) , where a qd ( @xmath9 ) is tunnel - coupled to both the sites residing at the extremities ( @xmath5 and @xmath6 in fig . [ cells](b ) ) of a @xmath3-bond . the final geometry describes a mixed connection ( mc ) , where two inter - coupled qds @xmath10 and @xmath11 are connected to the extremities of a @xmath3-bond ( i.e. to @xmath5 and @xmath6 sites ) as shown in fig . [ cells](c ) . in the two latter cases a uniform magnetic field is applied in a direction perpendicular to the plane of every closed loop . the system in each case is described by a tight - binding hamiltonian . we show that , for a particular algebraic relationship between the nearest neighbor hopping integrals @xmath12 along the backbone and the backbone - qd coupling @xmath13 , the infinite topologically disordered or quasiperiodic chain of scatterers yields absolutely continuous energy bands in the spectrum . in the case of lc ( fig . [ cells](a ) ) there will be two continuous subbands . in the nlc and mc cases ( fig . [ cells](b ) and ( c ) ) a single absolutely continuous band spans the entire energy spectrum when , in addition to the algebraic relationship between the hopping integrals @xmath12 and @xmath13 , the magnetic flux @xmath14 threading each elementary plaquette assumes a particular value . these two cases ( nlc and mc ) therefore represent situations where the spectral character can be grossly changed from pure point or singular continuous to absolutely continuous by tuning an external magnetic field . this may be useful from the standpoint of device technology . spinless , non - interacting electrons on the chain comprising the building blocks depicted in fig . [ cells ] are described by the hamiltonian , @xmath15 \label{hamilton}\ ] ] where , @xmath16 is the constant on - site potential , at every site including the qd ( marked @xmath9 ) . we have colored the atomic sites differently just to distinguish between their nearest neighbor bond configurations . these are marked as @xmath4 ( yellow circle ) , @xmath5 and @xmath6 ( blue circles ) respectively . the nearest - neighbor hopping integral @xmath17 ( double bond ) along the backbone on either side of an @xmath4-site , while it is @xmath8 ( denoted by red line segment ) between a @xmath5-@xmath6 pair . in the lc case ( fig . [ cells](a ) ) @xmath18 between the qd and the @xmath4-site . in the nlc and the mc situations ( fig . [ cells](b ) and ( c ) ) the presence of the magnetic flux breaks the time reversal symmetry along the edges of the loops . this is taken care of by incorporating the appropriate peierls phase factor in the hopping integrals , viz . , @xmath19 where , @xmath20 . @xmath21 is the perimeter of the plaquette and @xmath22 is the length of the bond connecting the @xmath23-th and the @xmath24-th sites of the loop . @xmath25 is the flux quantum . let us consider symmetric geometries only . this means that , in the nlc case , we assume that the qd is placed symmetrically above the @xmath5-@xmath6 cluster . in the mc case similarly , the @xmath5-@xmath10 , @xmath10-@xmath11 and the @xmath11-@xmath6 distances are equal . this just simplifies the mathematical expressions without sacrificing any physics that we are going to establish . thus , in the nlc ( fig . [ cells](b ) ) @xmath26 , @xmath27 , with @xmath28 and @xmath29 , @xmath30 and @xmath31 being the bond lengths between the @xmath5-@xmath6 pair , and the @xmath5-@xmath9 and @xmath6-@xmath9 pairs respectively . the asterisk denotes the complex conjugate . similarly , in the mc case ( fig . [ cells](c ) ) , @xmath32 , and @xmath33 . in this case however , @xmath34 and @xmath35 where , @xmath36 and @xmath37 are the bond lengths between the @xmath5-@xmath6 pair , and the @xmath5-@xmath10 , @xmath10-@xmath11 and @xmath11-@xmath6 pairs respectively . the respective complex conjugates are trivially understood . using the difference equation version of the schrdinger equation , viz . , @xmath38 we decimate out the vertices ( qds ) in each of the three cases to map the local , non - local and mixed clusters on to effective atomic sites with renormalized on - site potentials given by , @xmath39 in the lc ( fig . [ cells](a ) ) , @xmath40 in the nlc ( fig . [ cells](b ) ) , and @xmath41 in the mc case ( fig . [ cells](c ) ) , where , @xmath42 . the sites with renormalized on - site potential in each case are encircled with the red dotted lines in fig . [ cells ] . the hopping integrals are still @xmath7 and @xmath8 along the linear backbone for the lc , while they are , @xmath43 in the nlc case , and @xmath44 in the mc case . one can now build up an infinite chain of @xmath4 sites ( renormalized , in the lc case ) and the @xmath5-@xmath6 doublet ( renormalized in the nlc and mc cases ) in any desired order . the amplitude of the wave function at any remote site on such a chain is conveniently obtained by the transfer matrix technique . using the difference equation the amplitudes of the wave function at the neighboring sites along the effective one dimensional chain can be related using the @xmath45 transfer matrices , @xmath46 the hopping integrals @xmath47 will carry the appropriate phase factors when written for the nlc and mc cases . it is obvious that there are three kinds of transfer matrices , viz . , @xmath48 , @xmath49 , @xmath50 and which will differ in their matrix elements , depending on the respective on - site potentials and the nearest - neighbor hopping integrals . from the arrangement of the @xmath5-@xmath6 clusters and the isolated sites in the original chain it can be appreciated that the the wave function at a far end of the chain can be determined if one evaluates the product of the unimodular matrices @xmath48 and @xmath51 sequenced in the desired random or quasiperiodic fashion . the central result of this communication is that , in each of the three cases of lc , nlc and mc , the commutator @xmath52 $ ] can be made to vanish irrespective of the energy @xmath53 of the electron whenever the system parameters are inter - related in a certain algebraic fashion . let us look at the explicit expressions . we list below only one off diagonal element of the commutator for every configuration ( lc , nlc or mc ) , as the diagonal elements of the above commutator vanish identically in each case , and @xmath54_{21}= [ { \bm{m_{\alpha}}},{\bm{m_{\gamma\beta}}}]_{12}$ ] . * _ the local coupling : _ in this case , @xmath55_{12 } = \dfrac{\lambda^2-(t_{b}^{2}-t_{a}^{2})}{t_{a } t_{b } } \label{lc}\ ] ] * _ the non local coupling : _ here , @xmath56_{12 } = \nonumber \\ & \dfrac{(e-\epsilon ) e^{i 2\pi\phi/\phi_{0 } } ( t_{a}^{2}-t_{b}^{2}-\lambda^{2 } ) - 2 \lambda^{2 } t_{b } \cos ( \dfrac{2\pi\phi}{\phi_{0 } } ) } { t_{a } e^{i\theta_{1,nl } } [ \lambda^{2 } + ( e-\epsilon)t_{b } e^{i 2\pi\phi/\phi_{0 } } ] } \label{nlc}\end{aligned}\ ] ] and , * _ the mixed coupling : _ in this case , @xmath56_{12 } = \nonumber\\ & \dfrac{e^{i 2\pi\phi/\phi_{0 } } [ ( e-\epsilon)^{2}-\lambda^{2 } ] ( t_{a}^{2 } - t_{b}^{2 } - \lambda^{2 } ) - 2t_{b } \lambda^{3 } \cos ( \dfrac{2\pi\phi}{\phi_{0 } } ) } { t_{a } e^{i\theta_{1,m } } \left[t_{b } e^{i 2\pi\phi/\phi_{0 } } [ ( e-\epsilon)^{2}-\lambda^{2 } ] + \lambda^{3 } \right ] } \label{mc}\end{aligned}\ ] ] a look at eqs . - reveals that it is possible to make the commutator vanish independent of the energy @xmath53 . let us discuss case by case . shows that @xmath54_{12}$ ] ( and hence the full commutator ) vanishes if we set @xmath57 this implies that , with the above value of the tunnel hopping integral , the electronic energy spectrum will no longer be sensitive to the arrangement of the matrices @xmath48 and @xmath58 , that is , independent of the arrangement of the atomic site @xmath4 , and the pair @xmath5-@xmath6 . this happens independent of the energy @xmath53 of the electron . this result needs to be contrasted clearly with that in the rdm @xcite where the local structure of disorder could transform one subset of the transfer matrices into unit matrices , but only at special value of @xmath53 . in our case , with the commutation condition satisfied one can arrange the constituent elements @xmath4 and @xmath5-@xmath6 even in any kind of perfect periodic order . the wave functions as a result , will have to be of a _ perfectly extended _ , bloch - like character and the energy bands will exhibit absolutely continuous measure whenever @xmath59 . however , this condition is only necessary , and we discuss below the _ sufficient _ condition for observing extended eigenstates . taking advantage of the commutation of the transfer matrices we can shuffle any arrangement of the atoms into two infinite , periodic arrays of the effective renormalized @xmath9-@xmath4 cluster and @xmath5-@xmath6 clusters ( fig . [ order ] ) . sites ( yellow ) coupled to the qd ( red ) and its renormalized version where the renormalized @xmath4 sites are encircled by dotted red lines , and ( b ) periodic array of @xmath5-@xmath6 pairs . the strength of the hopping @xmath8 ( red line ) is taken to be greater than @xmath7 ( double lines).,width=321 ] the local density of states ( ldos ) at any site of these lattices can be worked out analytically , and for the @xmath4 and @xmath5 sites the results are , @xmath60^{2}}}\\ \rho_{\beta } = & \dfrac{1}{\pi } \dfrac{e - \epsilon}{\sqrt{4t_{a}^{2 } ( e-\epsilon)^{2 } - [ ( e-\epsilon)^{2 } - ( t_{b}^{2 } - t_{a}^{2})]^{2 } } } \end{aligned } \label{merge1}\ ] ] in each case , the ldos exhibits a continuous two - subband structure ( typical of a one dimensional binary ordered chain ) . it is obvious that , with the _ resonance _ condition @xmath61 the ldos in the two cases overlap . that is the bands formed by each individual periodic sublattices merge completely . so , a linear array of the structural units @xmath4-@xmath9 and the @xmath5-@xmath6 clusters , grown following any chosen pattern ( for example , completely disordered , or quasiperiodic geometry ) should also exhibit precisely these absolutely continuous sunbands . as extended and localized eigenstates can not coexist at the same energy , the electronic states must be of an _ extended character _ , a fact that is substantiated later by a flow of the hopping integrals under rsrg and a perfect two terminal transmission . this completes the proof that in the lc case , a suitable choice of the hopping integrals can generate absolutely continuous subbands populated only with _ extended _ single particle states . we now turn our attention to the cases of nlc and mc which essentially refer to an array of triangle shaped and square plaquettes threaded by a magnetic flux and single atomic sites ( fig . [ cells](b ) and ( c ) ) . the matrix elements @xmath54_{12}$ ] , as given by eqs . and become zero in either situation when , @xmath62 , and , in addition to it , @xmath63 in either case . it means that , even if we fix @xmath62 at the very outset , we still need to tune the magnetic flux @xmath14 through each plaquette to a particular value to have @xmath54=0 $ ] _ independent of the energy @xmath53 of the electron_. just as before , we can now , using the commutivity of @xmath48 and @xmath58 shuffle the building blocks to generate two infinite periodic chains corresponding to both the nlc and the mc cases , comprising of @xmath5-@xmath6 pairs , and isolated single sites @xmath4 ( @xmath64 , in both these cases ) . in terms of the parent lattices , in the nlc situation this means that the single @xmath4 sites and the @xmath5-@xmath9-@xmath6 triangle can be arranged in any desired pattern , while for the mc case its any arbitrary linear arrangement of the @xmath4 and the @xmath5-@xmath10-@xmath11-@xmath6 cluster . the @xmath4-lattice has the well known density of states , viz . , @xmath65^{-1/2}$ ] . to make things look algebraically simple , let us set @xmath66 , which just means that the side coupled qd is equispaced from the base sites , and that the phase acquired by the electron while hopping along an arm of a triangle as well as of a square is same for all the arms . the resonance condition now boils down to @xmath67 and of course , @xmath63 . the ldos at the @xmath5 site corresponding to the nlc case is given by , @xmath68 where , @xmath69 } { [ ( e-\epsilon)^{2 } - t_{b}^{2}]^{2 } } - \left[e - \dfrac{\epsilon ( e-\epsilon)^{2 } + 2t_{b}^{3 } \cos(2\pi\phi/\phi_0 ) + t_{a}^{2 } ( e-\epsilon ) + t_{b}^{2 } ( 2 e - 3 \epsilon ) } { ( e-\epsilon)^{2 } - t_{b}^{2}}\right]^{2}\ ] ] and , the same corresponding to the mc case is given by @xmath70 where , @xmath71 @xmath72 and @xmath73 are given by , @xmath74 \sin^{2}(\pi\phi/\phi_{0})\right ] } { ( e-\epsilon)^{2}[(e-\epsilon)^{2}-2t_{b}^{2}]^2}\\ \xi_{2}(e,\epsilon , t_{a},t_{b},\theta)&= \left [ \dfrac{(e-\epsilon)(\delta - t_{b}^{2})}{\delta}- \dfrac{\delta^2(t_{a}^{2}+t_{b}^{2 } ) + t_{b}^{4 } ( t_{b}^{2}+2\delta \cos(2\pi\phi/\phi_{0 } ) ) } { \delta ( e-\epsilon)(\delta - t_{b}^{2})}\right ] \end{aligned}\ ] ] with @xmath75 . it is interesting to note that the algebraic expressions in the nlc and mc cases reduce to the simple form @xmath76^{-1/2}$ ] as soon as we set @xmath67 and @xmath63 . this happens to be the ldos at the @xmath4-site of a pure @xmath4-chain . the band extends from @xmath77 to @xmath78 . thus , the same resonance condition , viz . , @xmath79 and @xmath80 results in a complete overlap of the energy bands at least in the energy range @xmath81 $ ] in both the cases . we have a single absolutely continuous band of extended eigenfunctions . as a specific example , we explicitly calculate the ldos at the @xmath5-sites in a golden mean fibonacci quasiperiodic chain . the chain is grown recursively following the usual fibonacci inflation rule @xmath82 and @xmath83 @xcite . the corresponding hopping integrals @xmath7 and @xmath8 follow a fibonacci arrangement . the local , non - local or the mixed attachments of the qds are shown in fig . [ lattices ] . the ` quasi one - dimensionality ' caused by the side coupled clusters are removed by decimating the attachments and creating an _ effective _ one dimensional chain in each case , as depicted in the same figure . the decimation results in renormalized values of the on - site potentials at the @xmath4-site in the lc case , and at the @xmath5 , and the @xmath6-sites in the nlc and the mc cases , as already mentioned . such a quasiperiodic fibonacci chain is , by construction , self similar and allows an exact implementation of the rsrg methods . renormalized versions of the fibonacci chain are obtained by the well known decimation scheme @xcite . for the sake of understanding and to facilitate a subsequent discussion on the flow in parameter space we present the explicit rsrg recursion relations connecting the @xmath84-th and the @xmath85-th stages of iteration for the three cases . * _ the local connection _ : @xmath86 with , @xmath87 , @xmath88 , @xmath89 and @xmath90 . * _ the non - local and the mixed coupling _ : in both these cases , the magnetic flux breaks the time reversal symmetry , but only locally , along the @xmath3 bonds connecting the @xmath5-@xmath6 vertices of the linear chain in the right panels of fig . [ lattices](b ) and ( c ) . for this we designate by @xmath91 and @xmath92 the _ forward _ and _ backward _ hopping respectively along the @xmath3 bond . this naturally takes care of the phase introduced by the field along this segment . the hopping @xmath93 along the @xmath2 bond , though free from any phase at the bare length scale , picks up phase on renormalization which needs to be taken care of . the recursion relations for both the chains are , @xmath94 the complex conjugate hopping integrals are defined appropriately . the initial values are of course different in these two cases , and are given by , @xmath95 , @xmath96 ; @xmath97 and @xmath98 in the nlc case , while , @xmath95 , @xmath99 ; @xmath97 and @xmath100 and @xmath42 in the mc case . the phase @xmath101 in the nlc case and it is @xmath102 in the mc one . site of an infinite fibonacci array for ( a ) the locally connected qds , ( b ) a single qd non - locally connected to every @xmath5-@xmath6 pair , and ( c ) the mixed case of directly and indirectly coupled qds to the @xmath5-@xmath6 pair . in each panel , the fragmented display represents the off - resonance case while the absolutely continuous sub - bands or band represent the cases when @xmath54=0 $ ] . we have set @xmath103 in all the cases . @xmath104 and @xmath105 in ( a ) while @xmath104 and @xmath106 in ( b ) and ( c).,width=264 ] at every stage of renormalization the renormalized _ forward _ and _ backward _ hopping integrals are , of course , complex conjugate of each other . the local green s function at any @xmath24-th site ( @xmath107 or @xmath6 ) is given by @xmath108 where , @xmath109 is the fixed point value of the corresponding on - site potential obtained by repeated application of the set of eq . and eq . for the local or the non - local and the mixed cases the ldos @xmath110 is obtained from the standard formula @xmath111 $ ] in the limit @xmath112 . we present the results in fig . [ density ] . in the top panel , the case of lc the ldos is obtained at a @xmath5-site . the off - resonance case is characterized by the sharp fragmented ldos profile that brings out the typical multifractal character of the wave functions in a quasiperiodic geometry . as the ` resonance condition ' @xmath113 ( with @xmath114 ) is satisfied , the fragmented spectrum turns into two absolutely continuous subbands . in the middle and the bottom panels the continuous band in the nlc and mc cases are illustrated by the shaded area . here we select @xmath115 . the resonance condition in either case is obtained by setting @xmath79 and @xmath80 . deviating away from this generates the characteristic fragmented spectral form of a fibonacci chain , as shown by the sharp blue lines ( for @xmath116 ) in each figure . the interesting difference with the lc case here is the existence of a single continuous band of states which will later be proven as extended , as shown by the shaded colored regions . it should be appreciated that our purpose has been only to demonstrate the appearance of absolutely continuous part(s ) in the energy spectrum . the ldos coming from any one kind of sites is enough for this purpose . the contribution to the full density of states coming from the side - coupled qd sites generally consists of delta like localized peaks some of which reside outside the continuum @xcite . these are of no concern in the present discussion , as the central motivation has always been to prove the generation of a band of extended states only as a result of some algebraic correlation between the numerical values of the parameters of the hamiltonian . the _ extended _ character of the eigenstates populating such continuous portions of the energy spectrum will subsequently be discussed in next sections . to substantiate the ldos profiles we also calculate the two terminal transport in the systems considered . the procedure is standard . the system is clamped between two perfectly periodic , semi - infinite leads on either side ( fig . [ transport ] ) . the sample trapped in between the leads is then decimated to a dimer by judiciously using the rsrg recurrence relations . finally , the transmission coefficient is obtained by the well known formula @xcite , @xmath117 \nonumber\\ & \text{and}\quad \mathcal{b}=[(p_{11}+p_{22})\sin ka]\nonumber\end{aligned}\ ] ] where , @xmath118 refer to the dimer - matrix elements , written appropriately in terms of the on - site potentials of the final renormalized _ left _ ( l ) and _ right _ ( r ) atoms @xmath119 and @xmath120 respectively , and the renormalized hopping between them @xcite . @xmath121 , @xmath122 and @xmath123 being the on - site potential and the hopping integral in the leads , and @xmath124 is the lattice constant in the leads which taken equal to unity throughout the calculation . in fig . [ trans ] we plot the transmission coefficient as a function of the energy of the electron in the three cases discussed so far . as a function of the energy @xmath53 of the electron for both the resonance and the off - resonance conditions . ( a ) represents the lc case , ( b ) represents the nlc case and ( c ) is for the mc case . the numerical values of the potentials and the hopping integrals are the same as in the ldos figures.,width=264 ] in each panel , again the resonance and off - resonance cases are plotted together for comparison . in the top panel , for the local coupling , when we set @xmath113 , the transmission coefficient attains very high values , achieving the limit unity in most cases for the entire regions of the continuous subbands . there is a clean gap between the two zones of high transmittivity . it is because one has gaps in the energy spectrum in this region , and any _ gap states _ arising out of the side coupled dots in this part must have a localized character . the perfect transmission under the resonance condition in the lc case brings out a variation over the recent studies of farchioni et al . @xcite , where it was rightly shown that , side - coupled dots in general suppress the transmission across a linear tight binding chain . in the central and the bottom panels , the energy spectrum exhibits a single continuous band spanning the entire energy range . to be consistent with the ldos figures we have preset @xmath79 . the resonance , or a deviation from resonance is now controlled only by controlling the external magnetic field only . when the flux is detuned from its resonance value , the spectrum represents a fragmented character typical of quasiperiodic lattices , while precisely at @xmath80 the transmission coefficient turns out to be unity for the entire range of the continuum confirming the extended character of the eigenstates . in this section we would like to draw the attention of the reader to an interesting flow pattern followed by the on - site potentials and the hopping integrals when the elemental building blocks are arranged in a quasiperiodic fibonacci chain , as discussed below . ( red ) , and @xmath5 and @xmath6 ( green ) , and the two nearest neighbor hopping integrals are @xmath7 and @xmath8 respectively.,width=321 ] first , it should be noted that , since the transfer matrices @xmath48 and @xmath58 corresponding to the the structural units depicted in fig . [ cells](a)-(c ) commute independent of energy @xmath53 under the appropriate _ resonance _ condition , the energy spectrum in this case should be the same for any uncorrelated disordered or quasiperiodic chains . as far as the quasiperiodic chains are concerned , though we have discussed the results specifically in terms of the golden mean fibonacci sequence , the idea and subsequent results hold true for any generalized fibonacci chain grown following the rule @xmath125 and @xmath83 , @xmath2 and @xmath3 representing the two bonds and @xmath126 . second important issue is the confirmation of the extended character of the eigenfunctions populating the continuous part of the ldos spectrum in all the cases . at least , for the deterministic fibonacci chain ( or it s generalizations ) an interesting answer to this question can be obtained by looking at how the on - site potentials and hopping integrals flow under successive rsrg iterations . let s try to understand . in the local coupling case for example , with reference to the eq . , it is found that , as soon as we set @xmath127 the parameter space follows the pattern @xmath128 and @xmath129 at every @xmath85-th stage of renormalization , whenever we select an eigenvalue @xmath53 arbitrarily from within the two continuous subbands in the ldos spectrum . this observation is substantiated by extensive numerical search throughout the observed continua scanned in arbitrarily small energy intervals . that such a pattern should correspond to extended bloch - like eigenfunctions can be justified by considering fig . [ rgflow ] where a perfectly periodic lattice of identical on - site potential @xmath16 and a constant nearest neighbor hopping @xmath130 is artificially converted into a golden mean fibonacci chain . on this _ artificial _ fibonacci chain @xmath131 , and different from @xmath132 . at the same time , @xmath133 , the latter being equal to @xmath130 . the _ flow pattern _ that we have been talking about therefore sets in at the very beginning . the artificial fibonacci chain in fig . [ rgflow ] can now be renormalized using the recursion relations eq . , and the density of states may be obtained from the appropriate green s function . as the parent lattice now is an ordered one , the typical one dimensional density of states is reproduced with the edges characterized by the van hove singularity . the spectrum is absolutely continuous , and all the wave functions are bloch functions . interestingly , with the initial set of values as given above , the on site potentials and the hopping integrals for the scaled version of the _ artificial _ fibonacci lattice ( fig . [ rgflow ] ) get locked into the flow pattern @xmath134 and @xmath135 at every @xmath85-th stage of renormalization , and for all energy eigenvalues within the range @xmath136 $ ] . in our actual case of fibonacci arrangement of the clusters in fig . [ cells](a ) as soon as such a flow pattern is set in for a special value of @xmath13 , it becomes impossible to judge whether the parent lattice was an ordered , perfectly periodic one , or a truly quasiperiodic fibonacci chain . thus the extendedness of the wave functions is firmly established whenever such an rsrg flow is observed . same flow pattern is also observed in the cases of fig . [ cells](b ) and ( c ) . in these cases , if we set beforehand @xmath79 , then the _ desired _ flow of the parameters can be achieved by tuning the external flux to @xmath80 . this refers to the interesting case of a _ flux driven crossover _ in the fundamental character of the wave functions in a non - locally coupled case or in the mixed case . in addition , for both the nlc case and the mc case , the ldos at the @xmath5 , @xmath6 or @xmath4 sites turn out to be exactly same whenever the resonance condition is satisfied . this is remarkable . the nlc and mc lattices are topologically different . an equality of the ldos for @xmath79 and @xmath80 implies that for both these cases the parameters @xmath137 , initially represented by two different _ points _ ( as their initial values are different ) in the five dimensional parameter space , are driven to the same _ fixed point _ following two different trajectories . as we have already mentioned , the result is independent of the order of arrangement of the triangles or the square boxes . thus , for the same resonance condition an indefinite number of geometrically different systems , beginning their ` journey ' at different locations in the five dimensional parameter space finally flow , following different trajectories , to the same fixed point , and thus come under the common umbrella . we are tempted to conceptualize a kind of _ universality class _ from this point of view . it is to be noted however , that the comment is based on the observed ldos at the sites on the backbone only . the average density of states can be different though . before we end , it should be mentioned that , the central idea presented in the present work is not restricted to only the geometries discussed here . for example , one can have an array of triangular or square plaquettes without any isolated @xmath4-site , where the plaquettes can ` touch ' each other giving rise to an additional site named @xmath138 and having a coordination number four . we refer to fig . [ ringlattice ] for a display of a disordered arrangement of such building blocks . the analysis proceeds in the same way and the one comes across a varied set of geometries for which the disorder - induced localization ( or , a quasiperiodicity driven power law localization ) can be suppressed and a full band ( or subbands ) of extended eigenfunctions can be generated . in conclusion , we have presented a class of topologically disordered array of building blocks described within a tight binding formalism , where delocalization of electronic eigenfunctions occur over either two subbands or over the entire range of allowed energies whenever the lattice parameters are inter - related through certain algebraic relation . we can have absolutely continuous spectrum even for such disordered or quasiperiodic arrangement of the unit cells in such cases . even an external magnetic field can be used to delocalize the electronic states over a continuous band of energy eigenvalues in certain cases . this aspect leads to the possibility of a flux driven state transition in such low dimensional systems .

we show that a discrete tight - binding model representing either a random or a quasiperiodic array of bonds , can have the entire energy spectrum or a substantial part of it absolutely continuous , populated by extended eigenfunctions only , when atomic sites are coupled to the lattice locally , or non - locally from one side . the event can be fine - tuned by controlling only the host - adatom coupling in one case , while in two other cases cited here an additional external magnetic field is necessary . the delocalization of electronic states for the group of systems presented here is sensitive to a subtle correlation between the numerical values of the hamiltonian parameters a fact that is not common in the conventional cases of anderson localization . our results are analytically exact , and supported by numerical evaluation of the density of states and electronic transmission coefficient .

the twist of the solar magnetic field plays an important role in transient phenomena such as solar flares and coronal mass ejections , and in the dynamo processes that cause the 11-year solar activity cycle . the magnetic twist can be measured in various ways . magnetic helicity is an integral that quantifies topological complexity of field lines , such as linking , twist , or kinking ( @xcite , @xcite ) . for a closed magnetic system it is defined by @xmath0 , and alternative definitions have been developed for open systems @xcite . in this letter we consider current helicity , which we define as @xmath1 where @xmath2 is the magnetic field and @xmath3 is the current density . the quantity @xmath4 has the advantage that it describes the _ local _ distribution of twist and shear in the magnetic field , and that it is more readily determined from limited observational data than @xmath5 which requires global information . for a force - free field ( @xmath6 ) we have @xmath7 and @xmath4 , which may be a function of space , is a fundamental parameter that describes the torsion of the field lines around one another . note that we shall not consider the _ integral _ current helicity @xmath8 because unlike @xmath5 it is not a near - conserved quantity in mhd @xcite , and it does not even in general take the same sign as @xmath5 ( except for linear force - free fields where @xmath4 is constant in space and @xmath4 , @xmath9 , and @xmath5 all have the same sign , * ? ? ? there are two main techniques for estimating @xmath4 from observed vector magnetograms , which so far only cover a small region of the solar surface such as a single active region : 1 . compute @xmath10 and hence @xmath11 , which should give @xmath4 exactly for a force - free field @xcite . 2 . compute a linear force - free extrapolation from @xmath12 and choose the overall value , @xmath13 , which best reproduces the observed @xmath14 , @xmath15 distribution over the region @xcite . the studies by @xcite and @xcite show that both techniques are generally consistent . the key result of these observations is a robust hemispheric rule whereby the average @xmath4 value is negative in the northern hemisphere and positive in the southern hemisphere , although there is significant scatter including a mixture of signs of @xmath4 within single active regions . this hemispheric pattern in @xmath4 has also been found by @xcite who reconstructed the radial and toroidal components of the global magnetic field under simplifying assumptions . a trans - equatorial sign change in helicity is supported by numerous proxy observations such as h@xmath4 images of active region structure @xcite , _ in situ _ heliospheric measurements @xcite , differential rotation @xcite , and filament / prominence magnetic fields @xcite . using newly - developed simulations of the global coronal evolution , we have recently been able to reproduce the filament hemispheric pattern including exceptions ( with 96% agreement ) , in a comparison with 109 observed filaments @xcite . in this letter we describe the distribution of current helicity in a 30-month simulation , which we hope to compare with new magnetic observations from the sdo ( nasa solar dynamics observatory ) mission . our simulations of the 3d coronal field evolution @xcite use the coupled flux transport and magnetofrictional model of @xcite , in a domain extending from @xmath16 to @xmath17 in longitude , @xmath18 to @xmath19 in latitude , and @xmath20 to @xmath21 in radius . the coronal magnetic field @xmath22 evolves _ via _ the non - ideal induction equation @xmath23 in response to flux emergence and advection by large - scale motions on the photospheric boundary . rather than solve the full mhd system we approximate the momentum equation by the magnetofrictional method @xcite , setting @xmath24 this artificial velocity ensures evolution through a sequence of near force - free states . the second term is a radial outflow imposed only near to the upper boundary , where it simulates the effect of the solar wind in opening up field lines in the radial direction @xcite . the diffusivity @xmath25 consists of a uniform background term and an enhancement in regions of strong current density @xmath26 ( see * ? ? ? the photospheric boundary conditions are described in @xcite ; the surface flux transport model includes newly emerging magnetic bipoles based on active regions observed in synoptic normal - component magnetograms from nso , kitt peak . the emerging bipoles take a simple mathematical form , with properties chosen to match the location , size , tilt , and magnetic flux of the observed regions . they are inserted in 3d with a non - zero twist ( magnetic helicity ) , chosen to match the observed sign of helicity in each hemisphere . the simulation illustrated in this letter models 30 months of continuous evolution during the rising phase of cycle 23 ( from 1997 april 9 to 1999 october 10 , rotations cr1921 to cr1954 ) . from an initial potential field extrapolation , the photospheric and coronal fields were evolved forward continuously for 914 days with 396 new bipoles inserted during this time . two example snapshots of the simulated magnetic field are shown in figure [ fig : field ] . to illustrate the sources of current helicity in our simulation within an individual active region , figure [ fig : single ] zooms in to a bipole in the northern hemisphere which emerged on day 136 ( as measured from the start of the simulation ) . there are three main sources of coronal currents and helicity in our model : 1 . the new bipoles emerge twisted . this twist is initially concentrated low down in the centre of the bipole , as seen from the field lines in figure [ fig : single](a ) which are skewed as they cross the bipole s central polarity inversion line ( pil ) . the sigmoidal concentration of negative @xmath4 at the centre of the bipole is clearly seen on day 140 in figure [ fig : single](b ) . when the bipoles emerge they displace older fields and produce currents at the interface between old and new flux systems ( see * ? ? ? * ) . in figure [ fig : single](b ) this is visible at the nw edge of the new bipole where it adjoins a pre - existing bipole , and a layer of positive @xmath4 has developed . note that this is opposite in sign to that from the twist of the new region , as seen in figure [ fig : single](a ) . this corresponds to field lines that are oppositely skewed at this edge of the new bipole , as compared to those across the central pil . this is just one example of how both signs of @xmath4 may naturally be produced within a single active region , as found in observations . 3 . over time , surface motions shear the coronal field generating further currents . this is visible in figure [ fig : single](c ) , which shows the distribution of @xmath4 for the same region on day 190 , after 50 days evolution . there is a significant build - up of negative @xmath4 , particularly at the north and south ends of the bipole where helicity was initially low . this build - up is caused by differential rotation and convergence ( due to supergranular diffusion ) . in addition to these sources of current helicity , it may also be locally reduced by diffusive cancellation and reconnection . also , helicity is periodically removed through the top boundary of the domain when excessive build - up of twist leads to localised temporary losses of equilibrium , and the ejection of twisted flux ropes @xcite . the global distribution of current helicity , @xmath4 , is shown in figure [ fig : global ] at days 10 , 100 , and 910 of the simulation . from the initial potential field on day 0 ( with @xmath27 everywhere ) , a pattern of intermixed positive and negative @xmath4 has developed by day 10 , simply due to photospheric shearing this is before the first active region emergence . after about 100 days , a clear latitudinal trend in @xmath4 emerges , although there is still significant local variation in both strength and sign . this pattern persists for the rest of the simulation , and up to medium heights in the 3d corona ( nearer the top of the computational box high values of @xmath4 become localized to closed field regions , with @xmath28 where the field is open ) . in figures [ fig : global](a ) , ( e ) , and ( f ) , it can be seen how the mean @xmath4 at low latitudes ( @xmath16 to about @xmath29 ) develops into the observed hemispheric trend , although with considerable scatter as observed on the real sun . however , at high latitudes the sign of @xmath4 is reversed . these polar reversals correspond to the east - west pils at the polar crown boundaries , and move steadily poleward through the simulation as the polar crowns reduce in size towards polar field reversal ( we are approaching solar maximum ) . this opposite sign of @xmath4 is caused by differential rotation of the predominantly north - south field lines at this latitude , and is a well - documented problem for theoretical models @xcite . at lower latitudes , as was illustrated by figure [ fig : single](c ) , differential rotation of north - south pils produces the observed hemispheric sign of helicity @xcite . figure [ fig : global ] shows mean values of @xmath4 at active latitudes of the order @xmath30 . the actual maximum and minimum values recorded on day 910 of the simulation were @xmath31 and @xmath32 . a key result of this study is that these values are much higher than those estimated from linear force - free extrapolations . such solutions suffer a constraint on the maximum @xmath4 in order to obtain a decay with height @xcite , requiring that @xmath33 ( the `` first resonant value '' ) , where @xmath34 is the horizontal length of the periodic box . the linear force - free model of an observed filament by @xcite has @xmath35 , and for the solutions of @xcite this first resonant value was at @xmath36 . by contrast , studies using nonlinear force - free extrapolations from vector magnetograms using the grad - rubin type method @xcite find locally higher values of @xmath4 ( e.g. , * ? ? ? * ) . they are also more realistic because they allow variable @xmath4 within a single region , as in our simulations . for a particular active region , @xcite found maximum values of the order @xmath37 , consistent with the results of our simulations . in this letter we have shown how our 3d simulations of the global coronal magnetic field evolution are able to model the development and transport of current helicity , @xmath4 , over many solar rotations . we find a clear latitudinal pattern of @xmath4 that persists throughout the simulation , although locally within single bipoles there is significant scatter and intermixing of both signs of @xmath4 , in agreement with observations . local values may be much higher than those predicted by linear force - free extrapolations . with existing measurements of @xmath4 limited to vector magnetograms of individual active regions , robust observations of the latitudinal distribution of @xmath4 await full - disk vector magnetograms . these will shortly be available from the nasa solar dynamics observatory ( sdo ) satellite . in particular the hmi ( helioseismic and magnetic imager ) instrument will provide synoptic full - disk vector magnetograms at 1 resolution and approximately @xmath38 cadence . this will offer an exciting opportunity to test and refine our theoretical model for the coronal magnetic field . in particular , consistent measurements over a large portion of the solar cycle will allow us to consider how the helicity distribution varies over both space and time . whether there is a systematic variation in the latitudinal trend of helicity over the solar cycle remains an unresolved issue @xcite , and has implications for the sub - surface origin of helicity @xcite . indeed @xcite showed that observations of @xmath4 in active regions provide important constraints on theories of the solar dynamo itself ( see also * ? ? ? ejection of helical fields from the corona , as included in our simulations , is also thought to play an important role in sustaining the solar cycle @xcite . a particular feature of our results is the sign reversal of current helicity at the high - latitude polar crowns . this would appear to be in conflict with observations of magnetic fields in polar crown filaments , which show no such reversal in their chirality pattern @xcite . we hope to address this outstanding issue in longer simulations covering a greater portion of the solar cycle . it is not at present clear whether longer - term poleward transport of the correct sign of helicity will be enough to counteract the effect of differential rotation on the north - south oriented field lines at these latitudes . observations of vector magnetic fields in the polar regions , such as those being made by _ hinode _ @xcite and soon the sdo mission , should help to constrain our models . financial support for ary and dhm was provided by the uk stfc . dhm and aavb would also like to thank the issi in bern for support . the simulations were performed on the ukmhd parallel computer in st andrews , funded jointly by srif / stfc . synoptic magnetogram data from nso / kitt peak was produced cooperatively by nsf / noao , nasa / gsfc , and noaa / sel and made publicly accessible on the world wide web .

current helicity quantifies the location of twisted and sheared non - potential structures in a magnetic field . we simulate the evolution of magnetic fields in the solar atmosphere in response to flux emergence and shearing by photospheric motions . in our global - scale simulation over many solar rotations the latitudinal distribution of current helicity develops a clear statistical pattern , matching the observed hemispheric sign at active latitudes . in agreement with observations there is significant scatter and intermixing of both signs of helicity , where we find local values of current helicity density that are much higher than those predicted by linear force - free extrapolations . forthcoming full - disk vector magnetograms from solar dynamics observatory will provide an ideal opportunity to test our theoretical results on the evolution and distribution of current helicity , both globally and in single active regions .

surrogate tests @xcite are examples of monte carlo hypothesis tests @xcite . taking the surrogate test of nonlinearity in a time series @xcite as an example , we first need to adopt a null hypothesis , which usually supposes the time series is generated by a linear stochastic process and potentially filtered by a nonlinear filter @xcite . based on this null hypothesis , a large number of data sets ( surrogates ) are to be produced from the original time series , which keeps the linearity of the original time series but destroys all other structures . we then calculate some nonlinear statistics ( discriminating statistics ) , for example , correlation dimension , of both the original time series and the surrogates . if the discriminating statistic of the original time series deviates from those of the surrogates , we can reject the null hypothesis we proposed and claim that the original time series is deterministic with certain confidence level ( depending on how many surrogates we have generated , to be shown later ) . in general , to apply the surrogate technique to test if a time series possesses the property @xmath0 we are interested , we first select a null hypothesis , which assumes the time series instead has a property @xmath1 opposite to @xmath0 . we then devise a corresponding algorithm to produce surrogates from the observed data set . in principle , these surrogates shall preserve the potential property @xmath1 while destroying all others . the next step is to choose a suitable discriminating statistic , which shall be an invariant measure for both the surrogates and the original time series if the null hypothesis is true . hence if the discriminating statistic of the original time series distinctly deviates from the distribution of the discriminating statistic of the surrogates , the null hypothesis is unlikely to be true , or in other words , the time series is much more likely to possess the property @xmath0 than @xmath1 . in this way , we can assess the statistical significance of our calculations through surrogate test technique even when we have only a very limited amount of observations . such assessments are important because in many practical situations statistical fluctuations are inevitable due to the presence of noise , hence the surrogate test is a proper tool to evaluate the reliability of our results in a statistical sense . in this communication , we are focused on discussing the algorithm to generate surrogates for pseudoperiodic time series . by pseudoperiodic time series we mean a representative of a periodic orbit perturbed by dynamical noise , or contaminated by observational noise , or with the combination of the both noises , whose states within one cycle are largely independent of those within previous cycles given a cycle length . note that , in our discussions we will always assume we have detected that the time series are produced from nonlinear deterministic systems , but they are also possibly contaminated by some noises . as we know , if an irregular time series comes from a nonlinear deterministic system , it shall be either chaotic or pseudoperiodic in most cases . in some situations , it might be important for us to discriminate between pseudoperiodicity and chaos . however , chaotic and pseudoperiodic time series often look similar , we might not be able to distinguish them from each other only through visual inspections , quantitative techniques are needed instead at this time . one choice is to apply the direct test techniques . for instance , we can calculate some characteristic statistics of the time series , such as the lyapunov exponent and the correlation dimension . however , a direct test usually will not give out the confidence level . if we find the lyapunov exponent of a time series is , for example , @xmath2 , it may be difficult for us to tell whether the time series is chaotic or the time series is pseudoperiodic , but the presence of noise causes the lyapunov exponent to be slightly larger than zero . as an alternative choice , we suggest one utilizes the surrogate test rather than the direct test , which can provide us the confidence level by calculating a large number of surrogates . through the surrogate tests , if we could exclude the possibility that the time series is pseudoperiodic , then the time series is more likely to be chaotic . this is the essential idea to apply our algorithm to distinguish chaos from pseudoperiodicity , as to be shown in section iii . first let us briefly review some of the algorithms to generate surrogates for pseudoperiodic time series . initially , to generate surrogates for pseudoperiodic time series , theiler @xcite proposed the cycle shuffling algorithm . the idea is to divide the whole data set into some segments and let each segment contain exactly an integer number of cycles . the surrogates are obtained by randomly shuffling these segments , which will preserve the intracycle dynamics but destroy the intercycle ones by randomizing the temporal sequence of the individual cycles . the difficulty in applying this algorithm is that it requires preknowledge of the precise periodicity , otherwise shuffling the individual cycles might lead to spurious results @xcite . recently , with the development of the cyclic theory of chaos @xcite , many authors have shown interest in searching unstable periodic orbits ( upos ) in noisy data sets from chaotic dynamical systems . the algorithms proposed in @xcite essentially deal with the unstable fixed points of the upos . but as observed , the presence of noise will reduce the statistical significance of these algorithms . one remedy is to introduce the surrogate test for reliability assessments , e.g. , dolan _ et.al _ @xcite claimed that the randomly shuffling surrogate algorithm @xcite together with the simple recurrence method @xcite correctly tests the appropriate null hypothesis . essentially , this approach is very similar to the cycle shuffling algorithm described previously . the simple recurrence algorithm is equivalent to applying a poincar map on the continuous dynamical systems and then studying only the data points falling on the cross - section plane , hence one does not need to consider the intracycle dynamics and no knowledge of the periodicity is required , while randomly shuffling these data points exactly aims to randomize the temporal sequence of the cycles . however , one potential problem of this algorithm is that it might generate spuriously high statistical significance due to the correlation between the cycles @xcite . later , small _ et.al _ @xcite _ _ _ _ proposed the pseudoperiodic surrogate ( pps ) algorithm from another viewpoint . they first apply the time delay embedding reconstruction @xcite to the original data set , then utilize a method based on local linear modelling techniques to produce surrogate data which approximate the behavior of the underlying dynamical system . as the authors pointed out , this algorithm works well even with very large dynamical noise , but it may incorrectly reject the null hypothesis if the intercycles of the pseudoperiodic orbit have a linear stochastic dependence induced by colored additive observational noise @xcite . in this communication we propose a new surrogate algorithm for continuous dynamical systems , which properly copes with linear stochastic dependence between the cycles of the pseudoperiodic orbits . the null hypothesis to be tested is that the stationary data set is pseudoperiodic with noise components which are ( approximately ) identically distributed and uncorrelated for sufficiently large temporal translations . note the constraints of the noise components in our null hypothesis are stronger than that of theiler s algorithm , which requires the noise distribution only periodically depends on the phase of the signal . however , under our hypothesis , we can produce the surrogates in a simple way through the algorithm to be described below . in addition , a large scope of noise processes often encountered in practical situations , including ( but not limited to ) linear colored additive observational noise described by the @xmath3 model @xcite , match the above constraints . the remainder of this communication is organized as follows . in sec . ii we will introduce the new algorithm to generate pseudoperiodic surrogates , while in sec . iii we will apply this algorithm to simulation data sets from the rssler system , which demonstrates the ability of the surrogate test based on this algorithm to distinguish chaotic orbits from pseudoperiodic ones . as one of the applications , we will use this surrogate technique to investigate whether a human electrocardiogram ( ecg ) record is possibly presentative of a chaotic dynamical system . finally , in sec . iv , we will have a summary of the whole communication . let @xmath4 be a data set with @xmath5 observations ( the form @xmath6 is adopted instead for convenience when causing no confusion ) , where @xmath7 means the observation measured at time @xmath8 with @xmath9 denoting the sampling time . we assume @xmath4 is stationary and can be decomposed into the deterministic components and the noise components , which are approximately independent of each other . similar to the surrogate test idea of time shifting to desynchronize two data sets @xcite , we assume the noise components ( approximately ) follow an identical distribution and are uncorrelated for sufficiently large temporal translations ( or time shifts ) . according to the null hypothesis we proposed in the previous section , if the deterministic components are periodic , then we can write a data point @xmath7 as @xmath10 , where @xmath11 and @xmath12 denote the periodic component and the noise component respectively . in many cases , we can set @xmath13 where @xmath14 is the expectation operator . since @xmath15 are roughly independent of @xmath16 , we have the autocovariance @xmath17 . let @xmath18with @xmath19 , where coefficients @xmath20 and @xmath21 satisfy @xmath22 and parameter @xmath23 is the temporal translation between subsets @xmath24 and @xmath25 , then the autocovariance function @xmath26 . now let us consider the noise components . if @xmath27 is sufficiently large , under our hypothesis , @xmath28 and @xmath29 are uncorrelated . we also note that @xmath30 and @xmath31 are drawn from ( approximately ) the same distribution , we have @xmath32 . for the deterministic component , if we require the translation @xmath23 to satisfy @xmath33 , then @xmath34 . hence by choosing a suitable temporal translation , the noise levels of @xmath35 , defined by @xmath36 , will be the same as that of @xmath4 , i.e. , @xmath37 . the reason to preserve the noise level is that , the presence of noise will affect the calculation of the correlation dimension , hence we would like to let the surrogates and the original time series ( roughly ) have the same noise level in order to make the results more conceivable . the above deduction leads to the central idea of our surrogate algorithm . from eq . ( [ linear combination ] ) , we note that if @xmath15 is periodic , the nonconstant deterministic components @xmath38 shall also be periodic . in addition , @xmath39 and @xmath35 shall have the same noise level if a suitable translation @xmath23 is selected . therefore by randomizing the coefficient @xmath20 or @xmath21 , we can generate many data sets @xmath35 as the surrogates of @xmath4 . note that @xmath15 and @xmath40 have the same degree - of - freedom , if both of them are periodic , their correlation dimensions @xcite will theoretically be the same . now let us consider the noise components . although the noise components @xmath41 may have a different distribution from that of @xmath30 , the noise level is preserved after the transform in eq . ( [ linear combination ] ) . as diks @xcite has reported , the gaussian kernel algorithm ( gka ) can reasonably estimate the correlation dimensions of noisy data sets with different noise distributions . this implies that , under the same noise level , the correlation dimensions of @xmath4 and @xmath42 , calculated by the gka , shall statistically be the same if @xmath4 and @xmath43 are both pseudoperiodic ( and satisfy the constraints we imposed ) . in contrast , if @xmath15 is chaotic , its linear combination , @xmath40 , may have a new dynamical structure with a different correlation dimension from that of @xmath44 , hence by adopting the correlation dimension as the discriminating statistic we might detect this difference . we shall also note that , for an unstable periodic orbit , even a small dynamical noise might drive the resultant orbit far away from the original position after a sufficiently long time , and the pseudoperiodicity might be broken . in such situations , our algorithm might fail to work . nevertheless , we suggest to apply our algorithm as the first step in pseudoperiodicity test , which is computationally fast and in principle deals well with a large scope of observational noise ( comparatively , the pps algorithm will sometimes fail for colored observational noise ) . if we can reject the null hypothesis proposed previously , the time series in test is possibly chaotic or pseudoperiodic perturbed by dynamical noise . then we can adopt the pps algorithm for further tests , which works well even under a large amount of dynamical noise . if the corresponding null hypothesis , i.e. , the time series is pseudoperiodic perturbed by dynamical noise , can be rejected again , then we may claim the time series is very likely to be chaotic . we now consider several computational issues in our algorithms . as described in eq . ( [ linear combination ] ) , to generate the surrogates @xmath45 , we select two subsets of @xmath46 , @xmath24 and @xmath47 , multiply them by the coefficients @xmath20 and @xmath21 respectively and then add them together . we shall emphasize that choosing the temporal translation @xmath23 is a crucial issue for our algorithm . from one aspect , we require the translation @xmath23 to satisfy the condition @xmath48 . the reason is that we want to keep the noise level for the original time series and the surrogates . in addition , we want the deterministic components @xmath49 to be orthogonal to @xmath50 for arbitrary coefficients @xmath20 and @xmath21 , otherwise the projection of @xmath51 onto @xmath50 might counteract @xmath50 under some situations , for example , if @xmath52 and @xmath53 , the deterministic components @xmath40 will almost vanish while the noise components @xmath54 remain . hence the correlation dimensions calculated are actually those of the noise components instead of the deterministic components , which will certainly cause the false rejection of the null hypothesis . from another aspect , we require @xmath23 to be sufficiently large to guarantee the decorrelation between the noise components . however , we expect @xmath24 and @xmath47 shall have at least some overlaps to make use of the information of the whole data set @xmath4 , which means @xmath23 shall not exceed @xmath55 . in addition , it is recommended the length of a data set shall not be too short in order to appropriately calculate its correlation dimension @xcite , which also implies @xmath23 shall not be too large . from eq . ( [ linear combination ] ) we see that the surrogates are generated from two segments @xmath56 and @xmath57 of the original time series @xmath58 . we want segments @xmath56 and @xmath59 to equivalently affect the generation of the surrogates , therefore we would like to let @xmath60 , @xmath61 and @xmath62 , where @xmath63 , @xmath64 and @xmath65 denote the maximal function , the minimal function and the probability function respectively . but note that the coefficient ratio @xmath66 ( or @xmath67 ) shall not be too large or too small , otherwise @xmath35 will be very close to @xmath68 or @xmath69 , which will lead to approximately the same correlation dimensions of @xmath4 and @xmath43 regardless of the dynamical behavior of @xmath46 , and thus decrease the discriminating power of the correlation dimension . in our calculations we let @xmath20 be uniformly drawn from the interval @xmath70 \cup \left [ 0.6,0.8\right ] $ ] and @xmath71 , which satisfies our requirements and provides moderate values for the ratio @xmath72 . in this section , through four examples from the rssler system , we demonstrate the ability of surrogate test based on our algorithm to discriminate chaotic orbits from pseudoperiodic ones . as an application , we will also employ the surrogate technique to investigate whether a recorded human electrocardiogram ( ecg ) data set is possibly chaotic . the equations of the rssler system are given by @xmath73with the initial conditions @xmath74 . we choose parameters @xmath75 , @xmath76 and the sampling time @xmath9 @xmath77 time units . for each example , the system is to be integrated @xmath78 times and the first @xmath79 data points are discarded to avoid including transient states . in the first example , we set parameter @xmath80 . the rssler system exhibits limit cycle behavior of period 6 . to obtain pseudoperiodic time series , we introduce @xmath81 observational noise into the periodic time series . although gaussian white observational noise is the most common choice in this situation , in order to demonstrate the ability of our surrogate algorithm to deal with colored noise , we will instead adopt the noise generated from the @xmath82 process @xcite @xmath83 with the variable @xmath84 following the normal gaussian distribution @xmath85 , which is the more difficult case due to the correlation between noise components . however , one shall note that , gaussian white noise and other colored noises satisfying the constraints in our null hypothesis , for example , those modelled by the @xmath86 processes , in principle can be dealt with in the same way . for convenience of observation and comparison , we plot the time series and the corresponding attractor in two dimensional state space ( or embedding space ) in panels @xmath87 and @xmath88 of fig . [ rosslerp5perobvdim4 ] respectively . to produce surrogate data , first we shall choose a suitable temporal translation . since it is impractical to separate noise from signal completely , in general it is difficult to estimate the correlation decay time between noise components . fortunately , to decorrelate noise components , all temporal translations are equivalent as long as they are large enough . in addition , in many real situations , it is often possible to observe the background noise and thus estimate the decay time . in our example , we think the @xmath82 noise to be uncorrelated when the temporal translation is larger than @xmath89 ( in units of the sampling time @xmath9 ) . as another requirement , temporal translation satisfying @xmath48 is desired . in practice , of course , this requirement is generally impractical due to the digitization and quantization in sampling process . recall the discussion in the previous section , by letting @xmath90 and @xmath91 , we have @xmath92 . function @xmath93 means we do not preserve the noise level . however , under the null hypothesis of pseudoperiodicity , there shall always be some temporal translations to make @xmath94 , hence the noise level will not deviate from the original one too much . besides , according to eq . ( [ linear combination ] ) , we generate the surrogates by uniformly drawing coefficient @xmath20 from interval @xmath95 \cup \left [ 0.6,0.8\right ] $ ] ( @xmath96 is always kept positive ) , the noise level of the surrogates will fluctuate around that of the original one due to the alternative signs of product @xmath97 . therefore , @xmath98 will only cause some fluctuations when to calculate the correlation dimension because of the fluctuations of noise level , however , generally such fluctuations will not affect our conclusion if we can select a temporal translation @xmath23 to let @xmath94 . since we have assumed the noise components are roughly independent of the deterministic components , then @xmath99 for a large enough temporal translation ( to decorrelate noise components ) , therefore in all of the examples , in order to let @xmath94 , we can equivalently require @xmath100 . in the first example , there are many temporal translations satisfying the two constraints discussed above , i.e. , @xmath101 and @xmath100 . to pick a value from all these candidates , we first select an interval @xmath102 $ ] , then search the temporal translation which makes the absolute value @xmath103 be the minimum ( most close to zero ) among all translations @xmath104 . one shall note that the choice of the interval @xmath102 $ ] is arbitrary , except that we have to make sure that the lower bound of the interval is larger than @xmath89 , and there exists temporal translations to let @xmath100 within the interval . after selecting the temporal translation , by randomizing the coefficient @xmath105 we will generate @xmath106 surrogates according to eq . ( [ linear combination ] ) . in order to calculate the correlation dimension , we adopt the time delay embedding reconstruction @xcite to recover the underlying system from the scalar time series . two parameters , i.e. , embedding dimension and time delay , shall be properly chosen to apply this technique . throughout this communication , we will use the false nearest neighbour criterion @xcite to determine the global optimal embedding dimension . using the program in tisean package @xcite , the embedding dimension @xmath107 of the original time series is selected at @xmath108 , which is the minimal value to make the fraction of false nearest neighbours be zero . to choose a suitable time delay , we will use the algorithm of redundancy and irrelevance tradeoff exponent ( rite ) proposed in @xcite . this algorithm selects the time delay by searching the optimal tradeoff between redundancy ( due to too small time delay ) and irrelevance ( due to too large time delay ) . as demonstrated , the rite algorithm can select equivalently suitable time delays compared to the average mutual information ( ami ) criterion @xcite , however , its implementation is much simpler and the computational cost is fairly low . therefore in case of large data sets , adopting the rite algorithm can facilitate our calculations . in the first example we generate @xmath106 surrogates , and for each surrogate we keep the embedding dimension @xmath109 and use the rite algorithm to choose the suitable time delay for time delay reconstruction . we will follow diks s method @xcite to calculate the correlation dimension , which is more robust against noise by extending the hard kernel function ( or the heaviside function ) @xcite in calculation of correlation integral to the general kernel functions . in his discussions , diks adopted the gaussian kernel function , hence this method is called gaussian kernel algorithm ( gka ) . here we will use the gka implemented in @xcite to calculate the correlation dimensions , which further enhances the computational speed . note that to speed up the calculation , only 2000 data points are used as the reference points for the gka . there are some statistical fluctuations even for the same data set when calculating its correlation dimension , therefore for the original time series , we will calculate @xmath106 times to estimate the mean correlation dimension and the standard deviation . as shown in panel @xmath110 of fig . [ rosslerp5perobvdim4 ] , there are three lines parallel to the abscissa . the middle line denote the estimation of the mean correlation dimension of the original time series , while the upper and lower lines indicate the positions twice the standard deviation away from the mean value . for the surrogates , however , we will calculate their correlation dimensions only once to save time . the results are illustrated as the asterisks in panel @xmath111 of fig . [ rosslerp5perobvdim4 ] . after the calculation of the correlation dimensions , we need to inspect whether the result is consistent with our null hypothesis . here we use the ranking criterion @xcite to determine whether the null hypothesis shall be rejected or not . the idea of this criterion is that , suppose the discriminating statistic of the original data set is @xmath112 , and those of @xmath113 surrogates are @xmath114 . rank the statistics @xmath115 in the increasing order and denote the rank of @xmath112 by @xmath116 , if the data set is consistent with the hypothesis ( i.e. , no evidence to reject ) , @xmath116 can have an equal possibility be any integer value between @xmath117 and @xmath118 . however , if the hypothesis is false , @xmath112 might deviate from the surrogate distribution @xmath119 , i.e , @xmath112 will be the smallest or largest amongst @xmath120 , hence we can reject the null hypothesis if @xmath121 or @xmath118 , the probability of a false rejection is @xmath122 for one - sided tests and @xmath123 for two - sided tests . for the first example , from panel @xmath111 of fig . rosslerp5perobvdim4 we can see that , the mean correlation dimension of the original time series falls within the dimension distribution of the surrogates , therefore we can not reject the null hypothesis as we expect , which means the original time series is possibly pseudoperiodic @xcite . now let us examine the other examples . in the second example , we still set parameter @xmath80 in eq . ( [ rossler ] ) . however , to obtain the pseudoperiodic time series , we first generate a data set by adding gaussian white noise with the standard deviation of @xmath124 to the @xmath125 component at each integration step , which simulates the system perturbed by additive dynamical noise , and then introduce @xmath81 observational @xmath82 noise into the previously obtained data set . the global optimal embedding dimension is chosen at @xmath109 . note in all of the four examples , we will generate @xmath106 surrogates , and parameter choices for surrogate generation will be the same , i.e. , we let the temporal translation be selected from @xmath126 $ ] and coefficient @xmath20 be uniformly drawn from @xmath70 \cup \left [ 0.6,0.8\right ] $ ] ( @xmath96 ) . for the second example , the correlation dimensions of the original time series and the surrogates are shown in panel @xmath111 of fig . rosslerp5perobv015ddim4 . under the ranking criterion , once again we can not reject our null hypothesis . therefore the time series is possibly pseudoperiodic , which is consistent with our knowledge . in the third example , we change parameter @xmath127 of eq . ( [ rossler ] ) to be @xmath128 . the rssler system exhibits chaotic behavior . we integrate eq . ( [ rossler ] ) to obtain a time series and then introduce @xmath81 observational @xmath82 noise . the optimal embedding dimension @xmath107 is selected at @xmath129 . from panel @xmath111 of fig . [ rosslech5perobvdim5_pps ] , we find that the mean correlation dimension of the original time series deviates from the distribution of the surrogate dimensions . using the ranking criterion , we can reject our null hypothesis . in order to exclude the possibility that the time series is generated from an unstable period orbit perturbed by dynamical noise , we also apply the pps algorithm for further test . from the pps algorithm we generate @xmath106 surrogates , and then use the gka to calculate their correlation dimensions . the results are shown in panel @xmath130 of fig . [ rosslech5perobvdim5_pps ] , as we can see , the mean correlation dimension of the original time series also falls outside the distribution of the surrogate dimensions , therefore we can reject the null hypothesis again . after the two surrogate tests for pseudoperiodicity , we can claim the time series is chaotic with a confidence level up to @xmath131 ( @xmath132 ) for two - sided test . the final example to be demonstrated is a chaotic time series also from the rssler system . to generate the time series , we keep parameter @xmath133 . similar to the way in the second example , we add gaussian white noise with the standard deviation of @xmath124 to the @xmath125 component at each integration step as the dynamical noise , and then introduce @xmath81 observational @xmath82 noise into the previously obtained data set . the global optimal embedding dimension is found to be @xmath109 . the results of surrogate tests based on the new algorithm and the pps algorithm are shown in panel @xmath134 and @xmath135 of fig . [ rosslech5perobv015ddim4_pps ] respectively , from which we can see that , surrogate tests based on both algorithms can detect the chaos in the time series . again we can claim the time series is chaotic with a confidence level up to @xmath131 for two - sided test . we have also investigated examples under different observational noise levels ( but keep the same dynamical noise if they have ) . for example , if we reduce the @xmath82 observational noise levels to @xmath136 in the above four examples , we can obtain the same results as we have reported . if we increase the observational noise levels to @xmath137 , for the pseudoperiodic time series we can still obtain the expected results , i.e. , we can not reject our null hypothesis . however , for the chaotic time series , we will falsely accept our null hypothesis due to the correlation dimension of the original time series marginally falling within the dimension distribution of the surrogates . the reason of false acceptance might be that , under large noise levels , the correlation dimension is not sensitive enough to detect the structure changes of the chaotic time series . for such cases , we will have to look for more powerful discriminating statistics @xcite . as an example of application , we employ the surrogate test based on our algorithm to investigate whether a human electrocardiogram ( ecg ) record ( with @xmath138 data points ) is likely to be chaotic . the ecg record was obtained by measuring from a resting healthy subject ( @xmath139 years old ) in a shielded room at the sampling rate of @xmath117 khz . the ecg record indicated in panel @xmath140 of fig . [ ecgtestem5 ] looks very regular and even possibly periodic , but we need a quantitative approach to verify the periodicity . here we choose the surrogate test technique . using the false nearest neighbour criterion , the global optimal embedding dimension is chosen at @xmath141 . the background noise is mainly from the measurement instruments , usually it is a blend of white and correlated noise . by observing the linear second order correlation function of the ecg data , we let the temporal translation be within the interval @xmath126 $ ] ( large enough to decorrelate the noise components ) , where there exists an integer temporal translation to make the correlation function almost be zero . then by uniformly drawing values from @xmath70 \cup \left [ 0.6,0.8% \right ] $ ] for coefficient @xmath20 in eq . ( [ linear combination ] ) ( @xmath142 ) , we generate @xmath106 surrogates . for demonstration , we plot in panel @xmath111 one surrogate generated from eq . ( [ linear combination ] ) with coefficient @xmath143 . we can see that the surrogate is different from the original ecg data in that there appear more spikes in the surrogate . however , as we can also find , the surrogate indicates the similar regularity to that in the original data , which implies that the surrogate preserves the potential periodicity in the original data as we expect ( although in a different pattern ) . with regards to the surrogate test , our calculation of the correlation dimensions is also based on the gka . the results are indicated in panel @xmath144 of fig . [ ecgtestem5 ] , from which we can see that the mean correlation dimension of the ecg data falls within the distribution of the correlation dimensions of the surrogates , therefore we can not reject our null hypothesis . hence the ecg record is possibly periodic . moreover , there is no evidence of chaos . to summarize , by imposing a few constraints on the noise process , we devise a simple but effective way to produce surrogates for pseudoperiodic orbits . the main idea of this algorithm is that a linear combination of any two segments of the same periodic orbit will generate another periodic orbit . by properly choosing the temporal translation between the two segments , under the same noise level we can obtain statistically the same correlation dimensions of the pseudoperiodic orbit and its surrogates . choosing the temporal translation is a crucial issue for our algorithm , which in principle shall guarantee the decorrelation between the noise components and preserve the noise level . another important issue is to select a proper discriminating statistic which helps determine whether to reject the null hypothesis . the correlation dimension is a suitable discriminating statistic in this case . it is possible there are other suitable discriminating statistics , we will leave the problem of finding such statistics for future study . the surrogate test technique based on our algorithm can be utilized to distinguish between chaotic and pseudoperiodic time series . initially , the pps algorithm was proposed to generate surrogates for a pseudoperiodic orbit driven by dynamical noise , but sometimes surrogate tests based on this algorithm will falsely reject the null hypothesis if the time series is also contaminated by colored observational noise . as a complement to the pps algorithm , our algorithm deals well with observational noise , but it might fail for large dynamical noise . however , due to the convenience in computation , we suggest to adopt surrogate test based on our algorithm as the first step for pseudoperiodicity detection . if we can reject the null hypothesis of our algorithm , then we shall use the pps algorithm for further tests . if we can reject the null hypotheses of both the algorithms , then the time series under investigation is very likely to be chaotic . in this communication , the concrete procedures of surrogate test for pseudoperiodicity are demonstrated through four simulation examples . as an application in practice , we also employ the surrogate technique based on our algorithm to investigate whether a human ecg record is possible to be chaotic . here we would like to elucidate that , the irregularity of a time series is usually brought by stochasticity or nonlinearity ( often chaos ) , therefore in the example of nonlinearity detection , if we can exclude ( most of ) the probability that the time series is generated by a stochastic process , then it is very likely that the time series is generated from a nonlinear deterministic system . d. pierson and f. moss , phys . 75 , 2124 ( 1995 ) ; x. pei and f. moss , nature ( lodon ) 379 , 618 ( 1996 ) ; p. so , e. ott , s.j . schiff , d.t . kaplan , t. sauer , and c. grebogi , phys . 76 , 4705 ( 1996 ) ; p. schmelcher and f.k . diakonos , phys . 78 , 4733 ( 1997 ) ; k. dolan , a. witt , m.l . spano , a. neiman , and f. moss , phys . e 59 , 5235 ( 1999 ) . we can not claim the time series is pseudoperiodic definitely , because if we can not reject a null hypothesis , there could be many interpretations other than that the data in test is consistent with our null hypothsis . for more details , see , for example , ref . @xcite .

in this communication a new algorithm is proposed to produce surrogates for pseudoperiodic time series . by imposing a few constraints on the noise components of pseudoperiodic data sets , we devise an effective method to generate surrogates . unlike other algorithms , this method properly copes with pseudoperiodic orbits contaminated with linear colored observational noise . we will demonstrate the ability of this algorithm to distinguish chaotic orbits from pseudoperiodic orbits through simulation data sets from the rssler system . as an example of application of this algorithm , we will also employ it to investigate a human electrocardiogram ( ecg ) record .

most modern speaker recognition systems are based on human - crafted acoustic features , for example the mel frequency cepstral coefficients ( mfcc ) . a problem of the mfcc ( and other primary features ) is that it involves plethora information besides the speaker identity , such as phone content , channels , noises , etc . these heterogeneous and noisy information convolve together , making it difficult to be used for either speech recognition or speaker recognition . all modern speaker recognition systems rely on a statical model to ` purify ' the desired speaker information . for example , in the famous gaussian mixture model - universal background model ( gmm - ubm ) framework @xcite , the acoustic space is divided into subspaces in the form of gaussian components , and each subspace roughly represents a phone . by formulating the speaker recognition task as subtasks on the phone subspaces , the gmm - ubm model can largely eliminate the impact of phone content and other acoustic factors . this idea is shared by many advanced techniques derived from gmm - ubm , including the joint factor analysis ( jfa ) @xcite and the i - vector model @xcite . in spite of the great success , the gmm - ubm approach and the related methods are still limited by the lack of discriminative capability of the acoustic features . some researchers proposed solutions based on discriminative models . for example , the svm approach for gmm - ubms @xcite and the plda approach for i - vectors @xcite . all these discriminative methods achieved remarkable success . another direction is to look for more task - oriented features , i.e. , features that are more discriminative for speaker recognition @xcite . although it seems to be straightforward , this ` feature engineering ' turns out to be highly difficult . a major reason , in our mind , is that most of the proposed delicate features are human - crafted and therefore tend to be fragile in practical usage . recent research on deep learning offers a new idea of ` feature learning ' . it has been shown that with a deep neural network ( dnn ) , task - oriented features can be learned layer by layer from very raw input . for example in automatic speech recognition ( asr ) , phone - discriminative features can be learned from spectra or filter bank energies ( fbanks ) . this learned features are very powerful and have defeated the mfcc that has dominated in asr for several decades @xcite . this capability of dnns in learning task - oriented features can be utilized to learn speaker - discriminative features as well . a recent study shows that this is possible at least on text - dependent tasks @xcite . the authors reported that reasonable performance can be achieved with the dnn - learned feature , and additional performance gains can be obtained by combining the dnn - based approach and the i - vector approach . although the dnn - based feature learning shows great potential , the existing implementation still can not compete with the i - vector baseline . there are at least two drawbacks with the current implementation : first , the dnn model does not use any information about the phone content , which leads to difficulty when inferring speaker - discriminative features ; second , the evaluation ( speaker scoring ) is based on speaker vectors ( so called ` d - vectors ' ) , which are derived by averaging the frame - wise dnn features . this simple average ignores the temporal constraint that is highly important for text - dependent tasks . note that for tasks with a fixed test phrase , the two drawbacks are closely linked to each other . this paper follows the work in @xcite and provides two enhancements for the dnn - based feature learning : first , phone posteriors are involved in the dnn input so that speaker - discriminative features can be learned easier by alleviating the impact of phone variation ; second , two scoring methods that consider the temporal constraint are proposed : segmentation pooling and dynamic time warping ( dtw ) @xcite . the rest of the paper is organized as follows . section [ sec : rel ] describes some related work , and section [ sec : theory ] presents the dnn - based feature learning . the new methods are proposed in section [ sec : improve ] and the experiments are presented in section [ sec : exp ] . finally , section [ sec : conl ] concludes this paper and discusses some future work . this paper follows the work in @xcite and provides several extensions . particularly , the speaker identity in @xcite is represented by a d - vector derived by average pooling , which is quite neat and efficient , but loses much information of the test signal , such as the distributional property and the temporal constraint . one of the main contribution of this paper is to investigate how to utilize the temporal constraint in the dnn - based approach . the dnn model has been studied in speaker recognition in several ways . for example , in @xcite , dnns trained for asr were used to replace the ubm model to derive the acoustic statistics for i - vector models . in @xcite , a dnn was used to replace plda to improve discriminative capability of i - vectors . all these methods rely on the generative framework , i.e. , the i - vector model . the dnn - based feature learning presented in this paper is purely discriminative , without any generative model involved . it is well - known that dnns can learn task - oriented features from raw input layer by layer . this property has been employed in asr where phone - discriminative features are learned from very low - level features such as fbanks or even spectra @xcite . it has been shown that with a well - trained dnn , variations irrelevant to the learning task can be gradually eliminated when the feature propagates through the dnn structure layer by layer . this feature learning is so powerful that in asr , the primary fbank feature has defeated the mfcc feature that was carefully designed by people and dominated in asr for several decades . this property can be also employed to learn speaker - discriminative features . actually researchers have put much effort in searching for features that are more discriminative for speakers @xcite , but the effort is mostly vain and the mfcc is still the most popular choice . the success of dnns in asr suggests a new direction , that speaker - discriminative features can be learned from data instead of being crafted by hand . the learning can be easily done and the process is rather similar as in asr , with the only difference that in speaker recognition , the learning goal is to discriminate different speakers . [ fig : dnn ] presents the dnn structure used in this work for speaker - discriminative feature learning . following the convention of asr , the input layer involves a window of 20-dimensional fbanks . the window size is set to @xmath0 , which was found to be optimal in our work . the dnn structure involves @xmath1 hidden layers , and each consists of @xmath2 units . the units of the output layer correspond to the speakers in the training data , and the number is @xmath3 in our experiment . the 1-hot encoding scheme is used to label the target , and the training criterion is set to cross entropy . the learning rate is set to @xmath4 at the beginning , and is halved whenever no improvement on a cross - validation ( cv ) set is found . the training process stops when the learning rate is too small and the improvement on the cv set is too marginal . once the dnn has been trained successfully , the speaker - discriminative features can be read from the last hidden layer . in the test phase , the features are extracted for all frames of the given utterance . to derive utterance - based representations , an average pooling approach was used in @xcite , where the frame - level features are averaged and the resultant vector is used to represent the speaker . this vector is called ` d - vector ' in @xcite , and we adopt this name in this work . the same methods used for i - vectors can be used for d - vectors to conduct the test , for example by computing the cosine distance or the plda score . the two kinds of speaker vectors , the d - vector and the i - vector , are fundamentally different . i - vectors are based on a linear gaussian model , for which the learning is unsupervised and the learning criterion is maximum likelihood on acoustic features ; in contrast , d - vectors are based on neural networks , for which the learning is supervised , and the learning criterion is maximum discrimination for speakers . this difference leads to several advantages with d - vectors : first , it is a ` discriminative ' vector , which represents speakers by removing speaker - irrelevant variance , and so sensitive to speakers and invariant to other disturbance ; second , it is a ` local ' speaker description that uses only local context , so can be inferred from very short utterances ; third , it relies on ` universal ' data to learn the dnn model , which makes it possible to learn from large amounts of data that are task - independent . there are several limitations in the implementation of the feature learning paradigm presented in the previous section . first , it does not involve any prior knowledge in model training , for example phone identities . second , the simple average pooling does not consider the temporal information which is particularly important for text - dependent recognition tasks . several approaches are proposed in this section to address these problems . a potential problem of the dnn - based feature learning described in the previous section is that it is a ` blind learning ' , i.e. , the features are learned from raw data without any prior knowledge . this means that the learning purely relies on the complex deep structure of the dnn model and a large amount of data to discover speaker - discriminative patterns . if the training data is abundant , this is often not a problem ; however in tasks with a limited amount of data , for instance the text - dependent task in our hand , this blind learning tends to be difficult because there are too many speaker - irrelevant variations involved in the raw data , particularly phone contents . a possible solution is to supply the dnn model extra information about which phone is spoken at each frame . this can be simply achieved by adding a phone indicator in the dnn input . however , it is often not easy to get the phone alignment in practice . an alternative way is to supply a vector of phone posterior probabilities for each frame , which is a ` soft ' phone alignment and can be easily obtained from a phone - discriminative model . in this work , we choose to use a dnn model that was trained for asr to produce the phone posteriors . [ fig : dnn ] illustrates how the phone posteriors are involved in the dnn structure . the training process does not change for the new structure . text - dependent speaker recognition is essentially a sequential pattern matching problem , but the current d - vector approach derives speaker identities as single vectors by average pooling , and then formulates speaker recognition as vector matching . this is certainly not ideal as the temporal constraint is totally ignored when deriving the speaker vector . a possible solution is to segment an enrollment / test utterance into several pieces , and derive the speaker vector for each piece . the speaker identity of the utterance is then represented by the sequence of the piece - wise speaker vectors , and speaker matching is conducted by matching the corresponding vector sequences . this paper adopts a simple sequence matching approach : the two sequences are assumed to be identical in length , and the matching is conducted piece by piece independently . finally the matching score takes the average of scores on all the pieces . note that for the i - vector approach , this segmentation method is not feasible , since i - vectors are inferred from feature distributions and so the piece - wised solution simply degrades the quality of the i - vector of each piece . for d - vectors , this approach is totally fine as they are inferred from local context and the segmentation does not impact quality of the piece - wise d - vectors very much . a more theoretical treatment is based on dynamic time warping ( dtw ) @xcite . the dtw algorithm is a principle way to measure similarities between two variable - length temporal sequences . in the most simple sense , dtw searches for an optimal path that matches two sequences with the lowest cost , by employing the dynamic programming ( dp ) method to reduce the search complexity . in our task , the dnn - extracted features of an utterance are treated as a temporal sequence . in test , the sequence derived from the enrollment utterance and the sequence derived from the test utterance are matched by dtw , where the cosine distance is used to measure the similarity between two frame - level dnn features . principally , segment pooling can be regarded as a special case of dtw , where the two sequences are in the same length , and the matching between the two sequences is piece - wise . the experiments are performed on a database that involves a limited set of short phrases . the entire database contains recordings of @xmath5 short phrases from @xmath6 speakers ( gender balanced ) , and each phrase contains @xmath7 chinese characters . for each speaker , every phrase is recorded @xmath8 times , amounting to @xmath9 utterances per speaker . the training set involves @xmath3 randomly selected speakers , which results in @xmath10 utterances in total . to prevent over - fitting , a cross - validation ( cv ) set containing @xmath11 utterances is selected from the training data , and the remaining @xmath12 utterances are used for model training , including the dnn model in the d - vector approach , and the ubm , the t matrix , the lda and plda model in the i - vector approach . the evaluation set consists of the remaining @xmath13 speakers . the evaluation is performed for each particular phrase . for each phrase , there are @xmath14 trails , including @xmath15 target trails and @xmath16 non - target trials . for the sake of neat presentation , we report the results with @xmath17 short phrases and use ` pn ' to denote the n - th phrase . the conclusions obtained here generalize well to other phrases . two baseline systems are built , one is based on i - vectors and the other is based on d - vectors . the acoustic features of the i - vector system are @xmath18-dimensional mfccs , which consist of @xmath19 static components ( including c0 ) and the first- and second - order derivatives . the number of gaussian components of the ubm is @xmath20 , and the dimension of the i - vector is @xmath2 . the d - vector baseline uses the dnn structure shown in fig . [ fig : dnn ] . the average pooling is used to derive d - vectors . the acoustic features are @xmath21-dimensional fbanks , with the left and right @xmath5 frames concatenated together . the frame - level features are extracted from the last hidden layer , and the dimension is @xmath2 . table [ tab : baseline ] presents the results in terms of equal error rate ( eer ) . it can be seen that the i - vector system generally outperforms the d - vector system in a significant way . particularly , the discriminative methods ( lda and plda ) clearly improves the i - vector system , however for the d - vector system , no improvement was found by these methods . this is not surprising , since the d - vectors have been discriminative by themselves . for this reason , lda and plda are not considered any more for d - vectors in the following experiments . .performance of baseline systems [ cols="<,^,^,^,^ " , ] following @xcite , we combine the best i - vector system ( plda ) and the best d - vector system ( dnn+pt+dtw ) . the combination is simply done by interpolating the scores obtained from the two systems : @xmath22 , where @xmath23 and @xmath24 are scores from the i - vector and d - vector systems respectively , and @xmath25 is the interpolation factor . the eer results with the optimal @xmath25 are shown in table [ tab : fusion ] . it can be seen that the combination leads to the best performance we can obtain so far . this paper presented several enhancements for the dnn - based feature learning approach in speaker recognition . we presented a phone - dependent dnn model to supply phonetic information when learning speaker features , and proposed two scoring methods based on a segment pooling and dtw respectively to leverage temporal constraints . these extensions significantly improved performance of the d - vector system . future work involves investigating more complicated statistical models for d - vectors , and use large amounts of data to learn more powerful speaker - discriminative features .

a deep learning approach has been proposed recently to derive speaker identifies ( d - vector ) by a deep neural network ( dnn ) . this approach has been applied to text - dependent speaker recognition tasks and shows reasonable performance gains when combined with the conventional i - vector approach . although promising , the existing d - vector implementation still can not compete with the i - vector baseline . this paper presents two improvements for the deep learning approach : a phone - dependent dnn structure to normalize phone variation , and a new scoring approach based on dynamic time warping ( dtw ) . experiments on a text - dependent speaker recognition task demonstrated that the proposed methods can provide considerable performance improvement over the existing d - vector implementation . * index terms * : d - vector , time dynamic warping , speaker recognition

massive stars are rare in the universe , due both to the low mass favored imf @xcite and their short life times . however , their high luminosities and mass loss rates make them important to the ecology of the universe . underpinning this understanding is the observational determination of intrinsic stellar parameters for the most massive stars . however , this is not a simple proposition ; except in rare cases where the distance is known reliably , it requires a detached , non - interacting binary system . we are aided by the fact that the binary fraction of o stars is nearly unity at birth , though this degrades some as the stars evolve ( e.g. mergers and dynamical interactions : @xcite ) . still , there are only @xmath3 50 o - star binaries in the milky way and magellanic clouds which meet these criteria and for which individual stellar characteristics have been established @xcite . this number may increase in the future owing to advances in interferometry , enabling one to determine stellar masses in systems such as hd 150136 ( o3 - 3.5 v((f*))+o5.5 - 6 v((f))+o6.5 - 7 v((f ) ) , @xcite , @xcite ) . right now , the small number of systems spanning the o spectral type makes calibrating models difficult , but an excellent effort to determine stellar parameters as a function of spectral type was made by @xcite . unfortunately , the problem of determining fundamental parameters becomes significantly worse for evolved o stars . it is extremely important to understand what effect leaving the main sequence has on the fundamental parameters of the star . however , there are currently only three such systems for which well - constrained masses have been derived , through the use of long baseline interferometry : @xmath4 ori a ( o9.2 ibvar nwk+o9.5 ii - iii(n ) , @xcite , @xcite , @xcite ) , hd 193322 ( o9.5vnn+o8.5iii+b2.5v , @xcite ) and my ser ( o7.5 iii + o9.5 iii+(o9.5-b0)iii i , @xcite ) . while increasing the total number of massive star systems with well known parameters is worthwhile , it is difficult as the number of candidate o stars is quite small . a much more realistic goal is finding a non - interacting example system with minimal mass loss which can be used as a template for understanding the evolution of o stars . mintaka ( @xmath0 ori ) , the westernmost belt star of orion , by virtue of its proximity ( @xmath5 pc , @xcite ) and brightness ( @xmath6 , @xcite ) , as well as encompassing the detached binary system aa , is an excellent choice . the @xmath0 ori system has been known to contain multiple components for well over 100 years @xcite . in fact , it is currently known to have at least 5 stellar components ( a full diagrammed layout of this system and its relevant parameters can be found in fig . 1 of @xcite ) . the embedded triple system , especially the short period binary @xmath0 ori aa , has received a significant amount of attention in the last 15 years . an artist s rendering of the orbits and lay out of the @xmath0 ori a can be found in fig 1 . of @xcite . * hereafter h02 ) , used spectra obtained with the international ultraviolet explorer to determine a putative secondary radial velocity curve . this , in combination with hipparcos photometry , led to a controversial maximum mass for the o - star of 11.2 @xmath7 , less than half the expected value based on its spectral type ( see @xcite ) . the system was revisited by ( * ? ? ? * hereafter m10 ) , in an attempt to understand this anomalous result . they worked under the assumption that since the tertiary component contributed @xmath3 25% of the total light ( @xcite , @xcite , @xcite , @xcite , @xcite ) , significantly more than the secondary , its presence might be able to explain the apparent discrepancy . they found that motion of the primary in combination with a wide tertiary component with spectral lines broader than that of the primary gave the false impression of orbital motion of the secondary . while m10 did improve the known orbital elements , the actual masses and radii are still poorly constrained . this is the result of a relatively weak secondary component as well as relatively poor light curve coverage and precision . this paper is part of a series which will attempt to characterize as rigorously as possible the triple system @xmath0 ori a , focusing on the o9.5ii primary aa1 . this includes analysis of the @xmath8ray spectrum by ( * ? ? * hereafter paper i ) and @xmath8ray variability by ( * ? ? ? * hereafter paper ii ) , as well as spectral modeling by ( * ? ? ? * hereafter paper iv ) . in this paper we focus primarily on the optical variability . this includes an analysis of high precision space based photometry obtained with the _ most _ space telescope as well as simultaneous spectroscopic observations , described in detail in section 2 . in section 3 , we discuss the spectroscopic variability , including a derivation of the primary radial velocity ( rv ) curve . in section 4 , we use this rv curve in conjunction with the _ most _ light curve to provide three possible binary fits at varying values of the primary mass . in section 5 , we provide a fit to the primary mass using apsidal motion confirming our results from section 4 . finally , we discuss and characterize previously unknown photometric variability within this system and report frequency and period spacings consistent with non - linear interactions between tidally excited and stellar g - mode oscillations . while we are still are unable to disentangle the secondary spectrum and its kinematics , we provide a more complete model than has been achieved up to this point . our optical photometry was obtained with the _ most _ microsatellite that houses a 15-cm maksutov telescope through a custom broad - band filter covering 35007500 . the sun - synchronous polar orbit has a period of 101.4 minutes ( @xmath9 ) , which enables uninterrupted observations for up to eight weeks for targets in the continuous viewing zone . a pre - launch summary of the mission is given by @xcite . ori taken in mid december 2012early january 2013 . error bars are smaller than the size of the points . the red line is the combination of the binary fit ( see section 4 ) and that of the secondary variations ( see section 6 ) . ] @xmath0 ori a was observed for roughly half of each _ most _ orbit from dec . 2012 through 7 jan . 2013 with the fabry - mode ( see fig . [ fig : full - lc ] ) at a cadence of 40.4 s. these data were then extracted using the technique of @xcite , and show a point - to - point standard deviation of @xmath10 mmag . we initiated a professional - amateur campaign in order to obtain a large number of high - quality optical spectra simultaneous with our _ most _ and _ chandra _ campaigns . this resulted in more than 300 moderate resolution ( @xmath11 ) spectra obtained over the 3 week period . the spectra were reduced by standard techniques utilizing bias , dark , and flat field images . wavelength calibration was accomplished through comparison with emission - lamp spectra taken before and/or after the object spectra . our main goal was originally to monitor the h@xmath12 line for wind variability and correlate any such emission with the x - ray spectra ( paper i , paper ii ) while simultaneously obtaining a measure of the orbital motion from the 6678 transition . however , no significant h@xmath12 variability was seen , except for the radial velocity changes related to the orbital motion of aa1 . furthermore , h@xmath12 could be diluted by wind emission , making rv measurements questionable . due to both the telluric absorption lines around h@xmath12 and the potential for wind emission , we focused on the he i 6678 transition for our radial velocity measurements . details of the telescopes and spectrographs used are given in table [ table : speclog ] . a typical s / n for any observation is @xmath3 100150 . the spectroscopic treatment of @xmath0 ori a from m10 and h02 are not in agreement . the m10 analysis assumed a tertiary contribution that was subtracted off , and left a single - lined binary , while the h02 analysis found evidence of the tertiary after iterating to remove first the primary and then the secondary spectrum from the combined one . the resulting masses are highly discrepant , with m10 assuming a standard mass based on spectral type of @xmath13 , while h02 found @xmath14 . treatment of the tertiary may lead to severe errors in the inferred stellar parameters . paper iv presents a quantitative photospheric and wind model for each of the three stars in the @xmath0 ori a system . the results of @xcite and @xcite show that the tertiary orbits the binary aa very slowly with a period of several hundred years , although the uncertainties in the tertiary s orbit are very large . as the period is so long , we expect no measurable kinematic motion in the tertiary component during one month of observations . therefore , if we take the model of the tertiary spectrum from the models of paper iv , then we can subtract its contribution a priori from the spectra in order to best constrain the binary properties of aa1,2 . this is similar to what was done by m10 . if the tertiary is not taken into account , then our results are in agreement with h02 . however , the optical spectrum consists of @xmath3 25 % contribution from the tertiary ( @xcite , @xcite , @xcite ) , so even with our s / n @xmath3 100150 , this contribution should always be seen in our spectra . we convolved the calculated spectrum of the tertiary to the resolution of our data , and can immediately see that it is the likely source of excess absorption . in figure [ tert_oplot ] , we show two example profiles of the he i 6678 line ( with large red and blue shifts from the primary s orbital motion ) along with the weighted contribution of the tertiary s spectrum calculated by paper iv . for the redshifted spectrum , we see that the red edge of the profile has an excess of absorption that matches the theoretical spectrum of the tertiary quite well . on the blue edge of the profiles , blending with he ii 6683 becomes more problematic . while the tertiary seems to reproduce some of the absorption , most of blue excess comes from the primary star s he ii line . after we subtracted the weighted contribution from the tertiary star , we cross correlated our observations against the model of the primary calculated by paper iv . from these velocities we calculated a single - lined orbit with the period derived by m10 , as the length of our data sets was insufficient for determining a more precise period . most velocities had uncertainties on the order of @xmath15 km s@xmath16 from this determination . data for all the velocities obtained are given in table 4 . we examined the residuals after removing the primary star s spectrum ( both through shift - and - add and from model subtraction using the spectrum calculated by paper iv ) . we found no residuals due to the secondary in the data at this point . this is consistent with the observational results of m10 as well as the results of the modeling of paper iv . in particular , the modeling results predict that only @xmath17 of the light in this wavelength region comes from the secondary , so the noise associated with the weak features would dominate , as the individual spectra have a signal - to - noise on the order 100150 on average . in order to reliably see the secondary with a s / n of 50 , we suspect that the combined spectrum of the three component stars needs to have a s / n @xmath18 1000 . a complete solution for a binary system requires a substantial amount of information . in general , it requires an eclipsing light curve and two radial velocity curves . despite being studied numerous times , no one has been able to reliably determine all three of these quantities for @xmath0 ori aa . this is primarily due to the faintness of the secondary with respect to both the primary and the tertiary . despite the lack of a secondary rv curve , data have two significant advantages over the previous works of h02 and m10 : high precision photometry along with simultaneous spectroscopy . since there is known apsidal motion in this system ( see section 5 ) having all photometry and spectra taken all within a month timespan mitigates this issue . for simulation and fitting of the @xmath0 ori system we used the alpha release of phoebe 2.0 ( @xcite , @xcite ) , an engine capable of simulating stars , binaries , and concurrent observational data . unlike the original phoebe , this is not built directly on the work @xcite and has been rewritten to take into account many second - order effects . this is especially relevant when dealing with spaced - based data . before this system was fit , some initial processing was required . as shown in fig . [ fig : full - lc ] , the light curve of @xmath0 ori is composed of an eclipsing light curve with second order variations superimposed . these underlying variations can not be easily or completely removed ( see section 6 ) so we must lessen their effects in order to get a more accurate representation of the binary variability . this can be done by first phasing on the binary period , which requires a precise ephemeris for the system . since our data span only 3 weeks ( about 3.5 orbits ) , the period determined by m10 is much more precise than what we are able to obtain , so we adopt their value @xmath19 d. for @xmath20 and @xmath21 we adopted our own derived values . for @xmath21 we determined a value from fits to the radial velocity curve of @xmath22 hjd . for @xmath20 we are actually able to obtain a value from o - c calculations of @xmath23 hjd . we should note that these numbers offer no specific improvement in the associated error over those derived in m10 , and in the case of @xmath20 the error is slightly worse . however , unlike m10 our method for determining @xmath20 is independent of the binary fit . this is important as second order stellar effects ( unknown in previous works ) make determinations from the binary fit less reliable . in addition , apsidal motion is significant in this system , and we did not want it to affect our results . using the revised ephemeris , the data were phased and then binned to provide a mild smoothing , which yields a template for the binary variations . in this template , the error is much smaller than the size of the points . however , incomplete removal of the secondary variations leads to scatter at the milli - mag level ( see fig . [ phlc ] ) . ori phased on the orbital period . it is then binned to 0.0125 in phase to smooth out remaining secondary variations . the bottom x - axis is phased to time of minimum light and the top is phased to periastron . ] in addition , modeling of stars and binary systems requires the presence of realistic atmosphere models . unfortunately , the kurucz atmosphere tables are the only ones which have been incorporated into phoebe 2.0 at this stage and they are insufficient for o stars . therefore , our best option is to use blackbody atmosphere models . in some cases this would not be adequate , but for o stars we sample the rayleigh jeans tail at optical wavelengths , making the differences from a blackbody minimal . in our specific case , a comparison with non - lte models calculated for the components of the system indicates a deviation of @xmath24% in the integrated flux across the _ most _ bandpass ( paper iv ) . therefore , this assumption will not significantly affect our solution as all three stars have similar temperatures and deviations from a blackbody . ) . in the bottom panels are the o - c residuals for the corresponding light and radial velocity curves . the intrinsic stellar variations that we are unable to completely remove are the main source of error . only one model is plotted as the differences are nearly impossible to distinguish at this scale . ] phoebe 2.0 has several built in fitting routines , but for the purposes of this project we decided to use the monte carlo markov chain method ( mcmc ) . the specific implementation used by phoebe 2.0 is emcee @xcite , an affine invariant mcmc ensemble sampler based on @xcite . while this routine requires a good initial fit to converge to a solution in a reasonable number of iterations , it gives the most comprehensive analysis of both the fit and the parameter correlations . mcmc also has become a standard in the fitting of photometric time series ( e.g. @xcite and @xcite ) . for a more general overview of this technique , see @xcite . however , this fitting method is not particularly useful without a full complement of constraints . since we have no secondary rv curve , we must put an extra constraint on the mass . we know the spectral type of the primary star quite well and can make a reasonable assumption that the mass is @xmath25 @xcite , further supported by non - lte modeling of the system by paper iv . unfortunately , as there are several viable sets of parameters , the fit was unable to converge to a single minimum . therefore , we divided this mass range further into three separate ranges : low mass primary ( 21 - 24 @xmath7 ; lm ) , medium mass primary ( 24 - 27 @xmath7 ; mm ) , and high mass primary ( 27 - 29 @xmath7 ; hm ) . these mass ranges were chosen based on previous results , as the system would converge to one of three possible solutions , with one falling in each bin . for each possible solution we iterated at least 800 times . while this may seem insufficient to converge to a true solution , we were not starting from a random point in the parameter space . parameters were adjusted so that the overall shape and amplitude were consistent with the observed light curve and rv curve before beginning mcmc . indeed , after about 200 iterations the log probability function defining our goodness of fit is virtually constant . a representation of the quality of these fits for the lm model is shown in fig . [ fig : bin - fit ] with the upper panels showing the phoebe fit to the light curve ( left ) and rv curve ( right ) . the lower panels show the corresponding residuals . while there are differences between the three models , they are impossible to distinguish on this scale and so only the lm model is shown . the only motivation for this model over the others is that its primary mass agrees best with what is derived from apsidal motion ( see section 5 ) . one can see a comparison of the light curve residuals of each of these models in fig . [ fig : res - comp ] . this figure demonstrates the differences inherent in the residuals of each model . these differences are small but still significant , especially within the eclipses . however , it is impossible to determine quantitatively which solution is best as the standard deviation of the residuals for each fit are the same to within @xmath26 . it should also be noted that phoebe 2.0 is still undergoing rigorous testing , so while our models seem reasonable , we also simulated our solutions with the original phoebe @xcite to ensure that the results were consistent . table [ table : bin - fit ] shows the complete solutions for each model as well as comparisons to previous solutions by h02 and m10 . while the spurious detection of the secondary rv biases the h02 results , there is good agreement between our work and that of m10 . one point to note is the improvement made to the inclination angle . while each of the previous works had fits at substantially different inclinations from 67@xmath27 - 81@xmath27 , our work shows that this inclination is quite firmly centered at 76@xmath28 regardless of the fit . in addition all of the orbital parameters are well constrained . it is very difficult to say if the high , low , or medium mass fit is the best . the probability function is virtually identical for the three fits , and the only noticeable differences are those for the masses and radii of the primary and secondary . in both the rv and light curve solutions the residuals are due more to the accuracy of the points than in any differences in the model ( see fig . [ fig : bin - fit ] and fig . [ fig : res - comp ] ) . therefore , we must allow for a family of solutions until we are able to retrieve a secondary rv curve . with no clear way to select a best fit model , we adopt the lm fit throughout this paper when we need to apply a model . in close binaries such as @xmath0 ori aa , the deformations of the stars in the elliptical orbit cause the longitude of periastron ( @xmath29 ) to precess with time , which is referred to as apsidal motion . the amount of this precession depends on the mass , internal structure of the stars , and the orbital eccentricity . @xcite and @xcite made assumptions for structure constants of massive stars and then used the measurements of apsidal motion to calculate stellar masses . @xmath0 ori a has been studied extensively for more than a century and shows signs of significant apsidal motion . the work of @xcite compiled all determinations of the spectroscopic orbit . this is combined with more recent studies of h02 and m10 along with our own work to retrieve orbital information spanning over 100 years . we use these derived orbital elements to calculate the apsidal motion , @xmath30 , using a weighted linear least - squares fit to @xmath29 as a function of time . the results shown in figure [ apsidal ] present a reasonable fit to the measurements and provide a value of @xmath31 ori . a weighted linear fit to the published values ( @xcite , h02 , m10 ) yields a measurement of @xmath31 ] with this measurement of the apsidal advance , we can make some assumptions about the internal structure constants of the primary star . this relies heavily on the age of the system . the age of @xmath0 ori is likely best constrained by the study of @xcite who find @xmath0 ori aa1 to be the brightest star in a small cluster . their study involved deep imaging to understand low - mass stars , and their color - magnitude diagrams imply an age of @xmath24 myr . in order to continue with the calculation we adopt our orbital parameters from 4.1 . combining these and the models of @xcite , we obtained the results shown in fig . [ apsmass ] , which give the stellar mass as a function of age for our derived apsidal motion . with an age of 5 myr , we see that the derived mass of the primary is @xmath33 . this value agrees with the spectral type / mass @xcite and our low mass model ( table [ table : bin - fit ] ) . @xcite estimate the age of the cluster containing @xmath0 ori , to be 5 myr , corresponding to a mass of @xmath33 consistent with our low mass model . the different lines represent the model determinations with the extremes of the apsidal motion from the errors ( fig . [ apsidal ] ) . ] it is clear from fig . [ fig : full - lc ] that the dominant signal in the light curve comes from binarity . it is also apparent from the non - phase - locked variations that additional signals are present in @xmath0 ori a. the presence of such variation is not completely unexpected as @xcite reported spectral variability in this system . however , the variation timescales seen in fig.[fig : full - lc ] are much longer than the @xmath3 4h periodicities they report . before addressing this variation , we first bin our light curve with the _ most _ orbital period ( 0.074 d ) . this allows us to mitigate the problem of light scattered into the _ most _ optics ( mainly due to the bright earth ) . as a consequence , we ignore variability on time scales shorter than a single orbit . once this is done , we must remove the eclipsing binary signal . this variability has already been analyzed in the previous section , and hinders our ability to study the additional variations present . there are two options available for removal of the binary signal . we could use the binary template derived earlier ( see fig . [ phlc ] ) or the lm model fit to this template . the decision of which binary light curve to use is not obvious , so the analysis was done using both subtractions . the results are largely the same , and we chose the lm model fit subtraction as the amplitude of the significant peaks was higher and there were slightly more significant frequencies . this choice is supported by the agreement between the subsequent analysis and the full light curve ( see fig . [ fig : full - lc ] , red line ) . the residuals are shown in fig . [ fig : resids - lc ] ( blue dots ) . ori a ( black).the spectral window is plotted in green and shows no significant aliasing . the fourier transform after pre whitening is shown in blue . the red line at 1.49 millimag represents the @xmath34 significance . the inset is a zoom in of the fourier transform from 02 c / d . whitened frequencies are labeled with arrows ( see section 6 ) . note that some frequencies are not immediately apparent without removal of larger amplitude peaks . ] a common method for finding and classifying periodic signals is known as pre - whitening . in this method peaks are detected in the fourier transform and then a sinusoid with the corresponding period , amplitude , and phase is subtracted from the light curve . this is repeated , and a combination of sinusoids representing each frequency are fitted and removed until there are no more significant peaks present . this method only works if the signals are sinusoidal ( or mostly sinusoidal ) and in the case of @xmath0 ori a , this is not obvious . the fourier transform of the @xmath0 ori a light curve is shown in fig . [ fig : del - ft ] ( black ) using period04 @xcite . we apply pre whitening to the transform and the end product is shown in blue . in order to determine which frequencies to include as significant , we use an empirically determined amplitude - to - noise ratio of 4 @xcite . there are 12 frequencies ( black arrows ) above this significance threshold @xmath35 ( red line in fig . [ fig : del - ft ] ) . the parameters associated with each frequency are given in table [ table : ft - peaks ] . the fit of these frequencies to the residuals , which is shown in fig . [ fig : resids - lc ] ( black line ) , does a reasonable job of fitting the variations . while @xcite attribute the variability they find to non - radial pulsations , we see no significant variability in the spectrum around 4 hrs ( 6 c / d ) . one possible explanation of extra periods is the presence of rotational modulation through starspots . while this is rarely considered in massive stars , the models of @xcite lend some credibility to this idea . moreover , the work of @xcite , @xcite , and @xcite show evidence for the occurrence of this phenomenon among o stars . if rotational modulation is the root cause of the variation , then the largest frequencies would occur near the rotation period of the star . the vast majority of the remaining peaks could be explained as spurious signals due to the variable nature of star spots ( e.g. lifetimes and number ) as explored extensively by @xcite for the o7.5 iii(n)((f ) ) star @xmath36 per . in our case the period of the largest amplitude signal is 4.61 d. if we assume this this is the rotation period of @xmath0 ori aa1 , then combined with the @xmath37 ( paper iv ) and the assumption of alignment between the orbital and rotation axis , this yields a radius of @xmath38 for the primary aa1 . this is smaller than the value found in our models ( see tab . [ fig : bin - fit ] ) and closer to that of a luminosity class iii star @xcite as opposed to class ii which @xmath0 ori aa1 is known to be . however , it is unclear what the uncertainty in radius would be in the models of @xcite and so this peak remains roughly consistent with rotational modulation . indeed , the phase folded light curves of the two strongest frequencies ( see fig . [ fig : period - lcs ] ) show evidence of non - sinusoidal behavior . this could be a consequence of a highly non - sinusoidal signal such as rotational modulation . while this hypothesis is plausible , similar variations can also be caused by pulsations . while rotational modulation would result in a forest of peaks that show no clear relationship to each other @xcite , pulsations should show evidence of coherent spacings . all the periodicities we observe are several hours to days long , therefore they would fall in the g - mode regime for all three massive stars seen in this system @xcite . these modes have the unique quality that when they have the same @xmath39 number they are equally spaced in period . further explanation can be found in any asteroseismology review ( e.g. @xcite . now , we can check for period spacings by creating an echelle diagram where the period of each variation is phased modulo a given spacing and plotted . equally spaced periods will lie roughly along a vertical line . the echelle diagram in fig . [ fig : echelle ] shows that at least half of the frequencies share a common period spacing . the results of this analysis provide evidence for pulsations , but this evidence is no more convincing than the idea of rotational modulation . however , there also appear spacings in the frequency domain . curiously , 10 of the 13 frequencies show a spacing at two or 3 times the orbital frequency ( see tab . [ table : spacings ] ) . this spacing suggests that there are tidal interactions associated with these signals . this is not unprecented as the work of @xcite , @xcite and @xcite also show some evidence of effects caused by tidal forces in massive binaries . this variability , though , is unlikely to arise from tidally excited oscillations , which occur at integer harmonics of the orbital frequency ( @xcite , @xcite , @xcite ) , in contrast to the oscillations we observe . however , frequency spacing at multiples of the orbital frequency has been seen in several other close binary stars ( @xcite , @xcite , @xcite ) , and indicates that the pulsations are tidally influenced . " similar to the papers cited above we speculate that the frequency spacing at multiples of the orbital frequency may arise from non - linear interactions between stellar oscillation modes and tidally excited oscillations . moreover , we interpret the frequency spacing as evidence that the observed variability is generated by stellar oscillations rather than by magnetic activity or rotational modulation . the main caveat to this hypothesis is that our frequency resolution is limited to 0.047 d@xmath16 due to the short span ( 3 weeks ) of the observations . this means that there are several spacings that could have been deemed significant . for instance @xmath40 and @xmath41 would both , within error , fit the same spacing because of how close the two peaks are in frequency . when this happens , we choose the peak that resulted in a spacing closest to a multiple of the orbital frequency . additionally , the two frequencies which do not appear in tab . [ table : spacings ] , @xmath42 and @xmath43 , are actually split by the orbital frequency . however , since the uncertainty is half the actual spacing , we choose not to include it . despite these issues , this frequency spacing is pervasive , appearing strongly between nearly every significant peak . while there is still uncertainty in this claim , it can be lifted with the addition of longer time baseline observations . another problem is that even if these are tidally influenced " oscillations , they still do not completely account for all the variation in this system . it is possible that these discrepancies could be due to amplitude modulation , though the ratio of period versus observation length makes this unlikely . it is also possible that there is also rotation modulation in this system from one of the components , just at a lower level than the pulsations . it is worth noting that the nature of the significant periodicities found in the lowest frequency range of the _ most _ data set is uncertain and could be a result of red noise. normally , this noise could be modeled and removed using autoregressive moving average techniques . however , our data is not stationary which is a requirement for this type of modeling . attempts at making the data stationary disrupt the relevant signals ; not only in our data , but also in the synthetic data using combinations of purely sinusoidal signals . therefore , it is impossible to determine unequivocally the root cause of this variation without the addition of longer baseline photometric data ( see section 8) . the system @xmath44 ori a remains an ideal candidate to define the intrinsic parameters of evolved o stars . the fundamental parameters of the primary , @xmath0 ori aa1 , appear consistent with models from paper iv as well as the results from modeling of the apsidal motion . our analysis removes ambiguity in the derived stellar parameters and makes the primary more consistent with stellar models , without having to appeal to non - conservative mass loss or mass transfer that might play a role in the evolution of a close binary like @xmath0 ori aa . despite these consistencies , @xmath0 ori aa has a rather larger eccentricity of @xmath45 , which is unusual for a close o - star binary system with a moderate age of @xmath3 5 myrs . in such a situation , circularization of the orbit should already have occurred . we can check this using the procedure outlined in @xcite with one key assumption : knowledge of the stellar structure constant @xmath46 . this constant , which is dependent on the interior composition and structure of the star , requires extensive effort to calculate precisely , but can be approximated by the fractional core radius to the eighth power . we have no evolutionary models for @xmath0 ori aa so we have no way to calculate or estimate @xmath46 directly . we do know from @xcite that @xmath46 tends to increase as a function of mass , so we will take the value of @xmath47 which is that of a 15 @xmath7 luminosity class v star . we choose this value as this is the largest mass star for which a value of @xmath46 was derived . this is clearly an underestimate of @xmath46 which is ameliorated slightly by the fact that @xmath0 ori aa is a slightly expanded luminosity class ii star ( implying a smaller fractional core radius ) . it is also clear based on the inverse dependence of the circularization time with @xmath46 that the value we obtain will be a maximum . a quick calculation reveals a circularization timescale of @xmath48myr , which is well short of the age estimate that has been derived for this system . this result shows that our knowledge of this system is incomplete and could indicate that our assumed primary mass is incorrect . however , this discrepancy could also be caused by incorrect age calculations , strong interactions with the tertiary component , or issues with our knowledge of stellar structure in massive stars . at the same time , there is currently no evidence which strongly supports any of these hypotheses , so we are confident in the assumptions made . the most intriguing aspect of this system is the pronounced second order variations in its light curve . such variations have never been seen before in this system , and the frequency spacings seem suggestive of tidally induced pulsations . if confirmed , this would be an important discovery as these modes have been seen rarely if ever in massive stars . this could lead to asteroseismic modeling and increase our understanding of the interiors of both massive stars in binaries , and massive stars as a whole , especially considering the extremely high fraction of these stars in binaries @xcite . while we have made significant progress towards understanding this system , it is clear that we are not at the level we need to be . our ultimate goal is for @xmath0 ori to be a test for both evolutionary and structural models of massive stars . for this objective to be realized , we must determine fundamental parameters to high precision . for this reason it is imperative that we identify the presence of the secondary component within the spectrum . in addition , we must improve our frequency resolution within the fourier spectrum . this will require a long time - series of high - precision photometry . as a result , we could identify with confidence the source of the variation , which could possibly allow for asteroseismic modeling . a secondary consequence of this would be an improved binary light curve with significantly reduced scatter from which to model our system . this would also lead to improved values of fundamental parameters , and place constraints on the apsidal advance of the system . this analysis to understand @xmath0 ori , in conjunction with papers i , ii , and iv , has highlighted the need to obtain the spectral characteristics and rv curve of @xmath0 ori aa2 . we will use both the _ hubble space telescope _ and _ gemini - north _ to obtain high - resolution spectra of the binary at opposite quadratures while spatially separating the tertiary and observing its spectrum . these observations are scheduled for the upcoming observing season , and we also plan to obtain higher signal - to - noise optical spectroscopy from the observatoire de mont mgantic . these observations will provide excellent constraints on the modeling of the system and reveal any unusual spectroscopic variability seen in the tertiary star . this will provide more constraints on the photometric variability seen with _ most _ . a better characterization of this variability as well as the binary parameters requires access to long time baseline photometry , which _ most _ is not able to provide . the _ brite - constellation _ project , however , is designed to provide 6 months of continuous coverage in both blue and red filters @xcite . _ brite - constellation _ began initial data acquisition of the orion constellation in october of 2013 . since this was the initial data taken , the full 6 months of continuous coverage was not achieved . however , there are nearly three continuous months of data in both filters . these data should be released to the collaboration in early 2015 . in addition , a second run with _ brite - constellation _ began in oct . 2014 that should provide a full 6-month time - series . mfc , jsn , wlw , and kh are grateful for support via chandra grant go3 - 14015a and go3 - 14015e . yn acknowledges support from the fonds national de la recherche scientifique ( belgium ) , the communaut franaise de belgique , the prodex xmm and integral contracts , and the action de recherche concerte ( cfwb - acadmie wallonie europe ) . ndr gratefully acknowledges his craq ( centre de recherche en astrophysique du qubec ) fellowship . afjm , dbg , jmm and smr are grateful for financial aid to nserc ( canada ) . afjm and hp would also like to frqnt ( quebec ) and the canadian space agency . jma acknowledges support from [ a ] the spanish government ministerio de economa y competitividad ( mineco ) through grants aya2010 - 15081 , aya2010 - 17631 , and aya2013 - 40611-p and [ b ] the consejera de educacin of the junta de andaluca through grant p08-tic-4075 . rk and ww acknowledge support by the austrian science fund ( fwf ) . nre is grateful for support from the chandra x - ray center nasa contract nas8 - 03060 . jlh acknowledges support from nasa award nnx13af40 g and nsf award ast-0807477 . lcccccc christian buil & 4 & c9 ( 0.23 m ) & eshel & atik460ex & 11,000 + christian buil & 6 & c11 ( 0.28 m ) & eshel & atik460ex & 11,000 + thierry garrel & 70 & meade lx200 14 ( 0.35 m ) & eshel & sbig st10xme & 11,000 + keith graham & 2 & meade lx200 12 ( 0.30 m ) & lhires iii & sbig st8xme & 15,000 + bernard heathcote & 32 & c11 ( 0.28 m ) & lhires iii & atik314l+ & 13,000 + thierry lemoult & 5 & c14 ( 0.36 m ) & eshel & st8xme & 11,000 + dong li & 16 & c11 ( 0.28 m ) & lhires iii & qhyimg2pro & 15,000 + benjamin mauclaire & 8 & sct 12 ( 0.30 m ) & lhires iii & kaf-1603me & 15,000 + mike potter & 201 & c14 ( 0.36 m ) & lhires iii & sbig st8 & 11,000 + jose ribeiro & 13 & c14 ( 0.36 m ) & lhires iii & sbig st10 & 15,000 + asiago & 4 & 1.82 m & reosc echelle & & 22,000 + observatory uc santa martina & 33 & 0.5 m & pucheros echelle & fli pl1001e & 20,000 + mcdonald & 5 & 2.7 m & cross - dispersed & & 60,000 + nordic optical telescope & 4 & 2.5 m & fies & & 46,000 + calar alto & 3 & 2.2 m & caf echelle & & 65,000 + [ table : speclog ] @xmath49(@xmath7 ) & & 23.81 & & 24.20 & & & & 27.59 & & 25 & & & & 11.2@xmath501.8 + @xmath51(@xmath7 ) & & 8.54 & & 8.55 & & & & 9.27 & & 9.91 & & & & 5.6@xmath500.4 + @xmath52(@xmath53 ) & & @xmath54 & & @xmath55 & & & & @xmath56 & & 15.6 & & & & 13 + @xmath57(@xmath53 ) & & @xmath58 & & @xmath59 & & & & @xmath60 & & @xmath61 & & & & @xmath62 + @xmath63(@xmath64 ) & & @xmath65 & & @xmath65 & & & & @xmath65 & & @xmath65 & & & & @xmath66 + @xmath67(@xmath64 ) & & @xmath68 & & @xmath69 & & & & @xmath70 & & 23835 & & & & + @xmath71+@xmath72(hjd ) & & @xmath73 & & @xmath73 & & & & @xmath73 & & @xmath74 & & & & + @xmath20+@xmath72(hjd ) & & @xmath75 & & @xmath75 & & & & @xmath75 & & @xmath76 & & & & + @xmath77(days ) & & @xmath78 & & @xmath78 & & & & @xmath78 & & @xmath79 & & & & @xmath80 + @xmath29 & & @xmath81 & & @xmath82 & & & & @xmath83 & & 140@xmath501.8 & & & & + @xmath84(@xmath53 ) & & @xmath85 & & @xmath86 & & & & @xmath87 & & 44@xmath500.3 & & & & + @xmath88 & & @xmath89 & & @xmath89 & & & & @xmath90 & & 0.0955@xmath500.0069 & & & & 0.075@xmath500.06 + @xmath91 & & @xmath92 & & @xmath93 & & & & @xmath94 & & 73.6@xmath500.3 & & & & 77 + @xmath95(km s@xmath16 ) & & @xmath96 & & @xmath97 & & & & @xmath98 & & 21.7@xmath500.5 & & & & + @xmath99(km s@xmath16 ) & & 96.02@xmath500.60 & & 96.02@xmath500.60 & & & & 96.02@xmath500.60 & & 106.33@xmath500.71 & & & & 94.9@xmath500.6 + @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & + @xmath105 & @xmath106 & @xmath107 & @xmath108 & @xmath109 & + @xmath110 & @xmath111 & @xmath112 & @xmath113 & @xmath114 & + @xmath42 & @xmath115 & @xmath116 & @xmath117 & @xmath118 & + @xmath43 & @xmath119 & @xmath120 & @xmath121 & @xmath122 & + @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & + @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath132 & + @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath137 & + @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath142 & + @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 & + @xmath148 & @xmath149 & @xmath150 & @xmath141 & @xmath151 & + @xmath152 & @xmath153 & @xmath154 & @xmath146 & @xmath155 & + @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & + [ table : ft - peaks ] @xmath128@xmath148 & 0.337 & @xmath161 + @xmath133@xmath156 & 0.347 & @xmath161 + @xmath105@xmath138 & 0.347 & @xmath161 + @xmath138@xmath110 & 0.358 & @xmath161 + @xmath133@xmath143 & 0.522 & @xmath162 + @xmath100@xmath110 & 0.520 & @xmath162 + [ table : spacings ]

we report on both high - precision photometry from the _ most _ space telescope and ground - based spectroscopy of the triple system @xmath0 ori a consisting of a binary o9.5ii+early - b ( aa1 and aa2 ) with @xmath1 5.7d , and a more distant tertiary ( o9 iv @xmath2 yrs ) . this data was collected in concert with x - ray spectroscopy from the chandra x - ray observatory . thanks to continuous coverage for 3 weeks , the _ most _ light curve reveals clear eclipses between aa1 and aa2 for the first time in non - phased data . from the spectroscopy we have a well constrained radial velocity curve of aa1 . while we are unable to recover radial velocity variations of the secondary star , we are able to constrain several fundamental parameters of this system and determine an approximate mass of the primary using apsidal motion . we also detected second order modulations at 12 separate frequencies with spacings indicative of tidally influenced oscillations . these spacings have never been seen in a massive binary , making this system one of only a handful of such binaries which show evidence for tidally induced pulsations .

in considering the connection between accretion and the formation of galaxy halos , perhaps nowhere is the process more dramatically illustrated than in the assembly of the most massive halos the extended bcg envelopes and diffuse intracluster light ( icl ) that is found in the centers of massive galaxy clusters . unlike quiescent field galaxies whose major accretion era lies largely in the past , under hierarchical accretion scenarios , clusters of galaxies are the most recent objects to form ( fakhouri 2010 ) ; their massive central galaxies continue to undergo active assembly and halo growth even at the current epoch , and may have accreted as much as half their mass since a redshift of @xmath0 ( de lucia & blaizot 2007 ) . thus the cluster environment presents an ideal locale for studying the accretion - driven growth of massive galaxy halos . as galaxy clusters assemble , their constituent galaxies interact with one another , first within infalling groups , then inside the cluster environment itself . over the course of time , a variety of dynamical processes liberate stars from their host galaxies , forming and feeding the growing population of intracluster stars . this complex accretion history is illustrated in figure [ sim ] , using the collisionless simulations of rudick ( 2011 ) . at early times , individual galaxies are strewn along a collapsing filament of the cosmic web . gravity quickly draws these galaxies into small groups , which then fall together to form larger groups . in the group environment , slow interactions between galaxies lead to strong tidal stripping and the formation of discrete tidal tails and streams . as the groups fall into the cluster , this material is efficiently mixed into the cluster icl ( rudick 2006 , 2009 ) . concurrently , mergers of galaxies in the cluster core expel more stars into intracluster space ( murante 2007 ) , as does ongoing stripping of infalling galaxies due to interactions both with other cluster galaxies and with the cluster potential itself ( conroy 2007 , purcell 2007 , contini 2014 ) . additionally , even _ in - situ _ star formation in the intracluster medium , from gas stripped from infalling galaxies , may contribute some fraction of the icl as well ( puchwein 2010 ) . all these processes lead to a continual growth of the intracluster light over time , as clusters continue to be fed by infalling groups and major cluster mergers . this evolution predicts that icl properties should be linked to the dynamical state of the cluster early in their formation history , clusters should be marked by a low _ total _ icl fraction but with a high proportion of light in cold ( and more easily visible tidal streams ) , while more evolved clusters would have higher icl fractions found largely in a smooth , diffuse , and well - mixed state . galaxy cluster . the panels run from @xmath1 ( upper left ) to the present day ( lower right ) . from rudick ( 2011).,width=504 ] the fact that these various processes all operate concurrently makes it difficult to isolate their individual contributions to the icl , and computational studies differ on whether group accretion , major mergers , or tidal stripping dominate the icl . fortunately , these processes imprint a variety of observable signatures in the icl . the morphology and color of the diffuse light as well as the spatial distribution and kinematics of discrete icl tracers ( red giant branch ( rgb ) stars , planetary nebulae ( pne ) , and globular clusters ( gcs ) ) all have potential to disentangle the icl formation channels . for example , the galaxy mass - metallicity relationship predicts that stripping of low mass satellites would deposit preferentially metal - poor stars into the icl , while mergers of massive galaxies would lead to more metal - rich icl . similarly the age distribution of icl populations may differentiate between stripping of old stellar systems versus that from star - forming galaxies , or even contributions from _ in - situ _ icl production . thus , observational studies of the morphology , colors , kinematics , and stellar populations in the icl are well - motivated to track the detailed accretion histories of massive clusters . ) wide - field imaging of virgo taken using cwru s burrell schmidt telescope ( mihos in prep ) ; the inset moon shows a 30@xmath2 scale . panels show m87 s extended halo ( upper left ; mihos 2005 ) , tidal streams in the virgo core ( upper right ; mihos 2005 , 2015 ) , m49 s system of accretion shells ( lower left ; janowiecki 2010 , mihos 2013 ) and the diffuse intragroup light surrounding ngc 4365 ( lower right ; bogdn 2012 , mihos in prep).,width=504 ] we can use the nearby virgo cluster to illustrate the wealth of information locked in the icl . figure [ virgo ] shows deep , wide - field imaging of the virgo cluster taken using cwru astronomy s burrell schmidt telescope ( mihos in prep ) . covering 16 square degrees down to a surface brightness of @xmath3 , the imaging reveals the complex web of diffuse light spread throughout the core of virgo . a number of tidal streams are visible , most notably two long ( @xmath4 100 kpc ) thin streams nw of m87 ( mihos 2005 , rudick 2010 ) . smaller streams are also found around the virgo ellipticals m86 and m84 , likely due to stripping of low mass satellite galaxies , as well as a system of shells and plumes around m89 suggestive of one or more major mergers ( malin 1979 , janowiecki 2010 ) . however , the total luminosity contained in these discrete streams is only @xmath5 l ; the bulk of the icl is likely found in more diffuse form , locked in the extended halo of m87 or strewn throughout the cluster at lower surface brightness . indeed the deep imaging reveals not only the thin icl streams but also the large radial extent of the halos of virgo ellipticals . in particular , m87 s halo is traced beyond 150 kpc , where a variety of signatures indicative of past accretion events can be seen . the outermost regions of m87 s halo are extremely boxy ( mihos in prep ) , a behavior reflected in the spatial distribution of its gc system as well ( durrell 2014 ) . this combination of boxy isophotes and low halo rotation ( romanowsky 2012 ) hints at a major merger event in m87 s past , and indeed , both the gc and pne systems around m87 show kinematic substructure ( romanowsky 2012 and longobardi 2015a respectively ) , suggesting the recent accretion of one or more @xmath6 lsystems . signatures of past accretion are also found in other virgo ellipticals as well . located south of the virgo core , m49 has long been known to have a dynamically complex halo , as traced by kinematic substructure in its gc system ( ct 2003 ) . the deep imaging in figure [ virgo ] reveals the cause : after subtraction of a smooth isophotal model for m49 , an extensive set of accretion shells ( janowiecki 2010 , arrigoni battaia 2011 , capaccioli 2015 ) can be seen , spanning @xmath7 150 kpc in extent and containing close to @xmath8 lof light ( janowiecki 2010 ) . the shells are morphologically similar to those formed during the radial accretion of a low mass satellite , and may be linked to the tidally disturbed dwarf companion vcc 1249 ( arrigoni battaia 2011 ) . the shells are also distinctly _ redder _ than m49 s surrounding halo ( mihos 2013 ) , suggesting that the accretion event is building up both the mass _ and _ metallicity of m49 s outer halo . figure [ virgo ] also illustrates the efficacy of the group environment in driving icl formation . lying 5.3@xmath9 to the sw of the virgo core ( and @xmath7 7 mpc behind ; mei 2007 ) is the infalling virgo w@xmath2 group , with the massive elliptical ngc 4365 at its core . our deep imaging shows an extended , diffuse tidal tail emanating sw from the galaxy ( bogdn 2012 ; mihos in prep ) , and gc kinematics clearly link the tail to an interaction with its companion ngc 4342 ( blom 2014 ) . the tail contains @xmath10 l , and a number of other streams are visible in ngc 4365 s halo as well ( including the loop visible to the ne of the galaxy ) , all indicative of cold tidal stripping in the group environment . once the w@xmath2 group eventually falls into the main body of virgo , this diffuse and extended intragroup light will be easily mixed into virgo s diffuse icl . finally , the imaging contains a dramatic example of the complex dynamical interplay between tidal stripping , icl formation , and the destruction and formation of cluster galaxy populations . lying at the center of the `` tidal streams '' panel of figure [ virgo ] is a large and extremely dim ultra - diffuse galaxy ; with a half light radius of 9.7 kpc and central surface brightness @xmath11 it is the most extreme ultradiffuse cluster galaxy yet discovered ( mihos 2015 ) . the galaxy also sports a long tidal tail arcing @xmath7 100 kpc to the north , as well as a compact nucleus whose photometric properties are well - matched to those of ultracompact dwarf galaxies ( ucds ) found in virgo ( zhang 2015 , liu 2015 ) . in this object , we are clearly seeing the tidal destruction of a low mass , nucleated galaxy which is both feeding virgo s icl population and giving rise to a new virgo ucd . to go beyond morphology and study the stellar populations in virgo s icl in detail , a variety of tools are available . the colors of the streams around m87 ( @xmath12 ; rudick 2010 ) are well - matched to those of the virgo de population and of m87 s halo itself , suggesting m87 s halo may be built at least in part from low mass satellite accretion . hst imaging of _ discrete _ rgb populations in virgo intracluster fields shows the icl to be predominantly old and metal - poor ( @xmath13 gyr , [ fe / h ] @xmath14 ; williams 2007 ) , but with an additional population of stars with intermediate ages and higher metallicities ( @xmath15 gyr , [ fe / h ] @xmath16 ) . these younger populations may arise either from stripped star forming galaxies or from icl formed _ in - situ_. the inference that stripping of late - type galaxies has contributed to the virgo icl is also supported by the luminosity function of pne in m87 s outer halo , which shows a `` dip '' characteristic of lower mass galaxies with extended star formation histories ( longobardi 2015b ) . the diversity of stellar populations seen in virgo s icl almost certainly reflects the diversity of processes that create diffuse light in clusters . while virgo s proximity gives us a detailed view of intracluster stellar populations , to gain a wider census of icl in galaxy clusters we must move beyond virgo . going to greater distances opens up the ability to study icl in a wider sample of clusters which span a range of mass , dynamical state , and redshift , allowing us to connect icl properties with cluster evolution . this comes at a cost , however ; beyond virgo , current generation telescopes can not directly image intracluster stars , and even studies of more luminous tracers such as pne and gc become more difficult . at higher redshifts , one becomes limited to broadband imaging , where the strong cosmological @xmath17 surface brightness dimming makes the already diffuse icl even more difficult to observe . aside from these observational difficulties , a second major problem is the ambiguous definition of intracluster light itself . since much of the icl is formed via the mergers that build up the central bcg , there is often no clear differentiation between the bcg halo and the extended icl the two components blend smoothly together ( and indeed may not be conceptually distinct components at all ) . in attempts to separate bcg halos from extended icl , a variety of photometric definitions have been proposed , which typically adopt different functional forms ( such as multiple @xmath18 or sersic profiles ) for each component when fitting the total profile ( gonzalez 2005 , krick & bernstein 2007 , seigar 2007 ) . however , such definitions are very sensitive to the functional forms adopted for the profiles . for example , m87 s profile is reasonably well - fit by either a single sersic or a double @xmath18 model ( janowiecki 2010 ) ; the former fit would imply little additional icl , while the latter fit puts equal light into the inner and outer profiles . to avoid this ambiguity , alternate non - parametric measures have also been employed to characterize the icl luminosity , defining the icl as diffuse light fainter than some characteristic surface brightness ( feldmeier 2004 , burke 2015 ) . while these present systematic uncertainties of their own , simulations suggest that thresholds of @xmath19 do a reasonable job of separating out an extended and perhaps unrelaxed icl component from the central bcg light ( rudick 2011 , cooper 2015 ) . , green squares ) , stars in low density intracluster space now ( open triangles ) or ever ( blue stars ) , stars unbound from galaxies ( filled red circles ) , including stars kinematically separated from the central cd galaxy ( open red circle ) . see rudick ( 2011 ) for details.,height=192 ] still other methods propose kinematic separation of the icl from the central galaxy light . dolag ( 2010 ) used simulated clusters to show that separate kinematic populations exist in cluster cores , well - characterized by distinct maxwellian distributions . these kinematic populations then separate out spatially into two sersic - like profiles plausibly identified as the bcg galaxy and the cluster icl ( perhaps reflecting different accretion events as well ; cooper 2015 ) . and indeed these definitions have some observational support . longslit spectroscopy of the bcg galaxy in abell 2199 shows a velocity dispersion profile that first falls with radius , then increases in the outer halo to join smoothly onto the cluster velocity dispersion ( kelson 2002 ) . meanwhile in virgo the velocities of the pne around m87 show a double gaussian distribution ( longobardi 2015b ) , suggesting distinct bcg and icl components . however , observational constraints make accessing kinematic information for the icl in distant clusters a daunting task . a comparison of these different metrics is shown in figure [ icldef ] ( from rudick 2011 ) , which shows that the inferred icl fraction in simulated clusters can vary by factors of 2@xmath203 depending on the adopted metric ( see also puchwein 2010 ) . kinematic separation leads to higher icl fractions , as a significant amount of starlight found within the bcg galaxy belongs to the high - velocity icl component . in contrast , density - based estimates yield systematically lower icl fractions , as material at high surface brightness is typically assigned to the cluster galaxies independent of its kinematic properties . given both the ambiguity in defining the icl and the observational difficulties in studying it , attempts to characterize icl in samples of clusters spanning a range of mass and redshift have led to varying results . an early compilation of results for local clusters by ciardullo ( 2004 ) showed icl fractions ranging from @xmath21% , with no clear dependence on cluster velocity dispersion or bautz - morgan type . recent imaging of more distant clusters probes the connection between cluster evolution and icl more directly , but again yields mixed results . while guennou ( 2012 ) find no strong difference between the icl content of clusters between at @xmath22 and today , burke ( 2015 ) find rapid evolution in the icl fraction of massive clusters over a similar redshift range . other studies of clusters at @xmath23 yield icl fractions of 10@xmath2025% ( presotto 2014 , montes & trujillo 2014 , giallongo 2014 ) , similar to @xmath24 results . however , these studies use different icl metrics and are limited to only a handful of clusters ; clearly a large sample of clusters with icl fractions measured in a consistent manner is needed to tackle the complex question of icl evolution . a similar story holds for recent attempts to constrain icl stellar populations as well . using hst imaging of distant clash clusters , demaio ( 2015 ) infer moderately low metallicities ( [ fe / h ] @xmath25 ) from the icl colors , in contrast to the case of abell 2744 , where montes & trujillo ( 2014 ) use colors to argue for a dominant population of intermediate age stars with solar metallicity . meanwhile , spectroscopic population synthesis studies show similarly diverse results . for example , in the hydra i cluster , coccato ( 2011 ) find old icl populations with sub - solar metallicities , while in the massive cluster rx j0054.0@xmath202823 , melnick ( 2012 ) find similarly old but metal - rich icl stars ( [ fe / h ] @xmath26 ) . however , while intriguing , all these studies are subject to strong photometric biases , limited largely to the brightest portions of the icl which may not be representative of the icl as a whole and may also include substantial fraction of what would normally be considered bcg light as well . the evolution shown in figure [ sim ] argues that interactions in the group environment should be particularly effective at stripping stars from galaxies and redistributing them into the diffuse intragroup light , an important precursor to the icl in massive clusters . curiously , though , these arguments have not always been borne out observationally . in the nearby leo i group , searches for intragroup light using both pne ( castro - rodriguez 2003 ) and broadband imaging ( watkins 2014 ) have come up empty , particularly notable given that the system is contains a large ( @xmath7 200 kpc ) hi ring thought to be collisional in origin ( michel - dansac 2010 ) . similarly , the m101 group also show little sign of extended diffuse light ( mihos 2013 ) , despite the tidal disturbances evident in m101 and its nearby companions . even in the clearly interacting m81/m82 group , early searches for orphaned rgb stars ( durrell 2004 ) and pne ( feldmeier 2003 ) could only place upper limits on the intragroup light fraction ( @xmath27 2% ) . in contrast , intragroup light is quite evident in dense , strongly interacting groups . the ability for these strong interactions to expel diffuse material to large distances is shown in recent deep imaging of the m51 system by watkins ( 2015 ; fig [ groups]a ) , where several extremely low surface brightness plumes extend nearly 50 kpc from the center . similarly , many compact groups are awash in diffuse light ( da rocha 2005 , 2008 ) , including the archetypal groups seyfert s quintet ( mendes de oliveira 2001 ) and seyfert s sextet ( durbala 2008 , figure [ groups]b ) . the contrast between the copious diffuse light seen in these dense systems and the dearth of light in loose groups is striking , arguing either that the tidal debris is rapidly dispersed to even lower ( undetectable ) surface brightnesses , or that close interactions in loose groups are relatively uncommon . more recently , the ability to probe discrete stellar populations in external galaxies provides a powerful new tool for studying intragroup light . probing stellar densities far below the capabilities of wide - area surface photometry , these techniques are now revealing the diffuse light contained even in loose groups . deep imaging by okamoto ( 2015 ) has uncovered the previously undetected and very extended stellar tidal debris field in the m81 group , while imaging of m31 and m33 by the pandas team ( ibata 2014 ) has mapped the myriad of tidal streams that characterize andromeda s extended stellar halo and trace its past interaction with m33 . while at low surface brightness and containing only a small amount of the total light of their parent groups , the diffuse starlight found in these studies of nearby loose groups represent the important first step in building the intragroup and intracluster light in dense galaxy environments .

the largest stellar halos in the universe are found in massive galaxy clusters , where interactions and mergers of galaxies , along with the cluster tidal field , all act to strip stars from their host galaxies and feed the diffuse intracluster light ( icl ) and extended halos of brightest cluster galaxies ( bcgs ) . studies of the nearby virgo cluster reveal a variety of accretion signatures imprinted in the morphology and stellar populations of its icl . while simulations suggest the icl should grow with time , attempts to track this evolution across clusters spanning a range of mass and redshift have proved difficult due to a variety of observational and definitional issues . meanwhile , studies of nearby galaxy groups reveal the earliest stages of icl formation : the extremely diffuse tidal streams formed during interactions in the group environment .

accelerating a thermonuclear flame to a large fraction of the speed of sound ( possibly supersonic ) is one of the main difficulties in modeling type ia supernovae ( see the recent review by @xcite and references therein ) . numerical results have shown that a prompt detonation does not yield the right abundance pattern to account for the observations @xcite ; consequently , the burning must begin as a subsonic flame . one dimensional simulations show that the flame needs to reach approximately one third of the speed of sound to properly account for the explosion energetics and nucleosynthetic yields @xcite , although , a transition from a deflagration to a detonation at some late stage of the explosion @xcite is not ruled out . laminar flame speeds are too slow by several orders of magnitude @xcite , so instabilities and the interaction with flame - generated turbulence are turned to in hopes of providing a mechanism to considerably accelerate the flame . in this paper , we focus on the effects of the rayleigh - taylor ( rt ) instability @xcite on the flame . after ignition , the subsonic flame moves outward from the center of the star . the pressure is essentially continuous across the flame . furthermore , since the star has time to expand in response to the burning , this pressure also remains constant in time . as the flame propagates , it leaves behind hot ash that is less dense than the cool fuel . gravity points toward the center of the star , so this flame front is rt unstable . bubbles of hot ash will rise and try to exchange places with spikes of cool fuel , increasing the surface area of the flame and leading to an enhancement of the flame speed . it is possible that this acceleration , operating on the stellar scale , can bring the flame speed up to a significant fraction of the speed of sound . furthermore , the shear layer between the fuel and ash which develops as the instability evolves is kelvin - helmholtz unstable , creating turbulence in the region of the flame . as the flame surface is wrinkled by the rt instability , it will interact with this turbulence . we look in detail at the interaction between the rt instability and the flame on small scales through 2-dimensional spatially resolved simulations in conditions appropriate to the late stages of a type ia supernova explosion . the difference in scale between the white dwarf and the flame width is enormous ( up to 12 orders of magnitude ) , making direct numerical simulations of the whole explosion process impossible . nevertheless , simulations of rt unstable flames have been attempted for some time , both on the full stellar scale and as smaller , constant density micro - physical studies . various techniques and approximations are used to follow an unresolved flame on these scales . some of the earliest such calculations @xcite used donor - cell advection to follow a combustion front through the star , burning most of it , and releasing enough energy to produce an explosion . in their model , the flame was propagated essentially through numerical diffusion . the rt instability was resolved only on the very largest scales of the star , missing the turbulence generated on the small scales . @xcite used a two - dimensional implicit lagrangian - remap hydrodynamics code with a mixing - length subgrid model to follow a rt unstable flame . the diffusion of reactants in the lagrangian - remap scheme is much lower than that used in @xcite , so a more physical flame speed could be used . the flame speed was chosen to be either the conductive ( laminar ) or turbulent speed which ever was larger . again , the whole star was modeled , with coarse resolution , showing that the rt instability sets up and greatly convolutes the flame front , but the resulting acceleration left the flame too slow to match observations . @xcite suggested that more resolution is needed to get the small scale physics right . the effects of turbulence on the burning were investigated by @xcite . two - dimensional , small scale rt simulations using a thickened - flame model , coupled to an explicit lagrange - remap ppm @xcite implementation to model a thin flame front were performed by @xcite and later extended to 3-d @xcite . in this model , a reaction - diffusion equation for the fuel concentration is solved along with the euler equations , with the conductivity and reaction rate modified to yield the desired laminar flame speed and a flame that was a few computational zones wide . the thermal structure of the true flame is stretched greatly , and , as a result , curvature effects of the flame are not properly included . energy is deposited behind the flame to couple this reaction - diffusion model to the hydrodynamics . similar techniques have also been used for terrestrial flames ( see for example @xcite ) . simulations of the rt instability at a density of @xmath1 lead the author to conclude that while the flame accelerates , it is too small of an acceleration ( when scaled up to the size of the star ) to account for the observations , and that a deflagration - detonation transition may be more promising . owing to the difference in behavior between two - dimensional and three - dimensional turbulence , @xcite concludes that the turbulence was more effective in increasing the flame speed in 3-d . @xcite demonstrated scaling relations for the 3-d rt unstable flame and concluded that the turbulent flame speed does not depend on the small scale burning or the laminar flame speed . the independence of the turbulent flame speed on the laminar speed was also postulated by @xcite , who performed a full - stellar model based on kolmogorov scaling of turbulence from the large stellar scales down to the flame scales . here , a turbulent scaling model using the gibson length ( the scale at which an eddy is completely burned during a single turnover ) as the small scale cutoff yields a turbulent flame velocity prediction for the grid scale . this was coupled with a subgrid model for the turbulent kinetic energy , and implemented within the ppm algorithm . their simulations of the flame propagating through the whole star showed the rt instability dominating the flow , but the flame did not accelerate enough to produce an explosion . more recently , large scale simulations of pure deflagration explosion have been performed using two different methods , level - sets @xcite and a thickened - flame model @xcite with a sharp - wheeler @xcite rt subgrid model . both groups using ppm as their underlying explicit hydrodynamics method . different numbers and locations of the ignition points are used , but the 3-d simulations by both groups release enough energy to explode the star , with the burning proceeding solely as a deflagration . these studies have demonstrated the tremendous effect the rt instability has as the dominant acceleration mechanism of the flame to the point where , in some simulations , a deflagration alone can unbind the star . however , in all the cases discussed above , some model for the flame or turbulence on scales smaller than the grid resolution was required . in this paper , we present fully resolved simulations of rt unstable flames at the densities appropriate to the late stages of the explosion no flame model is used . fully resolving the flame means that the effects of curvature and strain on the flame are implicitly accounted for . curvature effects on flames in conditions close to those studied here were found to be important in @xcite , where laminar flames were propagated in diverging and converging spherical geometries . here , the effects can be even larger as the flame will experience sharp kinks as it is distorted by the rt instability . to achieve converged steady - state laminar flames , approximately five grid point resolution in the thermal width is required . this limits the size of the domain that can be modeled significantly ( although the adaptive mesh refinement employed here helps greatly ) , but frees us from the need to deal with subgrid models . resolved simulations at the smallest scales complement the full star flame - model calculations , and it is hoped that progress from the small scales up will allow us in the future to develop and calibrate more accurate subgrid models . fully resolved multidimensional type ia - like flame simulations are rare @xcite , mainly because the flame moves so slowly , requiring excessive numbers of timesteps for compressible codes . resolved calculations of the reactive rt instability have been performed before @xcite , but using model flames ( a kpp - type reaction , @xcite ) and a boussinesq approximation , under conditions that are not directly applicable to type ia sne . when extrapolated to the astrophysical regime , they predict a flame speed independent of the laminar speed , similar to that found by @xcite . however , compressibility was neglected in these calculations . here we use a low mach number formulation of the equations of motion , which retains the compressibility , but frees us from the restrictive acoustic timestep constraint , and we use realistic input physics ( reaction rate , eos , etc . ) appropriate to the conditions in a type ia sne explosion . flame instabilities stretch and wrinkle the flame , increasing the bulk burning rate . the sne flame is subject to the landau - darrieus @xcite and rt instabilities , as well as the interaction with turbulence . in a previous paper , ( @xcite , henceforth referred to as paper i ) , we looked at landau - darrieus unstable flames , and validated the low mach number method for astrophysical flames , confirming the results of earlier studies that the acceleration of the laminar flame is quite small , while finding no evidence for the breakdown of the cusps in the non - linear regime . in the present paper , we look in detail at the interaction between the burning and the rt instability . a rt driven flame behaves very differently from its laminar counterpart . aside from looking at the increase in velocity over the laminar speed the wrinkling provides , we seek to determine whether there is a simple scaling relation that matches these results , and whether such a relation holds over the whole range of densities , where the character of the burning is expected to change dramatically . in [ sec : distributedburning ] , we discuss the effects of the burning on the rt instability when we enter the distributed burning regime . in [ sec : numerics ] , the low mach number numerical method and input physics are described , followed by the results of our simulations in [ sec : results ] . we conclude in [ sec : detonation ] by discussing the implications of the rt burning process on the explosion mechanism for type ia sne . the rt instability has been well studied analytically @xcite and numerically , both as an incompressible ( see recent results by @xcite ) and compressible ( see for example @xcite ) fluid . in the absence of burning , the growth of the multimode rt instability in the non - linear regime is characterized by two parameters , one describing the terminal velocity , and the other describing bubble merger @xcite . bubbles are assumed to merge when the difference in their heights exceeds some critical multiple of the smaller bubbles size . this leads to an expression for the average position of the interface , @xmath2 @xcite where @xmath3 is a function of the two parameters , and @xmath4 is the atwood number . much theoretical , computational , and experimental work aims to measure @xmath3 , and some evidence suggests that it may be constant for a wide range of initial conditions @xcite . the velocity of the merged bubbles can be written as @xmath5 where @xmath6 @xcite . these expressions were all formulated for the non - reactive rt instability and studies of rising bubbles . numerical experiments of reactive rt instabilities in sne conditions @xcite using a thickened - flame model found agreement with the above velocity expression , with @xmath7 . expressions of this form have been used as a subgrid model for the flame speed in full - star , pure deflagration calculations of type ia explosions in 1-d @xcite and 3-d @xcite , resulting in a successful explosion of the star . these results suggest that the rt instability is the dominant acceleration mechanism for the flame . @xcite and @xcite argued that , unlike a non - reactive rt instability , where all wavelengths can grow ( in the absence of surface tension ) , burning introduces a small scale cutoff to the instability . the growth rate for the pure rt instability is determined by the dispersion relation @xmath8 @xcite , where @xmath9 is the angular frequency and @xmath10 is the horizontal wave number for the dominant mode . we note that we have ignored contributions due to surface tension . by equating the period for the growth of the instability to the laminar flame propagation timescale , @xcite showed that there is a critical wavelength below which any perturbations will be washed out by burning . this wavelength is called the fire - polishing wavelength , @xmath11 : @xmath12 where @xmath13 is the effective gravitational acceleration . we note that this is a simple estimate , nevertheless , it provides a useful measure of the effectiveness of the rt instability on disrupting the flame . table [ table : flameproperties ] lists the flame properties , as computed by the code described in [ sec : numerics ] , including @xmath11 for the densities we are considering . stratification effects are negligible on the scales that we are considering , since we are modeling a tiny fraction of the star . the relative change in pressure over our domain is @xmath14 where we have expressed this ratio in terms of the mach number of the sharp - wheeler velocity for a rt flame encompassing our entire domain . since we are resolving the structure of the flame , the domain size restrictions limit the speed - ups we see to about 10 times the laminar speed , which gives a maximum @xmath15 of @xmath16 . as the density of the white dwarf decreases , the flame speed decreases sharply while the flame width , @xmath17 , increases . as table [ table : flameproperties ] shows , at the densities we are considering , we pass from the regime where @xmath18 ( low densities ) to @xmath19 ( high densities ) . when @xmath18 , the rt instability will dominate the flame , to the point where it may no longer be continuous , and one can no longer draw a simple curve from one side of the domain to the other marking the flame position . this increase in complexity of the flame surface signifies that we are leaving the flamelet regime . @xcite made estimates of the density where the burning transitions from the flamelet regime to the distributed burning regime , in which a distinguished local flame surface is not apparent . using the flame properties tabulated in @xcite and equating the gibson scale to the flame width , they found that for a 0.5 @xmath20c/0.5 @xmath21o flame , the burning will enter the distributed burning regime at a density of @xmath22@xmath23 . @xcite refined this estimate by suggesting that since the turbulence present in a type ia explosion is generated by the rt instability , it should follow bolgiano - obukhov rather than kolmogorov statistics . they found a slightly lower transition density of @xmath24 . at these densities , the turbulence is intense enough to disrupt the flame . we note that this transition density is right in the range where the fire polishing length becomes equal to the flame width , as discussed above . this density range is also interesting in the context of deflagration to detonation transitions ( ddt ) . arguments based on nucleosynthesis and one - dimensional modeling suggest that if the burning in a type ia supernova were to transition from a deflagration to a detonation , this transition would need to occur at a density around @xmath0 ( see for instance @xcite ) . the correlation between this ddt density and the density marking the transition to the distributed burning regime was pointed out by @xcite and @xcite . one - dimensional turbulence studies of the flame - turbulence interaction at these densities were performed by @xcite , where it was observed that the turbulence disrupts the flame sufficiently in some cases to create local explosions as the hot ash comes in contact with pockets of cool fuel . if these regions of fuel are large enough , these local explosions may be able to initiate a ddt . direct numerical simulations of thermonuclear flames at these densities do not exist . @xcite performed simplified flame model calculations in 3-d using a source term relevant to flames in type ia supernovae . they concluded that for densities larger than @xmath24 , scaling the flame speed in direct proportion to the increase in surface area is a valid subgrid model . for densities below @xmath0 however , they state that the effects of turbulence become important . as table [ table : flameproperties ] shows , the laminar flame speed in our density range is extremely subsonic our highest laminar flame mach number is @xmath25 . simulating a flame at these densities with a fully compressible code would be prohibitively expensive , requiring millions of timesteps . this in turn leads to a large accumulation of error . instead , a low mach number hydrodynamics code @xcite is employed for the present study . we discuss the numerical method in [ sec : numerics ] . together , the flame width and the fire - polishing length set strict limits on the densities and size of domains that can be addressed through fully resolved simulations . steady state laminar flame simulations show that we need about 510 zones inside the flames thermal width , @xmath26 we note that this width measurement is a factor of 23 narrower than the alternate measure defined as the width of the region where the temperature is 10% above the fuel temperature to 90% of the ash temperature that was used in @xcite . in order for the rt instability to develop , we need the computational domain to be at least one fire - polishing length wide , but in practice , we want 10 or more fire - polishing lengths in the domain for bubble merger to take place and the rt instability to become well developed . these restrictions limit us to consider densities @xmath27 in the present study . the lower limit of densities we can consider is set only by what is attained in the supernova explosion itself we stop at @xmath28 here . when the rt instability dominates the burning , it is the fluid motion mixing the fuel with the hot ash that controls the burning of the new fuel , rather than thermal conduction . thus , the flame has a very large thermal width , governed by the mixing , at the lowest densities . to initiate a detonation , the temperature in a sufficiently large region needs to rise in unison , such that the over - pressure generated by the burning is strong enough to drive a shock just ahead of the burning layer . the critical size of this region is called the matchhead size . if this rt thermal width becomes as large as the critical detonation matchhead size , as tabulated in @xcite , it is possible that a ddt can occur . furthermore , the size of the matchhead increases dramatically as the density decreases , and at low densities , it may be larger than the entire star . we note that recent simulations @xcite indicate that a pure deflagration can be successful in unbinding the star . we will explore the possible of a ddt , using the scaling behavior we find for flames in the distributed burning regime . the simulations were performed using an adaptive , low mach number hydrodynamics code , as described in @xcite . the state variables in the inviscid navier - stokes equations are expanded in powers of mach number , following @xcite . the result is that the pressure is decomposed into a dynamic and thermodynamic component , the ratio of which is of order mach number squared . only the dynamic component appears in the momentum equation . these equations are solved using an approximate projection formalism @xcite , breaking the time evolution into an advection and a projection step . the advection is computed using an unsplit , second - order godunov method that updates the species and enthalpy to the new time level and finds provisional velocities . a divergence constraint on the velocities is provided by the thermodynamics , and enforced in the projection step , where the provisional velocities are projected onto the space of vectors satisfying this constraint . burning is handled through operator splitting . the low mach number formulation allows us to follow the evolution of these slow moving flames in this density range , the laminar speeds are @xmath29without the constraint of the sound speed on the timestep . this affords us timesteps that are @xmath30 larger than a fully compressible code would take . we are still free however to have large density jumps as we cross the flame . this method was validated for astrophysical applications by comparing solutions of 1-d laminar flames to fully compressible results @xcite , and in our study of the landau - darrieus instability ( paper i ) , where the growth rate computed from our calculations matched the theoretical predictions across a range of wave numbers . the code is unchanged from the description in @xcite . we review the input physics used in the present simulations below . this method contrasts sharply with that used in the only previous study of the rt instability in type ia sne , as presented in @xcite . there , the fully compressible ppm algorithm was used , along with a model for a thickened flame to represent it on the small scales . the laminar flame speed matched the correct physical value , but the flame thickness was much larger than the physical value . in the present case , resolving the thermal structure frees us from the need for a flame model . the equation of state consists of a helmholtz free - energy tabular component for the degenerate / relativistic electrons / positrons , an ideal gas component for the ions , and a blackbody component for radiation , as described in @xcite . the conductivities contain contributions from electron - electron and electron - ion processes , and are described in @xcite . all of the flames are half carbon / half oxygen , but only the carbon is burned . a single reaction , @xmath20c(@xmath20c,@xmath31)@xmath32 mg , is followed , using the unscreened rate from @xcite . since we do not expect the burning to proceed up to the iron - group elements at these low densities , this reaction alone is sufficient to model the nuclear reactions . the reaction rate for oxygen burning is several orders of magnitude slower than the carbon rate at the ash temperatures we reach , so neglecting oxygen burning is valid on our scales . the calculations are initiated by mapping a one - dimensional steady state laminar flame ( computed with the same code ) onto our grid , shifting the zero point according to a random phase , 10 frequency sinusoidal perturbation . beginning with a steady - state flame in pressure equilibrium virtually eliminates any transients from disturbing the flow ahead of the flame . in all of the results presented , the flame is moving downward in our domain . the transverse boundaries are periodic , the upper boundary is outflow , and the lower boundary is inflow , with the inflow velocity set to the laminar flame speed . this keeps an unperturbed flame stationary on our grid . the solutions presented here were computed with an advective cfl number of @xmath33 and typically required 8000 timesteps . for all the runs we present , adaptive mesh refinement was used , with a base grid plus one finer level that has twice the resolution . this allows us to focus the resolution on the interesting parts of the flow , allowing larger domains to be studied . refinement triggered on the temperature gradient and the gradient of carbon mass fraction , ensuring that the flame s reaction zone was always at the finest resolution . table [ table : simparams ] lists the parameters used for all the simulations we present . for the lowest density case , @xmath34 , we actually present results for three different domain sizes and two resolutions and for the @xmath0 case , we present an additional simulation with the burning disabled . except where noted otherwise , all discussion of the @xmath28 results refer to the widest domain run . in the discussion below , we first concentrate on the change in character of the burning as the density is changed . next we look at integral quantities that help characterize the flame , followed by a measurement of the rt growth rate and a comparison of the rt instability with and without burning . finally , we consider resolution and domain size effects on the results we present . figures [ fig : rt_6.67e6_wide ] to [ fig : rt_1.5e7 ] show the results for the three densities we consider in this paper , @xmath34 , @xmath0 , and @xmath35 . the fuel is at the bottom of the domain ( and colored red ) , and the ash is at the top of the domain . gravity points up , toward increasing @xmath36 , in all simulations . for all runs , we set @xmath37 . this value is appropriate for the outer regions of the star , and assumes that some pre - expansion of the white dwarf has taken place . the domain width was chosen based on the flame width and fire - polishing length ( and considerations of how many zones we can reasonably compute ) to contain several unstable wavelengths . we see from the figures that as the density increases , the small scale structure is greatly diminished , as the burning is more effective in containing it . the calculations were run until the instability reached the top or bottom of the computational domain , or the size of the mixed region grew to be much larger than the width of the domain , leading to saturation of the instability which will be discussed later . in the descriptions below , ` bubble ' refers to the hot ash buoyantly rising into the cool fuel , and ` spike ' refers to the cool fuel falling into the hot ash . the results for @xmath38 , 768 cm wide simulation ( figure [ fig : rt_6.67e6_wide ] ) shows the flow quickly becoming mixed and the initial perturbations rapidly forgotten . a mixed region 5001000 cm wide forms , separating the fuel and ash . some smaller features begin to burn away as the flame evolves and the ash gets entrained by the fluid motions . at late times , it becomes impossible to draw a well defined flame surface separating the fuel and ash , as the rt mixing clearly dominates . in [ sec : res ] , we look at the effect of different domain widths on this growth . the @xmath0 run ( figure [ fig : rt_1.e7 ] ) shows a balance between the rt growth and the burning . spikes of fuel push well into the hot ash , forming well defined mushroom caps . before these caps can travel too far into the ash , they burn away . this is seen for several spikes in figure [ fig : rt_1.e7]the mushrooms gradually burn away , almost uniformly , turning yellow / green , and finally leaving a ephemeral outline in the ash as the burning completes . as the instability evolves , we see longer wavelength modes beginning to dominate , but they also begin to burn away . we are prevented from watching the final dominant mode that grows burn away because of it reaching the top of our domain . the highest density case , @xmath35 ( figure [ fig : rt_1.5e7 ] ) behaves qualitatively different than the two lower densities . a defining characteristic of this density is that the flame front is always sharply defined a trait that we expect to carry over to higher densities as well . at this density , the burning proceeds rapidly enough that it can suppress the rt instability significantly . this is as expected , since the fire - polishing length is larger than the flame width at this density ( see table [ table : flameproperties ] ) . as the spikes of fuel extend into the hot ash , we see them rapidly burning away from the outside inward . the rt mushroom caps hardly have time to form before their ends are burned away , giving them a hammerhead - like appearance . figure [ fig : carbon_destruction ] shows the carbon destruction rate , @xmath39 , at the midpoint of each calculation for the three densities . at the low density end , the burning is concentrated in small regions whereas at the high end , the burning is smoother . we would expect this trend to continue for densities outside the range we consider here . at higher densities , the burning will dominate even more , and we would expect the flame surface to be well defined on the small scales . at densities lower than @xmath28 , the mixed region will grow even larger , as the burning becomes even less effective at suppressing the rt instability . we will look at how the width of the reactive region grows in proportion to that of the mixed region at these low densities in [ sec : growth ] . having described the overall evolution of the flames , we now present some quantitative measures of flame behavior . we define the effective flame speed , @xmath40 , in terms of the carbon consumption in the domain ; namely , @xmath41 where @xmath42 is the spatial domain of the burning region , @xmath43 is the width of inflow face , @xmath44 is the inflow carbon mass fraction , and @xmath45 is the rate of consumption of carbon due to nuclear burning . accurate direct evaluation of this integral is problematic because of the operator split treatment of reactions in the numerical method . a robust and accurate estimate of @xmath40 can be obtain by integrating the conservation equation for carbon mass fraction @xmath46 over @xmath42 and a time interval @xmath47 $ ] to obtain @xmath48 @xmath49 is the inflow velocity . note that this formula is only valid when essentially all carbon is being consumed in the domain . another quantity of interest is the area of the flame , which for our two - dimensional studies is a length . computing a flame length for the lower density cases is problematic since there is often not a distinct flame surface . in this paper we define a flame length as the number of zone edges where the carbon mass fraction passes through 0.25 . we normalize the length to the initial flame length so we can measure the growth of the burning surface . this approach , while crude , is robust and defines a reasonable method for computing the length of a well mixed flame where other approaches are infeasible . in particular , this approach systematically overestimates the flame length ; however , this can be compensated for by normalizing by the initial flame length . we have validated that for well - defined flames , this method agrees well with the length of a contour as computed with the commercial idl package , after normalization . figures [ fig : rt_6.67e6_speeds ] to [ fig : rt_1.5e7_length ] show the speeds and lengths of the flames at the three densities as a function of time ( the additional curves on the @xmath28 plots will be discussed in [ sec : res ] ) . the maximum speedups we observe are 3 times the laminar speed for the @xmath28 run , 5 times for the @xmath0 run , and 2.5 times for the @xmath50 run . these speedups are much larger than any seen in our companion study on the landau - darrieus instability ( paper i ) , as expected , based on the outcome of the large scale simulations of the explosion . the flame length increases by a factor of 50 throughout the simulation for the two lowest densities , but only increases to by a factor of 8 for the @xmath35 case , due to the strong moderation of the rt instability by the burning . comparing the velocity curves to the length curves , we see that , in general , the velocity is not strictly proportional to the flame surface area . this is not unexpected , since the stretch and curvature that the flame experiences as well as the complicated motions of the rt instability pushing the fuel and ash together will modify the local burning rate . evidence of this variability is seen in figure [ fig : carbon_destruction ] where considerable variation in the carbon destruction rate is observed , even for the highest density case . to further quantify this effect , in figures [ fig : rt_6.67e6_v_vs_a ] to [ fig : rt_1.5e7_v_vs_a ] we plot the velocity divided by flame surface area normalized to @xmath51 , where @xmath43 is the width of the box . with the caveat that there are ambiguities in defining flame surface area for the `` distributed '' flames , we observe that , after an initial transient , the flame speed versus area relaxes to an essentially statistically stationary value . for the highest density case , the effective flame velocity is approximately 30% of the laminar flame speed @xmath52 the area enhancement . for the lower density cases , the value drops to approximately 10% . thus , at these densities , we find @xmath53 where @xmath54 is a proportionality constant not equal to 1 . this proportionality constant increases with density , reflecting the fact that the fire - polishing wavelength , which sets the smallest scale on which we can wrinkle the flame , grows as well . thus at higher densities , the localized curvature is smaller , and as a result , the localized velocity is less affected . this has strong implications for subgrid models , particularly in the flamelet regime , where it is normally assumed that the velocity scales directly as the increase in area of the flame surface . it appears that this is not the case , and is further complicated by the density dependence of the proportionality constant . @xcite introduced a fractal model to describe the growth of the flame surface in a supernova when the burning is in the flamelet regime . the self - similar range through which a fractal description applies is bounded by the fire - polishing length on the small scales and the sharp - wheeler description for the rt instability growth on the large scales . in two - dimensions , a fractal model for the growth of the flame surface would be @xmath55 where we expect @xmath56 in the self - similar scaling regime . the offset in time , @xmath57 , is chosen such that @xmath58 at @xmath59 . the value of @xmath3 is taken to be @xmath60 . as we will see in [ sec : growth ] , the sharp - wheeler model may not be the proper description of the reactive rt instability . with that caveat , figures [ fig : rt_1.e7_length ] and [ fig : rt_1.5e7_length ] , we plot the predictions of the fractal scaling model using @xmath61 and @xmath62 , which give area growing as @xmath63 and @xmath64 respectively . from these figures , it seems that , if this fractal model is correct , the growth in the flame surface is closely represented by a fractal dimension of 1.7 . this agrees with the value computed from two - dimensional rt studies by @xcite . some evidence suggests that the fractal dimension of rt induced turbulence may increase with time @xcite . we caution however that we only span about a decade between @xmath65 and @xmath66 in these simulations . if such a scaling holds , it could serve as a subgrid model for large scale simulations , however , the evolution of the proportionality constant in equation ( [ eq : v_vs_a ] ) needs to be understood . additional calculations that encompass a larger range of spatial scales are needed to further validate this model . not all of the energy release by the flame goes into driving the expansion of the star . the rt unstable flame and associated kelvin - helmholtz instabilities also generate turbulence . turbulence diagnostics can be difficult to define here we use an integral quantity , the favre average turbulent kinetic energy which plays a role in the @xmath67 subgrid model for turbulence . the favre average turbulent kinetic energy ( for a discussion of which see @xcite ) , can be written @xmath68 where @xmath69 denotes the horizontal spatial average of the quantity enclosed in the brackets . figure [ fig : rt_6.67e6_tke_y ] shows @xmath70 as a function of height at several instances in time for the @xmath28 flame . we see that , within the mixed region , the turbulent kinetic energy is roughly constant , and this plateau rises quadratically with time , as shown in figure [ fig : rt_6.67e6_tke_peak ] . the integral of @xmath71 over the vertical extent of the domain is the total turbulent kinetic energy @xmath72 . the generation of turbulent kinetic energy as a function of time is shown for the three densities in figures [ fig : rt_6.67e6_tke ] to [ fig : rt_1.5e7_tke ] . the data suggest that the increase in kinetic energy as a function of time is well approximated by a power law of the form @xmath73 are shown in figures [ fig : rt_6.67e6_tke ] to [ fig : rt_1.5e7_tke ] , and summarized in table [ table : tke ] . the power law fits , summarized in table [ table : tke ] match the data quite well , giving an exponent , @xmath74@xmath75 , depending on the density , with the power increasing with increasing density . in the lowest density run , there is considerable turbulent energy , and , as figure [ fig : rt_6.67e6_tke_y ] shows , @xmath76 is reached after @xmath77 s. this is on a scale of @xmath78 cm . @xcite presumed pre - existing turbulent kinetic energy of this level on scales of @xmath79 cm in their main calculation . assuming a one - third power scaling of the energy cascade , on their length scales the rt generate turbulence already overwhelms this pre - existing turbulence , and it is continuing to grow , apparently quadratically with time . these results suggest that the rt instability alone can provide the turbulent motions presumed to exist in the star . in the absence of any reactions , the extent of the mixed region should increase as @xmath80 , as predicted by the sharp - wheeler model , equation ( [ eq : sw ] ) . this relation is only expected to hold during the phase where bubbles are merging , which excludes the initial linear growth range . also , because we are resolving the flame structure , the interface between the fuel and ash is not infinitesimally thin , which complicates the definition of the atwood number . furthermore , at late times , once the bubble merging in our domain has stopped and the size of the mixed region rivals the width of the domain , this relation will also break down . this relation was derived for the purely hydrodynamical rt instability . burning can have large consequences here . the sharp - wheeler model describes the growth of this mixed region , which will contain fuel and ash , but that fuel is burning , and as it turns into ash , we would expect to find the mixed region smaller than that of the purely hydrodynamical case . for the present simulations , since we have a limited number of modes in our box ( typically 10 ) , we are only going to be able to do the fits on a small subset of the data . a future study will focus on scaling in the flamelet regime , using much larger domains . figures [ fig : rt_6.67e6_width ] to [ fig : rt_1.5e7_width ] show the extrema of the mixed region , computed by laterally averaging the carbon mass fractions and finding the positions where it first exceeds 0.05 and 0.45 . this definition is consistent with that used in @xcite . the curves are measured with respect to the initial position of the interface . the top curve measures the position the spikes of fuel pushing into the region of hot ash . the lower curve is the position of the bubbles of hot ash floating into the fuel . the sharp kinks in the spike curves represent the instances where the plume of fuel , having pushed far into the ash region , burned away . at the highest density ( figure [ fig : rt_1.5e7_width ] ) , the bubble curve is very smooth . we can make fits of the growth of the mixed region to time by fitting to a power law , @xmath81 we do not attempt to fit to the sharp - wheeler scaling and find a value of @xmath3 because of the limited range of wavelengths we follow , and the uncertainty as to whether that relation holds in the reactive case . we exclude the initial linear phase of the growth of the rt instability in our fits . figures [ fig : rt_6.67e6_width_fit ] to [ fig : rt_1.5e7_width_fit ] shows the results . in all cases , we find @xmath82 less than 2.0 , indicating that we are not in the sharp - wheeler regime . furthermore , @xmath82 increases with increasing density . we find @xmath83 for @xmath28 , @xmath84 for @xmath0 , and @xmath85 for @xmath35 . larger scale studies , with a greater range of wavelengths are needed to determine whether this is a general result for reactive rayleigh - taylor , or if this is because of the small size domains considered here . one way we can get some insight into this is to rerun one of these simulations without burning . ( note that in this case , we shift to a non - moving frame of reference whereas the reacting cases are performed in a reference frame moving at the laminar flame speed . ) the growth rate for the @xmath0 run with burning disabled is over - plotted in figure [ fig : rt_1.e7_width ] . we see that it is much smoother than the corresponding reactive case , since the spikes of fuel which push into the hot ash never burn away . the only wiggles in the mixing region growth curves result from bubble mergers . the fit to a power law for this non - reactive simulation is presented in figure [ fig : rt_1.e7_noburn_width_fit ] . the carbon mass fraction for several points in time are shown in figure [ fig : rt_1.e7_noburn ] , which compares directly to figure [ fig : rt_1.e7 ] . here , @xmath86 , higher than the reactive case , but still not equal to 2.0 . the difference between this value and the reactive case shows that the burning does have some influence . in the non - reactive case we really do expect @xmath87 , and therefore the difference we see is very likely because we also are probably not following enough modes to see the sharp - wheeler scaling . for the @xmath35 run , we were closest to growing as @xmath80 . the burning here is quite vigorous , compared to the rt growth , so as argued above , we would expect to see its influence on the extent of the mixed region . however , we would expect it to affect the position of the spikes of fuel only , since the bubbles of ash do not react . therefore , we can fit the growth of the bubbles and spikes from the initial interface separately . this is shown in figure [ fig : rt_1.5e7_bubble_spike_fit ] . again , we exclude the initial linear growth phase of the instability . we find the bubble position growing as @xmath88 and the spike region growing as @xmath89 . based on the trends we see here and above , we would expect that once we have enough unstable modes in the box , the width of the mixed region during the bubble merger phase will scale as slightly smaller @xmath80 , with the bubble position itself growing as @xmath80 . therefore , more studies , in larger domains with more unstable modes are needed . it is interesting to look at how the size of the reactive region scales with that of the mixed region , especially at the lowest density . as discussed in [ sec : rt ] , if the reactive region grows to the size of the detonation matchhead , then it may be possible for a deflagration - detonation transition to occur . measuring the scaling of the reactive region to the mixed region is not easy , because the flame is so wrinkled . we will use a volume weighted definition here . the collection of zones , @xmath90 , where the carbon mass fraction , @xmath91 , is between @xmath92 and @xmath93 is @xmath94 similarly , we can define the collection of zones , @xmath95 , where the density - weighted carbon destruction rate , is within @xmath31 of the peak , @xmath96 then , the ratio of the size of the reactive region to the mixed region is @xmath97 where @xmath98 is the volume of the zones in set @xmath90 . once the mixed region develops , we expect this quantity to be always less than unity . because of the freedom to choose the parameters @xmath92 , @xmath93 , and @xmath31 , the numerical value of @xmath99 itself does not have much meaning , but the trend with time does . we pick @xmath100 , @xmath101 , and two values of @xmath31 : 0.1 and 0.8 . we note that the trend is relatively insensitive to the choice of these parameters , as illustrated by figure [ fig:6.67e6_scale ] , which shows @xmath99 as a function of time for the @xmath28 rt unstable flame for the two choices of @xmath31 . after an initial transient , it appears that these curves level off , indicating that the reactive region continues to growth in size proportional to the mixed region as the flame evolves . we note that this shows that the reactive region is a small fraction of the mixed region , but this absolute scaling is dependent on the choice of the three parameters . this is illustrated in figure [ fig:6.67e6_scale_thresh ] , where the mixed and reactive regions are shown , midway through the calculation . at late times , the peak nuclear energy generation comes from regions where the carbon mass fraction is @xmath102 , as shown in figure [ fig : rt_6.67e6_yc12_dydt ] . this is also apparent by looking at the laterally averaged flame profile and comparing to the laminar state see figure [ fig : rt_6.67e6_wide_averages ] . if this reactive region can growth to a critical matchhead size , before the entire star is consumed , it may be possible for a deflagration to detonation transition to proceed . however , since at late times the peak burning is occurring at mass fractions of 0.15 rather than 0.5 , this will greatly increase the required matchhead size , further complicating the possibility of a deflagration - detonation transition . in this subsection , we examine the role of domain size and resolution on the computational results . figures [ fig : rt_6.67e6 ] and [ fig : rt_6.67e6_small2 ] show the evolution of the @xmath103 rt simulation in narrower domains ( 96 cm and 384 cm wide respectively ) . other than the domain size , the parameters are identical to those used in the results described shown in figure [ fig : rt_6.67e6_wide ] . in the narrowest case , once the size of the mixed region reaches a fixed size , comparable to the width of the domain , the mixing and reacting processes saturate and the flow enters a quasi - steady - state , with vortices that are remnants of the early mixing driving transverse shear flow in the mixed region . once this saturation is reached , the burning rate begins to slowly decay . this behavior is not seen in the corresponding wider domain calculation , although , we would expect a similar pattern if we use a taller domain and ran out to longer times . figure [ fig : rt_6.67e6_speeds ] shows three additional curves corresponding to these narrower domains , the narrowest at two different resolutions ( see table [ table : simparams ] ) . in the 96 cm wide domain , the peak velocity is much smaller , reaching only about 3 times the laminar speed . this is to be expected from equation ( [ eq : sw2])the wider domain allows longer wavelength modes to go unstable . the decrease in velocity at late times in the narrow domain begins when the size of the mixed layer becomes comparable to the width of the domain , preventing any further modes from growing . this is reflected in the later panels of figure [ fig : rt_6.67e6 ] . in the 384 cm wide domain , the velocity continues to grow , beyond 6 times the laminar speed we would expect our widest domain run to continue to accelerate as well , but it began to interact with the top of our computational domain and was not run out as long . the velocity of the low resolution run tracks that of the high resolution case very closely , suggesting that , for the large scale diagnostics , we have converged . this lowest resolution run has approximately 5 points in the thermal width ( equation [ eq : thermalwidth ] ) , a value found acceptable in the convergence study performed in @xcite for landau - darrieus unstable astrophysical flames . we note again that this definition of thermal width is smaller than some other commonly used definitions ( see , for example @xcite ) . finally , in all calculations , the initial perturbations are well resolved , with typically 50100 zones per wavelength ( with the only exception being the 96 cm wide , @xmath28 run , where there are only 20 zones per wavelength ) . this zoning is at or exceeds the resolution determined to give acceptable convergence of single - mode rt growth rates in the 3-d compressible study presented in @xcite . we also can look at how the scaling of the turbulent kinetic energy depends on the domain size . for the 384 cm wide , @xmath34 run , this is shown in figure [ fig : rt_6.67e6_tke_small2 ] . two fits are shown . when all of the data is included , the fit is quite poor , since it is biased by by data at the end of the run when the mixing has saturated and the turbulent kinetic energy stops increasing . excluding this data , the fit to the first @xmath104 s is quite good , and with the kinetic energy @xmath105 quite close to the 2.267 power obtained from the 768 cm wide run . thus , it appears that the scaling of the turbulent kinetic energy prior to saturation is insensitive to the domain size . based upon the observed systematics of flame propagation in the distributed regime , we can begin to speculate on the potential for a transition to detonation . as discussed by @xcite and @xcite , there are two possible modes for a delayed transition to detonation : a ) a `` local '' transition to detonation because a fluid element of critical mass burns with a supersonic phase velocity and b ) a `` macroscopic '' transition because some appreciable fraction of the white dwarf volume develops such complex topology that burning briefly consumes more fuel than could a spherical front encompassing that region and moving at supersonic speed . the former , also known as the zeldovich mechanism , is the basis for the `` delayed detonation model '' by @xcite ; the latter is the basis for a similar model by @xcite . @xcite has argued that the volume detonation model requires special preconditioning , but because we study only the small scales on which the flame is resolved , we can not comment in a meaningful way on this issue . also , because we have not included turbulence cascading down from scales larger than our grid , we can not conclusively argue about the local transition to detonation either ( a large amount of turbulence could , for example , induce a transition to distributed burning at a higher density ) . still some scaling relations are observed that , if they can be generalized by larger scale studies , argue against a transition to detonation . as is well recognized ( e.g. * ? ? ? * ) , transition to detonation can never occur so long as a well defined flame exists . the width of a steady flame is always thinner than the critical mass required for detonation . however , a mixture of hot fuel and cold ash is potentially explosive as calculations in the sn ia context @xcite have suggested . figure [ fig : carbon_destruction ] shows the existence , in the distributed regime , of localized hot spots of unsteady burning . the high temperature sensitivity of the @xmath20c + @xmath20c reaction itself makes it difficult for one of these hot spots to become supercritical in mass . for the conditions we consider , carbon fusion dominates the energetics , and the energy production is @xmath106 where @xmath91 is the mass fraction of carbon , and @xmath82 given by the coulomb barrier between two carbons nuclei , @xmath107 @xmath108 with @xmath109 the temperature in 10@xmath110 k. for our calculations with @xmath111 = @xmath112 , the temperature of the hot ash is @xmath113 . anticipating that the temperature of a combustible fuel - ash mixture will not be far from that , we find @xmath114 . the carbon mass fraction in the fuel is 0.5 and in the ash it is zero , and the temperature of the unburned fuel is negligible . in any mixed mass , @xmath115 , composed of fractions @xmath116 of ash and @xmath117 of fuel , the temperature will be @xmath118 and the carbon mass fraction @xmath119 the dependence of the heat capacity on temperature is important and mitigates somewhat the extreme sensitivity of the reaction rate . in the vicinity of @xmath109 = 2.5 , the heat capacity is predominantly due to the electron gas ( with @xmath120 ) , but with a non - negligible contribution from radiation ( @xmath121 ) @xcite . to good approximation , near @xmath109 = 2.5 , @xmath122 . the maximum in energy generation will then occur for @xmath123 \ = \ 0 \enskip , \ ] ] or for @xmath124that is , in the burning mixture , @xmath125 and @xmath126 . a factor of two variation in the energy generation occurs for @xmath127 , or @xmath91 from @xmath128 to @xmath129 . this agrees well with what is observed in the numerical simulation ( fig . [ fig : rt_6.67e6_yc12_dydt ] ) . coupled with the fact that burning to magnesium or silicon releases less energy than burning to the iron group , this low carbon mass fraction implies that the overpressure from burning will be small in the regions where ash and fuel are mixed . burning from @xmath130 to @xmath131 , the typical ash temperature , will only raise the pressure by 8% . since this overpressure determines the temperature reached in a detonation and the burning rate depends on this to a high power , the critical mass for initiating a detonation will be large . the size will certainly be greater than the distance a sound wave can travel during the time it takes the mixture to burn . at @xmath109 = 2.2 and @xmath132 , the nuclear time scale is @xmath133 or about 0.07 s. the speed of sound is approximately 5000 km s@xmath134 . hence a critical mass for these conditions must be at least 300 km in size . on the other hand , figure [ fig:6.67e6_scale ] shows that , in steady state , the size of the reactive region is approximately a small constant times the size of the mixed region . even allowing a range of burning time scales of a factor of 10 across the burning region , the explosive mixture only has dimensions about 10% that of the region that has been mixed by the rt instability . taking @xmath135 as an upper bound to that ( see [ sec : growth ] ) , and realizing that the temperature will decline still further in a few tenths of a second because of expansion , the largest mixed , potentially explosive region is less than 5 km . thus , though a transition to detonation is in principal possible , there simply may not be room , nor time enough to develop one in an exploding white dwarf . we presented direct numerical simulations of reactive rt instabilities in conditions appropriate to the late stages of a type ia supernova explosion . fully resolving the flame frees us from the need to specify any model parameters , such as the flame speed , markstein length , etc . in the density range we consider , the flame transitions from having a distinct , well defined interface at the high density end to being a chaotic , well mixed burning region at the lower density end , with a transition density of @xmath136 . this transition is expected based on the arguments presented in [ sec : rt ] . furthermore , this would suggest that at the even higher densities characteristic of the early stage of the explosion , the flame surface continues to be well defined , consistent with the conclusions of @xcite and the 2-d flame - model simulations at @xmath1 presented in @xcite . the effects of the burning were further made concrete by presenting the @xmath0 case both with and without reactions . at the lowest density , the mixed region becomes very large , although , still smaller than the critical carbon detonation matchhead size @xcite . it appears that the reactive region grows in direct proportion to the size of the mixed region , making it possible in theory that such a transition can occur , however , at these low densities , much of the star is already consumed , and , based on scaling predictions from the present simulations , the time required for the reactive region to grow to the matchhead size seems to be longer than the total explosion time . in effect , the flame runs out of star before a critical mass can be built up . future simulations will explore this in more depth . in all cases presented here , the flames accelerated considerably , reaching speeds between 2 and 6 times the laminar flame speed . this is significantly larger than accelerations seen in small - scale / resolved landau - darrieus instability studies ( paper i ) . the growth of the flame surface appeared to be well described by a fractal model , with a fractal dimension of @xmath137 . in the cases considered , the effective flame speed was asymptotically proportional to the increase in flame surface area but the constant of proportionality was about 0.1 for the low density cases and 0.3 for the higher density case . thus , the flame acceleration was considerably less than would be predicted by the increase in flame surface area alone . this proportionality constant may approach unity as the density is increased , because the smallest scale on which the rt instability can bend the flame ( the fire - polishing length ) increases dramatically with density , and therefore the magnitude of the curvature that the flame experiences locally decreases . we would expect that on larger domains the flame would continue to accelerate , further supporting the already widely held view that the rt instability provides most of the acceleration to the flame in a pure deflagration type ia sne explosion . the growth of the mixed region for the reactive rt instability appears to be slower than the non - reactive case . the present results do not seem to support the sharp - wheeler model , but larger scale studies are needed , with more bubbles merging over a longer period of time to better understand the growth of the mixed region . understanding the growth of the mixed region is critical to providing an accurate subgrid model . future studies will focus on the three - dimensional counterparts to the flames we discussed here . turbulence behaves very differently in two and three dimensions , so we expect to see some differences in the evolution of the instability . numerical results indicate that the growth rate of the pure rt instability is faster in 3-d than in 2-d @xcite . furthermore , in 3-d the surface to volume ratio of the spikes of fuel is larger than in 2-d , exposing more fuel to the hot ash , so we may expect them to burn away more quickly . we still expect , based on the arguments presented in [ sec : rt ] and the results of the simulations , that the flame will undergoing the same transition in behavior as the density decreases . larger two - dimensional studies in the flamelet regime are also needed , moving toward higher densities . capturing a greater range of scales on the grid will allow for validation of the fractal scaling model and a better understanding of the moderation of the sharp - wheeler prediction for the growth of the mixed region . finally , these fully resolved simulations can serve as the basis for testing various models to represent the flame on subgrid scales , which become necessary to address the larger scale type ia physics . the authors thank f. x. timmes for making his equation of state and conductivity routines available online . support for this work was provided by the doe grant no . de - fc02 - 01er41176 to the supernova science center / ucsc and the applied mathematics program of the doe office of mathematics , information , and computational sciences under the u.s . department of energy under contract no . de - ac03 - 76sf00098 . sew acknowledges nasa theory award nag5 - 12036 . some calculations were performed on the ibm sp ( seaborg ) at the national energy research scientific computing center , which is supported by the office of science of the doe under contract no . de - ac03 - 76sf00098 , the ibm power4 ( cheetah ) at ornl , sponsored by the mathematical , information , and computational sciences division ; office of advanced scientific computing research ; u.s . doe , under contract no . de - ac05 - 00or22725 with ut - battelle , llc , and the ucsc upsand cluster supported by an nsf mri grant ast-0079757 . rdrddr & & @xmath138 & & & + & & ( @xmath139 ) & & & + @xmath140 & 0.529 & @xmath141 & 5.6 & 0.026 & @xmath142 + @xmath143 & 0.482 & @xmath144 & 1.9 & 0.23 & @xmath145 + @xmath146 & 0.436 & @xmath147 & 0.54 & 1.8 & @xmath148 + rddddddl & & & & & & & comments + & & & + @xmath140 & 96.0 & 3072.0 & 1.0 & 17.1 & 3290 & 5.6 & + & 96.0 & 1536.0 & 0.5 & 17.1 & 3290 & 11.2 & + & 384.0 & 2304.0 & 1.0 & 68.6 & 14800 & 5.6 & + & 768.0 & 2304.0 & 1.0 & 137 & 29500 & 5.6 & + @xmath143 & 163.84 & 327.7 & 0.16 & 86.2 & 712 & 11.9 & + & 163.84 & 327.7 & 0.16 & 86.2 & 712 & 11.9 & no burning + @xmath146 & 53.5 & 107.0 & 0.0522 & 99.1 & 29.7 & 10.3 & + rrrl & & & comments + + & @xmath149 & 1.626 & 384 cm wide ; all data + & @xmath150 & 2.325 & 384 cm wide ; initial 0.005 s + & @xmath151 & 2.267 & 768 cm wide + @xmath143 & @xmath152 & 2.591 & + @xmath146 & @xmath153 & 2.867 & +

a type ia supernova explosion likely begins as a nuclear runaway near the center of a carbon - oxygen white dwarf . the outward propagating flame is unstable to the landau - darrieus , rayleigh - taylor , and kelvin - helmholtz instabilities , which serve to accelerate it to a large fraction of the speed of sound . we investigate the rayleigh - taylor unstable flame at the transition from the flamelet regime to the distributed - burning regime , around densities of @xmath0 , through detailed , fully resolved simulations . a low mach number , adaptive mesh hydrodynamics code is used to achieve the necessary resolution and long time scales . as the density is varied , we see a fundamental change in the character of the burning at the low end of the density range the rayleigh - taylor instability dominates the burning , whereas at the high end the burning suppresses the instability . in all cases , significant acceleration of the flame is observed , limited only by the size of the domain we are able to study . we discuss the implications of these results on the potential for a deflagration to detonation transition .

in quantum mechanics non - hermitian hamiltonians with imaginary potentials have become an important tool to describe systems with loss or gain effects @xcite . non - hermitian @xmath0 symmetric hamiltonians , i.e. hamiltonians commuting with the combined action of the parity ( @xmath2 : @xmath3 , @xmath4 ) and time reversal ( @xmath5 : @xmath6 , @xmath4 , @xmath7 ) operators , possess the interesting property that , in spite of the gain and loss , they can exhibit stationary states with real eigenvalues @xcite . when the strength of the gain and loss contributions is increased typically pairs of these real eigenvalues pass through an exceptional point , i.e. a branch point at which both the eigenvalues and the wave functions are identical , and become complex and complex conjugate . a promising candidate for the realisation of a @xmath0 symmetric quantum system are bose - einstein condensates . at sufficiently low temperatures and densities they can in the mean - field limit be described by the nonlinear gross - pitaevskii equation @xcite . if a condensate is trapped in a double - well potential it is possible to add atoms to one side of the double well and remove atoms from the other . this leads to a gain or loss to the condensate s probability amplitude . if the strength of both contributions is equal , the process can effectively be described by a complex external potential @xmath8 rendering the hamiltonian @xmath0 symmetric @xcite . the experimental realisation of an open quantum system with @xmath0 symmetry will be an important step since the experimental verification has only been achieved in optics so far @xcite . the nonlinearity @xmath9 of the gross - pitaevskii equation introduces new interesting properties such as @xmath0 broken states which appear for gain / loss contributions lower than those at which the corresponding @xmath0 symmetric states vanish . in optics the same effect can be observed for wave guides with a kerr nonlinearity . they are described by an equation being mathematically equivalent to the gross - pitaevskii equation . it has been shown that the additional features appearing in the presence of the nonlinearity might be exploited to create uni - directional structures @xcite or solitons . since @xmath0 symmetry in optics is extensively studied and has been experimentally realised @xcite these approaches seem to be very promising . from the mathematical point of view a new type of bifurcation appearing in the nonlinear @xmath0 symmetric gross - pitaevskii equation is of special interest . as in linear quantum systems two @xmath0 symmetric stationary states merge in an exceptional point if the strength of a parameter describing the gain and loss processes is increased . however , in contrast to its linear counterpart the gross - pitaevskii equation possesses no @xmath0 broken states emerging at this exceptional point . they already appear for lower strengths of the gain / loss parameter and bifurcate from one of the @xmath0 symmetric states in a pitchfork bifurcation . the new bifurcation point has been identified to be a third - order exceptional point @xcite . for attractive nonlinearities one finds that the @xmath0 broken solutions bifurcate from the ground state . in this scenario the @xmath0 symmetric ground state is the only state which exists on both sides of the bifurcation and always possesses a real energy eigenvalue . the pitchfork bifurcation is expected to entail a change of its stability . however , it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different value of the gain / loss parameter . the discrepancy between the points of bifurcation and stability change seems to be surprising and does not appear in all similar systems . the mean - field limit of a two mode approximation with a bose - hubbard hamiltonian @xcite does not show this effect . the model has , however , two crucial differences to the treatment of bose - einstein condensates with the full gross - pitaevskii equation in ref . @xcite . the latter system contains a harmonic trap in which infinitely many stationary states can be found , whereas the nonlinear two - mode system exhibits only four states , viz . the two @xmath0 symmetric and the two @xmath0 broken states mentioned above . furthermore the nonlinearity derived in @xcite is slightly different from that of the gross - pitaevskii equation . it has the form @xmath10 , and hence does not depend on the norm of the wave function . thus , there might be two reasons for the appearance of the discrepancy . it could have its origin in the existence of higher modes influencing the ground state s stability or in the norm - dependency of the gross - pitaevskii nonlinearity . it is the purpose of this article to clarify this question . to do so we study a bose - einstein condensate in an idealised double-@xmath1 trap @xcite , a system of which already its linear counterpart helped to understand basic properties of @xmath0 symmetric structures . this system is described by the gross - pitaevskii equation , i.e. the contact interaction has the norm - dependent form @xmath9 . however , it exhibits only four stationary states of which two are @xmath0 symmetric and two are @xmath0 broken as in the two - mode model @xcite . additionally , in a numerical study the structure of the nonlinearity can easily be changed such that the system s mathematical properties can be brought in agreement with the mean - field limit of the bose - hubbard dimer . the article is organised as follows . we will introduce and solve the gross - pitaevskii equation of a bose - einstein condensate in a double-@xmath1 trap for an attractive atom - atom interaction in section [ sec : bec_delta ] . some properties of the stationary solutions which are important for the following discussions are recapitulated . then we will investigate the ground state s stability in the vicinity of the bifurcation in section [ sec : stability ] . the bogoliubov - de gennes equations are solved for both types of the nonlinearity , and the origin of the discrepancy between the bifurcation and the stability change is discussed . conclusions are drawn in section [ sec : conclusion ] . we assume the condensate to be trapped in an idealised trap of two delta functions , i.e. the potential has the shape @xcite @xmath11 where the units are chosen such that the real part due to the action of the @xmath1 functions has the value -1 . it describes two symmetric infinitely thin potential wells at positions @xmath12 . in the left well we describe an outflux of atoms by a negative imaginary contribution @xmath13 , and on the right side an influx of particles is described by a positive imaginary part @xmath14 of the same strength . this leads to the time - independent gross - pitaevskii equation in dimensionless form @xcite , @xmath15 \psi(x ) = -\kappa^2 { \psi}(x ) \label{eq : gpe}\end{gathered}\ ] ] with the energy eigenvalue or chemical potential @xmath16 . the parameter @xmath17 is determined by the s - wave scattering length , which effectively describes the van der waals interaction for low temperatures and densities . physically it can be tuned via feshbach resonances . throughout this article we assume @xmath17 to be negative , i.e. the atom - atom interaction is attractive . solutions to the gross - pitaevskii equation are found with a numerical exact integration . the wave function is integrated outward from @xmath18 in positive and negative direction . to do so the initial values @xmath19 , and @xmath20 have to be chosen . the arbitrary global phase is exploited such that @xmath21 is chosen to be real . together with @xmath22 and @xmath23 we have five parameters which have to be set such that physically relevant wave functions are obtained . these have to be square - integrable and normalised in the nonlinear system . the five conditions @xmath24 , @xmath25 , and @xmath26 ensure that these conditions are fulfilled , and , together with the five initial parameters , define a five - dimensional root search problem , which is solved numerically . figure [ fig : stationary ] ( red solid lines ) , which is compared with its linear counterpart @xmath27 ( green dashed lines ) and the weaker interaction @xmath28 ( blue dotted lines ) . in the nonlinear case we observe two complex conjugate states bifurcating from the ground state in a pitchfork bifurcation . for @xmath29 this occurs at @xmath30 . ] shows a typical example for all stationary states found in the case @xmath29 ( red solid lines ) . two states with purely real eigenvalues vanish for increasing @xmath31 in an exceptional point . since @xmath20 is plotted and @xmath16 is the energy eigenvalue , the upper line corresponds to the ground state . two complex and complex conjugate eigenvalues bifurcate from this ground state in a pitchfork bifurcation at a critical value @xmath32 . the same plots for @xmath28 ( blue dotted lines ) and @xmath27 ( green dashed lines ) demonstrate how the pitchfork bifurcation is introduced by the nonlinearity . this eigenvalue structure is very generic for @xmath0 symmetric systems with a quadratic term in the hamiltonian . it is found for bose - einstein condensates in a true spatially extended double well in one and three dimensions @xcite which contain the nonlinearity @xmath33 stability analysis of the ground state -------------------------------------- the linear stability is analysed with the bogoliubov - de gennes equations . they are derived under the assumption that a stationary state @xmath34 is perturbed by a small fluctuation @xmath35 , i.e. @xmath36 \ ; , \label{eq : theta}\ ] ] where @xmath37 with this ansatz and a linearisation in the small quantities @xmath38 and @xmath39 one obtains from the gross - pitaevskii equation the coupled system of the bogoliubov - de gennes differential equations , @xmath40 u(x ) + g \psi_0(x)^2 v(x ) \ ; , \\ \frac{\mathrm{d}^2}{\mathrm{d}x^2 } & v(x ) = \bigl [ -\left ( 1 - \mathrm{i } \gamma \right ) \delta \left(x + b\right ) - \left ( 1 + \mathrm{i } \gamma \right ) \delta \left(x - b\right ) \notag \\ & + ( \kappa^2)^ * + \omega + 2 g |\psi_0(x)|^2 \bigr ] v(x ) + g \psi_0^*(x)^2 u(x ) \ ; . \end{aligned}\ ] ] [ eq : bdge ] in equation it can be seen that @xmath41 decides on the temporal evolution of the fluctuation . real values of @xmath41 describe stable oscillations , whereas imaginary parts lead to a growth or decay of the fluctuation s amplitude . thus , @xmath41 measures the stability of the stationary solution @xmath42 against small fluctuations . numerically the bogoliubov - de gennes equations are solved with the same method as the stationary states , i.e. the wave functions @xmath38 and @xmath39 are integrated outward from @xmath43 . it can easily be seen that the bogoliubov - de gennes equations are invariant under the transformation @xmath44 , @xmath45 with a real phase @xmath46 . similarly to the procedure for the integration of the stationary states this symmetry can be exploited . the remaining initial values with which the integration has to be started are @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 . in a nine - dimensional root search they have to be chosen such that the conditions @xmath52 , @xmath53 , and @xmath54 are fulfilled @xcite . two further symmetries can be exploited to reduce the number of independent stability eigenvalues which have to be calculated . the replacement @xmath55 leaves the ansatz invariant . thus , if @xmath41 is a stability eigenvalue also @xmath56 is a valid solution . furthermore for every eigenvalue @xmath41 there is also one solution with the eigenvalue @xmath57 if the stationary state @xmath42 is @xmath0 symmetric . this can be verified by applying the @xmath0 operator to the bogoliubov - de gennes equations . due to these two symmetries it is sufficient to search for stability eigenvalues with @xmath58 and @xmath59 . the relevant question which has to be answered by our calculation is whether or not the discrepancy between the @xmath31 values of the pitchfork bifurcation and the stability change appears for the double-@xmath1 potential . thus we calculated the stability eigenvalue with @xmath58 , @xmath60 for a range of @xmath31 around the bifurcation , which is shown in figure [ fig : stability_normdep ] for the stationary ground state in the case @xmath29 . the stability change occurs at @xmath61 . to illustrate the pitchfork bifurcation the real parts of @xmath20 of all stationary states are also shown ( blue dotted lines ) . the value @xmath62 is marked by the black dashed - dotted line . obviously there is a discrepancy between both values . ] for @xmath29 . for increasing @xmath31 the eigenvalue @xmath41 switches from real to imaginary at @xmath61 marking the stability change . the pitchfork bifurcation is visible in the real parts of @xmath20 of all stationary states of the system . it is marked by the black dashed - dotted line . obviously there is a discrepancy between @xmath63 and @xmath64 . the difference @xmath65 is @xmath66 . the system does not possess any further stationary states besides those shown in figure [ fig : stationary ] . three of these four states are participating in the pitchfork bifurcation . only the excited @xmath0 symmetric solution would be able to influence the dynamics of the ground state at a value @xmath67 . however , it stays real for all parameters @xmath31 shown in figure [ fig : stability_normdep ] and can not cause any qualitative different behaviour of the ground state s dynamics . thus , an influence of further states can be ruled out to be the reason for the discrepancy in the double - well system of refs . @xcite . the remaining difference between the gross - pitaevskii equation and the two mode system of refs . @xcite is the norm - dependency of the nonlinearity . indeed , an influence of the norm is already present if the study done in figure [ fig : stability_normdep ] is repeated for different values of the nonlinearity parameter @xmath17 . figure [ fig : gap_g ] and @xmath64 as defined in equation as a function of @xmath17 . a strong dependency is clearly visible . ] shows @xmath68 as a function of @xmath17 . a strong dependency is visible . even the sign changes . for @xmath69 the ground state becomes unstable at @xmath70 . for @xmath71 the discrepancy vanishes as expected . an even clearer identification of @xmath68 with the norm - dependency of the nonlinearity in the gross - pitaevskii equation can be given with a small modification . the replacement @xmath72 makes the gross - pitaevskii equation norm - independent . note that this is exactly the form of the mean - field limit of refs . @xcite . since the stationary states are normalised to 1 they are not influenced by the replacement . however , it influences the dynamics and also the linear stability . the bogoliubov - de gennes equations have to be adapted . assuming again a small perturbation of the form and a linearisation in @xmath38 and @xmath39 leads us to @xmath73 u(x ) + g \psi_0(x)^2 v(x ) \\ + g |\psi_0(x)|^2 \psi_0(x ) s \ ; , \label{eq : mod_bdge_u}\end{gathered}\ ] ] @xmath74 v(x ) + g \psi_0^\ast(x)^2 u(x ) \\ + g |\psi_0(x)|^2 \psi_0^*(x ) s \label{eq : mod_bdge_v}\end{gathered}\ ] ] with the integral @xmath75 dx \ ; . \label{eq : mod_bdge_int}\ ] ] for a numerical solution of the modified bogoliubov - de gennes equations the value of @xmath76 is included in the root search . i.e. for the integration of the equations and a value for @xmath76 is guessed and subsequently compared with the result of equation . since @xmath76 is in general a complex value this increases the dimension of the root search to 11 . an example for a typical result is shown in figure [ fig : stability_normindep ] . but for the modified bogoliubov - de gennes equations - . now the @xmath31 values at which the pitchfork bifurcation and the stability change occur agree perfectly . ] the discrepancy between the @xmath31 values at which the pitchfork bifurcation and the stability change occur vanishes . both values agree as it is expected for a pitchfork bifurcation . this is true for all values of @xmath17 . thus , we conclude that the discrepancy appearing in figure [ fig : stability_normdep ] for the gross - pitaevskii equation is solely a result of the norm - dependent nonlinearity @xmath77 in the hamiltonian and is not a consequence of the interaction with higher excited states . the reason why the stability change does not occur exactly at the bifurcation can also be understood intuitively . figure [ fig : stationary ] indicates that the value @xmath63 of the pitchfork bifurcation is not equal for all values of @xmath17 . for @xmath78 there is no pitchfork bifurcation . the two real eigenvalues @xmath20 vanish in a tangent bifurcation and two complex eigenvalues emerge . only for nonvanishing @xmath17 the pitchfork bifurcation exists and moves to smaller values of @xmath31 for increasing @xmath79 . if now the norm of a wave function changes , this can also be understood as a variation of @xmath17 , i.e. a wave function with a norm @xmath80 has the same effect as a wave function with the norm 1 and a modified value @xmath81 . the values @xmath63 of the pitchfork bifurcation is obviously different for @xmath17 and @xmath82 . with this relation in mind it is not surprising that a fluctuation changing the norm of the wave function may cause a qualitative change of the condensate s stability properties in the vicinity of @xmath63 . we investigated the origin of the discrepancy between the value of the gain / loss parameters , at which the ground state of a bose - einstein condensate in a double - well trap passes through a pitchfork bifurcation and at which its stability changes . in a naive expectation these @xmath31 values should be identical . however , it is found that this is not exactly fulfilled . since this discrepancy does not occur for a similar system , viz . the mean - field limit of a bose - hubbard dimer @xcite , we investigated the differences between the equations describing both systems . it was found that the norm - dependency of the nonlinearity in the hamiltonian @xmath77 is unambiguously the origin of the discrepancy . it can be completely removed with the replacement @xmath83 . an intuitive explanation can be given . fluctuations which change the norm of a stationary state are able to shift the position @xmath63 of the bifurcation . since the dynamical properties of the wave functions are crucial for an experimental observability of a @xmath0 symmetric bose - einstein condensate it is important to know about all processes introducing possible instabilities . as has been shown in this article the stability relations are nontrivial close to branch points in condensate setups with gain and loss . the gross - pitaevskii has a norm - dependent nonlinearity , and therefore it should be clarified in future work , which type of fluctuation influences the wave function s norm such that additional instabilities appear . in particular , it would be interesting to see , how the amplitude of a fluctuation is related to the size of the difference @xmath68 . also a deeper understanding , how this effect influences realistic setups generating the @xmath0 symmetric external potential @xcite , would be of high value .

a bose - einstein condensate trapped in a double - well potential , where atoms are incoupled to one side and extracted from the other , can in the mean - field limit be described by the nonlinear gross - pitaevskii equation ( gpe ) with a @xmath0 symmetric external potential . if the strength of the in- and outcoupling is increased two @xmath0 broken states bifurcate from the @xmath0 symmetric ground state . at this bifurcation point a stability change of the ground state is expected . however , it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different strength of the in-/outcoupling effect . we investigate a bose - einstein condensate in a @xmath0 symmetric double-@xmath1 potential and calculate the stationary states . the ground state s stability is analysed by means of the bogoliubov - de gennes equations and it is shown that the difference in the strength of the in-/outcoupling between the bifurcation and the stability change can be completely explained by the norm - dependency of the nonlinear term in the gross - pitaevskii equation . 1 1 1 1 1

periodic arrangements are quite ubiquitous in various physical systems . from naturally occurring crystal structures in solid state systems , to artificially made periodic arrangements such as photonic crystals and optical lattices , both theoretical and experimental research in these systems has provided us with an understanding of fundamental physical phenomena , as well as with the possibility of achieving technological applications . there has been a growing interest in recent years in understanding the fundamental properties of photonic crystals @xcite and of optical lattices @xcite , particularly concerning their possible applications in quantum optics and condensed matter systems . photonic crystals are periodic arrays of dielectric media in which light propagates linearly in various ways , depending on the geometry of the system . on the other hand , optical lattices are periodic arrays of electromagnetic field arising from the interference pattern of concurrent laser fields . in this case , ultracold atoms propagate nonlinearly in ways which depend also on the specific geometry of the system . both systems share important properties , namely the band structure of their energy spectrum as well as the possibility of producing confined and extended modes of light in photonic crystals @xcite and of matter in optical lattices@xcite . however , there are important physical differences which render each system unique . interaction between atoms in optical lattices has led to research into the nature of nonlinear matter waves in these systems , revealing a vast spectrum of nonlinear behaviour(see e.g. @xcite ) . by contrast , light waves in photonic crystals obey maxwell s equations and , hence , always satisfy the superposition principle . it is of conceptual importance to understand how light and matter waves in these two systems are comparable , and to investigate the possibility of using the light fields in photonic crystals to control the matter waves in optical lattices . such a study can be carried out in two regimes : in the high - particle - density and coherent regime ( in the case of photonic crystals , we mean photons , and in the case of optical lattices , we mean atoms ) , a description using a classical field in both systems is appropriate ; on the other hand , in the low particle density regime , quantum phenomena arise , and the description of both systems will require a microscopic study ( for photonic crystals see , @xcite , and for optical lattices , see @xcite ) . in the present paper , we address the first regime of high particle density . we focus on two interesting connected possibilities : first , one particular type of wave manipulation , namely the generation of confined modes produced by defects in both periodic structures ; and second , the propagation of nonlinear matter waves in the presence extended light fields produced in photonic crystals . we will see that , although defects are just one example of wave manipulation that would be interesting to implement experimentally to create elaborate optical lattices , there are technical difficulties that have to be met before a successful realization . the present paper is organized as follows : first , we review some important properies of defects in photonic crystals and in optical lattices . second , we explain how an optical lattice can be induced by extended modes in photonic crystals . finally , we exemplify this new way of obtaining optical lattices by using a photonic crystal with a two - dimensional square geometry . we conclude by suggesting the potential of using defect modes of photonic crystals to manipulate nonlinear matter waves embedded in them , and comment on the difficulties that must be resolved before realizing it experimentally . photonic crystals ( pc ) are periodic arrays of different dielectric materials whose dispersion relation presents bandgaps depending on specific structural parameters . this is their most important feature , because it allows us to control the flow of light through the material with low energy loss . the dielectric contrast of a photonic crystal is what produces a gap in the energy spectrum , for a given value of the propagation vector @xmath0 of the light field that propagates in the structure . periodicity allows us to use the bloch theorem to obtain with ease the band diagram and electric and magnetic field modes . @xcite by introducing defects , it is possible to confine a single mode of the electromagnetic field in a very narrow space of the lattice . furthermore , we can introduce an active medium , such as a quantum dot or a two level atom , inside the defect , thus making it possible to analize radiation - matter interactions @xcite . in order to obtain an analytical solution for the master equation , use is made of maxwell s equations in free space in cartesian coordinates for a square array of dielectric rods embedded in air ; only the linear term in the electric displacement is taken into account . the solution consists on diagonalizing the resulting matrix which is obtained when we expand the dielectric function and the electric field in a basis of plane waves and introducing them in the corresponding maxwell equation . the dielectric function has the form @xmath1 where the @xmath2 are the expansion coefficients of the dielectric function . these contain the information about the geometrical distribution of the dielectric rods and have the form @xmath3 where @xmath4 is the diameter of the rods , @xmath5 is the dielectric constant and @xmath6 is the length of the supercell ( @xmath7 is its area ) ; @xmath8 is the number of layers around the defect . we have obtained equation ( [ a1 ] ) analytically . the unitary cell for @xmath9 is shown in fig.([diel1 ] ) . the electric field has the form @xmath10 where @xmath11 is the periodic function that takes into account the periodicity of the field in the @xmath12 plane and the @xmath13 are the reciprocal vectors of the square array . the periodic function has the form @xmath14 introducing eq.([die21 ] ) and eq.([ezp ] ) in maxwell s equations we obtain the resulting eigenvalue equation @xmath15=k^{2}\sum_{p , q}{\alpha_{nm}\beta_{n - p , m - q}},\label{auto2}\ ] ] where the @xmath16 are the expansion coefficients of the electric field . we can note that this is a generalized eigenvalue problem whose eigenfunctions correspond to the expansion coefficients of the electric field with eigenvalues that correspond to the allowed energies for a fixed pair @xmath17 and @xmath18 . with this solution , we construct the band diagram of the pc with a defect and we analyze the effect of introducing it in the crystal . ultracold atoms in optical lattices are described in three different regimes depending on the particle density in the system @xcite . we focus on the limit of high - particle - density , in which the system remains in a macroscopic superfluid state . in this limit , we can make use of a classical field to describe the macroscopic coherent state of the atoms in the optical lattice , which is reminiscent of the classical field that describes light in photonic crystals . however , as has been noted , there is a marked difference between both systems : in the case of ultracold atoms , the waves behave nonlinearly , due to the interparticle interaction , whereas light always propagates linearly . this nonlinear description of the atomic case can be modeled using the well - known gross - pitaevskii ( gp ) equation : @xmath19\psi(\mathbf{r})=\mu \psi(\mathbf{r}),\ ] ] where @xmath20 denotes the nonlinear coupling . this equation can be derived using a hartree - type ansatz to minimize the energy of the system , asuming @xmath21wave scattering between the atoms . the wavefunction @xmath22 describes a macroscopic coherent state . in the case of optical lattices , the periodic potential is given by @xmath23 where @xmath24 is proportional to the intensity of the laser light . such a periodic potential can be generated by setting up standing waves with two lasers at perpendicular angles . the two - dimensional nature of this optical lattice can be achieved by setting up a tightly confining field in the third perpendicular direction , thus eliminating to a large extent the corresponding degree of freedom . the gp equation was numerically integrated with a procedure consisting of two stages . first , we perform a pseudospectral interpolation of the gp equation without the nonlinearity , i.e. of the schr@xmath25dinger single - particle equation . since the usual bloch argument concerning periodic lattices in solid state physics applies also in optical lattices , we make use of the bloch form of the wave function . once we obtain the solution in @xmath26-space we reconstruct the spatial wave function as a superposition of plane waves , which amounts to making the plane - wave expansion that what was done in the photonic crystal case . afterwards , we propagate this single - particle wavefunction in imaginary time , using the gp equation . the evolution in imaginary time generates a superposition of eigenstates of the gp equation with exponentially decaying coefficients . by using this procedure , the term in the superposition with the lowest energy , i.e. the ground - state , will be the least suppressed in the evolution and , thus , will be predominant after some time has passed . in this way we will have obtained the gp ground state . it is important to notice that , because the gp equation is determined by the spatial density of the system , we need to normalize the wave function continuously in the time evolution . otherwise , the system will not evolve with the correct effective gp hamiltonian . using this same procedure , we can introduce a gaussian defect in the optical lattice . such defects have been studied in one - dimensional optical lattices , and have been shown to exhibit transport properties with dynanic solitonic behaviour due mainly to the nonlinearity inherent in the system @xcite . however , here we focus mainly on the ground state of the system . it should be noted that there is an important difference between defects in optical lattices and in photonic crystals concerning confined states . in experimental setups in optical lattices , it is the ground state which can be achieved by lowering the temperature of the system . as a result , in the band diagram of the optical lattice , we always choose the point of lowest energy . this contrasts to what is done in photonic crystals , where the interest focuses on confined modes that have an energy in the bandgap of the spectrum . in this case , it is not the lowest energy state which is chosen in order to confine light waves , but rather a higher energy state . as we have discussed previously , optical lattices are stationary periodic light fields . such lattices are usually generated through the use of interfering laser fields . however , it is not necessary to use laser fields . what matters is that we have a light field that couples to the atoms we are using through an effective external dipole potential . such a dipole potential is proportional to the intensity as @xmath27 where @xmath28 denotes the field intensity . in particular , photonic crystals are suitable structures in which various types of distributions can be generated . as we reviewed in the previous section , by a suitable combination of geometry , it is possible to mold the field distribution by making use of , for example , defects . if we could take advantage of the vast set of intensity distributions in photonic crystals , we could achieve much more exotic behaviour than what has been achieved up until now in conventional optical lattices . before discussing an example of an optical lattice induced by a photonic crystal , let us first show some results of both systems independently , in which we can witness the use of defects as important means of generating more elaborate light fields in photonic crystals , and of studying nonlinear behaviour in optical lattices . for the pc case , we have taken a finite basis of plane waves to expand eq.([auto2 ] ) and to obtain a set of linear equations . after diagonalizing the resulting coefficient matrix , we have obtained the eigenfunctions corresponding to the electric field modes whose spatial distribution is shown in fig . ( [ elecfield ] ) . point defects in pc confine light modes with a definite energy in a very narrow space . this fact can be used to construct cavities with a very high quality factor and obtain a strong coupling regime wich is important , for example , in the study of polaritonic systems . furthermore , in these systems it is important to have a single mode of the field for simulating certain theoretical models in quantum optics such as the jaynes - cummings model . we have obtained the tm modes of the electric field because the square geometry only presents bandgap for such modes . these are precisely the modes which can be confined when we introduce the defect in the pc , modifying their band diagram and allowing modes into the gap . the band diagram is shown in fig.([band1 ] ) . it can be noticed that the field is highly , although not totally , concentrated in the defect region . the reason the field is largely concentrated inside the defect is due to the fact that point defects introduce defective modes in the bandgap . however , there is a region outside the defect where the intensity is not zero . this is due to the evanescent nature of the light in point defects that do not have quality factor high enough as to confine the field completely . on the other hand , for the optical lattice case , we can see the resulting ground state in fig.([ground ] ) . this can be compared to the ground state without defect , shown in fig . ( [ density ] ) . the localization effect of the wave function comes out of the minimization of the average energy of the system . since the gaussian defect serves as a deeper quantum well than other parts of the optical lattice , it becomes more energetically favourable to occupy it , even though this effect competes with the increased positive interaction between the atoms when they are brought close together . the manipulations we show here of both light and matter waves are interesting on their own right . the molding of both types of waves through the use of appropriate geometries seems to offer a large number of possibilities . in particular , it would be interesting if we could use the molding of light in photonic crystals in order to further accomplish molding of matter waves . we now discuss a simplified example of this possibility using a system with perfect square periodicity for proof of principle . however , we do not introduce defects , or any special geometry , in this example because there is a technical difficulty in photonic crystals that must first be resolved before taking advantage of the actual molding of light to manipulate atoms . we will discuss this difficulty shortly . in the following example , we have considered a square photonic crystal which has very thin dielectric posts ( of the order of @xmath29 , with @xmath30 being the lattice constant ) , so that the cold atoms we use can be affected by the the periodic optical potential without being significantly affected by the material obstacles . we chose an extended mode to illustrate a simple example in which the potential is perfectly periodic . we show this in fig.([elecfield ] ) : the electric field of the extended mode is mainly concentrated at the dielectric posts , further inhibiting the atoms to come into close proximity of the dielectric posts . the gross - pitaevskii ground state was computed using this potential , and the resulting distribution is shown in fig . ( [ ground2 ] ) . this distribution is markedly different from that of the usual optical lattice . the new effective potential is such that each potential minimum is not completely separated by a tunneling barrier from the neighboring minima but , instead , the potential tends to be flat . this facilitates the motion of the atoms between minima of the optical potential . because of this , we see that the height of the wave function does not change significantly in the region between the posts , which contrasts with the ground state found in fig.([density ] ) . the solution shown in fig.([ground2 ] ) exhibits the basic effect of having an extended mode of the pc as an optical lattice . in an actual experimental setup it would be necessary to have an external parabolic potential to provide a confinement for the atoms ; otherwise they would just escape from the photonic crystal . we must address the deterrents which might make it difficult to use these types of optical lattices . the most important one is that if we use , for example , periodic defects in a photonic crystal to induce an optical lattice , we must take into account that the kinds of modes we can produce have some field minima inside the dielectric regions , which would promote collisions of atoms with the posts . although this is not an actual disadvantage , it does require a careful theoretical study in order to quantify such effects . furthermore , even if we can manage to have the light intensity concentrate at the posts , there will inevitably be some residual scattering of the atoms from the posts . this would make it difficult to model the system theoretically , as opposed to relatively simple hamiltonians that arise in the standard optical lattices . finally , there is the experimental difficulty of mounting atoms to the photonic crystal and of performing , for example , time - of - flight measurements to probe the state of the system . however , we feel these experimental difficulties , although important , could be eventually resolved through suitable modifications of the usual experimental setups for optical lattices . in the present work we have explored the possibility of using photonic crystals to induce an optical lattice . by using geometrical constructs such as defects , it is possible to manipulate and mold the intensity distribution of light waves in photonic crystals and the nonlinear dynamics of matter waves in optical lattices . we have argued that a suitable combination of both will provide a new source of many - body quantum behaviour . furthermore , using an extended mode in photonic crystals in order to generate the periodic potential for an optical lattice could be more favourable in reducing costs and in exploring new geometries which can not be obtained easily with the conventional lasers used today . thus , there is a practical , besides fundamental , justification for considering such new types of optical lattices . the authors acknowledge financial support from the codi - universidad de antioquia . we aknowledge j. p. vasco , b. rodriguez and p. soares guimaraes for enlightening discussions . 10 yamamoto y , machida s and bjrk g 1991 _ phys . rev . a _ * 44 * 657 - 668 abdel - aty m 2005 _ applied physics b : lasers and optics _ * 81 * 193 akahane y , asano t , song b and noda s 2003 _ nature _ * 425 * 944 vignolini s , riboli f , intonti f , belotti m , gurioli m and chen y 2008 _ phys . e _ * 78 * 045603 badolato a , winger m , hennessy k j , hu e l and imamoglu a 2008 _ comptes rendus physique _ * 9 * 850 - 856 tuchman a k and kasevich m a 2009 _ phys . lett . _ * 103 * 130403 catani j , barontini g , lamporesi g , rabatti f , thalhammer g , minardi f , stringari s and inguscio m 2009 _ phys . lett . _ * 103 * 140401 pepino r a , cooper j , anderson d z and holland m j 2009 _ phys . lett . _ * 103 * 140405 mishmash r v and carr l d 2009 _ phys . * 103 * 140403 joannopoulos j d , meade r d , johnson s g and winn j n 2008 _ photonic crystals : molding the flow of light , second edition _ ( new jersey : princeton university press ) morsch o and oberthaler m 2006 _ rev . * 78 * 179 stanescu t d , galitski v , vaishnav j y , clark c w and das sarma s 2009 _ phys . a _ * 79 * 053639 peterson m r , zhang c , tewari s and das sarma s 2008 _ phys . lett . _ * 15 * 150406 wang b , chen h , das sarma s 2009 _ phys . rev . a _ * 79 * 051604 wu c and das sarma s 2008 _ phys . * 235107 greentree a d , tahan c , cole j h and hollenberg l c l 2006 _ nature _ * 2 * 856 - 861 jaksch d , bruder c , cirac j i , gardiner c w and zoller p 1998 _ phys . rev . * 81 * 3108 - 3111 yablonovitch e 1987 _ phys . lett . _ * 58 * 2059 - 2026 brazhnyi v a , konotop v v and prez - garca v m 2006 _ phys lett . _ * 96 * 060403

we propose a way of generating optical lattices embedded in photonic crystals . by setting up extended modes in photonic crystals , ultracold atoms can be mounted in different types of field intensity distributions . this novel way of constructing optical lattices can be used to produce more elaborate periodic potentials by manufacturing appropriate geometries of photonic crystals . we exemplify this with a square lattice and comment on the possibility of using geometries with defects .

the understanding of spin dynamics in ferromagnets is widely based on radio frequency spectroscopy @xcite and inelastic scattering techniques employing photons @xcite or neutrons @xcite . these experimental methods provide frequencies and decay rates of magnetic excitations by the line position and line width , respectively . more recently time domain techniques that employ short laser pulses have been established in a considerable number of laboratories and provide a complementary approach to analyze spin dynamics @xcite . in pump - probe experiments the magnetization dynamics is driven by an external stimulus like a laser or magnetic field pulse and is probed by a second , time - delayed laser pulse . using femtosecond laser pulses , which are nowadays routinely provided by ti : sapphire laser systems , the field of femtosecond magnetization dynamics has developed enormously . one reason for the interest in this approach might be that it is not only complementary to the established tools for magnetization dynamics , but that the non - equilibrium state and the respective dynamical magnetization processes have become accessible for systematic experiments @xcite . a number of novel phenomena like laser - induced demagnetization on sub - picosecond time scales @xcite , laser - driven formation of ferromagnetic order @xcite , and magnetization reversal by individual laser pulses @xcite demonstrates the potential for spin manipulation in solid state materials on ultrafast time scales . at the same time , these phenomena challenge our understanding of ferromagnetism and ask for an appropriate description of the observed dynamics . beside the investigation of multi - constituent materials with a potential for novel phenomena , it is essential to develop insight into the more general laser - driven dynamics to establish this understanding . finally , the meanwhile well established sub - picosecond demagnetization of elemental metallic ferromagnets is still controversial because the underlying elementary interactions remain under discussion . it has been shown recently that descriptions based on phonon - mediated spin - flip scattering ( elliott - yafet type ) @xcite , electron - spin - flip scattering ( stoner excitation ) @xcite , and ballistic transport of spin - polarized charge carriers @xcite can be used to describe rather similar experimental data of femtosecond demagnetization in ni films . in the phonon - mediated process the magnetic moment is considered to be transferred directly to the lattice through phonons with appropriate symmetry obeying angular momentum conservation . in the electronic spin - flip scattering process the magnetic moment is not transferred within the elementary scattering process itself , but is dissipated by secondary low energy spin excitations , which in their sum lead to demagnetization @xcite . in the transport concept @xcite the spin polarization is basically taken out of the ferromagnet into a substrate or , alternatively , out of the experimental observation window . therefore , the current understanding is not conclusive and a discrimination of these above processes is important at this stage . our efforts have concentrated on rare earth ferromagnets and in particular on gadolinium which is considered to be a model system for a heisenberg ferromagnet with a magnetic moment localized at the ion core . we have employed in earlier studies time - resolved surface sensitive techniques which are optical magneto - induced second harmonic generation ( shg ) and photoelectron spectroscopy , to analyze the excited state and its relaxation . the response to the laser excitation is characterized by a coherent phonon - magnon mode localized at the surface and incoherent dynamics of optically excited electron - hole pairs , lattice vibrations , and magnetic excitations . a comprehensive overview is given in refs . @xcite . here we present experimental results obtained by the femtosecond time - resolved magneto - optical kerr effect ( moke ) on gd(0001 ) films to analyze the magnetization dynamics in the bulk part of the gd film , which we compare with the surface sensitive shg signal . we find pronounced differences during the first 2 ps which we attribute to transfer of spin - polarized carriers between the surface and bulk part of the film . moreover , we study the fluence dependence of the demagnetization process and compare it with predictions based on the two - temperature model . the experiments were carried out at a setup that combines an ultrahigh vacuum ( uhv ) chamber and a femtosecond ( fs ) ti : sapphire laser . for details see @xcite . gd was evaporated from an electron beam heated crucible under uhv conditions onto a w(110 ) substrate at 300 k. annealing to 700 k subsequent to the deposition results in smooth epitaxial films @xcite . here we studied films with 20 nm thickness . pump - probe experiments were performed at 50 k equilibrium temperature of the sample which was cooled by a liquid he cryostat . laser pulses of 35 fs duration and 40 nj energy were generated by a cavity dumped ti : sapphire oscillator at 800 nm central wave length with a repetition rate of 1.52 mhz . for pump - probe experiments the laser output was divided at a ratio 4:1 . to detect the p - polarized second harmonic ( sh ) generated by the p - polarized probe pulse a monochromator selecting 400 nm and a photomultiplier were used . to analyze the moke a balance detection scheme was used to measure the polarization rotation of the probe pulse @xmath1 in the longitudinal moke geometry . the magneto - optical signals were analyzed by a lock - in amplifier for moke and a single photon counter for shg to detect the differential signals for open and blocked pump in opposite saturation magnetic fields ( @xmath2 ) as a function of pump - probe delay @xmath3 . formally negative delays @xmath4 denote the state before the optical excitation has occurred . for moke we show below @xmath5 where @xmath6 is the angle between the linear polarization of the reflected probe pulse and a fixed reference polarization , @xmath1 is the magneto - optical kerr rotation . @xmath7 is the time - dependent change in the kerr rotation which represents the magnetization of the bulk part of the gd film . shg is detected in the transversal geometry leading to magneto - induced changes in the sh intensity . @xmath8 here @xmath9 and @xmath10 are the shg optical fields which behave as even or odd with respect to reversal of the magnetization @xmath11 . the phase between these two field contributions is @xmath12 which is smaller than @xmath13 and weakly time - dependent @xcite . since @xmath14 we plot below @xmath15 which represents the time - dependence of the surface sensitive magneto - optical signal . the article presents results of time - resolved shg and moke . the first section focuses on the comparison of surface and bulk sensitive detection of the transient magnetic state in gd(0001 ) . the second section contains a detailed study of time - resolved moke and reports on the fluence dependence of the bulk demagnetization . we start by comparing the transient shg and moke signals that probe the surface and bulk of the gd(0001 ) film , respectively . in fig . [ fig1 ] , top panel , we show @xmath16 ( solid line , left axis , surface ) and @xmath17 ( dotted line , right axis , bulk ) taken at nominally identical experimental conditions . both signals indicate a pronounced laser - induced demagnetization , but their transient behavior is very different . the magnetic shg signal is reduced within the laser pulse duration of 35 fs by 0.5 and exhibits on that level oscillations that are damped out within 3 ps . the signal starts to recover after 1.5 ps and returns to its value before optical excitation after several 100 ps ( not shown , see ref . the oscillatory part has been explained before as a coupled phonon - magnon mode localized to the surface @xcite . it is therefore expected , that it is not found in the bulk signal . the moke data can be described by a continuous reduction that will be fitted by a single exponential decay below and tends to saturate at a delay of 3 ps , before it demagnetizes further , not shown see ref . overall , the bulk sensitive signal is reduced on a much slower time scale than the surface one and is in agreement with femtosecond x - ray magnetic dichroism studies @xcite if the different excitation densities are considered . we therefore conclude that the time - resolved moke signal probes the time - dependent magnetization of the film . we note here that a detailed comparison of moke ellipticity and rotation points to time - dependent variations of the magneto - optical constants at delays where the excess energy resides predominantly in the electronic system , as will be discussed in a forthcoming publication @xcite . to explain the difference between the surface and bulk demagnetization it is informative to inspect the electronic structure at the surface and to take transfer processes into account . [ fig1 ] , bottom , depicts several data sets which in their combination represent the electronic valence states of epitaxial gd(0001 ) films . for a ferromagnetically ordered situation the states are exchange - split due to intra - atomic exchange interaction with the strong magnetic moment of the half filled @xmath18 shell . at the @xmath19-point the bulk @xmath20-states appear at 1.4 ev ( minority ) and 2.4 ev ( majority ) binding energy and disperse towards @xmath21 with increasing @xmath22 @xcite . the majority component of the exchange - split @xmath23-surface state is dominantly occupied and the minority one unoccupied . the system also contains unoccupied exchange - split bulk states @xcite . top panel : time - dependent magneto - optical signals measured on 20 nm thick gd(0001 ) films , which were grown epitaxially on w(110 ) . the solid line shows the pump - induced change of the magnetic shg contribution @xmath24 sensitive to the surface . similar results were published before @xcite . the dotted line depicts the pump - induced moke polarization rotation @xmath25 normalized to the static rotation @xmath26 . the bottom panel is reprinted from ref . @xcite and shows the valence electron states of gd(0001 ) taken from normal direction photoemission ( circles ) , inverse photoemission ( solid lines ) , and scanning tunneling spectroscopy data above the fermi level ( solid line , filled ) @xcite . the exchange - split surface state ( filled area ) appears around the fermi level . vertical arrows represent the majority and minority character of the electronic states . indicated are the two main absorption channels for 1.5 ev pump photons and the resonant second harmonic probing scheme . ] for linear polarized light , within the dipole approximation , optical transitions proceed among occupied and unoccupied electronic states in ( bulk ) valence bands with the same spin character . in case of the excitation by the 800 nm ( or 1.55 ev ) pump laser pulses , an additional excitation channel becomes available at the surface because of resonant transitions coupling the bulk and the localized surface electronic states , see fig . [ fig1 ] , bottom . therefore a difference of the surface sensitive shg response with respect to the bulk sensitive moke can be expected . similarly to a charge - transfer excitation across molecular interfaces @xcite , such surface - bulk resonances lead to effective electron transfer between surface and bulk which modify the transient population of the minority and majority components of the surface state . the population of the initially occupied majority surface state component is reduced due to the surface - to - bulk spin - up electron transfer , see fig . [ fig1 ] , bottom . at the same time , the population of the initially unoccupied minority surface state component increases due to the bulk - to - surface spin - down electron transfer , which can alternatively be viewed as a surface - to - bulk spin - down hole transfer . the carriers in the surface state which are transfered to bulk bands represent initially a wave packet at the surface which spreads and propagates into the bulk material with its fermi velocity @xmath27 of about 1 nm / fs @xcite . thus , a spin - polarized current which propagates from the surface to the bulk is optically excited . in the vicinity of the surface this current propagates ballistically and consists of electron and hole contributions . spin - down holes have the same spin polarization as spin - up electrons . therefore , the spin components of these two contributions will add up . if the charge of electrons and holes is in sum zero we conclude that a net spin current between surface and bulk is excited . it is likely that the efficiency of electronic transitions in spin - up and spin - down channels are different ( fig . [ fig1 ] , bottom ) which will lead to a prevalence of either the electron or hole component of the spin - polarized current . note that this does not necessarily imply charging of the surface because a transient charge imbalance will be screened by electron rearrangement on the time scale of the inverse plasma frequency @xcite . note that on the basis of these considerations an effect due to optically excited transfer of spin polarization between surface and bulk should proceed within few femtoseconds , i.e. well within the time resolution of the shg experiment . as seen in fig . [ fig1 ] , top panel , the pronounced initial drop in the surface sensitive magneto - optical signal occurs within the experimental time resolution and clearly faster than the bulk demagnetization . therefore , we explain the pronounced difference in the bulk and surface demagnetization times by spin transfer between surface and bulk of the film . the time scale at which the magnetic shg signal @xmath28 changes from 0 to -0.5 at time zero is in agreement with a ballistic character of these transport effects . the pump - induced reduction of the relative magneto - optical signals differ by a factor of six . we consider the resonant surface excitation among bulk and the surface states near @xmath19 , see fig . [ fig1 ] , as essential for an explanation of the pronounced effect at the surface . symbols indicate experimental data of time - resolved moke for different relative pump fluences @xmath29 , with @xmath30 mj/@xmath0 . the solid lines are fits considering a single exponential decay . ] the setup used for the present study allowed in addition a more detailed analysis of the bulk demagnetization phenomenon . using a combination of half wave plate and glan - thomson polarizer the absorbed fluence has been reduced step wise from the maximum value @xmath31 , @xmath32 was determined to be @xmath33 mj/@xmath0 . [ fig2 ] shows representative time - resolved moke curves for different @xmath29 up to delay times of 4 ps . the values at 4 ps decrease linearly with @xmath29 . this ensures that we are analyzing a low excitation density regime reasonably far away from a full demagnetization of the sample , where magnetic fluctuations would contribute to the ultrafast magnetization dynamics @xcite . near time zero the data feature a less clear behavior and an effectively positive contribution in @xmath34 . the time delay at which the signal crosses the zero line shifts with increasing fluence closer to time zero . while for @xmath35 the amplitude of the positive and negative contributions are comparable to each other , the negative one dominates for larger @xmath29 . in this study we focus on the pronounced demagnetization dynamics , i.e. at the fluences where the negative contribution dominates . the origin of the positive contributions are currently under discussion since they could originate from transient changes in the magneto - optical constants or the spin polarization of the @xmath20 electrons . all data were fitted by a single exponential time dependence with fixed time zero and variable @xmath17 at 0 and 4 ps . the obtained fits are shown in fig . [ fig2 ] and describe the experimental data well . [ fig3 ] depicts the time constants @xmath36 determined by the fitting procedure . the numbers range from about 0.5 to 0.8 ps with a trend towards larger times for higher @xmath29 . the smallest @xmath36 are obtained for the lowest @xmath37 and 0.27 and the respective @xmath36 are out of the weak linear increase observed for @xmath38 . we already mentioned that for such small @xmath29 the positive signal near time zero is comparable in size to the demagnetization observed at later delays . it is therefore well possible that the obtained @xmath36 is influenced by the processes that are responsible for the positive @xmath17 and we refrain from further conclusions based on @xmath36 obtained for @xmath39 . characteristic times determined from the decay times in the single exponential fitting of @xmath40 as a function of relative fluence . the line is a linear fit to the data with @xmath41 we fitted the values for @xmath38 by a linear dependence . the result is shown in fig . [ fig3 ] by a line . we find that the demagnetization time increases by @xmath42 fs within @xmath31 with a nominal zero fluence limit of @xmath43 fs . for itinerant ferromagnets like ni or co an observed increase in @xmath36 with fluence in combination with a finite value at zero fluence was explained by model descriptions based on elliott - yafet scattering @xcite and stoner excitations @xcite . the case of gd has been discussed in ref . it features two separate demagnetization times in agreement with the experimental observation @xcite . these two time scales are attributed ( i ) to cooling of the hot electron system through e - ph coupling for the fast time scale , which is discussed in the present article , and ( ii ) to the demagnetization determined by the equilibrium spin - flip probability , which is weaker in gd than for @xmath44 transition metal ferromagnets and sets the slower timescale observed in refs . @xcite . this description predicts a fluence dependence of the ultrafast demagnetization of gd that is determined by the increase of the transient lattice temperature @xmath45 . we have calculated @xmath46 and @xmath47 for different fluence by the well known two - temperature model @xcite in refined versions , which were published earlier @xcite ; regarding its application to gd(0001 ) see @xcite . considering a variation in fluence following the experimentally investigated range we find a pronounced variation of the time scale on which @xmath46 and @xmath47 equilibrate ( not shown ) . this change in time scale is at a first glance more pronounced than the changes found for the demagnetization time in fig . [ fig3 ] . however , a systematic analysis of the time scale at which @xmath46 and @xmath47 are changed is non - trivial since they vary in a non - exponential way . therefore , we turn to a discussion of the excess energy @xmath48 of the electronic system which is related to the electron temperature through @xmath49 , with @xmath50 being the linear parameter in the temperature dependent specific heat of the electron system . [ fig4 ] shows the results obtained for @xmath51 with @xmath52 varying between 0.2 and 1 . the transient behavior can be described by a single exponential that represents energy transfer from the electron system to the lattice . the exponential time scales were determined with 100800 fs and are plotted in the inset of fig . the energy transfer time shifts to larger values for higher fluence and changes by three times in the investigated fluence range . consequently the energy content of the lattice changes with these varying time constants . main panel : time - dependent excess energy density for different relative fluence , which was calculated by the two - temperature model . the inset shows the characteristic times @xmath53 of energy transfer from the electronic system to the lattice mediated by electron - phonon scattering which was determined by single exponential decay to the excess energy density in time interval from 100 to 800 fs . ] the comparison between time - dependent demagnetization curves and the time evolution of excess energy emphasizes two points which lead to the conclusion that the explanation of the ultrafast demagnetization in gd solely through the time - dependent electron or lattice temperature is incomplete . at first we argue that the range of the energy transfer times @xmath53 found in the investigated fluence range is with 0.20.6 ps considerably broader than the range observed for @xmath36 which is about 0.650.8 ps ( see figs . [ fig3],[fig4 ] ) . second , and maybe more general , the demagnetization can be described by a simple exponential time dependence similar to the transient energy density . in contrast , the transient electron and lattice temperatures follow a more complicated evolution . moreover , spin fluctuations which present a sizeable energy content of the system in particular in excited magnetic systems @xcite , might be essential to take into account . our pump - probe analysis of fs laser - induced magnetization dynamics in gd(0001 ) revealed two qualitatively different mechanisms , which both change transiently the magnetic moment per atom . the one mechanism is transfer of angular momentum from the spin system to another degree of freedom of the sample , which is finally the lattice . in this case the magnetic moment of the sample as a whole is reduced . such mechanisms are discussed in the literature @xcite to offer consistent descriptions of ultrafast demagnetization as observed in several time - resolved magneto - optical studies , see e.g. ref . @xcite , and x - ray magnetic - circular dichroism ( xmcd ) experiments @xcite for ni . the observed demagnetization time of about 700 fs determined for gd(0001 ) can be explained by relaxation of the optically excited electrons by interactions with phonons or spin waves which subsequently interact with the lattice . on a more microscopic level the demagnetization process in ni and gd must be different , because in a transition metal the magnetic moment is formed in the same band which is also optically excited . in a lanthanide like gd the dominant part of the magnetic moment is localized near the ion core and resides in the @xmath18 level , which is not primarily optically excited @xcite . to achieve significant demagnetization the excess energy generated by the laser excitation must be transfered to the @xmath18 electrons . while details of this process will be discussed in a forthcoming publication , we note here that optically excited @xmath20 electrons are coupled to the localized @xmath18 electrons through intraatomic exchange interaction . ultrafast changes of the magnetization , which are investigated by moke , can be expected to be dominated by the @xmath20 contribution to the magnetic moment . the femtosecond xmcd study performed at the gd m@xmath54 edge probes the magnetic moment of the @xmath18 levels directly @xcite . a comparison of the time scales of demagnetization obtained in both these experiments suggests that the @xmath18 and @xmath20 contribution to the magnetic moment are essentially strongly coupled on ultrafast time scales accessible here and demagnetize as a total magnetic moment . the second mechanism investigated here is spin transfer which we probe by means of the exchange - split spin polarized 5@xmath55 surface state of gd(0001 ) . here the magnetic moment integrated over the sample remains constant , but is redistributed between different parts of the sample , in our case among surface and bulk electronic states . we explain this process to originate from the optical transitions that couple the exchange - split surface and bulk states . in the literature a similar scenario has been discussed recently as a potential explanation for bulk demagnetization in ni @xcite . in this theoretical study the authors consider the optically excited transport of hot carriers in metallic samples which redistribute the magnetic moment from the ferromagnetic layer into a para- or diamagnetic substrate . such transport effects of hot carriers are known since early investigations of femtosecond electron dynamics in metallic layers @xcite , however , to what fraction they are indeed responsible for the ultrafast demagnetization remains an interesting question which is to be clarified in future investigations . by combining ultrafast bulk and surface sensitive magneto - optical techniques we have analyzed the femtosecond laser - induced demagnetization of epitaxial gd(0001 ) films . in the bulk the demagnetization occurs with a characteristic time of about 0.7 ps . considering that the surface sensitive signal changes within the laser pulse duration of 35 fs and taking resonant optical transitions between valence electronic surface and bulk states into account , we attribute this ultrafast change to transfer of spin - polarized charge carriers between surface and bulk states . variation of the pump fluence up to 1 mj/@xmath0 shows a weak increase in the bulk demagnetization time of about 70 fs/(mj/@xmath0 ) . a comparison with the fluence dependence expected from the energy transfer among electrons and the lattice revealed that albeit there is qualitative agreement in the increasing trend of demagnetization time such description remains incomplete due to pronounced quantitative variations . we are grateful for the continuous support by m. wolf and acknowledge fruitful discussions with o. chubykalo - fesenko . this work was supported by the deutsche forschungsgemeinschaft through me3570/1 and by the hec - daad . c. stamm and t. kachel and n. pontius and r. mitzner and t. quast and k. holldack and s. khan and c. lupulescu and e. f. aziz and m. wietstruk and h. a. drr and w. eberhardt , nature mater . * 6 * , 740 ( 2007 )

ultrafast laser - induced demagnetization of gd(0001 ) has been investigated by magneto - induced optical second harmonic generation and the magneto - optical kerr effect which facilitate a comparison of surface and bulk dynamics . we observe pronounced differences in the transient changes of the surface and bulk sensitive magneto - optical signals which we attribute to transfer of optically excited , spin - polarized carriers between surface and bulk states of the gd(0001 ) film . a fluence dependent analysis of the bulk magnetization dynamics results in a weak variation of the demagnetization time constant , which starts at about 700 fs and increases by 10% within a fluence variation up to 1 mj/@xmath0 . we compare these results with fluence dependent changes in the transient energy density calculated by the two temperature model . the determined characteristic times of excess energy transfer from the electron system to the lattice , which is mediated by e - ph scattering , range from 0.2 - 0.6 ps . such a more pronounced fluence dependent change in the characteristic time compared to the observed rather weakly varying demagnetization times suggests a more advanced description of the optically excited state than by the two - temperature model .

an open universe , with @xmath0 , seems to fit most astronomical observations . in connection with cdm , it gives the best fit to the observed clustering ( see e.g. ref . @xcite ) ; a similar value is requested for explaining the dynamics of bound objects on relatively small scales ( see e.g. ref . @xcite ) ; it also increases the age of the universe , alleviating the conflict with the age of the globular clusters . further , @xmath0 is in better agreement with direct geometrical estimates from number counts ( see e.g. ref . @xcite @xcite ) . however , as it is well known , inflation predicts @xmath1 . this contradiction is one of the most interesting problems in modern cosmology . it is certainly possible that the observations in favour of an open universe are too limited to be representative of the whole universe , and that higher values of @xmath2 will be found at larger scales . on the other hand , it is also possible to choose the initial conditions in inflation so as to give @xmath3 today , either by starting with an extremely small density parameter at the beginning of inflation , or by assuming that inflation lasted less than 60 @xmath4-foldings or so . both possibilities , however , introduce that fine - tuning of the initial conditions that inflation itself tried to overcome ; moreover , there would be also a conflict with the microwave background isotropy @xcite . two ways out of the enigma have been proposed so far . the first , the single- bubble scenario @xcite @xcite @xcite , assumes that a single giant bubble nucleated from a false vacuum ( fv ) state , when the universe was already flat because of previous inflationary expansion , and inflated for about 60 @xmath4-foldings afterward : our horizon is contained inside the bubble , and appears to be locally open . the weak point of this model is the old graceful exit problem : inflation never ends outside the bubble , and there is no reason why we should live inside that infinitesimal fraction of space which nucleated out of the false vacuum , unless one invokes anthropic arguments . other problems of the single - bubble model have been discussed in ref . @xcite . in the second proposal , the many - bubbles scenario @xcite , one has a two - field potential : while a field drives the inflationary slow - rolling , the second one performs a phase transition , generating bubble - like open universes , with all possible density parameters , from zero to unity , and all possible sizes . here the phase transition completes , all the universe eventually nucleates out of the fv state , but again there is no reason to expect nor a preferential value of @xmath2 , nor a bubble size big enough to contain our horizon . actually , it is difficult to avoid the conclusion that most of the volume is inside bubbles nucleated at the end of inflation , thus exponentially smaller than our observed universe . linde @xcite argues that eventually quantum cosmology will explain why we live in a open universe ( if we really do ) . the model we propose in this work implements a many - bubble scenario in which the nucleation rate varies in such a way to give a @xmath3 universe with maximal probability . in our model , the peak of the bubble nucleation can be chosen to occur early enough to have super - horizon - sized bubbles , approximating a @xmath3 universe by today , and narrow enough to consider our universe as typical . in other words , a flat , huge ( or infinite ) universe appears to be composed of locally open super - horizon - sized bubbles . it is remarkable that local observations , @xmath2 and the amplitude of perturbations , and the `` natural '' assumption that our universe is typical , put strong constraints on the theory parameters , i.e. the fundamental scales of the inflation and of the primordial phase transition . our model , outlined in the next section , has been already introduced in ref . @xcite to produce large scale power out of the remnants of the primordial phase transition . all what we have to do here is to determine the parameters so to tune the nucleation peak at super - horizon scales . to realize our scenario we need two pre - requisites . first , we need two channels , a false vacuum channel , to drive the inflation in the parent universe , and a true vacuum channel , to drive the shorter inflation inside the bubbles . second , we need a tunneling rate tunable in time , so as to produce a nucleation peak at the right time . it is remarkable that the same model that we introduced in ref . @xcite has just these features . this is certainly not the unique possibility , but we will show that it is a rather simple one . the model works in fourth order gravity @xcite , and exploits two fields : one , the scalaron @xmath5 ( i.e. the ricci scalar ) drives the slow - rolling inflation ; the second , @xmath6 , performs the first - order phase transition . the phase transition dynamics is governed both by the potential of @xmath7 and by its coupling to @xmath5 ; the dynamics of the slow - roll is `` built - in '' in the fourth order lagrangian . we already presented our model in detail in ref . @xcite ; here we sketch its main features . our starting point is the lagrangian density @xmath8 , where ( @xmath9 ) @xmath10 and @xmath11 the coupling of the scalaron with @xmath6 can be thought of as a field - dependent effective mass @xmath12 just like in brans - dicke gravity the coupling is a field - dependent planck mass . putting @xmath13 , one finds in the slow - rolling regime @xmath14 which is essentially the only dynamical equation we need . in the following , the instant labelled _ in _ will correspond to the beginning of the last @xmath15 @xmath4-foldings of inflation . the theory ( [ lagra ] ) can be conformally transformed @xcite into canonical gravity with the new metric @xmath16 once written in the conformal frame , our model becomes indistinguishable from a ordinary gravity theory with two fields governed by a specific potential . in the slow - rolling approximation useful relations link @xmath17 and the number @xmath18 of @xmath4-foldings to the end of inflation : @xmath19 correspondingly , @xmath20 . the value @xmath21 is fixed by the request that the largest observable scale , @xmath22 , was crossing out the horizon @xmath21 @xmath4-foldings before the end of inflation , and it is close to 60 for standard cosmological values @xcite . from ( [ lagra ] ) and ( [ conformal ] ) we obtain then einstein gravity with two scalar fields @xmath6 and @xmath23 , coupled by a potential given by @xmath24\,.\ ] ] the choice of a quartic for @xmath25 and a mass term for @xmath26 realizes the two conditions discussed above : @xmath27 this carves in fact in ( [ conpot ] ) two parallel channels of different height , separated by a peak at @xmath28 . the degeneracy of @xmath29 in @xmath30 and @xmath31 is indeed removed by @xmath32 ; the true vacuum ( tv ) channel remains at @xmath33 , while the false vacuum ( fv ) channel is slightly displaced from @xmath34 . in ref . @xcite we evaluated the tunneling rate @xmath35 for our model , defined as @xmath36 where @xmath37 is of the order of the energy of the false vacuum , and @xmath38 is the minimal euclidean action , i.e. the action for the so - called bounce solution of the euclidean equation of motion . the calculation of @xmath38 is simplified in the limit @xmath39 , i.e. for @xmath40 . in this case in fact @xmath41 and we can directly use coleman s formulas @xcite to evaluate @xmath38 , provided we are in the thin wall limit ( twl ) . if @xmath42 ( @xmath43 ) is the potential energy of the false vacuum ( true vacuum ) state , and @xmath44 is the energy at the top of the barrier , the twl is guaranteed if @xmath45 , i.e. if @xmath46 the result is ( ref . @xcite ) @xmath47 the eulidean action decreases as @xmath48 decreases ; the bubble nucleation is then more likely to occur during the last stages of inflation than at earlier times . finally , we can write the relevant parameter @xmath49 , i.e. the number of bubbles per horizon volume per hubble time ( quadrihorizon , for short ) , as @xmath50 where we have introduced a new parameter @xmath51 to mark the time at which there is on average one bubble per quadrihorizon , roughly corresponding to the end of the phase transition . it is useful to keep in mind that , as we will show later , @xmath52 . @xmath51 , or @xmath37 , in principle , can be derived in terms of the potential parameters @xcite ; the derivation is , however , very difficult : as customarily done in extended inflation ( see e.g. @xcite ) , we will determine @xmath51 by requesting that the transition eventually ends . to summarize , our model has four characteristic quantities : the slow - rolling inflationary rate , set by @xmath53 ; the difference in energy between vacua states , set by @xmath54 ; the barrier height , set by @xmath55 ; and finally the separation between the vacua states , set by @xmath34 . these constants completely define the slow - rolling and the phase transition dynamics . in the next section we proceed to the evaluation of the tunneling probability , and show that we can tune the parameters to give @xmath3 today with maximal probability . let the number of bubbles per horizon nucleated in the time interval @xmath56 be @xmath57 , where @xcite @xmath58 \ , , \label{ourspe}\ ] ] where @xmath59 is the nucleation rate , and where @xmath60 is the horizon volume at @xmath21 . the quantity @xmath61 is proportional to the tunneling rate per volume , @xmath35 , and to the fv volume left at the time @xmath62 ( the volume not already occupied by bubbles ) . if after a certain time the exponential term in ( [ ourspe ] ) decreases faster than @xmath63 increases , the fv volume fraction decreases ; if this decrease is faster than @xmath35 increases , then @xmath61 will have a turnaround somewhere , indicating that the transition is being completed . the rest of this paper is essentially devoted to find the condition for this maximum to occur at the right time . the @xmath4-folding time @xmath48 is defined as @xmath18 , where @xmath64 , so that we can put @xmath65 since in all what follows we have @xmath66 ( i.e. , the nucleation occurs around @xmath21 ) , we can write @xmath67\,,\ ] ] where @xmath68 we will neglect the mild dependence of @xmath69 on @xmath48 compared to the exponential dependence of @xmath35 . during inflation , @xmath70 , so that we can integrate easily the argument of the exponential in eq . ( [ ourspe ] ) , and obtain finally @xmath71 \,,\ ] ] where @xmath72 and @xmath73 and where @xmath74 expanding the argument of the exponential to the second order , we can approximate ( [ dndt ] ) as a gaussian curve . the second order term in ( [ qq ] ) should be included as well ; however , for the values of interest of @xmath51 and @xmath75 , it is irrelevant . the result is @xmath76 \,,\ ] ] where @xmath77 and where the preexponential factor is @xmath78 the instant @xmath79 marks the peak of the nucleation process ; this will be fixed by the request to have bubbles ( slightly ) larger than the present horizon . we must have nucleation before @xmath21 , or simply @xmath80 , and it is clear that a necessary condition is in any case @xmath81 . it is now useful to use @xmath48 as time variable . it follows then that @xmath82\,,\ ] ] with @xmath83\over3n_t}\right)\,,\qquad % \sigma_n^2=e^{r_t - r_0 } { n_t\over 12r_t}={n_{t}^{2}\over 16r_{t}^{2}b}\ , . n_p = n_t\left(1-{ht_p\over n_t}\right)\,,\quad \sigma_n^2={n_{t}^{2}\over 16r_{t}^{2}b}\,,\ ] ] where we require that @xmath84 . we need now the relation between the present curvature @xmath85 and the nucleation time @xmath48 . we know that if a bubble nucleates at @xmath48 its density parameter is @xmath86^{-1}\approx 1-\gamma n^{-2}\,,\ ] ] where the constant @xmath87 has been taken small ( compared to @xmath88 ) for simplicity of calculation , even if it is not necessary . now , from the friedmann equation , in the limit of small deviation from flatness , it is easy to see that the relation between the curvature today and the curvature during inflation is @xmath89 . however , when the curvature deviates sensibly from zero , this relation is no longer acceptable . neglecting the rde phase , it is possible to derive from the standard solution of an open universe ( see e.g. @xcite ) the following expression @xmath90 where @xmath91 thus , we get finally @xmath92 it appears then that , as we would expect , @xmath93 if the bubble nucleates at @xmath94 , and @xmath95 if the nucleation occurs at small @xmath48 . only a value @xmath66 would result in intermediate values of @xmath96 . from eq . ( [ finally ] ) , we can express the nucleation rate directly in terms of the present curvature @xmath96 : @xmath97 ^ 2\over \sigma_c^2 } \right\}\,,\ ] ] with @xmath98 @xmath99 and @xmath100 we can then define the probability @xmath101 that we live in a @xmath2 universe as @xmath102 where @xmath103 is the comoving size of a bubble which today has curvature @xmath96 . as we will see shortly , @xmath101 is normalized to unity . ( [ dn - c ] ) is the central result of this work ; it gives the probability density to live today in a bubble with given @xmath2 . as we have seen , we have obtained it using the standard tools of inflationary cosmology , without employing arguments from quantum cosmology . most importantly , we have obtained @xmath101 _ without assuming special initial conditions_. we now impose three observational constraints on @xmath101 : first , we must impose the condition that the bubbles fill the universe , i.e. that the transition completes ; second , we require the nucleation peak to occur for @xmath3 , the present observational value ; and third , we require a narrow distribution , say @xmath104 , to ensure our universe is typical . this implies three conditions on our parameters @xmath105 and @xmath106 . the first constraint amounts to requiring that the volume contained in bubbles nucleated from @xmath107 down to @xmath108 be equal to , or larger than , the present horizon : @xmath109 where @xmath110 is the comoving scale of a bubble nucleated at @xmath48 . in terms of @xmath101 , eq . ( [ norm ] ) is simply the condition that the probability be normalized to unity ( or to a value larger than unity , which simply indicates that the bubbles are too densely packed to assume spherical symmetry ) . since @xmath111 we have the normalization condition @xmath112 neglecting again the time dependence of @xmath69 during inflation . using directly the form ( [ dndt ] ) where we express @xmath62 in terms of @xmath48 with ( [ nt ] ) it follows @xmath113 we display in fig . 1 the region of the plane ( @xmath114 ) which satisfies the constraint above ( with @xmath115 ) , for which the curves ( [ dn - n ] ) are sharply peaked around @xmath116 @xmath117 , and for which @xmath84 ; as one can see , there is a vast region in which our theory is successful . finally , the condition that @xmath118 gives @xmath119 from fig . 1 one sees that @xmath75 has to be roughly larger than @xmath120 , and therefore @xmath121 ( from ( [ tp ] ) with @xmath122 ) and , by the condition ( [ sen1 ] ) , @xmath123 let us summarize the constraints to which the parameters are subject . the tunneling function has to be peaked at @xmath48 slightly larger than @xmath21 ; the value of @xmath124 at that time has to be such that today @xmath0 ; the peak should be narrow , so that the probability to live in a @xmath3 universe is high ; and the transition should be intense enough to fill the universe with bubbles . a further condition is that the slow roll do not generate too strong inhomogeneities ; roughly , this implies @xmath125 in planck units @xcite . finally , we have to be in the thin wall limit ( [ twl ] ) . it is remarkable that our model meets easily all these requirements . for instance , we can have @xmath126 and @xmath127 , so that @xmath128 ( from ( [ c - cond ] ) ) ; then , fixing @xmath129 and @xmath130 ( respecting ( [ twl ] ) ) , we have @xmath131 ( the planck scale ) , @xmath132 ( near the scale of @xmath53 ) . in fig . 2 we plot @xmath101 for this set of parameters . we presented a scenario in which the flat , inflationary universe is filled by super - horizon - sized underdense bubbles , which approximate open universes . this reconcile the astronomical observations in favour of @xmath3 with inflation . our own bubble - universe is one of an infinite number of similar bubbles . our model differs from the single - bubble scenario @xcite @xcite @xcite in which one must invoke anthropic arguments to explain our position , and is different also from the model presented by linde @xcite , in which there is no reason to expect preferential nucleation for any given value of @xmath124 . as we showed in the previous section , we can tune the parameters to achieve maximal probability for the nucleation of @xmath3 bubbles , without assuming special initial conditions , and satisfying all other constraints . it is worth remarking again that the measure of @xmath2 along with the assumption that the universe had an inflationary epoch , and that our position is typical , put strong constraints on the fundamental parameters of the primordial potential . it is interesting to observe that the phase transition parameters @xmath133 would be unobservable either if the nucleation occurred much earlier , because then the subsequent expansion would have again flattened the space , or much later , because then the very small sub - horizon bubbles would have thermalized , recovering again a @xmath134 universe . inside the bubbles one has the usual mechanism of generation of inflationary perturbations @xcite @xcite @xcite . in ref . @xcite we presented a scenario in which extra power is provided by the bubble - like structure of a primordial phase transition . it is possible that reducing the local @xmath2 to @xmath135 is enough to reconcile canonical cdm with large scale structure , so that no further phase transitions need to be invoked . however , evidences are increasing toward the presence of huge voids in the distribution of matter in the present universe , and for velocity fields that are difficult to explain without a new source of strong inhomogeneities . if this is the case , the possibility of an additional primordial phase transition occurred around 50 @xmath4-foldings before the end of inflation should be seriously taken into account . park , c. , vogeley , m.s . , geller , m. , huchra , j. astrophys . j. * 431 * , 569 ( 1994 ) p.j.e . peebles , principle of physical cosmology , ( princeton univ . press , 1993 ) b. ratra & p.j.e . peebles , astrophys . j. * 432 * l5 ; preprint pupt-1444 a. kashlisnsy , i. tkachev , & j. frieman phys . rev . lett . * 73 * ( 1994 ) a. guth & e. weinberg b. , nucl . * b212 * 321 ( 1983 ) m. sasaki , t. tanaka , yamamoto k. & j. yokoyama , pjys . , * b317 * 510 ( 1993 ) k. yamamoto , m. sasaki , & t. tanaka , preprint kuns 1309 m. bucher , a.s . goldhaber , & n. turok , princeton university preprint , hep - th/9411206 a. linde , preprint su - itp-95 - 5 , hep - th/9503097 f. occhionero and l. amendola , phys . d , * 50 * 4846 ( 1994 ) a.a . starobinsky , sov . jetp letters * 30 * , 682 ( 1979 ) . b. whitt , phys . 145b * , 176 ( 1984 ) . kolb & m. turner , the early universe ( addison- wesley , 1990 ) s.coleman , phys . d * 15 * 2929 ( 1977 ) ; c. callan and s. coleman , _ ibid . _ * 16 * 1762 ( 1977 ) ; s. coleman and f. de luccia , _ ibid . _ * 21 * 3305 ( 1980 ) . a.r . liddle and d. wands , mon . not . . soc . * 253 * , 637 ( 1991 ) .

it is already understood that the increasing observational evidence for an open universe may be reconciled with inflation if our horizon is contained inside one single huge bubble nucleated during the inflationary phase transition . in the scenario we present here , the universe consists of infinitely many superhorizon bubbles , like our own , the distribution of which can be made to peak at @xmath0 . therefore , unlike the existing literature , we do not have to rely upon the anthropic principle nor upon special initial conditions .

recently , our knowledge on background cosmology has improved dramatically due to new supernovae and cosmic microwave background data . current observations favor a flat universe with @xmath13 ( see e.g. harun - or - rashid & roos 2001 ; krauss 2003 and references therein ) and the remaining contribution in the form of cosmological constant or some other form of dark energy . a class of models that satisfy these observational constraints has been proposed by caldwell , dave , & steinhardt ( 1998 ) where the cosmological constant is replaced with an energy component characterized by the equation of state @xmath14 . the component can cluster on largest scales and therefore affect the mass power spectrum ( ma et al . 1999 ) and microwave background anisotropies ( doran et al . 2001 ; balbi et al . 2001 ; caldwell & doran 2003 ; caldwell et al . 2003 ) . a considerable effort has gone into attempts to put constraints on models with quintessence and presently the values of @xmath15 seem most feasible observationally ( wang et al . 2000 ; huterer & turner 2001 ; jimenez 2003 ) . another direction of investigations is into the physical basis for the existence of such component with the oldest attempts going back to ratra & peebles ( 1988 ) . one of the promising models is based on so - called tracker fields " that display an attractor - like behaviour causing the energy density of quintessence to follow the radiation density in the radiation dominated era , but dominate over matter density after matter - radiation equality ( zlatev , wang , & steinhardt 1999 ; steinhardt , wang , & zlatev 1999 ) . it is still debated , however , how @xmath1 should depend on time , and whether its redshift dependence can be reliably determined observationally ( barger & marfatia 2001 ; maor , brustein , & steinhardt 2001 ; weller & albrecht 2001 ; jimenez 2003 ; majumdar & mohr 2003 ) . from the gravitational instability point of view , the quintessence field and the cosmological constant play a very similar role : both can be treated as ( unclustered ) dark energy components that differ by their equation of state parameter , @xmath1 . technically , the equations governing the expansion of the universe and the growth of density perturbations in the two models differ only by the value of @xmath1 . given the growing popularity of models with quintessence , in this paper we generalize the description of the mass functions to include the effect of dark energy with constant @xmath1 . the mass function of clusters of galaxies has been used extensively to estimate cosmological parameters , especially @xmath5 and the rms density fluctuation @xmath6 , usually yielding ( due to degeneracy ) a constraint on some combination of the two . it has been shown that this degeneracy can be significantly decreased by using the data at different redshifts ( carlberg et al . 1997a ; bahcall , fan & cen 1997 ; bahcall & bode 2003 ) due to different growth rates of density fluctuations in different models . the growth rate also varies among models with different @xmath1 ; one may therefore hope to distinguish between them by analyzing mass function data at high redshift . it is worth noting that the power spectra of models differing only by @xmath1 are different only at very large scales , so the rms density fluctuations at scales of interest are almost identical when normalized to @xmath6 . the paper is organized as follows . in section 2 we briefly summarize the properties of the cosmological model with quintessence , including the linear growth factor of density fluctuations . section 3 presents the @xmath0-body simulations and the mass function measured from their output . in section 4 we describe the transformation between mass functions in terms of fof masses and in terms of abell masses . section 5 is devoted to the comparison of the theoretical mass functions to the data for clusters at different redshifts to obtain constraints on cosmological parameters . the discussion follows in section 6 . quintessence obeys the following equation of state relating its density @xmath16 and pressure @xmath17 @xmath18 the case of @xmath2 corresponds to the usually defined cosmological constant . the evolution of the scale factor @xmath19 ( normalized to unity at present ) in the quintessential universe is governed by the friedmann equation @xmath20 where @xmath21^{-1/2}\ ] ] and @xmath22 is the present value of the hubble parameter . the quantities with subscript @xmath23 here and below denote the present values . the parameter @xmath24 is the standard measure of the amount of matter in units of critical density and @xmath25 measures the density of quintessence in the same units : @xmath26 the einstein equation for acceleration @xmath27 shows that @xmath28 is needed for the accelerated expansion to occur . solving the equation for the conservation of energy @xmath29 with condition ( [ q1 ] ) , we get the following evolution of the density of quintessence in the general case of @xmath30 : @xmath31\ ] ] in agreement with caldwell et al . ( 1998 ) . for @xmath32 , the case considered in this paper , the formula reduces to @xmath33 the evolution of @xmath24 and @xmath25 with scale factor ( or equivalently redshift ) is given by @xmath34 while the hubble parameter itself evolves so that @xmath35 $ ] . the linear evolution of the matter density contrast @xmath36 is governed by equation @xmath37 , where dots represent derivatives with respect to time . for flat models and arbitrary @xmath1 an analytical expression for @xmath38 was found by silveira & waga ( 1994 , corrected for typos ) . with our notation and the normalization of @xmath39 for @xmath40 and @xmath41 , it becomes @xmath42\ ] ] where @xmath43 is a hypergeometric function . the solutions ( [ th8 ] ) for @xmath44 and @xmath4 are plotted in figure [ doda1 ] for the cosmological parameters @xmath45 and @xmath46 . in order to study the cluster mass functions in models with quintessence we have used a subset of the gpc@xmath47 dark matter @xmath0-body simulations of bode et al . ( 2001 ) . among the family of cosmological models they considered two are of interest for us here : @xmath48cdm ( with @xmath2 , @xmath45 , @xmath46 , @xmath49 , @xmath50 , @xmath51 ) and qcdm ( with @xmath52 , @xmath53 , and all other parameters the same ) . both models had the primordial spectral index @xmath54 . the @xmath2 and @xmath52 simulations were carried out using the tree - particle - mesh ( tpm ) code described in bode , ostriker & xu ( 2000 ) . for the purpose of this study , a new simulation with @xmath55 ( and @xmath56 , with all the remaining parameters unchanged ) was run using a newer , publicly available version of tpm code which includes a number of improvements over the earlier code ( bode & ostriker 2003 ) . the changes in the code which would affect the numerical results include improvements in the time stepping ( adding individual particle time steps within trees , and a stricter time step criterion ) and domain decomposition ( adding an improved treatment of tidal forces , and a new selection criterion for regions of full force resolution ) . the changes in time step only affect the innermost cores of collapsed objects , where the relaxation time is short ( bode & ostriker 2003 ) , leaving the mass function unchanged . the changes in domain decomposition affect the mass function , but only at the low - mass end : in the @xmath2 and @xmath52 simulations , the mass function is complete only above @xmath57 ( 100 particles ) , while in the @xmath55 run it extends down to @xmath58 ( 16 particles ) . in this paper we concentrate on the high - mass end of the mass function , where these numerical differences have no impact . the identification of halos in the final particle distribution has been performed with the standard friends - of - friends ( fof ) halo - finding algorithm with the linking parameter @xmath59 ( particles are linked if their separation is smaller than @xmath60 times mean interparticle distance ) . as an alternative , we have also used the hop regrouping algorithm ( eisenstein & hut 1998 ) . this procedure employs several parameters ; however , the resulting mass function is sensitive to only one of them , the density ( in units of the background density ) at the outer boundary of the halo , @xmath61 . as discussed by eisenstein & hut ( 1998 ) , assuming @xmath62 is equivalent to using the linking parameter of @xmath59 in the fof halo - finding algorithm . indeed , we verified that the mass functions measured in the simulations are almost identical when using these two halo - finding schemes . = 8.5 cm figure [ jesyz ] shows in solid lines the mass functions measured with fof from @xmath0-body simulations with @xmath2 ( upper panel ) , @xmath52 ( middle panel ) and @xmath55 ( lower panel ) at redshifts @xmath63 and @xmath64 . the quantity plotted is the cumulative mass function ( the comoving number density of objects of mass grater than @xmath65 ) @xmath66 , where @xmath67 is the number density of objects with mass between @xmath65 and @xmath68 . jenkins et al . ( 2001 ) established that when the masses are identified with fof(0.2 ) the simulated mass function in different cosmologies and at different epochs can be well approximated by the following fitting formula @xmath69 where @xmath70 in the above formulae @xmath71 is the background density , @xmath72 is the rms density fluctuation at top - hat smoothing scale @xmath73 @xmath74 where @xmath38 is the linear growth factor given by equation ( [ th8 ] ) , @xmath75 is the top - hat filter in fourier space and the mass is related to the smoothing scale by @xmath76 . @xmath77 in equation ( [ q13 ] ) is the power spectrum of density fluctuations , which we assume here to be given in the form proposed by ma et al . ( 1999 ) for flat models . for the present time @xmath78 the power spectrum is @xmath79 where @xmath80 is the primordial power spectrum index ( we will assume @xmath54 ) and @xmath81 is the transfer function . for the case of cosmological constant ( @xmath48cdm ) we take the transfer function @xmath82 in the form proposed by sugiyama ( 1995 ) @xmath83^{-1/2 } \nonumber\end{aligned}\ ] ] where @xmath84 and @xmath85 $ ] . for models with quintessence the transfer function is @xmath86 , where @xmath87 can be approximated by fits given in ma et al . however , for @xmath88 , @xmath89 differs from unity only at very large scales , i.e. very small wavenumbers . thus if the spectra are normalized to @xmath6 ( the rms density fluctuation smoothed with @xmath90 mpc ) , then there is no need to introduce the correction from @xmath91 to @xmath92 in ( [ q13 ] ) because it does not affect the calculation of @xmath72 . the predictions for @xmath93 obtained from equations ( [ q16])-([v4c ] ) for different @xmath1 and redshifts are shown in figure [ jesyz ] as dotted lines . one can see that the jenkins et al . formula accurately reproduces the simulated mass function , especially at @xmath94 . at higher redshifts the formula of jenkins et al . slightly overpredicts the number density of haloes ; a similar trend has been found recently by reed et al . therefore , we conclude that although originally designed on basis of other cosmological models , it can be considered valid also in the presence of quintessence . we therefore confirm the result of linder & jenkins ( 2003 ) who also found a good agreement between the simulations and the predictions of jenkins et al . ( 2001 ) formula in a different quintessence model . the masses of clusters of galaxies are usually measured as the so - called abell masses , i.e. the masses inside the abell radii of @xmath95 mpc . for the purpose of comparison with observations we need to transform the mass functions expressed in terms of the fof masses to abell masses . this can be done assuming a density distribution inside the cluster . since clusters are believed to be dominated by dark matter ( e.g. carlberg et al . 1997b ; okas & mamon 2003 ) their density distribution can be well approximated by the universal profile proposed by navarro , frenk & white ( 1997 , nfw ) for dark matter haloes @xmath96 where the radius has been expressed in units of the virial radius @xmath97 , @xmath98 . the virial radius is defined as the distance from the centre of the halo within which the mean density is @xmath99 times the critical density , @xmath100 . the value of the virial overdensity @xmath99 is estimated from the spherical collapse model and depends on the cosmological model . for flat models with constant quintessence parameter @xmath1 , its behaviour can be well approximated as ( weinberg & kamionkowski 2003 ) @xmath101 with @xmath102 and @xmath103 $ ] , @xmath104 $ ] . = 8.5 cm the quantity @xmath105 introduced in equation ( [ c6 ] ) is the concentration parameter , @xmath106 , where @xmath107 is the scale radius ( at which the slope of the profile is @xmath108 ) . the function @xmath109 in equation ( [ c6 ] ) is @xmath110 $ ] . from cosmological @xmath0-body simulations ( nfw ; jing 2000 ; jing & suto 2000 ; bullock et al . 2001 ) , we know that @xmath105 depends on the mass and redshift of formation of the object , as well as the initial power spectrum of density fluctuations . we will approximate this dependence using the toy model of bullock et al . ( 2001 ) ( their equations ( 9)-(13 ) with parameters @xmath111 and @xmath112 as advertised for masses @xmath113 ) . the predictions of the model at @xmath94 for cosmological models as in our simulations ( but all normalized to @xmath51 ) are shown in figure [ cm ] for masses @xmath114 . for higher redshifts we assume @xmath115 , following bullock et al . the dependence of the concentration on @xmath1 is rather weak but there is a trend of larger @xmath105 values for less negative @xmath1 . a similar trend was observed in the properties of dark haloes obtained in the @xmath0-body simulations by klypin et al . ( 2003 ) , although for smaller masses . since the nfw profile describes the halo properties in terms of the virial radius and the corresponding virial mass @xmath116 , we have to transform the fof masses first to virial masses . as previously noted , the fof parameter @xmath59 corresponds to the local overdensity at the border of the halo @xmath117 . therefore we will assume that the fof masses are equal to masses enclosed by such isodensity contour , @xmath118 . using the nfw distribution we find that for our simulated @xmath48cdm model at @xmath94 we have @xmath119 , while for higher redshifts @xmath120 . in the case of models with @xmath52 we instead get @xmath121 , and more so for @xmath55 . the differences between @xmath122 and @xmath123 for the models , redshifts and mass range considered here are typically of the order of a few percent and do not exceed 20% . they depend somewhat on mass , and are due mainly to the dependence of @xmath99 on cosmology . once the fof masses are translated to virial masses we can obtain the abell masses @xmath124 using the nfw mass distribution following from equation ( [ c6 ] ) @xmath125\ ] ] where @xmath126 is the abell radius in units of the virial radius . = 8.5 cm figure [ asymza ] shows the cumulative mass functions in terms of the abell masses as measured in the simulations ( solid lines ) and as calculated with the jenkins et al . formula ( [ q16])-([q17 ] ) with the proper mass transformation ( dotted lines ) . the agreement is very good for @xmath94 . in general , the analytic predictions match the @xmath0-body results as long as the virial radius is near or greater than the abell radius . however , at higher redshifts the predictions for lower masses are overestimated . several effects might lead to this disagreement . abell masses were measured in the simulations in the manner described in bode et al . ( 2001 ) . in this method , clusters can not be closer together than 1 @xmath127mpc ; also , if two abell radii overlap , particles in the overlap region are only included in one of the clusters ( based on binding energy ) . these two factors may lead to fewer objects lower mass objects being subsumed into larger ones . as discussed by bode et al . ( 2001 ) higher resolution simulations produce somewhat steeper abell mass functions which would agree better with the predictions . we must also keep in mind that the result was obtained with the assumption of nfw profile , while for smaller haloes this may not be the case . the haloes of mass @xmath128 have only 160 particles in our simulations and can hardly be expected to have a well defined density profile . we have also extrapolated the model for concentration of bullock et al . ( 2001 ) both for quintessence and very large masses , two regimes where it has never been tested , while even in well studied models concentration shows substantial scatter . given the number of approximations involved in the result we conclude that the fits are satisfactory , especially in the mass range between the two vertical dashed lines in each panel of figure [ asymza ] , covered by the data , which will be used in the next section . = 8.5 cm for comparison of the theoretical mass functions with observations we used the data for cluster mass function from carlberg et al . ( 1997a ) . the data consist of seven data points of @xmath93 for different abell masses @xmath129 , for redshift bins in the range @xmath130 . there are four data points coming from different surveys in the low redshift bin @xmath131 and three data points in higher redshift bins @xmath132 , @xmath133 and @xmath134 , respectively ( see table 1 of carlberg et al . the data for @xmath45 are shown in figure [ data ] with the filled circles corresponding to the low - redshift samples and open symbols to the higher - redshift ones . together with the data we show in figure [ data ] our model abell mass functions discussed in the previous section . all models have @xmath45 , @xmath46 , @xmath51 and differ only in the value of @xmath1 . the three upper lines shown in the figure are for @xmath94 , while the three lower ones for @xmath135 . we note that the small difference between the models with different @xmath1 at @xmath94 is due only to our mapping procedure between the fof and abell masses which involves @xmath1-dependent parameters @xmath99 and @xmath105 . ( the predictions of the jenkins et al . 2001 formula for fof masses applied without any corrections would be identical for the given set of cosmological parameters since the differences in the power spectrum for different @xmath1 are negligible . ) the differences between predicted abell mass functions for different @xmath1 start to be more pronounced at higher redshifts ( three lower curves in figure [ data ] ) due to different growth factors of density perturbations in these models ( see figure [ doda1 ] ) . we performed a standard @xmath136 fit to the data points in terms of @xmath137 with three free parameters : @xmath5 , @xmath6 and @xmath1 . since the data are in the form of a @xmath137 value per redshift bin , for the predictions we take the mean redshift of the bin ( it is not necessary to average over redshift because @xmath137 is approximately linear in redshift ) . when considering different @xmath5 in models we adjust the data point by linearly interpolating between the data given in carlberg et al . ( 1997a ) for @xmath138 and @xmath139 . the analysis has been done only for flat models , i.e. we kept @xmath140 . we also assumed hubble constant @xmath141 and the primordial spectral index @xmath54 . 100= figure [ contours ] shows the @xmath142 , @xmath143 and @xmath144 probability contours in the @xmath145 ( left column ) , @xmath146 ( middle column ) and @xmath147 ( right column ) parameter planes respectively . for each plane the cuts through the confidence region are done for three values of the third parameter ; the value is indicated at the corner of each panel . the contours correspond to @xmath148 , where the minimum value @xmath149 is obtained for @xmath150 , @xmath151 and @xmath7 . the contours in the @xmath145 plane presented in the left column of figure [ contours ] are the cuts through the confidence region at ( from top to bottom ) @xmath2 , @xmath152 and @xmath153 . they show a typical shape obtained in this kind of analyses . however , the three cuts through the confidence space shown in this column of figure [ contours ] actually move significantly when the assumed value of @xmath1 is changed ( note that the axes scales in the left column of the figure are the same in each panel ) . taking into account the dependence on @xmath1 and the variability of the contours in the whole range @xmath8 , we find @xmath10 and @xmath11 at @xmath12 confidence level . while for @xmath153 the best - fitting values of the remaining parameters are @xmath45 , @xmath51 , for lower @xmath1 the contours move towards lower @xmath5 and higher @xmath6 . for @xmath2 they are centered on @xmath154 , @xmath155 . the middle and right columns of figure [ contours ] show the constraints on @xmath1 in two planes , @xmath146 and @xmath147 respectively . the middle column has the cuts for @xmath138 , @xmath45 and @xmath156 from the top to the bottom panel , while in the right column the values of the third parameter are @xmath157 and @xmath158 . as can be seen in the plots , there is a strong degeneracy between @xmath1 and any of the two remaining parameters . the particular shape of the contours can be understood by referring back to figure [ doda1 ] , showing the growth rate of density fluctuations for different @xmath1 . by changing the normalization of the curves in figure [ doda1 ] to give the same value of @xmath38 at present ( @xmath159 ) , it is easily seen that the magnitude of density fluctuations drops faster with redshift for more negative @xmath1 . thus in order to reproduce a given redshift dependence of the cluster mass function data , for a more negative @xmath1 and a given @xmath5 ( @xmath6 ) a higher value of @xmath6 ( @xmath5 ) is needed ( as both these parameters enhance the growth of structure ) . the confidence regions we obtained in figure [ contours ] in the @xmath145 plane are qualitatively similar to the results of the analysis of the same data by carlberg et al . ( 1997a ) , which differed from our approach in that they used the press & schechter ( 1974 ) approximation , a power - law distribution of mass in clusters , and did not consider the dependence on @xmath1 . while for @xmath153 the best - fitting values of the remaining parameters are @xmath45 , @xmath51 in very good agreement with other estimates ( e.g. from cmb , see spergel et al . 2003 or type ia supernovae , see tonry et al . 2003 ) , for lower @xmath1 the contours move towards lower @xmath5 and higher @xmath6 . at @xmath2 they are centered on @xmath154 , @xmath155 . a very similar methodology is provided by using sunyaev - zeldovich cluster surveys to probe the evolution of the surface density of clusters as a function of redshift ( battye & weller 2003 and references therein ) . the constraints derived by the sunyaev - zeldovich surveys are based on the same physical processes and models as those employed here , namely the mass spectrum of haloes in the quintessential cosmology . battye & weller ( 2003 ) studied the constraining power of sunyaev - zeldovich surveys by analyzing mock catalogs of future surveys . comparing the results obtained here and by battye & weller we find that apart from the future planck survey the carlberg et al . ( 1997a ) data provide tighter or comparable constraints to the sunyaev - zeldovich surveys . the planck cluster survey will reduce the uncertainties in @xmath6 and @xmath5 by roughly a factor of 2 in comparison with the present results . assuming the @xmath48cdm cosmological model , bahcall et al . ( 2003 ) used data from the sdss collaboration to estimate that for @xmath154 , @xmath160 , a lower value than found here . however , the current result includes higher redshift data , which leads to higher estimates for @xmath6 , independent of @xmath5 ( bahcall & bode 2003 ) . it seems that assuming @xmath2 when considering cluster abundances leads to rather low values of @xmath5 ( or alternatively , high values of @xmath6 ) ; as pointed out by oguri et al . ( 2003 ) , these are in mild conflict with most other estimates . oguri et al . ( 2003 ) suggested that decaying cold dark matter may resolve this discrepancy . our results offer another possibility : they show that the constraints on @xmath5 and @xmath6 from cluster mass functions are in better agreement with other estimates if the assumption of @xmath2 is relaxed and a less negative value of @xmath1 is adopted . we have demonstrated the existence of a strong degeneracy between @xmath1 and any of the two remaining parameters , @xmath5 and @xmath6 , which is not broken in spite of using high redshift data : reproducing the evolution of the cluster mass function data requires a higher value of @xmath5 or @xmath6 for more negative @xmath1 . a similar behaviour of the confidence regions , including the degeneracies , in the @xmath146 and @xmath147 planes was recently observed also by schuecker et al . ( 2003 , see the lower panels of their fig . 3 ) who used the reflex x - ray cluster sample . as discussed by douspis et al . ( 2003 ) and crooks et al . ( 2003 ) the degeneracies between @xmath1 and other cosmological parameters also plague the constraints obtained from other , e.g. cmb , data sets . the best fit to the cluster mass function data is obtained for a surprisingly high value of @xmath161 , but the dependence on @xmath1 is rather weak and no value in the entire considered range @xmath8 can actually be excluded : for every value in this range a reasonable combination of @xmath5 and @xmath6 can be found which places the point in @xmath142 confidence region . it is interesting to note , however , that depending on the values of @xmath5 and @xmath6 , we get upper or lower limit on @xmath1 : for high @xmath5 and low @xmath6 the contours tend to provide a lower limit on @xmath1 , while for low @xmath5 and high @xmath6 we have an upper limit . combined with additional constraints on @xmath5 or @xmath6 from other data sets , cluster mass functions can therefore prove useful in estimating the value of @xmath1 . such an analysis has been recently performed by schuecker et al . ( 2003 ) who combined the data from x - ray clusters with the data for snia . since the supernova data show a strong preference for negative @xmath1 the resulting best - fitting value of @xmath1 is very close to @xmath162 . with the estimates of the cosmological parameters presently available the analysis presented here tends to provide a lower limit on @xmath1 . for example , for the best estimates from wmap ( spergel et al . 2003 ) , @xmath163 and @xmath51 , the constraint from our analysis is @xmath164 at 95% confidence level . the values of @xmath5 and @xmath6 , however , are not yet known exactly so a proper combination with other data sets would have to be performed . typically , other data sets give an upper limit on @xmath1 , e.g. wmap in combination with other astronomical data gives @xmath165 ( spergel et al . 2003 ) while from the snia data tonry et al . ( 2003 ) obtain @xmath166 . this means that the combination of these limits with the cluster mass function data will probably result in a preferred range of @xmath1 instead of only an upper or lower limit . the high values of @xmath1 preferred by the cluster mass function data may also turn out to be in conflict with other estimates . this would point towards some unknown systematics in the data or inconsistencies in the models . then the cosmological model with constant @xmath1 would have to be rejected and replaced with one involving some time - dependence of @xmath1 . we thank ofer lahav , gary mamon and the anonymous referee for their comments on the paper . this work was supported in part by the polish kbn grant 2p03d02726 . computer time to perform @xmath0-body simulations was provided by ncsa . pb received support from ncsa nsf cooperative agreement asc97 - 40300 , paci subaward 766 . yh acknowledges support from the israel science foundation ( 143/02 ) and the sheinborn foundation . bahcall n. a. , bode p. , 2003 , apjl , 588 , 1 bahcall n. a. , fan x. , cen r. , 1997 , apjl , 485 , 53 bahcall n. a. et al . , 2003 , apj , 585 , 182 balbi a. , baccigalupi c. , matarrese s. , perrotta f. , vittorio n. , 2001 , apjl , 547 , 89 barger v. , marfatia d. , 2001 , phys . b , 498 , 67 battye r. a. , weller j. , 2003 , phys . d , 68 , 083506 bode p. , ostriker j. p. , 2003 , apjs , 145 , 1 bode p. , ostriker j.p . , xu g. , 2000 , apjs , 128 , 561 bode p. , bahcall n. a. , ford e. b. , ostriker j. p. , 2001 , apj , 551 , 15 bullock j. s. , kolatt t. s. , sigad y. , somerville r. s. , kravtsov a. v. , klypin a. a. , primack j. r. , dekel a. , 2001 , mnras , 321 , 559 caldwell r. r. , doran m. , 2003 , astro - ph/0305334 caldwell r. r. , dave r. , steinhardt p. j. , 1998 , phys . , 80 , 1582 caldwell r. r. , doran m. , mller c. m. , schfer g. , wetterich c. , 2003 , apjl , 591 , 75 carlberg r. g. , morris s. l. , yee h. k. c. , ellingson e. , 1997a , apj , 479 , l19 carlberg r. g. et al . 1997b , apj , 485 , l13 crooks j. l. , dunn j. o. , frampton p. h. , norton h. r. , takahashi t. , 2003 , astro - ph/0305495 doran m. , lilley m. j. , schwindt j. , wetterich , c. , 2001 , apj , 559 , 501 douspis m. , riazuelo a. , zolnierowski y. , blanchard a. , 2003 , a&a , 405 , 409 eisenstein d. j. , hut p. , 1998 , apj , 498 , 462 harun - or - rashid s. m. , roos m. , 2001 , a&a , 373 , 369 huterer d. , turner m. s. , 2001 , phys . d , 64 , 123527 jenkins a. , frenk c. s. , white s. d. m. , colberg j. m. , cole s. , evrard a. e. , couchman h. m. p. , yoshida n. , 2001 , mnras , 321 , 372 jimenez r. , 2003 , new astron . rev . , 47 , 761 jing y. p. , 2000 , apj , 535 , 30 jing y. p. , suto y. , 2000 , apjl , 529 , 69 klypin a. a. , maccio a. v. , mainini r. , bonometto s. a. , 2003 , submitted to apj , astro - ph/0303304 krauss l. m. , 2003 , proc . eso - cern - esa symposium on astronomy , cosmology and fundamental physics , p. 50 , astro - ph/0301012 linder e. v. , jenkins a. , 2003 , mnras , 346 , 573 okas e. l. , mamon g. a. , 2003 , mnras , 343 , 401 ma c. p. , caldwell r. r. , bode p. , wang l. , 1999 , apj , 521 , l1 maor i. , brustein r. , steinhardt p. j. , 2001 , phys . 86 , 6 majumdar s. , mohr j. j. , 2003 , submitted to apj , astro - ph/0305341 navarro j. f. , frenk c. s. , white s. d. m. , 1997 , apj , 490 , 493 oguri m. , takahashi k. , ohno h. , kotake k. , 2003 , apj , 597 , 645 press w. h. , schechter p. , 1974 , apj , 187 , 425 ratra b. , peebles p. j. e. , 1988 , phys . d , 37 , 3406 reed d. , gardner j. , quinn t. , stadel j. , fardal m. , lake g. , governato f. , 2003 , mnras , 346 , 565 schuecker p. , caldwell r. r. , bhringer h. , collins c. a. , guzzo l. , weinberg n. n. , 2003 , a&a , 402 , 53 silveira v. , waga i. , 1994 , phys . rev . d , 50 , 4890 spergel d. n. et al . , 2003 , apjs , 148 , 175 steinhardt p. j. , wang l. , zlatev i. , 1999 , phys . rev . d , 591 , 270 sugiyama n. , 1995 , apj , 471 , 542 tonry j. l. et al . , 2003 , apj , 594 , 1 wang l. , caldwell r. r. , ostriker j. p. , steinhardt p. j. , 2000 , apj , 530 , 17 weinberg n. n. , kamionkowski m. , 2003 , mnras , 341 , 251 weller j. , albrecht a. , 2001 , phys . lett . , 86 , 1939 zlatev i. , wang l. , steinhardt p. j. , 1999 , phys . lett . , 82 , 896

we use @xmath0-body simulations to measure mass functions in flat cosmological models with quintessence characterized by constant @xmath1 with @xmath2 , @xmath3 and @xmath4 . the results are compared to the predictions of the formula proposed by jenkins et al . at different redshifts , in terms of fof masses as well as abell masses appropriate for direct comparison to observations . the formula reproduces quite well the mass functions of simulated haloes in models with quintessence . we use the cluster mass function data at a number of redshifts from carlberg et al . to constrain @xmath5 , @xmath6 and @xmath1 . the best fit is obtained in the limit @xmath7 , but none of the values of @xmath1 in the considered range @xmath8 can actually be excluded . however , the adopted value of @xmath1 affects significantly the constraints in the @xmath9 plane . taking into account the dependence on @xmath1 we find @xmath10 and @xmath11 ( @xmath12 c.l . ) . since less negative @xmath1 push the confidence regions toward higher @xmath5 and lower @xmath6 we conclude that relaxing the assumption of @xmath2 typically made in such comparisons may resolve the discrepancy between recent cluster mass function results ( yielding rather low @xmath5 and high @xmath6 ) and most other estimates . the fact that high @xmath1 values are preferred may however also point towards some unknown systematics in the data or the model with constant @xmath1 being inadequate . methods : @xmath0-body simulations methods : analytical cosmology : theory cosmology : dark matter galaxies : clusters : general large - scale structure of universe

diffractive scattering through strong interactions without any large momentum transfer has historically been described in terms of the exchange of a pomeron , a virtual hadron - like object with vacuum quantum numbers . the regge approach @xcite of the pre - qcd era provides a working phenomenology to describe such processes in a hadron basis since no parton structure is resolved . the idea @xcite to introduce a hard scale in a diffractive process opened the possibility to examine these processes at the level of quarks and gluons in the modern framework of qcd . the discovery of such _ diffractive hard scattering _ was made by the ua8 experiment @xcite by observing high-@xmath3 jets in single diffractive events in @xmath4 collisions at cern . many other hard processes in diffractive events have been observed later on , for a review see e.g.@xcite . of special importance are the measurements of diffractive deep inelastic scattering ( ddis ) at the electron - proton collider hera @xcite , where the well understood point - like electromagnetic interaction probes the parton structure in the diffractive reaction mechanism . this has resulted in theoretical descriptions of data based on essentially two different approaches , one being pomeron exchange in regge phenomenology and the other colour screening via soft gluon exchange in qcd . the pomeron approach starts with the initial proton state fluctuating in a soft non - perturbative process into a proton and a pomeron . the latter is assumed to have a partonic structure which is probed by the momentum transfer @xmath5 of the deep inelastic photon exchange to produce the hadronic final state , well separated in rapidity from the final state proton carrying most of the longitudinal momentum of the beam proton . hera data on ddis can then be described @xcite as a product of an effective pomeron flux factor from regge phenomenology and deep inelastic scattering on the pomeron having parametrised parton density functions ( pdf ) @xcite . alternatively , one may parametrise diffractive structure functions without an explicit pomeron flux but instead being conditionally dependent on the momentum of the final proton . however , this approach is not universal in the sense that such parametrisations do not reproduce diffractive hard scattering data in hadron - hadron collisions . for example , such pomeron pdfs overestimate substantially the cross - sections for diffractive hard scattering processes , such as production of jets or @xmath6 , at the tevatron @xcite . this has called for introducing a gap survival suppression factor @xmath7 , which can be given a qualitative theoretical motivation but which is difficult to calculate quantitatively . the colour exchange approach starts instead with the hard scattering process and then adds softer gluon exchanges to achieve the effective colour singlet exchange in diffractive processes . thus , the underlying hard process is assumed to be the same as in the corresponding non - diffractive process and its momenta naturally not affected by other exchanges at much lower momentum transfer scales . however , the formation of confining colour fields may well be affected by the softer gluon exchanges and thereby the hadronisation process such that a different distribution of the final state hadrons emerges . for example , when different colour singlet string - fields emerge separated in rapidity they will hadronise into two hadron systems separated by a rapidity gap with no hadrons . a simple , but phenomenologically rather successful model of this kind is the soft colour interaction ( sci ) model @xcite added to the lepto @xcite and pythia @xcite monte - carlo event generators . a large variety of diffractive data could then be reproduced with essentially the same value of a single new parameter , introduced to give the probability for exchange a soft colour octet gluon between any pair of partons . this colour exchange alter the formation of the colour string - fields and hence the application of the conventional lund string hadronisation model results in a different topology of the hadronic final state . the model gives an essentially correct description of diffractive dis @xmath8 scattering at hera @xcite as well as diffractive events at the tevatron having jets or quarkonia @xcite or gauge bosons @xcite . it has therefore been applied for predictions , e.g. of diffractive higgs production in double gap processes at lhc @xcite . the gap survival factor @xmath7 often used in other kinds of models , is not necessary here since the full event simulation accounts for such effects resulting in correct rates for the investigated diffractive processes in @xmath8 and @xmath9 and @xmath10 collisions . other models of a similar nature , with different forms of colour exchange , have been developed . the gal model @xcite for example considers soft colour exchanges between strings with a probability that favours minimization of the strings area in energy - momentum coordinates . a recent development of such colour string reconnection models makes a more elaborate account for @xmath11 colour statistics @xcite . a theoretical qcd basis for basic colour exchanges has been proposed in @xcite and , in a more elaborated form , in @xcite . the basic hard scattering process is treated by conventional perturbative qcd ( pqcd ) . its large momentum scale implies that it occurs on a small space - time scale compared to the bound state proton and is thus embedded in the proton . therefore , the emerging hard - scattered partons propagate through the proton s colour field and may interact with it . the amplitude for such multiple gluon exchanges is calculated in the eikonal approximation to all orders in perturbation theory resulting in an analytic expression for such a colour screening effect . the theoretical approach @xcite used here to resum multigluon exchange has similarities with the one in @xcite giving a similar eikonal factor @xmath12 in the amplitude , but there are also differences as will be discussed below . it is this amplitude that we here develop into a probabilistic model . we implement it for two different monte carlo event generators : lepto for general deep inelastic lepton - nucleon scattering including first order qcd matrix elements and parton showers based on conventional @xmath13 evolution from the dglap equations @xcite , and as well for cascade @xcite specialised on small-@xmath0 electron - proton scattering based on the off - shell @xmath14 first order matrix element and @xmath2 factorisation with ccfm evolution @xcite and unintegrated gluon density of the proton . the paper is organized as follows . in section [ sec : dynamic - rescattering ] we describe the basic process with the colour screening . section [ sec : ddis - via - dcs ] discusses the resulting cross - section for diffractive deep inelastic scattering and its monte carlo implementation . section [ sec : res ] shows our results in comparison to hera data . finally , section [ sec : conclusions ] presents our conclusions . in the virtue of the sci model , the skeleton of both inclusive and diffractive ( with rapidity gaps and/or leading proton ) dis process is provided by the same perturbative qcd diagram illustrated in fig . [ fig : ddis ] . a parton with longitudinal momentum fraction @xmath15 in the initial proton at the starting scale @xmath16 for pqcd is evolved to smaller momentum fractions , but higher transverse momenta and virtualities up to a hard scale @xmath17 . here , a virtual photon @xmath18 with momentum @xmath19 and virtuality @xmath20 resolves a quark at bjorken @xmath0 @xmath21 at small @xmath0 the process will dominantly be initiated by a gluon , which can radiate and splits into a @xmath22 pair . the total momentum fraction taken from the proton is @xmath23 , and the total mass @xmath24 of the parton system denoted as @xmath25 in fig . [ fig : ddis ] is @xmath26 for a @xmath27 pair ( without additional gluons ) having with quark transverse momentum @xmath2 and longitudinal momentum fraction @xmath28 , the corresponding invariant mass is @xmath29 with a subsequent rescattering of the @xmath30 dipole off the target colour field ( left panel ) . schematic diagram of the diffractive dis process @xmath31 accounting for final state rescattering by multiple gluon exchange at @xmath32 and perturbative parton shower off initial state parton which builds up the diffractive system @xmath25 ( right panel ) . the latter can be separated from the leading proton ( or small - mass system @xmath33 ) by a rapidity gap . the final state radiation is not shown as it does not affect the overall kinematics of the @xmath25 system.,title="fig:",scaledwidth=50.0% ] with a subsequent rescattering of the @xmath30 dipole off the target colour field ( left panel ) . schematic diagram of the diffractive dis process @xmath31 accounting for final state rescattering by multiple gluon exchange at @xmath32 and perturbative parton shower off initial state parton which builds up the diffractive system @xmath25 ( right panel ) . the latter can be separated from the leading proton ( or small - mass system @xmath33 ) by a rapidity gap . the final state radiation is not shown as it does not affect the overall kinematics of the @xmath25 system.,title="fig:",scaledwidth=35.0% ] at large @xmath34 , the parton distribution functions ( pdfs ) of the incident proton are dominated by valence quarks leaving practically no chance for the proton to survive such an interaction , and hence resulting in a non - diffractive event . at small @xmath34 , however , the pdfs are dominated by gluons , and the partonic system @xmath25 is created in photon - gluon fusion @xmath35 as depicted in fig . [ fig : ddis ] . in this case , the momentum exchange via multiple soft gluons with a small net fraction @xmath36 between the proton and the perturbative @xmath25 systems does not significantly change the momenta of partons emerging from the hard scattering . however , they do change the colour structure of the resulting @xmath25 and @xmath33 systems . the original sci model captures the main effects in many processes as ddis at hera and diffractive hard scattering at the tevatron . despite this success , it is not derived from a perturbative qcd amplitude . for the case of ddis , a derivation of the amplitude for the colour screened process has been done based on perturbative qcd in the large @xmath37 limit @xcite and provides a theoretical basis for the ddis process in terms of colour exchanges . it improves on the previous description by introducing a dependence on the kinematical details of the event which also leads to colour transparency . we start with the outline of the resummed colour screening amplitude and the derivation of probability for an event - by - event treatment in an event generator . consider the ddis amplitude in impact parameter representation in the target rest frame which corresponds to the colour dipole picture of the process . the lowest fock component of the virtual photon @xmath38 corresponds to its fluctuation to a @xmath30 dipole with transverse separation @xmath39 in the colour background field of the traget proton at impact distance @xmath40 from its centre . the prepared fock component then propagates through the field in the proton and softly interacts with it such that it can , in principle , change its colour but not kinematics ( the dipole size is frozen at the time scale of its propagation through the colour medium ) . we consider the forward limit where the total transverse momentum @xmath41 of gluon exchanges in the @xmath42-channel is small , @xmath43 . in this limit , as a straightforward consequence of the optical theorem in the limit of large @xmath44 c.m . energy , the ddis amplitude @xmath45 can be written with the ordinary gluon - initiated inclusive dis amplitude @xmath46 and the dynamical colour screening ( dcs ) amplitude @xmath47 as @xmath48 where @xmath49 is the relative quark transverse momentum in the @xmath30 dipole in the lowest order subprocess @xmath50 . the screening amplitude @xmath51 accounts for the soft gluon exchanges between the proton remnant @xmath33 and the rest of the final state commonly denoted as @xmath25 , with @xmath52 at the lowest order . these exchanges carry a small longitudinal fraction @xmath53 and the transverse momentum transfer @xmath54 is at a soft scale @xmath55 . @xmath51 is resummed to all orders in the large-@xmath56 limit where it acquires a simple eikonal form @xcite . @xmath57 here , @xmath58 is the colour factor for the single gluon exchange amplitude and @xmath59 is the effective coupling constant at the soft hadronic scale @xmath55 . the effective qcd coupling is not small in this case . several approaches dealing with the landau singularities at low momentum transfers were proposed in the literature , e.g. @xcite . in practice , we use the infrared - stable analytic perturbation theory ( apt ) approach @xcite . for our study we use similarly the inclusive amplitude @xmath60 in impact parameter space the fraction of the cross section with colour screening between the systems @xmath25 and @xmath33 is obtained from the ratio @xmath61 which defines the probability function for the overall colour singlet exchange . with @xmath62 , this leads to @xmath63 to apply this as the probability in a monte - carlo generated event at the parton level , we will associate @xmath64 and @xmath65 , and approximate by taking the average over the relative angle @xmath66 @xmath67 which is motivated by the fact that @xmath66 will be uniformly distributed in a sample of many collisions . for different values of @xmath68 ( upper to lower curve ) ] the resulting colour screening probability is shown in fig . [ fig : rescatter - prob - for - angles ] , for different choices of @xmath59 which enters as a normalisation factor . two important characteristic properties can here be observed . first , infrared safety with the probability levelling off at large @xmath69 , which resembles the saturation feature of the dipole scattering amplitude . second , vanishing colour screening probability for small dipoles @xmath70 , which is compatible with the colour transparency property . our basic theoretical approach @xcite for resumming multiple gluon exchanges has similarities with the theory developed in @xcite and resulting in an amplitude with an eikonal factor of the form @xmath12 in similarity with eq . ( [ eik - formula ] ) above . that approach can also be applied to both inclusive and diffractive dis starting from the same hard matrix element and employing factorization and resummation of soft @xmath42-channel gluons . despite these similarities , there are significant differences . in @xcite all exchanged gluons are treated on the same footing via a wilson line resumming them into a color singlet exchange . our approach separates one `` leading '' gluon , carrying the largest @xmath0-fraction , from the rest of the exchanged gluons which are much softer , carrying @xmath71 . these softer gluons are resummed to all orders and are required to be in a color octet state that matches the leading gluon to an overall color singlet exchange that provide the mechanism for rapidity gap formation . our leading gluon is treated via conventional @xmath2-factorisation in terms of the unintegrated gluon distribution in the proton , thus providing a model which is explicitly the same for diffractive and inclusive scattering without introducing any new kind of parton distribution functions , cf eq . ( [ sigmad ] ) below . this is in contrast with the formalism in @xcite which introduces diffractive parton distribution functions ( pdf ) containing non - perturbative dynamics and interpreted as the probability to find a parton with a momentum fraction @xmath72 in the proton , provided the proton emerges intact with momentum fraction @xmath73 . such diffractive pdfs can be analyzed theoretically @xcite which may introduce model parameters to be determined from data . naturally , one should study these , and other , theoretical approaches and via data find the optimal description of data to understand the rather complex diffractive processes in terms of basic theory and few free parameters . our approach has only two new parameters , which both have physical meanings that constrain their values to a rather narrow range . the following sections specify the monte carlo implementation of the model and show detailed numerical comparisons to data . the ddis cross section is then obtained using the inclusive cross section and standard inclusive parton densities together with the probability @xmath74 in eq . ( [ eq : prob - from - rb ] ) for dynamic colour screening resulting in @xmath75 where @xmath76 are the standard inclusive parton distributions . @xmath77 represents the differential distribution of the standard dis cross section in @xmath78 which is obtained from the parton evolution event - by - event in the monte carlo . since @xmath78 represents the transverse size of the @xmath22 together with the pqcd radiation and the amplitude for colour screening is dominated by a rescattering off large dipoles , we use the smallest @xmath2 difference within the partonic @xmath25-system and let @xmath79 . although this @xmath80 is typically related to the pqcd cutoff , the monte carlo simulation can give very small such relative @xmath2 due to random angular orientations of the momentum vectors . we therefore introduce a cut - off @xmath81 to avoid a spurious divergence and transverse sizes @xmath78 that are not perturbatively small . thus , we let @xmath82 . the impact parameter @xmath83 is related to the soft transverse momentum of the screening multiple gluon exchange , which is expected to be well below the factorisation scale for the pqcd processes . on the other hand @xmath84 is expected to be somewhat larger than the confining energy - momentum scale @xmath85{mev}$ ] in order for the screening process to occur fast enough that the proton state can stay quantum mechanically coherent into the final state . the colour screening probability therefore depends on the ratio @xmath69 given by @xmath86 where @xmath81 , as mentioned , regulates the divergence . the values of @xmath84 and @xmath81 constitute the two free parameters of the model and are to be determined from a comparison with experimental data . from the construction , we expect their values to be approximately between @xmath87 and the perturbative cutoff @xmath88 in the gluon pdf . using eq . ( [ eq : def - rb - ratio ] ) in eq . ( [ eq : prob - from - rb ] ) results in a probability @xmath89 for the effective colour screening that depends on the internal kinematics of the system @xmath25 . we calculate the diffractive reduced cross section @xmath90 within the same kinematic limits as applied by the experiment @xcite . in addition , we adopt two different notions of the diffractive cross section . the first definition @xmath91 is based on a forward remnant system with a mass @xmath92{gev}$ ] and proton quantum numbers . the other definition @xmath93 requires a large rapidity gap ( lrg ) of two units in pseudorapidity . this choice is potentially sensitive to the inner radiation structure of the system @xmath25 because the lrg is defined in terms of pseudorapidity . the emissions in the ccfm evolution @xcite are not strongly ordered in virtuality as they are by assumption in the dglap evolution @xcite . therefore , the parton in the hard interaction can no longer be approximated as on - shell and instead an off - shell matrix element is used for the hard interaction . likewise , the branching gluons in the ccfm evolution are described by an unintegrated gluon density function ( ugdf ) . in the ccfm evolution , one effectively resums the leading logarithms in the energy splitting @xmath94 and @xmath95 , as well as the leading logarithms in @xmath5 . experimental data on the ddis cross section @xmath96 covers a wide kinematic range where , in particular , one can have @xmath97 . because of such potentially very different hard scales , we expect corrections from large logarithms to become important in certain parts of the phase space . because of eq . ( [ mxdef ] ) we expect the ccfm evolution to be better suited for the ddis observables , at least , in the case of @xmath98 when the large leading logs @xmath99 are properly treated . because of the soft divergence in the first order qcd matrix element , the hard process @xmath100 will favour an uneven splitting of the energy between the quarks . specifically , if we define the fraction of energy taken by the quark as @xmath28 , then the matrix elements have soft divergences as @xmath94 and @xmath95 for @xmath101 or @xmath102 , respectively . one aspect of the soft divergence is that it favors a rather large ratio between @xmath0 and the energy fraction @xmath103 of the parton entering the matrix element . specifically , in the matrix element @xmath100 one can have a large ratio @xmath104 without any additional radiation from the quark propagator . such a large @xmath105 is not accounted for in the initial state parton evolution . in the case of the matrix elements @xmath106 and @xmath107 plus dglap evolution this phase space is in part taken into account , but the dglap evolution does not resum potentially large @xmath108 . another aspect of the uneven splitting is that one of the quarks will be very forward in pseudorapidity in the lab frame when @xmath28 is close to the divergence . in this case , the quark can have a large enough forward momentum @xmath109 to populate the gap region and the event therefore does not contribute to the diffractive cross section as defined in terms of a lrg in spite of having a leading proton . the divergent behaviour is unphysical and is usually avoided with a cutoff . still , an inclusion of higher order effects may be important for the cross section and also for the inner structure within the @xmath25 system . in particular , the energy of the @xmath22 system may be shared by additional gluons and therefore significantly reduce the rapidity range of the final @xmath25-system and therefore can have an influence on the lrg observable . in comparison with the h1 data @xcite . the model prediction uses dynamic colour screening with the parameters and and the cascade event generator with ccfm evolution . @xmath89 for the fitted parameters is shown in the upper - left corner . diffractive events in the model are defined as having a remnant system @xmath33 with proton quantum numbers and invariant mass @xmath92{gev}$ ] . rows for different values of @xmath110 are offset by a factor @xmath111 as indicated on the figure . ] in order to study the dynamic colour screening in more detail , we interface the model with different monte carlo event generators . in particular , we employ the program lepto @xcite which offers first order qed and first order qcd matrix elements combined with dglap @xcite parton showering and collinear pdfs . as a second program we use cascade @xcite which offers @xmath112 with @xmath2-factorised off - shell matrix elements and ccfm evolution @xcite which is intended to account for potentially large logarithms of incident momentum fractions of radiated partons . the photon - gluon fusion matrix element @xmath113 illustrated also in fig . [ fig : ddis ] is the dominant contribution to the diffractive dis cross section at small @xmath0 . lepto includes this process as a first order qcd matrix element , as well as via a combination of the qed hard process @xmath114 augmented by a @xmath115 dglap splitting . cascade provides this process as an off - shell first order qcd matrix element , but not as a first order qed matrix element with parton splitting . after generating events on parton level using matrix elements augmented with initial and final state parton showers , we apply the colour screening model before any special treatment of the remnant . in particular , we do not allow any cluster fragmentation of systems with a small invariant mass because the dynamic screening will potentially change the colour topology of the event before the scale of hadronization is reached and therefore change the possible outcomes of the fragmentation . this is understood as colour rescattering to happen on the scale between the perturbative cutoff at @xmath116{gev}$ ] and @xmath87 of hadronisation . the remnant system is in a monte carlo program treated by a non - perturbative model . for the case that the perturbative interaction resolves a gluon , which is the class of events which potentially leads to diffraction , the remnant is usually split into a @xmath117 pair with a certain sharing of momenta . this splitting typically introduces a relative transverse momentum representing the fermi motion in the bound state proton which is given by a gaussian distribution with a width @xmath118 . this relative @xmath2 affects the later hadronisation which introduces an uncertainty for the prediction of a forward proton spectrum . in this work , we are interested in diffraction defined by a forward small - mass system with proton quantum numbers or a large rapidity gap , which is insensitive to whether the hadronisation model maps the small - mass forward remnant state to a proton state or a resonance . all plots for the diffractive cross section in this paper show the reduced cross section @xmath119 which is related to the cross section via @xmath120 with @xmath121 given as @xmath122 with the negligible nucleon mass @xmath123 . , @xmath124 and for different values of @xmath5 . as the hard scale increases , there is more phase space for initial state radiation available and we see more gluon branchings . as the number of gluon branchings increases , a larger part of the full perturbative event and specifically its @xmath110-value is described using ccfm evolution which leads to a more accurate differential cross section . ] as in fig . [ fig : cascade_cutsnone_m01alphas70 ] but model results using the lepto event generator based on matrix elements for all hard processes to order @xmath125 and parton showers based on dglap @xmath1 evolution . the parameters for the dynamic colour screening model obtained from the fit against data @xcite are and . ] but with a constant probability of @xmath126 fitted from this data . we observe that especially at small @xmath110 the resulting diffractive cross section has a significantly different slope with respect to @xmath5 and the description of data is not as good as in the case of the dynamic rescattering model . ] [ fig : cascade_cutsnone_m01alphas70 ] shows the diffractive cross section @xmath127 obtained with dynamic colour screening and the process @xmath128 from the cascade event generator with ccfm evolution . events are selected according to the forward small - mass system prescription which requires a remnant system @xmath33 with proton quantum numbers and invariant mass @xmath92{gev}$ ] . the two parameters of the rescattering model are fitted and we obtain and . these values are physically reasonable in the sense that both are between @xmath87 and @xmath88 , and that the typical transverse momentum scale @xmath84 of the proton background is smaller than the minimal transverse momentum scale @xmath81 of the partonic @xmath25 system . we note that there is an overall good agreement with experimental data over a very wide region of the kinematical space . this agreement is remarkable because the model does not introduce specialised diffractive parton distributions , but uses standard proton ugdfs as input and introduces only two new physically motivated parameters . nevertheless , we note that one specific kinematic region is not very well described , namely , where @xmath110 and @xmath5 are both small and @xmath34 is large . the discrepancy develops for @xmath129 and increases for decreasing @xmath5 in the range of a few @xmath130 . a qualitative understanding of the problem can here be obtained from the principles of parton evolution via gluon radiation . at small @xmath110 , meaning large @xmath131 compared to @xmath5 , large logarithms of @xmath132 become important . on the other hand , at small @xmath5 scales , the event is mainly described by the matrix element , whereas the shower activity is low , which leads to a relative damping of the cross section with respect to the data . this lower radiative activity is illustrated in fig . [ fig : branchings-3em2 - 17em3 ] in terms of the number of branchings in the parton evolution for different @xmath5 at a representative value of @xmath124 . at very small @xmath133 there is a very long evolution path from the initial gluon with momentum fraction @xmath34 to a much smaller bjorken-@xmath0 at the quark - photon vertex . the cascade event generator has the advantage to resum both @xmath1 and @xmath134 contributions in its ccfm - based treatment of initial - state gluon radiation off the incoming gluon . however , gluon emissions from the quark propagator between the @xmath115 vertex and the quark - photon vertex ( _ cf . [ fig : ddis]b ) can also generate large @xmath134 contributions , in particular for the longest total evolution paths at very small @xmath110 . this radiation from the quark propagator can not be generated in cascade since the ccfm equation only includes gluon radiation from gluons and not from quarks . the simulation process starts by using the hard matrix element for @xmath135 , and then the initial gluon is evolved down to the starting scale of the proton state . thus , at very small @xmath110 in particular , there can be a substantial phase space available for gluon radiation off the quark propagator connecting to the virtual photon , which is not taken into account . gluon emission from the quark propagator can be taken into account by instead using the lepto monte carlo generator . it includes not only the hard matrix element for @xmath136 but also the other first order qcd process @xmath137 , where the gluon can be emitted from the incoming or outgoing quark , as well as the zeroth - order process @xmath114 . for both these processes , additional gluon radiation from the initial quark may occur through the initial - state parton showering , which in lepto is generated through conventional dglap evolution in @xmath1 , but without a resummation of @xmath134 contributions . [ fig : lepto_cutsnone ] shows the results of using lepto with the same dynamic colour screening model , with fitted parameter values and of expected magnitudes . the description of data is good in the inner region of the covered kinematic space , but with substantial discrepancies at small @xmath34 and very large @xmath110 . in the discussed problematic region , however , the agreement with data is better than that for cascade at large @xmath34 , small @xmath5 and small @xmath110 around 0.01 , although not at the very smallest @xmath138 . this indicates that , as discussed , emissions from the quark propagator is of some importance , but accounting for large @xmath134 contributions is also needed . no presently available event generator include all the mentioned effects that seems necessary in order to describe the diffractive cross - section over the entire kinematic region . the indication is , however , that combining all available theoretical formalisms in perturbative qcd , i.e. matrix elements and parton showers including @xmath1 and @xmath134 resummations , with the dynamical colour screening model presented here should provide a working description of the observed diffractive deep inelastic scattering process . constrained only by the requirement to be in the physically sensible range @xmath139 $ ] and to be fairly smooth . solid line : the probability from the fit of the colour screening model as used in fig . [ fig : cascade_cutsnone_m01alphas70 ] ( upper left corner ) . we note that both methods result in very similar functions for @xmath89 . ] it is interesting to compare the dynamic colour screening model with the results from having a fixed colour screening probability while keeping all other parameters equal . [ fig : cascade_cutsnone_const ] shows the diffractive cross section with @xmath126 , obtained by a fit to data . we note that the constant probability results in a significantly worse description of the data . the overall normalisation as well as the shape of @xmath96 with respect to @xmath5 is better described by the dynamic screening model . in the results from the dynamic model of fig . [ fig : cascade_cutsnone_m01alphas70 ] , we have fitted the two parameters @xmath81 and @xmath84 of the dynamic screening model to data . the parameters determine the overall normalisation and essentially the position of the slope where the screening probability @xmath89 falls off to zero . on the other hand , the general shape of this probability is given by the underlying model itself . it is interesting though to investigate what form of @xmath89 would result in a good fit without assuming an underlying model . to this end , we fit a mapping @xmath140 which is only constrained by the fact that it should lie in the physically sensible range @xmath141 $ ] and that it should be reasonably smooth . the result of such a fit is shown in fig . [ fig : prob - casc - m01-alphas70 ] . we observe that even though we did not place any particular constraint on the functional form , one obtains essentially the same result as in the dynamic screening model and therefore support to its qcd - basis . at the scales 1 , 4 and @xmath142{gev}$ ] for the solid , dashed and dotted lines , respectively , is shown in the left panel . the gluon density a1 for the same scales is shown in the right panel . the distribution starts out with a steeper slope at small scales and influences the observable as shown in fig . [ fig : cascade_gdf1013_cutsnone_m01alphas70 ] . ] at the scales 1 , 4 and @xmath142{gev}$ ] for the solid , dashed and dotted lines , respectively , is shown in the left panel . the gluon density a1 for the same scales is shown in the right panel . the distribution starts out with a steeper slope at small scales and influences the observable as shown in fig . [ fig : cascade_gdf1013_cutsnone_m01alphas70 ] . ] [ fig : gdf - plot-1013 ] but using a gluon density @xmath143 that increases stronger towards low-@xmath0 already at the starting scale ( a0 in @xcite ) . only the interesting subset of the kinematic plane is shown . by comparison with fig . [ fig : cascade_cutsnone_m01alphas70 ] it is seen that @xmath96 is at large @xmath110 and large @xmath34 sensitive to the shape of the gluon density at the starting scale . ] the result in fig . [ fig : cascade_cutsnone_m01alphas70 ] is obtained using the unintegrated gluon density @xmath144 illustrated in fig . [ fig : gdf - plot-1010 ] ( left ) . this density starts out flat at a low scale @xmath145 which can be compared with the @xmath146 behavior of the pomeron flux in regge - based models . the diffractive cross section in our model is sensitive to the slope especially in the kinematic region where @xmath110 is large . we can compare the main result in fig . [ fig : cascade_cutsnone_m01alphas70 ] with the result obtained by using a parton density which has a stronger increase towards small @xmath0 already at low scales , shown in fig . [ fig : gdf - plot-1010 ] ( right ) . the corresponding @xmath127 is shown in fig . [ fig : cascade_gdf1013_cutsnone_m01alphas70 ] . we note that especially the dependence of the cross section on @xmath5 is sensitive to the gluon density @xmath144 at small scales @xmath145 . by including diffractive data into the fit of a gluon density , this dependence could be used to further constrain the shape of the gluon distribution at low scales . but showing @xmath147 where the diffractive cross section is defined in terms of a large rapidity gap between @xmath148 and @xmath149 in pseudorapidity . ] range available in the experimental data is indicated by a horizontal thinner line for each row in @xmath110 . the fraction of events with a gluon in the lrg region is shown in the two rightmost plots . ] range available in the experimental data is indicated by a horizontal thinner line for each row in @xmath110 . the fraction of events with a gluon in the lrg region is shown in the two rightmost plots . ] [ fig : cascade_lrg40t60_m01alphas70 ] shows the diffractive cross section @xmath150 as defined by the presence of a large rapidity gap in the range @xmath151 in pseudorapidity . we note the close similarity to fig . [ fig : cascade_cutsnone_m01alphas70 ] . when requiring a minimum gap size @xmath152 in the event , the non - diffractive cross - section is exponentially suppressed by @xmath152 . while the presence of a lrg of size @xmath152 in the final state is a clear indication of diffraction , the exact location of the gap is sensitive to the description of the kinematics of the central system @xmath25 , and @xmath150 depends therefore additionally on the treatment of dynamics within the @xmath25-system . the dynamic colour screening model describes the probability of a colour screened interaction , but the additional soft exchanges do not change the kinematics of final state partons . we know on the other hand that our description in terms of matrix elements plus showering does not cover the full phase space . especially , the lack of additional gluon radiation from the quark propagator in the matrix element @xmath112 could be the reason why this @xmath22-system extends up to too high rapidity and can hadronize into the gap region . we therefore keep the gap size @xmath153 as in the data , but shift its starting value from 3.3 in data to 4 . this modification is irrelevant for our result at @xmath154 , but noticeable at @xmath155 where it causes at most a correction of @xmath156 at small @xmath110 and small @xmath5 , but becomes important at @xmath157 . we expect @xmath150 to be sensitive to higher order corrections especially close to the soft divergence @xmath94 where partons from the matrix element cause activity in the lrg region . this is illustrated in fig . [ fig : lrg - parton - veto ] where the fraction of events having a @xmath19 or @xmath158 with momentum vector into the lrg is shown . we note that the fraction of quarks in the lrg increases towards low @xmath5 which can be understood from the correlation between @xmath5 and the transverse momentum . on the other hand , the fraction of gluons in the lrg depends more strongly on @xmath34 and weaker on @xmath5 . this can be understood from the fact that gluons arise from the parton shower in contrast to the quarks which are defined by the matrix element . at the leading order of our computation , we note that the result is only mildly sensitive to the cuts employed to regulate the @xmath94 divergence . also , a comparison with massive matrix elements at leading order suggests that typical quark masses lead effectively to the same cuts on the energy sharing variable @xmath28 as used in our results . on the other hand , higher order corrections could significantly alter the internal event structure , especially in the region of the phase space where it is likely to have a large step @xmath28 between @xmath0 and @xmath103 . there , an additional final state gluon could modify the extension of the @xmath25 system in rapidity . an improved parton evolution based on ccfm with additional splittings @xmath159 and @xmath160 could therefore in principle improve the description further because the diffractive process at a low scale could be described with a qed matrix element and low @xmath0 resummed parton evolution . similarly , @xmath150 is also sensitive to the distribution of the transverse momenta and energy splitting in the parton shower . we have developed the probability for dynamic colour screening in dis in a way that can be used in monte carlo event generators and applied it with cascade and lepto . the resulting model predicts the diffractive dis cross section based on perturbative qcd matrix elements and standard inclusive parton densities in both collinear and @xmath2-factorisation approaches . this facilitates practical applications of previously obtained theoretical derivations of the amplitude for colour screening through semi - soft multiple gluon exchanges calculated in the eikonal approximation to all orders in perturbative qcd . the basic formalism gives a theoretical understanding why the soft colour interaction ( sci ) model has been phenomenologically successful , but goes beyond that model by leading to a colour screening probability that depends on the dynamics of the perturbative qcd parton dynamics . this dynamical screening probability exhibits a saturated behaviour at small transverse momenta of the emerging parton system as well as colour transparency at large transverse momentum . the monte carlo model has only two , physically motivated parameters . their values are obtained by fitting the hera diffractive cross section and found to be of the expected magnitude . the model successfully describes the data over a large kinematic range , significantly better than with a constant screening probability . interestingly , a fit of a free - form probability function results in the same shape as in our model , and hence gives support for our account of the basic qcd dynamics of relevance . it is noteworthy that we have not introduced any diffractive parton distribution functions to describe soft dynamics through unknown functions fitted to data . this explicitly demonstrates that diffractive and inclusive scattering can be described by the same basic qcd processes when explicit account is taken of color degrees of freedom in gluon exchanges at scales below normal cut - offs of @xmath161 gev of hard , perturbative qcd processes and @xmath162 of the hadronisation phase transition . in some kinematic regions there are two very different scales present , namely the invariant mass of the diffractive system @xmath131 and the photon virtuality @xmath5 . this calls for a resummation of large logarithms @xmath163 , or equivalently @xmath164 . to address this issue , we take the cross section in the @xmath2-factorisation approach from off - shell matrix elements and unintegrated gluon densities together with the ccfm evolution which provides a resummation of leading logarithms in @xmath165 . we show that this significantly improves the description of the diffractive hera data at @xmath166 corresponding to @xmath97 . nevertheless , there are residual deviations in the region where both @xmath110 and @xmath5 are at their lowest values . this may be attributed to the extreme region of very small @xmath110 when the quark propagator connected to the virtual photon have a significant phase space available for gluon radiation , which is not accounted for ; neither in the leading order matrix element for @xmath167 nor in the ccfm evolution that does not include the @xmath160 splitting . the dglap evolution does include this and also shows a slightly better result in this case , but is still not sufficient since here the effects of large @xmath168 is not included . to conclude and connect to the discussion in the introduction , our study has shown that the phenomenon of diffractive deep inelastic scattering can be described using a basic qcd - framework . the hard subprocess is treated in the same way as for non - diffractive events but a colour screening process occurs as a result of multiple gluon exchanges that are resummed to all orders . significant deviations from data occur in a special kinematic region , where potentially large logarithmic corrections are not yet fully included in available evolution equations for gluon radiation . still , the overall results show that gluonic colour screening in qcd is a viable approach to understand diffraction . j. r. forshaw and d. a. ross , cambridge lect . notes phys . * 9 * , 1 ( 1997 ) . g. ingelman and p. e. schlein , phys . b * 152 * , 256 ( 1985 ) . r. bonino _ et al . _ [ ua8 collaboration ] , phys . b * 211 * , 239 ( 1988 ) . g. ingelman , int . j. mod . a * 21 * , 1805 ( 2006 ) [ hep - ph/0512146 ] . s. chekanov _ et al . _ [ zeus collaboration ] , nucl . b * 816 * , 1 ( 2009 ) ; t. ahmed _ et al . _ , nucl . b * 429 * , 477 ( 1994 ) ; nucl . b * 435 * , 3 ( 1995 ) . f. d. aaron _ et al . _ [ h1 and zeus collaborations ] , eur . j. c * 72 * , 2175 ( 2012 ) [ arxiv:1207.4864 [ hep - ex ] ] . g. ingelman and k. prytz , z. phys . c * 58 * , 285 ( 1993 ) . t. affolder _ et al . _ [ cdf collaboration ] , phys . lett . * 84 * , 5043 ( 2000 ) . a. edin , g. ingelman and j. rathsman , phys . b * 366 * , 371 ( 1996 ) ; z. phys . c * 75 * , 57 ( 1997 ) . g. ingelman , a. edin and j. rathsman , comput . commun . * 101 * , 108 ( 1997 ) . t. sjstrand , s. mrenna and p. z. skands , jhep * 0605 * , 026 ( 2006 ) [ hep - ph/0603175 ] . a. edin , g. ingelman and j. rathsman , phys . d * 56 * , 7317 ( 1997 ) . r. enberg , g. ingelman and n. timneanu , phys . d * 64 * , 114015 ( 2001 ) . g. ingelman , r. pasechnik , j. rathsman and d. werder , phys . d * 87 * , no . 9 , 094017 ( 2013 ) [ arxiv:1210.5976 [ hep - ph ] ] . r. enberg , g. ingelman , a. kissavos and n. timneanu , phys . lett . * 89 * , 081801 ( 2002 ) . j. rathsman , phys . b * 452 * , 364 ( 1999 ) [ hep - ph/9812423 ] . s. j. brodsky , r. enberg , p. hoyer and g. ingelman , phys . d * 71 * , 074020 ( 2005 ) . r. pasechnik , r. enberg and g. ingelman , phys . d * 82 * , 054036 ( 2010 ) ; phys . b * 695 * , 189 ( 2011 ) . f. hautmann , z. kunszt and d. e. soper , phys . lett . * 81 * , 3333 ( 1998 ) ; nucl . b * 563 * , 153 ( 1999 ) . f. hautmann and d. e. soper , phys . d * 63 * , 011501 ( 2001 ) ; phys . d * 75 * , 074020 ( 2007 ) . v. n. gribov and l. n. lipatov , sov . j. nucl . * 15 * , 438 ( 1972 ) [ yad . fiz . * 15 * , 781 ( 1972 ) ] ; + g. altarelli and g. parisi , nucl . b * 126 * , 298 ( 1977 ) ; + y. l. dokshitzer , sov . jetp * 46 * , 641 ( 1977 ) [ zh . fiz . * 73 * , 1216 ( 1977 ) ] . h. jung _ et al . _ , eur . j. c * 70 * , 1237 ( 2010 ) . s. catani , f. fiorani and g. marchesini , nucl . b * 336 * , 18 ( 1990 ) ; + s. catani , f. fiorani and g. marchesini , phys . b * 234 * , 339 ( 1990 ) ; + s. catani , m. ciafaloni and f. hautmann , nucl . b * 366 * , 135 ( 1991 ) ; + g. marchesini , nucl . b * 445 * , 49 ( 1995 ) [ hep - ph/9412327 ] . b. andersson , g. gustafson , g. ingelman and t. sjstrand , phys . rept . * 97 * , 31 ( 1983 ) . s. j. brodsky , g. f. de teramond and a. deur , phys . d * 81 * , 096010 ( 2010 ) . d. v. shirkov and i. l. solovtsov , phys . * 79 * , 1209 ( 1997 ) . f. d. aaron _ et al . _ [ h1 collaboration ] , eur . j. c * 72 * ( 2012 ) 2074 [ arxiv:1203.4495 [ hep - ex ] ] .

we present a novel monte - carlo implementation of dynamic colour screening via multiple exchanges of semi - soft gluons as a basic qcd mechanism to understand diffractive electron - proton scattering at the hera collider . based on the kinematics of individual events in the standard qcd description of deep inelastic scattering at the parton level , which at low @xmath0 is dominantly gluon - initiated , the probability is evaluated for additional exchanges of softer gluons resulting in an overall colour singlet exchange leading to a forward proton and a rapidity gap as the characteristic observables for diffractive scattering . the probability depends on the impact parameter of the soft exchanges and varies with the transverse size of the hard scattering subsystem and is therefore influenced by different qcd effects . we account for matrix elements and parton shower evolution either via conventional dglap @xmath1-evolution with collinear factorisation or ccfm small-@xmath0 evolution with @xmath2-factorisation and discuss the sensitivity to the gluon density distribution in the proton and the importance of large log@xmath0-contributions . the overall result is that , with only two model parameters which have theoretically motivated values , a satisfactory description of the observed diffractive cross - section at hera is obtained in a wide kinematical range . lu tp 15 - 51 + april 2016 1.5 cm

the computational effort needed to deal with large combinatorial structures considerably varies with the task to be performed and the resolution procedure used@xcite . the worst case complexity of a task , more precisely an optimization or decision problem , is defined as the time required by the best algorithm to treat any possible inputs to the problem . for instance , the sorting problem of a list of @xmath0 numbers has worst - case complexity @xmath1 : there exists several algorithms that can order any list in at most @xmath2 elementary operations , and none with asymptotically less operations . unfortunately , the worst - case complexities of many important computational problems , called np - complete , is not known . partitioning a list of @xmath0 numbers in two sets with equal partial sums is one among hundreds of such np - complete problems . it is a fundamental conjecture of theoretical computer science that there exists no algorithm capable of partitioning any list of length @xmath0 , or of solving any other np - complete problem with inputs of size @xmath0 , in a time bounded by a polynomial of @xmath0 . therefore , when dealing with such a problem , one necessarily uses algorithms which may takes exponential times on some inputs . quantifying how ` frequent ' these hard inputs are for a given algorithm is the question answered by the analysis of algorithms . in this paper , we will present an overview of recent works done by physicists to address this point , and more precisely to characterize the average performances , called hereafter complexity , of a given algorithm over a distribution of inputs to an optimization problem . the history of algorithm analysis by physical methods / ideas is at least as old as the use of computers by physicists . one well - established chapter in this history is , for instance , the analysis of monte carlo sampling algorithms for statistical mechanics models . in this context , it is well known that phase transitions , _ i.e. _ abrupt changes in the physical properties of the model , can imply a dramatic increase in the time necessary to the sampling procedure . this phenomenon is commonly known as critical slowing down . the physicists insight in this problem comes mainly from the analogy between the dynamics of algorithms and the physical dynamics of the system . this analogy is quite natural : in fact many algorithms mimick the physical dynamics itself . a quite new idea is instead to abstract from physically motivated problems and use statistical mechanics ideas for analyzing the dynamics of algorithms . in effect there are many reasons which suggest that analysis of algorithms and statistical physics should be considered close relatives . in both cases one would like to understand the asymptotic behavior of dynamical processes acting on exponentially large ( in the size of the problem ) configuration spaces . the differences between the two disciplines mainly lie in the methods ( and , we are tempted to say , the style ) of investigation . theoretical computer science derives rigorous results based on probability theory . however these results are sometimes too weak for a complete characterization of the algorithm . physicists provide instead heuristic results based on intuitively sensible approximations . these approximations are eventually validated by a comparison with numerical experiments . in some lucky cases , approximations are asymptotically irrelevant : estimates are turned into conjectures left for future rigorous derivations . perhaps more interesting than stylistic differences is the _ point of view _ which physics brings with itself . let us highlight two consequences of this point of view . first , a particular importance is attributed to `` complexity phase transitions '' _ i.e. _ abrupt changes in the resolution complexity as some parameter defining the input distribution is varied@xcite . we shall consider two examples in the next sections : * random satisfiability of boolean constraints ( sat ) . in @xmath3-sat one is given an instance , that is , a set of @xmath4 logical constraints ( clauses ) among @xmath0 boolean variables , and wants to find a truth assignment for the variables which fulfill all the constraints . each clause is the logical or of @xmath3 literals , a literal being one of the @xmath0 variables or its negation e.g. @xmath5 for 3-sat . random @xmath3-sat is the @xmath3-sat problem supplied with a distribution of inputs uniform over all instances having fixed values of @xmath0 and @xmath4 . the limit of interest is @xmath6 at fixed ratio @xmath7 of clauses per variable@xcite . * vertex cover of random graphs ( vc ) . an input instance of the vc decision problem consists in a graph @xmath8 and an integer number @xmath9 . the problem consists in finding a way to distribute @xmath9 covering marks over the vertices in such a way that every edge of the graph is covered , that is , has at least one of its ending vertices marked . a possible distribution of inputs is provided by drawing random graphs @xmath8 _ la _ erds - reny _ i.e. _ with uniform probability among all the graphs having @xmath0 vertices and @xmath10 edges . the limit of interest is @xmath11 at fixed ratio @xmath12 of edges per vertex . the algorithms for random sat and vc we shall consider in the next sections undergo a complexity phase transition as the input parameter @xmath13 ( @xmath14 for sat , @xmath15 for vc ) crosses some critical threshold @xmath16 . typically resolution of a randomly drawn instance requires linear time below the threshold @xmath17 and exponential time above @xmath18 . the observation that most difficult instances are located near the phase boundary confirms the relevance of the phase - transition phenomenon . secondly , a key role is played by the intrinsic ( algorithm independent ) properties of the instance under study . the intuition is that , underlying the dramatic slowing down of a particular algorithm , there can be some _ qualitative _ change in some structural property of the problem e.g. the geometry of the space of solutions . while there is no general understanding of this question , we can further specify the above statements case - by - case . let us consider , for instance , a local search algorithm for a combinatorial optimization problem . if the algorithm never increases the value of the cost function @xmath19 where @xmath20 is the configuration ( assignment ) of variables to be optimized over , the number and geometry of the local minima of @xmath19 will be crucial for the understanding of the dynamics of the algorithm . this example is illustrated in sec . [ gradxor ] . while the `` dynamical '' behavior of a particular algorithm is not necessarily related to any `` static '' property of the instance , this approach is nevertheless of great interest because it could provide us with some ` universal ' results . some properties of the instance , for example , may imply the ineffectiveness of an entire class of algorithms . while we shall mainly study in this paper the performances of search algorithms applied to hard combinatorial problems as sat , vc , we will also consider easy , that is , polynomial problems as benchmarks for these algorithms . the reason is that we want to understand if the average hardness of resolution of solving np - complete problems with a given distribution of instances and a given algorithm truly reflects the intrinsic hardness of these combinatorial problems or is simply due to some lack of efficiency of the algorithm under study . the benchmark problem we shall consider is random xorsat . it is a version of a satisfiability problem , much simpler than sat from a computational complexity point of view@xcite . the only but essential difference with sat is that a clause is said to be satisfied if the exclusive , and not inclusive , disjunction of its literals is true . xorsat may be recast as a linear algebra problem , where a set of @xmath4 equations involving @xmath0 boolean variables must be satisfied modulo 2 , and is therefore solvable in polynomial time by various methods e.g. gaussian elimination . nevertheless , it is legitimate to ask what are the performances of general search algorithms for this kind of polynomial computational problem . in particular , we shall see that some algorithms requiring exponential times to solve random sat instances behave badly on random xorsat instances too . a related question we shall focus on in sec . [ codesection ] is decoding , which may also , in some cases , be expressed as the resolution of a set of boolean equations . the paper is organized as follows . in sec . [ dpllsection ] we shall review backtracking search algorithms , which , roughly speaking , work in the space of instances . we explain the general ideas and then illustrate them on random sat ( sec . [ dpllsatsection ] ) and vc ( sec . [ dpllvcsection ] ) . in sec . [ dpllflucsection ] we consider the fluctuations in running times of these algorithms and analyze the possibility of exploiting these fluctuations in random restart strategies . in sec . [ localsection ] we turn to local search algorithms , which work in the space of configurations . we review the analysis of such algorithms for decoding problems ( sec . [ codesection ] ) , random xorsat ( sec . [ gradxor ] ) , and sat ( sec . [ walksatsection ] ) . finally in the conclusion we suggest some possible future developments in the field . in this section , we briefly review the davis - putnam - loveland - logemann ( dpll ) procedure @xcite . a decision problem can be formulated as a constrained satisfaction problem , where a set of variables must be sought for to fulfill some given constraints . for simplicity , we suppose here that variables may take a finite set of values with cardinality @xmath21 e.g. @xmath22 for sat or vc . dpll is an exhaustive search procedure operating by trials and errors , the sequence of which can be graphically represented by a search tree ( fig . [ trees ] ) . the tree is defined as follows : * ( 1 ) * a node in the tree corresponds to a choice of a variable . * ( 2 ) * an outgoing branch ( edge ) codes for the value of the variable and the logical implications of this choice upon not yet assigned variables and clauses . obviously a node gives birth to @xmath21 branches at most . * ( 3 ) * implications can lead to : * ( 3.1 ) * a violated constraint , then the branch ends with @xmath20 ( contradiction ) , the last choice is modified ( backtracking of the tree ) and the procedure goes on along a new branch ( point 2 above ) ; * ( 3.2 ) * a solution when all constraints are satisfied , then the search process is over ; * ( 3.3 ) * otherwise , some constraints remain and further assumptions on the variables have to be done ( loop back to point 1 ) . a computer independent measure of computational complexity , that is , the amount of operations necessary to solve the instance , is given by the size @xmath23 of the search tree _ i.e. _ the number of nodes it contains . performances can be improved by designing sophisticated heuristic rules for choosing variables ( point 1 ) . the resolution time ( or complexity ) is a stochastic variable depending on the instance under consideration and on the choices done by the variable assignment procedure . its average value , @xmath24 , is a function of the input distribution parameters @xmath13 e.g. the ratio @xmath25 of clauses per variable for sat , or the average degree @xmath15 for the vc of random graphs , which can be measured experimentally and that we want to calculate theoretically . more precisely , our aim is to determine the values of the input parameters for which the complexity is linear , @xmath26 or exponential , @xmath27 , in the size @xmath0 of the instance and to calculate the coefficients @xmath28 as functions of @xmath13 . the dpll algorithm gives rise to a dynamical process . indeed , the initial instance is modified during the search through the assignment of some variables and the simplification of the constraints that contain these variables . therefore , the parameters of the input distribution are modified as the algorithm runs . this dynamical process has been rigorously studied and understood in the case of a search tree reducing to one branch ( tree a in figure [ trees])@xcite . study of trees with massive backtracking e.g. trees b and c in fig . [ trees ] is much more difficult . backtracking introduces strong correlations between nodes visited by dpll at very different times , but close in the tree . in addition , the process is non markovian since instances attached to each node are memorized to allow the search to resume after a backtracking step . the study of the operation of dpll is based on the following , elementary observation . since instances are modified when treated by dpll , description of their statistical properties generally requires additional parameters with respects to the defining parameters @xmath13 of the input distribution . our task therefore consists in 1 . identifying these extra parameters @xmath29@xcite ; 2 . deriving the phase diagram of this new , extended distribution @xmath30 to identify , in the @xmath30 space , the critical surface separating instances having solution with high probability ( satisfiable phase ) from instances having generally no solution ( unsatisfiable phase ) , see fig . [ schemoins ] . 3 . tracking the evolution of an instance under resolution with time @xmath31 ( number of steps of the algorithm ) , that is , the trajectory of its characteristic parameters @xmath32 in the phase diagram . whether this trajectory remains confined to one of the two phases or crosses the boundary inbetween has dramatic consequences on the resolution complexity . we find three average behaviours , schematized on fig . [ schemoins ] : * if the initial instance has a solution and the trajectory remains in the sat phase , resolution is typically linear , and almost no backtracking is present ( fig . [ trees]a ) . the coordinates of the trajectory @xmath32 of the instance in the course of the resolution obey a set of coupled ordinary differential equations accounting for the changes in the distribution parameters done by dpll . * if the initial instance has no solution , solving the instance , that is , finding a proof of unsatisfiability , takes exponentially large time and makes use of massive backtracking ( fig . [ trees]b ) . analysis of the search tree is much more complicated than in the linear regime , and requires a partial differential equation that gives information on the population of branches with parameters @xmath30 throughout the growth of the search tree . * in some intermediary regime , instances have solutions but finding one requires an exponentially large time ( fig . [ trees]c ) . this may be related to the crossing of the boundary between sat and unsat phases of the instance trajectory . we have therefore a mixed behaviour which can be understood through combination of the two above cases . we now explain how to apply concretly this approach to the cases of random sat and vc . and @xmath29 are scalar and not vectorial parameters . vertical axis is the instance distribution defining parameter @xmath13 . instances are almost always satisfiable if @xmath33 , unsatisfiable if @xmath34 . under the action of dpll , the distribution of instances is modified and requires another parameter @xmath29 to be characterized ( horizontal axis ) , equal to , say , zero prior to any action of dpll . for non zero values of @xmath29 , the critical value of the defining parameter @xmath13 obviously changes ; the line @xmath35 defines a boundary separating typically sat from unsat instances ( bold line ) . when the instance is unsat ( point u ) , dpll takes an exponential time to go through the tree trajectory . for satisfiable and easy instances , dpll goes along a branch trajectory in a linear time ( point s ) . the mixed case of hard sat instances ( point ms ) correspond to the branch trajectory crossing the boundary separating the two phases ( bold line ) , which leads to the exploration of unsat subtrees before a solution is finally found . ] clauses involving @xmath36 variables @xmath37 , which can be assigned to true ( t ) or false ( f ) . @xmath38 means ( not @xmath39 ) and @xmath40 denotes the logical or . the search tree is empty . dpll randomly selects a clause among the shortest ones , and assigns a variable in the clause to satisfy it , e.g. @xmath41 t ( splitting with the generalized unit clause guc heuristic @xcite ) . a node and an edge symbolizing respectively the variable chosen ( @xmath39 ) and its value ( t ) are added to the tree . the logical implications of the last choice are extracted : clauses containing @xmath39 are satisfied and eliminated , clauses including @xmath38 are simplified and the remaining ones are left unchanged . if no unitary clause ( _ i.e. _ with a single variable ) is present , a new choice of variable has to be made . * * splitting takes over . another node and another edge are added to the tree . * * same as step 2 but now unitary clauses are present . the variables they contain have to be fixed accordingly . the propagation of the unitary clauses results in a contradiction . the current branch dies out and gets marked with c. * 6 . * dpll backtracks to the last split variable ( @xmath42 ) , inverts it ( f ) and creates a new edge . * * same as step 4 . the propagation of the unitary clauses eliminates all the clauses . a solution s is found and the instance is satisfiable . for an unsatisfiable instance , unsatisfiability is proven when backtracking ( see step 6 ) is not possible anymore since all split variables have already been inverted . in this case , all the nodes in the final search tree have two descendent edges and all branches terminate by a contradiction c. ] the input distribution of 3-sat is characterized by a single parameter @xmath13 , the ratio @xmath25 of clauses per variable . the action of dpll on an instance of 3-sat , illustrated in fig . [ algo ] , causes the changes of the overall numbers of variables and clauses , and thus of @xmath25 . furthermore , dpll reduces some 3-clauses to 2-clauses . we use a mixed 2+p - sat distribution@xcite , where @xmath43 is the fraction of 3-clauses , to model what remains of the input instance at a node of the search tree . using experiments and methods from statistical mechanics@xcite and rigorous calculations@xcite , the threshold line @xmath44 , separating sat from unsat phases , may be estimated with the results shown in fig . [ sche ] . for @xmath45 , _ i.e. _ left to point t , the threshold line is given by @xmath46 , and saturates the upper bound for the satisfaction of 2-clauses . above @xmath47 , no exact value for @xmath44 is known . the phase diagram of 2+p - sat is the natural space in which the dpll dynamics takes place . an input 3-sat instance with ratio @xmath25 shows up on the right vertical boundary of fig . [ sche ] as a point of coordinates @xmath48 . under the action of dpll , the representative point moves aside from the 3-sat axis and follows a trajectory in the @xmath49 plane . in this section , we show that the location of this trajectory in the phase diagram allows a precise understanding of the search tree structure and of complexity as a function of the ratio @xmath25 of the instance to be solved ( inset of fig . [ sche ] ) . in addition , we shall present an approximate calculation of trajectories accounting for the case of massive backtracking , that is for unsat instances , and slightly below the threshold in the sat phase . our approach is based on a non rigorous extension of works by chao and franco who first studied the action of dpll ( without backtracking ) on easy , sat instances@xcite as a way to obtain lower bounds to the threshold @xmath50 , see @xcite for a recent review . let us emphasize that the idea of trajectory is made possible thanks to an important statistical property of the heuristics of split we consider @xcite , * unit - clause ( uc ) heuristic : pick up randomly a literal among a unit clause if any , or any unset variable otherwise . * generalized unit - clause ( guc ) heuristic : pick up randomly a literal among the shortest avalaible clauses . * short clause with majority ( sc@xmath51 ) heuristic : pick up randomly a literal among unit clauses if any , or pick up randomly an unset variable @xmath21 , count the numbers of occurences @xmath52 of @xmath21 , @xmath53 in 3-clauses , and choose @xmath21 ( respectively @xmath53 ) if @xmath54 ( resp . @xmath55 ) . when @xmath56 , @xmath21 and @xmath53 are equally likely to be chosen . these heuristics do not induce any bias nor correlation in the instances distribution@xcite . such a statistical `` invariance '' is required to ensure that the dynamical evolution generated by dpll remains confined to the phase diagram of fig . [ sche ] . in the following , the initial ratio of clauses per variable of the instance to be solved will be denoted by @xmath57 . let us consider the first descent of the algorithm , that is the action of dpll in the absence of backtracking . the search tree is a single branch ( fig . [ trees]a ) . the numbers of 2 and 3-clauses are initially equal to @xmath58 respectively . under the action of dpll , @xmath59 and @xmath60 follow a markovian stochastic evolution process , as the depth @xmath61 along the branch ( number of assigned variables ) increases . both @xmath59 and @xmath60 are concentrated around their average values , the densities @xmath62 $ ] ( @xmath63 ) of which obey a set of coupled ordinary differential equations ( ode)@xcite , @xmath64 where @xmath65 is the probability that dpll fixes a variable at depth @xmath31 through unit - propagation . function @xmath66 depends upon the heuristic : @xmath67 , @xmath68 ( if @xmath69 ) , @xmath70 with @xmath71 and @xmath72 denotes the @xmath73 modified bessel function . to obtain the single branch trajectory in the phase diagram of fig . [ sche ] , we solve the odes ( [ ode ] ) with initial conditions @xmath74 , and perform the change of variables @xmath75 results are shown for the guc heuristics and starting ratios @xmath76 and 2.8 in fig . [ sche ] . trajectories , indicated by light dashed lines , first head to the left and then reverse to the right until reaching a point on the 3-sat axis at a small ratio . further action of dpll leads to a rapid elimination of the remaining clauses and the trajectory ends up at the right lower corner s , where a solution is found . frieze and suen @xcite have shown that , for ratios @xmath77 ( for the guc heuristics ) , the full search tree essentially reduces to a single branch , and is thus entirely described by the odes ( [ ode ] ) . the number of backtrackings necessary to reach a solution is bounded from above by a power of @xmath78 . the average size @xmath24 of the branch then scales linearly with @xmath0 with a multiplicative factor @xmath79 that can be analytically computed @xcite . the boundary @xmath80 of this easy sat region can be defined as the largest initial ratio @xmath57 such that the branch trajectory @xmath81 issued from @xmath57 never leaves the sat phase in the course of dpll resolution . for ratios above threshold ( @xmath82 ) , instances almost never have a solution but a considerable amount of backtracking is necessary before proving that clauses are incompatible . as shown in fig . [ trees]b , a generic unsat tree includes many branches . the number of branches ( leaves ) , @xmath83 , or the number of nodes , @xmath84 , grow exponentially with @xmath0@xcite . it is convenient to define its logarithm @xmath85 through @xmath86 . contrary to the previous section , the sequence of points @xmath87 characterizing the evolution of the 2+p - sat instance solved by dpll does not define a line any longer , but rather a patch , or cloud of points with a finite extension in the phase diagram of fig . [ schemoins ] . we have analytically computed the logarithm @xmath85 of the size of these patches , as a function of @xmath88 , extending to the unsat region the probabilistic analysis of dpll . this is , _ a priori _ , a very difficult task since the search tree of fig . 1b is the output of a complex , sequential process : nodes and edges are added by dpll through successive descents and backtrackings . we have imagined a different building up , that results in the same complete tree but can be mathematically analyzed : the tree grows in parallel , layer after layer ( fig . [ struct ] ) . denotes the depth in the tree , that is the number of variables assigned by dpll along each ( living ) branch . at depth @xmath61 , one literal is chosen on each branch among 1-clauses ( unit - propagation , grey circles not represented on figure 1 ) , or 2,3-clauses ( split , black circles as in figure 1 ) . if a contradiction occurs as a result of unit - propagation , the branch gets marked with c and dies out . the growth of the tree proceeds until all branches carry c leaves . the resulting tree is identical to the one built through the usual , sequential operation of dpll . ] a new layer is added by assigning , according to dpll heuristic , one more variable along each living branch . as a result , a branch may split ( case 1 ) , keep growing ( case 2 ) or carry a contradiction and die out ( case 3 ) . cases 1,2 and 3 are stochastic events , the probabilities of which depend on the characteristic parameters @xmath89 defining the 2+p - sat instance carried by the branch , and on the depth ( fraction of assigned variables ) @xmath31 in the tree . we have taken into account the correlations between the parameters @xmath90 on each of the two branches issued from splitting ( case 1 ) , but have neglected any further correlation which appear between different branches at different levels in the tree@xcite . this markovian approximation permits to write an evolution equation for the logarithm @xmath91 of the average number of branches with parameters @xmath90 as the depth @xmath31 increases , @xmath92 \qquad . \label{croi}\ ] ] @xmath93 incorporates the details of the splitting heuristics . in terms of the partial derivatives @xmath94 , @xmath95 , we have for the uc and guc heuristics @xmath96 \nonumber \\ { h } _ { guc } % ( c_2 , c_3 , y_2 , y_3 , t ) & = & \log _ 2 \nu ( y_2 ) + \frac{1}{\ln2 } \left [ \frac { 3\ , c_3}{1-t}\ ; \left ( e^{y_3 } \frac{1+e^{-y_2}}{2 } -1 \right)+ \frac{c_2}{1-t } \ ; \left ( \nu(y_2 ) -2 \right ) \right ] \nonumber \\ \hbox{\rm where } & & \nu ( y_2 ) = \frac 12\ ; e^{y_2}\left ( 1 + \sqrt{1 + 4 e^{-y_2 } } \right)\qquad .\end{aligned}\ ] ] partial differential equation ( pde ) ( [ croi ] ) is analogous to growth processes encountered in statistical physics @xcite . the surface @xmath85 , growing with `` time '' @xmath31 above the plane @xmath97 , or equivalently from ( [ change ] ) , above the plane @xmath87 ( fig . [ dome ] ) , describes the whole distribution of branches . the average number of branches at depth @xmath31 in the tree equals @xmath98 , where @xmath99 is the maximum over @xmath100 of @xmath101 reached in @xmath102 . in other words , the exponentially dominant contribution to @xmath103 comes from branches carrying 2+p - sat instances with parameters @xmath102 , which define the tree trajectories on fig . [ sche ] . the hyperbolic line in fig . [ sche ] indicates the halt points , where contradictions prevent dominant branches from further growing . each time dpll assigns a variable through unit - propagation , an average number @xmath104 of new 1-clauses is produced , resulting in a net rate of @xmath105 additional 1-clauses . as long as @xmath106 , 1-clauses are quickly eliminated and do not accumulate . conversely , if @xmath107 , 1-clauses tend to accumulate . opposite 1-clauses @xmath42 and @xmath108 are likely to appear , leading to a contradiction @xcite . the halt line is defined through @xmath109 . as far as dominant branches are concerned , the equation of the halt line reads @xmath110\;\frac 1{1-p } \simeq \frac{1.256}{1-p}\qquad .\ ] ] along the tree trajectory , @xmath99 grows from 0 , on the right vertical axis , up to some final positive value , @xmath111 , on the halt line . @xmath112 is our theoretical prediction for the logarithm of the complexity ( divided by @xmath0 ) . values of @xmath113 obtained for @xmath114 by solving equation ( [ croi ] ) compare very well with numerical results @xcite . .5 cm we have plotted the surface @xmath85 above the @xmath87 plane , with the results shown in fig . it must be stressed that , though our calculation is not rigorous , it provides a very good quantitative estimate of the complexity . furthermore , complexity is found to scale asymptotically as @xmath115 ^ 2 \simeq \frac{0.292}{\alpha _ 0 } \qquad ( \alpha _ 0 \gg \alpha _ c ) .\ ] ] this result exhibits the expected scaling@xcite , and could indeed be exact . as @xmath57 increases , search trees become smaller and smaller , and correlations between branches , weaker and weaker . the interest of the trajectory approach proposed in this paper is best seen in the upper sat phase , that is ratios @xmath57 ranging from @xmath80 to @xmath116 . this intermediate region juxtaposes branch and tree behaviors , see fig . [ trees]c . the branch trajectory starts from the point @xmath117 corresponding to the initial 3-sat instance and hits the critical line @xmath118 at some point g with coordinates ( @xmath119 ) after @xmath120 variables have been assigned by dpll ( fig . [ sche ] ) . the algorithm then enters the unsat phase and generates 2+p - sat instances with no solution . a dense subtree , that dpll has to go through entirely , forms beyond g till the halt line ( fig . [ sche ] ) . the size of this subtree , @xmath121 , can be analytically predicted from our theory . g is the highest backtracking node in the tree ( fig . [ trees]c ) reached back by dpll , since nodes above g are located in the sat phase and carry 2+p - sat instances with solutions . dpll will eventually reach a solution . the corresponding branch ( rightmost path in fig . [ trees]c ) is highly non typical and does not contribute to the complexity , since almost all branches in the search tree are described by the tree trajectory issued from g ( fig . [ sche ] ) . we have checked experimentally this scenario for @xmath122 . the coordinates of the average highest backtracking node , @xmath123 ) , coincide with the analytically computed intersection of the single branch trajectory and the critical line @xmath118@xcite . as for complexity , experimental measures of @xmath85 from 3-sat instances at @xmath124 , and of @xmath125 from 2 + 0.78-sat instances at @xmath126 , obey the expected identity @xmath127 and are in very good agreement with theory@xcite . therefore , the structure of search trees for 3-sat reflects the existence of a critical line for 2+p - sat instances . we now consider the vc problem , where inputs are random graphs drawn from the @xmath128 ensemble@xcite . in other words , graphs have @xmath0 vertices and the probability that a pair of vertices are linked through an edge is @xmath129 , independently of other edges . when the number @xmath130 of covering marks is lowered , the model undergoes a cov / uncov transition at some critical density of covers @xmath131 for @xmath132 . for @xmath133 , vertex covers of size @xmath134 exist with probability one , for @xmath135 the available covering marks are not sufficient . the statistical mechanics analysis of ref . @xcite gave the result @xmath136 where @xmath137 solves the equation @xmath138 . this result is compatible with the bounds of refs . @xcite , and was later shown to be exact @xcite . for @xmath139 , eq . ( [ critical_vc ] ) only gives an approximate estimate of @xmath131 . more sophisticated calculations can be found in ref . @xcite . , high-@xmath15 uncov phase is separated by the dashed line , cf . ( [ critical_vc ] ) , from the high-@xmath42 , low-@xmath15 cov phase . the symbols ( numerics ) and continuous lines ( analytical prediction , cf . ( [ eqtrajvc ] ) ) refer to the simple search algorithm described in the text . the dotted line is the separatrix between two types of trajectories . ] let us consider a simple implementation of the dpll procedure for the present problem . during the computation , vertices can be _ covered _ , _ uncovered _ or just _ free _ , meaning that the algorithm has not yet assigned any value to that vertex . at the beginning all the vertices are set _ free_. at each step the algorithm chooses a vertex @xmath140 at random among those which are _ free_. if @xmath140 has neighboring vertices which are either _ free _ or _ uncovered _ , then the vertex @xmath140 is declared _ covered _ first . in case @xmath140 has only covered neighbors , the vertex is declared _ uncovered_. the process continues unless the number of covered vertices exceeds @xmath9 . in this case the algorithm backtracks and the opposite choice is taken for the vertex @xmath140 unless this corresponds to declaring _ uncovered _ a vertex that has one or more _ uncovered _ neighbors . the algorithm halts if it finds a solution ( and declares the graph to be cov ) or after exploring all the search tree ( in this case it declares the graph to be uncov ) . , is plotted versus the number of covering marks . here we consider random instances with average connectivity @xmath141 . the phase transition is at @xmath142 and corresponds to the peak in computational complexity . ] of course one can improve over this algorithm by using smarter heuristics @xcite . one remarkable example is the `` leaf - removal '' algorithm defined in ref . @xcite . instead of picking any vertex randomly , one chooses a connectivity - one vertex , declare it _ uncovered _ , and declare _ covered _ its neighbor . this procedure is repeated iteratively on the subgraph of _ free _ nodes , until no connectivity - one nodes are left . in the low - connectivity , cov region @xmath143 , it stops only when the graph is completely covered . as a consequence , this algorithm can solve vc in linear time with high probability in all this region . no equally good heuristics exists for higher connectivity , @xmath139 . under the action of one of the above algorithms , the instance is progressively modified and the number of variables is reduced . in fact , at each step a vertex is selected and can be eliminated from the graph regardless whether it is declared _ covered _ or _ uncovered_. the analysis of the first algorithm is greatly simplified by the remark that , as long as backtracking has not begun , the new vertex is selected randomly . this implies that the modified instance produced by the algorithm is still a random graph . its evolution can be effectively described by a trajectory in the @xmath144 space . if one starts from the parameters @xmath145 , @xmath146 , after @xmath147 steps of the algorithm , he will end up with a new instance of size @xmath148 and parameters @xcite @xmath149 some examples of the two types of trajectories ( the ones leading to a solution and the ones which eventually enter the uncov region ) are shown in fig . [ traj_vc ] . the separatrix is given by @xmath150 and corresponds to the dotted line in fig . [ traj_vc ] . above this line the algorithm solves the problem in linear time . for more general heuristics the analysis becomes less straightforward because the graph produced by the algorithm does not belong to the standard random - graph _ ensemble_. it may be necessary to augment the number of parameters which describe the evolution of the instance . as an example , the leaf - removal algorithm mentioned in the previous section is conveniently described by keeping track of three numbers which parametrize the degree profile ( i.e. the fraction of vertices @xmath151 having a given degree @xmath152 ) of the graph @xcite . below the critical line @xmath131 , cf . ( [ critical_vc ] ) , no solution exists to the typical random instance of vc . our algorithm must explore a large backtracking tree to prove it and this takes an exponential time . the size of the backtracking tree could be computed along the lines of sec . however a good result can be obtained with a simple `` static '' calculation @xcite . as explained in sec . ii.b.2 , we imagine the evolution of the backtracking tree as proceeding `` in parallel '' . at the level @xmath4 of the tree a set of @xmath4 vertices has been visited . call @xmath153 the subgraph induced by these vertices . since we always put a covering mark on a vertex which is surrounded by vertices declared _ uncovered _ , each node of the backtracking tree will carry a vertex cover of the associated subgraph @xmath153 . therefore the number of backtracking nodes is given by @xmath154 where @xmath155 is the number of vc s of @xmath153 using at most @xmath9 marks . a very crude estimate of the right - hand side of the above equation is : @xmath156 where we bounded the number of vc s of size @xmath157 on @xmath153 with the number of ways of placing @xmath157 marks on @xmath4 vertices . the authors of @xcite provided a refined estimate based on the _ annealed approximation _ of statistical mechanics . the results of this calculation are compared in fig . [ time_vc ] with the numerics . if the parameters which characterize an instance of vc lie in the region between @xmath131 , cf . ( [ critical_vc ] ) , and @xmath158 , cf . ( [ separatrix_vc ] ) , the problem is still soluble but our algorithm takes an exponential time to solve it . in practice , after a certain number of vertices has been visited and declared either _ covered _ or _ uncovered _ , the remaining subgraph @xmath159 can not be any longer covered with the leftover marks . this happens typically when the first descent trajectory ( [ eqtrajvc ] ) crosses the critical line ( [ critical_vc ] ) . it takes some time for the algorithm to realize this fact . more precisely , it takes exactly the time necessary to prove that @xmath159 is uncoverable . this time dominates the computational complexity in this region and can be calculated along the lines sketched in the previous section . the result is , once again , reported in fig . [ time_vc ] , which clearly shows a computational peak at the phase boundary . finally , let us notice that this mixed behavior disappears in the entire @xmath160 region if the leaf - removal heuristics is adopted for the first descent . up to now we have studied the typical resolution complexity . the study of fluctuations of resolution times is interesting too , particularly in the upper sat phase where solutions exist but are found at a price of a large computational effort . we may expect that there exist lucky but rare resolutions able to find a solution in a time much smaller than the typical one . due to the stochastic character of dpll complexity indeed fluctuates from run to run of the algorithm on the same instance . in fig . [ historun ] we show this run - to - run distribution of the logarithm @xmath85 of the resolution complexity for four instances of random 3-sat with the same ratio @xmath161 . the run to run distribution are qualitatively independent of the particular instances , and exhibit two bumps . the wide right one , located in @xmath162 , correspond to the major part of resolutions . it acquires more and more weight as @xmath0 increases and corresponds to the typical behavior analysed in section ii.b.3 . the left peak corresponds to much faster resolutions , taking place in linear time . the weight of this peak ( fraction of runs with complexities falling in the peak ) decreases exponentially fast with @xmath0 , and can be numerically estimated to @xmath163 with @xmath164 . therefore , instances at @xmath161 are typically solved in exponential time but a tiny ( exponentially small ) fraction of runs are able to find a solution in linear time only . a systematic stop - and - restart procedure may be introduced to take advantage of this fluctuation phenomenon and speed up resolution . if a solution is not found before @xmath0 splits , dpll is stopped and rerun after some random permutations of the variables and clauses . the expected number @xmath165 of restarts necessary to find a solution being equal to the inverse probability @xmath166 of linear resolutions , the resulting complexity scales as @xmath167 . to calculate @xmath168 we have analyzed , along the lines of the study of the growth of the search tree in the unsat phase , the whole distribution of the complexity for a given ratio @xmath25 in the upper sat phase . calculations can be found in @xcite . linear resolutions are found to correspond to branch trajectories that cross the unsat phase without being hit by a contradiction , see fig . results are reported in fig . [ histolin ] and compare very well with the experimentally measured number @xmath165 of restarts necessary to find a solution . in the whole upper sat phase , the use of restarts offers an exponential gain with respect to usual dpll resolution ( see fig . [ histolin ] for comparison between @xmath168 and @xmath85 ) , but the completeness of dpll is lost . a slightly more general restart strategy consists in stopping the backtracking procedure after a fixed number of nodes @xmath169 has been visited . a new ( and statistically independent ) dpll procedure is then started from the beginning . in this case one exploits lucky , but still exponential , stochastic runs . the tradeoff between the exponential gain of time and the exponential number of restarts , can be optimized by tuning the parameter @xmath170 . this approach has been analyzed in ref . @xcite taking vc as a working example . in fig . [ time_rvc ] we show the computatonal complexity of such a strategy as a function of the restart parameter @xmath170 . we compare the numerics with an approximate calculation @xcite . the instances were random graphs with average connectivity @xmath171 , and @xmath172 covering marks per vertex . the optimal choice of the parameter seems to be ( in this case ) @xmath173 , corresponding to polynomial runs . the analytical prediction reported in fig . [ time_rvc ] requires , as for 3-sat , an estimate of the execution - time fluctuations of the dpll procedure ( without restart ) . it turns out that one major source of fluctuations is , in the present case , the location in the @xmath144 plane of the highest node in the backtracking tree . in the typical run this coincides with the intersection @xmath174 between the first descent trajectory ( [ eqtrajvc ] ) and the critical line ( [ critical_vc ] ) . one can estimate the probability @xmath175 for this node to have coordinates @xmath144 ( obviously @xmath176 ) . when an upper bound @xmath170 on the backtracking time is fixed , the problem is solved in those lucky runs which are characterized by an atypical highest backtracking node . roughly speaking , this means that the algorithm has made some very good ( random ) choices in its first steps . in fig . [ root_rvc ] we plot the position of the highest backtracking point in the ( last ) successful runs for several values of @xmath170 . once again the numerics compare favourably with an approximate calculation . we now turn to the description and study of algorithms of another type , namely local search algorithms . as a common feature , these algorithms start from a configuration ( assignment ) of the variables , and then make successive improvements by changing at each step few of the variables in the configuration ( local move ) . for instance , in the sat problem , one variable is flipped from being true to false , or _ vice versa _ , at each step . whereas complete algorithms of the dpll type give a definitive answer to any instance of a decision problem , exhibiting either a solution or a proof of unsatisfiability , local search algorithms give a sure answer when a solution is found but can not prove unsatisfiability . however , these algorithms can sometimes be turned into one - sided probabilistic algorithms , with an upper bound on the probability that a solution exists and has not been found after @xmath61 steps of the algorithm , decreasing to zero when @xmath177@xcite . local search algorithms perform repeated changes of a configuration @xmath20 of variables ( values of the boolean variables for sat , status marked or unmarked of vertices for vc ) according to some criterion , usually based on the comparison of the cost function @xmath178 ( number of unsatisfied clauses for sat , of uncovered edges for vc ) evaluated at @xmath20 and over its neighborhood . it is therefore clear that the shape of the multidimensional surface @xmath179 , called cost function landscape , is of high importance . on intuitive grounds , if this landscape is relatively smooth with a unique minimum , local procedures as gradient descent should be very efficient , while the presence of many local minima could hinder the search process ( fig . [ landscape ] ) . the fundamental underlying question is whether the performances of the dynamical process ( ability to find the global minimum , time needed to reach it ) can be understood in terms of an analysis of the cost function landscape only . this question was intensively studied and answered for a limited class of cost functions , called mean field spin glass models , some years ago@xcite . the characterization of landscapes is indeed of huge importance in physical systems . there , the cost function is simply the physical energy , and local dynamics are usually low or zero temperature monte carlo dynamics , essentially equivalent to gradient descent . depending on the parameters of the input distribution , the minima of the cost functions may undergo structural changes , a phenomenon called clustering in physics . clustering has been rigorously shown to take place in the random 3-xorsat problem@xcite , and is likely to exist in many other random combinatorial problems as 3-sat@xcite . instances of the 3-xorsat problem with @xmath180 clauses and @xmath0 variables have almost surely solutions as long as @xmath181@xcite . the clustering phase transition takes place at @xmath182 and is related to a change in the geometric structure of the space of solutions , see fig . [ landscape ] : * when @xmath183 , the space of solutions is connected . given a pair of solutions @xmath184 , _ i.e. _ two assignments of the @xmath0 boolean variables that satisfy the clauses , there almost surely exists a sequence of solutions , @xmath185 , with @xmath186 , @xmath187 , @xmath188 , connecting the two solutions such that the hamming distance ( number of different variables ) between @xmath189 and @xmath190 is bounded from above by some finite constant when @xmath132 . * when @xmath191 , the space of solutions is not connected any longer . it is made of an exponential ( in @xmath0 ) number of connected components , called clusters , each containing an exponentially large number of solutions . clusters are separated by large voids : the hamming distance between two clusters , that is , the smallest hammming distance between pairs of solutions belonging to these clusters , is of the order of @xmath0 . from intuitive grounds , changes of the statistical properties of the cost function landscape e.g. of the structure of the solutions space may potentially affect the search dynamics . this connection between dynamics and static properties was established in numerous works in the context of mean field models of spin glasses @xcite , and subsequently also put forward in some studies of local search algorithms in combinatorial optimization problems@xcite . so far , there is no satisfying explanation to when and why features of _ a priori _ algorithm dependent dynamical phenomena should be related to , or predictable from some statistical properties of the cost function landscape . we shall see some examples in the following where such a connection indeed exist ( sec . iii.b ) and other ones where its presence is far less obvious ( sec . iiic , d ) . , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] 0 00 coding theory is a rich source of computational problems ( and algorithms ) for which the average case analysis is relevant @xcite . let us focus , for sake of concreteness , on the decoding problem . codewords are sequences of symbols with some built - in redundancy . if we consider the case of linear codes on a binary alphabet , this redundancy can be implemented as a set of linear constraints . in practice , a codeword is a vector @xmath192 ( with @xmath193 ) which satisfies the equation @xmath194 where @xmath195 is an @xmath196 binary matrix ( _ parity check matrix _ ) . each one of the @xmath4 linear equations involved in eq . ( [ paritycheckmatrix ] ) is called a _ parity check_. this set of equation can be represented graphically by a _ tanner graph _ , cf . [ tanner ] . this is a bipartite graph highlighting the relations between the variables @xmath197 and the constraints ( parity checks ) acting on them . the decoding problem consists in finding , among the solutions of eq . ( [ paritycheckmatrix ] ) , the `` closest '' one @xmath198 to the output @xmath199 of some communication channel . this problem is , in general , np - hard @xcite . the precise meaning of `` closest '' depends upon the nature of the communication channel . let us make two examples : * the binary symmetric channel ( bsc ) . in this case the output of the communication channel @xmath199 is a codeword , i.e. a solution of ( [ paritycheckmatrix ] ) , in which a fraction @xmath200 of the entries has been flipped . `` closest '' has to be understood in the hamming - distance sense . @xmath198 is the solution of eq . ( [ paritycheckmatrix ] ) which minimizes the hamming distance from @xmath199 . * the binary erasure channel ( bec ) . the output @xmath199 is a codeword in which a fraction @xmath200 of the entries has been erased . one has to find a solution @xmath198 of eq . ( [ paritycheckmatrix ] ) which is compatible with the remaining entries . such a problem has a _ unique _ solution for small enough erasure probability @xmath200 . there are two sources of randomness in the decoding problem : @xmath201 the matrix @xmath195 which defines the code is usually drawn from some random _ ensemble _ ; @xmath202 the received message which is distributed according to some probabilistic model of the communication channel ( in the two examples above , the bits to be flipped / erased were chosen randomly ) . unlike many other combinatorial problems , there is therefore a `` natural '' probability distribution defined on the instances . average case analysis with respect to this distribution is of great practical relevance . recently , amazingly good performances have been obtained by using low - density parity check ( ldpc ) codes @xcite . ldpc codes are defined by parity check matrices @xmath195 which are large and sparse . as an example we can consider gallager _ regular _ codes @xcite . in this case @xmath195 is chosen with flat probability distribution within the family of matrices having @xmath203 ones per column , and @xmath204 ones per row . these are decoded using a suboptimal linear - time algorithm known as `` belief - propagation '' or `` sum - product '' algorithm @xcite . this is an iterative algorithm which takes advantage of the locally tree - like structure of the tanner graph , see fig . [ tanner ] , for ldpc codes . after @xmath205 iterations it incorporates the information conveyed by the variables up to distance @xmath205 from the one to be decoded . this can be done in a recursive fashion allowing for linear - time decoding . belief - propagation decoding shows a striking threshold phenomenon as the noise level @xmath200 crosses some critical ( code - dependent ) value @xmath206 . while for @xmath207 the transmitted codeword is recovered with high probability , for @xmath208 decoding will fail almost always . the threshold noise @xmath206 is , in general , smaller than the threshold @xmath209 for optimal decoding ( with unbounded computational resources ) . the rigorous analysis of ref . @xcite allows a precise determination of the critical noise @xmath206 under quite general circumstances . nevertheless some important theoretical questions remain open : can we find some smarter linear - time algorithm whose threshold is greater than @xmath206 ? is there any `` intrinsic '' ( i.e. algorithm independent ) characterization of the threshold phenomenon taking place at @xmath206 ? as a first step towards the answer to these questions , ref . @xcite explored the dynamics of local optimization algorithms by using statistical mechanics techniques . the interesting point is that `` belief propagation '' is by no means a local search algorithm . for sake of concreteness , we shall focus on the binary erasure channel . in this case we can treat decoding as a combinatorial optimization problem within the space of bit sequences of length @xmath210 ( the number of erased bits , the others being fixed by the received message ) . the function to be minimized is the _ energy density _ @xmath211 where we denote as @xmath212 the hamming distance between two vectors @xmath213 and @xmath214 , and we introduced the normalizing factor for future convenience . notice that both arguments of @xmath215 in eq . ( [ costfunction ] ) are vectors in @xmath216 . we can define the @xmath217-neighborhood of a given sequence @xmath218 as the set of sequences @xmath219 such that @xmath220 , and we call @xmath217-stable states the bit sequences which are optima of the decoding problem within their @xmath217-neighborhood . one can easily invent local search algorithms @xcite for the decoding problem which use the @xmath217-neighborhoods . the algorithm start from a random sequence and , at each step , optimize it within its @xmath217-neighborhood . this algorithm is clearly suboptimal and halts on @xmath217-stable states . let us consider , for instance , a @xmath221 regular code and decode it by local search in @xmath222-neighborhoods . in fig . [ glauber ] we report the resulting energy density @xmath223 after the local search algorithm halts , as a function of the erasure probability @xmath200 . we averaged over 100 different realizations of the noise and of the matrix @xmath195 . for sake of comparison we recall that the threshold for belief - propagation decoding is @xmath224 @xcite , while the threshold for optimal decoding is at @xmath225 @xcite . it is evident that local search by @xmath222-neighborhoods performs quite poorly . a natural question is whether ( and how much ) , these performances are improved by increasing @xmath217 . it is therefore quite natural to study _ metastable _ states . these are @xmath217-stable states for any @xmath226 if @xmath227 . ] . there exists no completely satisfying definition of such states : here we shall just suggest a possibility among others . the tricky point is that we do not know how to compare @xmath217-stable states for different values of @xmath0 . this forbids us to make use of the above asymptotic statement . one possibility is to count without really defining them . this can be done , at least in principle , by counting @xmath217-stable states , take the @xmath228 limit and , at the end , the @xmath229 limit@xcite . on physical grounds , we expect @xmath217-stable states to be exponentially numerous . in particular , if we call @xmath230 the number of @xmath217-stable states taking a value @xmath223 of the cost function ( [ costfunction ] ) , we have @xmath231 we can therefore define the so called ( physical ) complexity @xmath232 as follows , @xmath233 roughly speaking we can say that the number of metastable states is @xmath234 . of course there are several alternative ways of taking the limits @xmath229 , @xmath228 , and we do not yet have a proof that these procedures give the same result for @xmath232 nevertheless it is quite clear that the existence of an exponential number of metastable states should affect dramatically the behavior of local search algorithms . statistical mechanics methods @xcite allows to determine the complexity @xmath232 @xcite . in `` difficult '' cases ( such as for error - correcting codes ) , the actual computation may involve some approximation , e.g. the use of a variational ansatz . nevertheless the outcome is usually quite accurate . in fig . [ complexity ] we consider a @xmath235 regular code on the binary erasure channel . we report the resulting complexity for three different values of the erasure probability @xmath200 . the general picture is as follows . below @xmath206 there is no metastable state , except the one corresponding to the correct codeword . between @xmath206 and @xmath209 there is an exponential number of metastable states with energy density belonging to the interval @xmath236 ( @xmath232 is strictly positive in this interval ) . above @xmath209 , @xmath237 . the maximum of @xmath232 is always at @xmath238 . the above picture tell us that any local algorithm will run into difficulties above @xmath206 . in order to confirm this picture , the authors of ref . @xcite made some numerical computations using simulated annealing as decoding algorithm for quite large codes ( @xmath239 bits ) . for each value of @xmath200 , we start the simulation fixing a fraction @xmath240 of spins to @xmath241 ( this part will be kept fixed all along the run ) . the remaining @xmath242 spins are the dynamical variables we change during the annealing in order to try to satisfy all the parity checks . the energy of the system counts the number of unsatisfied parity checks . the cooling schedule has been chosen in the following way : @xmath243 monte carlo sweeps ( mcs ) proposed spin flips . each proposed spin flip is accepted or not accordingly to a standard metropolis test . ] at each of the 1000 equidistant temperatures between @xmath244 and @xmath245 . the highest temperature is such that the system very rapidly equilibrates . typical values for @xmath243 are from 1 to @xmath246 . notice that , for any fixed cooling schedule , the computational complexity of the simulated annealing method is linear in @xmath0 . then we expect it to be affected by metastable states of energy @xmath238 , which are present for @xmath208 : the energy relaxation should be strongly reduced around @xmath238 and eventually be completely blocked . some results are plotted in fig . [ annealing ] together with the theoretical prediction for @xmath238 . the good agreement confirm our picture : the algorithm gets stucked in metastable states , which have , in the great majority of cases , energy density @xmath238 . both `` belief propagation '' and local search algorithms fail to decode correctly between @xmath206 and @xmath209 . this leads naturally to the following conjecture : no linear time algorithm can decode in this regime of noise . the ( typical case ) computational complexity changes from being linear below @xmath206 to superlinear above @xmath206 . in the case of the binary erasure channel it remains polynomial between @xmath206 and @xmath209 ( since optimal decoding can be realized with linear algebra methods ) . however it is plausible that for a general channel it becomes non - polynomial . in this section the local procedure we consider is gradient descent ( gd ) . gd is defined as follows . * ( 1 ) * start from an initial randomly chosen configuration of the variables . call @xmath10 the number of unsatisfied clauses . * ( 2 ) * if @xmath247 then stop ( a solution is found ) . otherwise , pick randomly one variable , say @xmath248 , and compute the number @xmath249 of unsatisfied clauses when this variable is negated ; if @xmath250 then accept this change _ i.e. _ replace @xmath248 with @xmath251 and @xmath10 with @xmath249 ; if @xmath252 , do not do anything . then go to step 2 . the study of the performances of gd to find the minima of cost functions related to statistical physics models has recently motivated various studies@xcite . numerics indicate that gd is typically able to solve random 3-sat instances with ratios @xmath253 @xcite close to the onset of clustering @xcite . we shall rigorously show below that this is not so for 3-xorsat . let us apply gd to an instance of xorsat . the instance has a graph representation explained in fig . [ xorgr ] . vertices are in one to one correspondence with variables . a clause is fully represented by a plaquette joining three variables and a boolean label equal to the number of negated variables it contains modulo 2 ( not represented on fig . [ xorgr ] ) . once a configuration of the variables is chosen , each plaquette may be labelled by its status , s or u , whether the associated clause is respectively satisfied or unsatisfied . a fundamental property of xorsat is that each time a variable is changed , _ i.e. _ its value is negated , the clauses it belongs to change status too . this property makes the analysis of some properties of gd easy . consider the hypergraph made of 15 vertices and 7 plaquettes in fig . [ bi ] , and suppose the central plaquette is violated ( u ) while all other plaquettes are satisfied ( s ) . the number of unsatisfied clauses is @xmath254 . now run gd on this special instance of xorsat . two cases arise , symbolized in fig . [ bi ] , whether the vertex attached to the variable to be flipped belongs , or not , to the central plaquette . it is an easy check that , in both cases , @xmath255 and the change is not permitted by gd . the hypergraph of fig . [ bi ] will be called hereafter island . when the status of the plaquettes is u for the central one and s for the other ones , the island is called blocked . though the instance of the xorsat problem encoded by a blocked island is obviously satisfiable ( think of negating at the same time one variable attached to a vertex @xmath256 of the central plaquette and one variable in each of the two peripherical plaquettes joining the central plaquette at @xmath256 ) , gd will never be able to find a solution and will be blocked forever in the local minimum with height @xmath254 . the purpose of this section is to show that this situation typically happens for random instances of xorsat . more precisely , while almost all instances of xorsat with a ratio of clauses per variables smaller than @xmath257 have a lot of solutions , gd is almost never able to find one . even worse , the number of violated clauses reached by gd is bounded from below by @xmath258 where @xmath259 in other words , the number of clauses remaining unsatisfied at the end of a typical gd run is of the order of @xmath0 . our demonstration , inspired from @xcite , is based on the fact that , with high probability , a random instance of xorsat contains a large number of blocked islands of the type of fig . [ bi ] . to make the proof easier , we shall study the following fixed clause probability ensemble . instead of imposing the number of clauses to be equal to @xmath260 , any triplet @xmath243 of three vertices ( among @xmath0 ) is allowed to carry a plaquette with probability @xmath261 . notice that this probability ensures that , on the average , the number of plaquettes equals @xmath262 . let us now draw a hypergraph with this distribution . for each triplet @xmath243 of vertices , we define @xmath263 if @xmath243 is the center of a island , 0 otherwise . we shall show that the total number of islands , @xmath264 , is highly concentrated in the large @xmath0 limit , and calculate its average value . the expectation value of @xmath265 is equal to @xmath266 = \frac{(n-3)\times ( n-4 ) \times \ldots \times ( n-13)\times ( n-14)}{8\times 8\times 8 } \times \mu ^a \ , ( 1-\mu ) ^b \ , \ ] ] where @xmath267 is the number of plaquettes in the island , and @xmath268 is the number of triplets with at least one vertex among the set of 15 vertices of the island that do not carry plaquette . the significance of the terms in eq . ( [ expitau ] ) is transparent . the central triplet @xmath243 occupying three vertices , we choose 2 vertices among @xmath269 to draw the first peripherical plaquette of the island , then other 2 vertices among @xmath270 for the other peripherical plaquette having a common vertex with the latter . the order in which these two plaquettes are built does not matter and a factor @xmath271 permits us to avoid double counting . the other four peripherical plaquettes have multiplicities calculable in the same way ( with less and less available vertices ) . the terms in @xmath272 and @xmath273 correspond to the probability that such a 7 plaquettes configuration is drawn on the 15 vertices of the island , and is disconnected from the remaining @xmath274 vertices . the expectation value of the number @xmath275 of islands per vertex thus reads , @xmath276 = \lim _ { n\to\infty } \frac 1n \ , { n \choose 3}\ , e[i_{\tau } ] = \frac{729}{8 } \ , \alpha^7 \ , e^{-45\ , \alpha } \quad .\ ] ] chebyshev s inequality can be used to show that @xmath140 is concentrated around its above average value . let us calculate the second moment of the number of islands , @xmath277= \sum _ { \tau , \sigma } e [ i_{\tau } i_{\sigma } ] $ ] . clearly , @xmath278 $ ] depends only on the number @xmath279 of vertices common to triplets @xmath243 and @xmath280 . it is obvious that no two triplets of vertices can be centers of islands when they have @xmath281 or @xmath282 common vertices . if @xmath283 , @xmath284 and @xmath285=e [ i_{\tau}]$ ] has been calculated above . for @xmath286 , a similar calculation gives @xmath287 finally , we obtain @xmath288 = \frac 1{n^2 } \left [ { n \choose 3 } e_{\ell =3 } + { n \choose 3}{n -3\choose 3 } e_{\ell = 0 } \right ] = e[i]^2 + o\left ( \frac 1n \right ) \quad .\ ] ] therefore the variance of @xmath140 vanishes and @xmath140 is , with high probability , equal to its average value given by ( [ eq89 ] ) . to conclude , an island has a probability @xmath289 to be blocked by definition . therefore the number ( per vertex ) of blocked islands in a random xorsat instance with ratio @xmath25 is almost surely equal to @xmath290 given by eq . ( [ teh ] ) . since each blocked island has one unsatisfied clause , this is also a lower bound to the number of violated clauses per variable . notice however that @xmath291 is very small and bounded from above by @xmath292 over the range of interest , @xmath293 . therefore , one would in principle need to deal with billions of variables not to reach solutions and be in the true asymptotic regime of gd . the proof is easily generalizable to gradient descent with more than one look ahead . to extend the notion of blocked island to the case where gd is allowed to invert @xmath217 , and not only 1 , variables at a time , it is sufficient to have @xmath294 , and not 2 , peripherical plaquettes attached to each vertex of the central plaquette . the calculation of the lower bound @xmath295 to the number of violated clauses ( divided by @xmath0 ) reached by gd is straightforward and not reproduced here . as a consequence , gd , even with @xmath217 simultaneous flips permitting to overcome local barriers , remains almost surely trapped at an extensive ( in @xmath0 ) level of violated clauses for any finite @xmath217 . actually the lower bound @xmath296 tends to zero only if @xmath217 is of the order of @xmath78 . we stress that the statistical physics calculation of physical ` complexity ' @xmath297 ( see sec . [ codesection ] ) predicts there is no metastable states for @xmath298@xcite , while gd is almost surely trapped by the presence of blocked islands for any @xmath299 . this apparent discrepancy comes from the fact that gd is sensible to the presence of configurations blocked for finite @xmath217 , while the physical ` complexity ' considers states metastable in the limit @xmath229 only@xcite . and @xmath300 . each clause or equation is represented by a plaquette whose vertices are the attached variables . when the variables are assigned some values , the clauses can be satisfied ( s ) or unsatisfied ( u ) . ] the pure random walksat ( prwsat ) algorithm for solving @xmath3-sat is defined by the following rules@xcite . 1 . choose randomly a configuration of the boolean variables . 2 . if all clauses are satisfied , output `` satisfiable '' . 3 . if not , choose randomly one of the unsatisfied clauses , and one among the @xmath3 variables of this clause . flip ( invert ) the chosen variable . notice that the selected clause is now satisfied , but the flip operation may have violated other clauses which were previously satisfied . 4 . go to step 2 , until a limit on the number of flips fixed beforehand has been reached . then output `` do nt know '' . what is the output of the algorithm ? either `` satisfiable '' and a solution is exhibited , or `` do nt know '' and no certainty on the status of the formula is achieved . papadimitriou introduced this procedure for @xmath301 , and showed that it solves with high probability any satisfiable 2-sat instance in a number of steps ( flips ) of the order of @xmath302@xcite . recently schning was able to prove the following very interesting result for 3-sat@xcite . call ` trial ' a run of prwsat consisting of the random choice of an initial configuration followed by @xmath303 steps of the procedure . if none of @xmath61 successive trials on a given instance has been successful ( has provided a solution ) , then the probability that this instance is satisfiable is lower than @xmath304 . in other words , after @xmath305 trials of prwsat , most of the configuration space has been ` probed ' : if there were a solution , it would have been found . though this local search algorithm is not complete , the uncertainty on its output can be made as small as desired and it can be used to prove unsatisfiability ( in a probabilistic sense ) . schning s bound is true for any instance . restriction to special input distributions allows to strengthen this result . alekhnovich and ben - sasson showed that instances drawn from the random 3-satisfiability ensemble described above are solved in polynomial time with high probability when @xmath25 is smaller than @xmath306@xcite . in this section , we briefly sketch the behaviour of prwsat , as seen from numerical experiments @xcite and the analysis of @xcite . a dynamical threshold @xmath307 ( @xmath308 for 3-sat ) is found , which separates two regimes : * for @xmath309 , the algorithm finds a solution very quickly , namely with a number of flips growing linearly with the number of variables @xmath0 . figure [ wsat_phen]a shows the plot of the fraction @xmath310 of unsatisfied clauses as a function of time @xmath31 ( number of flips divided by @xmath4 ) for one instance with ratio @xmath311 and @xmath312 variables . the curve shows a fast decrease from the initial value ( @xmath313 in the large @xmath0 limit independently of @xmath25 ) down to zero on a time scale @xmath314 . fluctuations are smaller and smaller as @xmath0 grows . @xmath315 is an increasing function of @xmath25 . this _ relaxation _ regime corresponds to the one studied by alekhnovich and ben - sasson , and @xmath316 as expected@xcite . * for instances in the @xmath317 range , the initial relaxation phase taking place on @xmath318 time scale is not sufficient to reach a solution ( fig . [ wsat_phen]b ) . the fraction @xmath319 of unsat clauses then fluctuates around some plateau value for a very long time . on the plateau , the system is trapped in a _ metastable _ state . the life time of this metastable state ( trapping time ) is so huge that it is possible to define a ( quasi ) equilibrium probability distribution @xmath320 for the fraction @xmath319 of unsat clauses . ( inset of fig . [ wsat_phen]b ) . the distribution of fractions is well peaked around some average value ( height of the plateau ) , with left and right tails decreasing exponentially fast with @xmath0 , @xmath321 with @xmath322 . eventually a large negative fluctuation will bring the system to a solution ( @xmath323 ) . assuming that these fluctuations are independant random events occuring with probability @xmath324 on an interval of time of order @xmath222 , the resolution time is a stochastic variable with exponential distribution . its average is , to leading exponential order , the inverse of the probability of resolution on the @xmath325 time scale : @xmath326 \sim \exp ( n \zeta)$ ] with @xmath327 . escape from the metastable state therefore takes place on exponentially large in@xmath0 time scales , as confirmed by numerical simulations for different sizes . schning s result@xcite can be interpreted as a lower bound to the probability @xmath328 , true for any instance . the plateau energy , that is , the fraction of unsatisfied clauses reached by prwsat on the linear time scale is plotted on fig . [ wsat_plateau ] . notice that the `` dynamic '' critical value @xmath329 above which the plateau energy is positive ( prwsat stops finding a solution in linear time ) is strictly smaller than the `` static '' ratio @xmath330 , where formulas go from satisfiable with high probability to unsatisfiable with high probability . in the intermediate range @xmath331 , instances are almost surely satisfiable but prwsat needs an exponentially large time to prove so . interestingly , @xmath332 and @xmath330 coincides for 2-sat in agreement with papadimitriou s result@xcite . furthermore , the dynamical transition is apparently not related to the onset of clustering taking place at @xmath333 . a b .5 cm .5 cm when prwsat finds easily a solution , the number of steps it requires is of the order of @xmath0 , or equivalently , @xmath4 . let us call @xmath334 the average of this number divided by the number of clauses @xmath4 . by definition of the dynamic threshold , @xmath315 diverges when @xmath335 . assuming that @xmath336 can be expressed as a series of powers of @xmath25 , we find the following expansion@xcite @xmath337 around @xmath338 . as only a finite number of terms in this expansion have been computed , we do not control its radius of convergence , yet as shown in fig . [ wsat_fig_tres_q1 ] the numerical experiments provide convincing evidence in favour of its validity . the above calculation is based on two facts . first , for @xmath339 the instance under consideration splits into independent subinstances ( involving no common variable ) that contains a number of variables of the order of @xmath78 at most . moreover , the number of the connected components containing @xmath340 clauses , computed with probabilistic arguments very similar to those of section [ gradxor ] , contribute to a power expansion in @xmath25 only at order @xmath341 . secondly , the number of steps the algorithm needs to solve the instance is simply equal to the sum of the numbers of steps needed for each of its independent subinstances . this additivity remains true when one averages over the initial configuration and the choices done by the algorithm . one is then left with the enumeration of the different subinstances with a given size and the calculation of the average number of steps for their resolution . a detailed presentation of this method has been given in a general case in @xcite , and applied more specifically to this problem in @xcite ; the reader is referred to these previous works for more details . equation ( [ dev_cluster_tresk ] ) is the output of the enumeration of subinstances with up to three clauses . the above small @xmath25 expansion does not allow us to investigate the @xmath342 regime . we turn now to an approximate method more adapted to this situation . let us denote by @xmath20 an assignment of the boolean variables . prwsat defines a markov process on the space of the configurations @xmath20 , a discrete set of cardinality @xmath343 . it is a formidable task to follow the probabilities of all these configurations as a function of the number of steps @xmath61 of the algorithm so one can look for a simpler description of the state of the system during the evolution of the algorithm . the simplest , and crucial , quantity to follow is the number of clauses unsatisfied by the current assignment of the boolean variables , @xmath344 . indeed , as soon as this value vanishes , the algorithm has found a solution and stops . a crude approximation consists in assuming that , at each time step @xmath61 , all configurations with a given number of unsatisfied clauses are equiprobable , whereas the hamming distance between two configurations visited at step @xmath61 and @xmath345 of the algorithm is at most @xmath204 . however , the results obtained are much more sensible that one could fear . within this simplification , a markovian evolution equation for the probability that @xmath346 clauses are unsatisfied after @xmath61 steps can be written . using methods similar to the ones in section [ dpllsatsection ] , we obtain ( see @xcite for more details and @xcite for an alternative way of presenting the approximation ) : * the average fraction of unsatisfied clauses , @xmath347 , after @xmath348 steps of the algorithm . for ratios @xmath349 , @xmath319 remains positive at large times , which means that typically a large formula will not be solved by prwsat , and that the fraction of unsat clauses on the plateau is @xmath350 . the predicted value for @xmath351 , @xmath352 , is in good but not perfect agreement with the estimates from numerical simulations , around @xmath353 . the plateau height , @xmath354 , is compared to numerics in fig . [ wsat_plateau ] . * the probability @xmath321 that the fraction of unsatisfied clauses is @xmath310 . it has been argued above that the distribution of resolution times in the @xmath355 phase is expected to be , at leading order , an exponential distribution of average @xmath356 , with @xmath357 . predictions for @xmath358 are plotted and compared to experimental measures of @xmath168 in fig . [ wsat_fig_zeta ] . despite the roughness of our markovian approximation , theoretical predictions are in qualitative agreement with numerical experiments . a similar study of the behaviour of prwsat on xorsat problems has been also performed in @xcite , with qualitatively similar conclusions : there exists a dynamic threshold @xmath329 for the algorithm , smaller both than the satisfiability and clustering thresholds ( known exactly in this case @xcite ) . for low values of @xmath25 , the resolution time is linear in the size of the formula ; between @xmath329 and @xmath330 resolution occurs on exponentially large time scales , through fluctuations around a plateau value for the number of unsatisfied clauses . in the xorsat case , the agreement between numerical experiments and this approximate study ( which predicts @xmath359 ) is quantitatively better and seems to improve with growing @xmath3 . in this article , we have tried to give an overview of the studies that physicists have devoted to the analysis of algorithms . this presentation is certainly not exhaustive . let us mention that use of statistical physics ideas have permitted to obtain very interesting results on related issues as number partitioning@xcite , binary search trees @xcite , learning in neural networks @xcite , extremal optimization @xcite ... it may be objected that algorithms are mathematical and well defined objects and , as so , should be analysed with rigorous techniques only . though this point of view should ultimately prevail , the current state of available probabilistic or combinatorics techniques compared to the sophisticated nature of algorithms used in computer science make this goal unrealistic nowadays . we hope the reader is now convinced that statistical physics ideas , techniques , ... may be of help to acquire a quantitative intuition or even formulate conjectures on the average performances of search algorithms . a wealth of concepts and methods familiar to physicists e.g. phase transitions and diagrams , dynamical renormalization flow , out - of - equilibrium growth phenomena , metastability , perturbative approaches ... are found to be useful to understand the behaviour of algorithms . it is a simple bet that this list will get longer in next future and that more and more powerful techniques and ideas issued from modern theoretical physics will find their place in the field . open questions are numerous . variants of dpll with complex splitting heuristics , random backtrackings@xcite or applied to combinatorial problems with internal symmetries@xcite would be worth being studied . as for local search algorithms , it would be very interesting to study refined versions of the pure walksat procedure that alternate random and greedy steps @xcite to understand the observed existence and properties of optimal strategies . one of the main open questions in this context is to what extent performances are related to intrinsic features of the combinatorial problems and not to the details of the search algorithm@xcite . this raises the question of how the structure of the cost function landscape may induce some trapping or slowing down of search algorithms@xcite . last of all , the input distributions of instances we have focused on here are far from being realistic . real instances have a lot of structure which will strongly reflect on the performances of algorithms . going towards more realistic distributions or , even better , obtaining results true for any instance would be of great interest . d. mitchell , b. selman , h. levesque , hard and easy distributions of sat problems , _ proc . of the tenth natl . conf . on artificial intelligence ( aaai-92 ) _ , 440 - 446 , the aaai press / mit press , cambridge , ma ( 1992 ) . j. gu , p.w . purdom , j. franco , b.w . wah , algorithms for satisfiability ( sat ) problem : a survey . _ dimacs series on discrete mathematics and theoretical computer science _ * 35 * , 19 - 151 , american mathematical society ( 1997 ) . for a review on the analysis of search heuristics in the absence of backtracking , see : + d. achlioptas , lower bounds for random 3-sat via differential equations , _ theor . sci . _ * 265 * , 159 - 186 ( 2001 ) . kaporis , l.m . kirousis , e.g. lalas , the probabilistic analysis of a greedy satisfiability algorithm , _ proceedings of sat 2002 _ , pp 362376 ( 2002 ) , available at http://gauss.ececs.uc.edu/conferences/sat2002/abstracts/kaporis.ps a.c . kaporis , l.m . kirousis , y.c . stamatiou , how to prove conditional randomness using the principle of deferred decisions , technical report , computer technology institute , greece ( 2002 ) , available at http://www.ceid.upatras.fr/faculty/kirousis/kks-pdd02.ps r. monasson , r. zecchina , s. kirkpatrick , b. selman , l. troyansky , 2+p - sat : relation of typical - case complexity to the nature of the phase transition , _ random structure and algorithms _ * 15 * , 414 ( 1999 ) . s. cocco , r. monasson , trajectories in phase diagrams , growth processes and computational complexity : how search algorithms solve the 3-satisfiability problem , _ phys . lett . _ * 86 * , 1654 ( 2001 ) ; analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms , _ eur . phys . j. b _ * 22 * , 505 ( 2001 ) . s. cocco , r. monasson , exponentially hard problems are sometimes polynomial , a large deviation analysis of search algorithms for the random satisfiability problem , and its application to stop - and - restart resolutions , _ phys . e _ * 66 * , 037101 ( 2002 ) ; restart method and exponential acceleration of random 3-sat instances resolutions : a large deviation analysis of the davis putnam loveland logemann algorithm , to appear in amai ( 2003 ) . s. boettcher , m. grigni , jamming model for the extremal optimization heuristic , _ j. phys . a _ * 35 * , 1109 - 1123 ( 2002 ) . + s. boettcher , a.g . percus , extremal optimization : an evolutionary local - search algorithm , in computational modeling and problem solving in the networked world , eds . h. m. bhargava and n. ye ( kluver , boston , 2003 ) . l. baptista , j.p . marques - silva , using randomization and learning to solve hard real - world instances of satisfiability , _ in international conference of principles and practice of contsraint programming _ , 489491 ( 2000 ) .

an overview of some methods of statistical physics applied to the analysis of algorithms for optimization problems ( satisfiability of boolean constraints , vertex cover of graphs , decoding , ... ) with distributions of random inputs is proposed . two types of algorithms are analyzed : complete procedures with backtracking ( davis - putnam - loveland - logeman algorithm ) and incomplete , local search procedures ( gradient descent , random walksat , ... ) . the study of complete algorithms makes use of physical concepts such as phase transitions , dynamical renormalization flow , growth processes , ... as for local search procedures , the connection between computational complexity and the structure of the cost function landscape is questioned , with emphasis on the notion of metastability .

the discovery of millisecond oscillations in the persistent emission and also in thermonuclear x - ray bursts from neutron star low - mass x - ray binaries ( lmxbs ) has opened a new window on the dynamics of accreting neutron stars @xcite . the time scales of the khz quasiperiodic oscillations found in the persistent emission ( hereafter khz qpos ) match those expected from accretion dynamics very close to the neutron star in a region of strong gravity , and the possibility that the khz qpos could be used as probes of strong gravity is very intriguing . the burst oscillations are in a range expected for the neutron star spin periods @xcite , and have been interpreted as evidence for millisecond spin periods in accreting neutron stars @xcite . however , this interpretation has recently been questioned because of the large frequency shifts seen in some bursts . observations of bursts from new sources would be useful in determining the correct interpretation of the burst oscillations . here , we describe observations made with the rossi x - ray timing explorer ( rxte ; bradt , rothschild , & swank 1993 ) following the detection of x - ray bursts from the transient source sax j1750.8 - 2900 . we report the discovery of khz qpos in the persistent emission and of oscillations in one x - ray burst . we describe the source and our observations in 2 , our results on x - ray bursts in 3 , our results on the persistent emission in 4 , and discuss our results in 5 . sax j1750.8 - 2900 was discovered in march 1997 with the wide field cameras ( wfcs ) on bepposax as a faint and short duration transient 12 from the galactic center showing type - i x - ray burst activity @xcite . analysis of data from the wfcs and the all - sky monitor ( asm ) on rxte shows a peak flux of 0.12 crab ( where the unit ` crab ' is defined as the flux of the crab nebula in the 2 - 10 kev band , @xmath1 ) and activity lasting approximately 3 weeks above 0.01 crab @xcite . the wfcs detected nine type - i x - ray bursts , with peak fluxes between 0.4 and 1.0 crab and @xmath2-folding decay times between 4 and 11 s in the 2 - 8 kev band . no counterparts in other wavelength regimes were identified . the type - i bursts indicate that the source is an accreting neutron star . sax j1750.8 - 2900 turned on again four years later , on 1 march 2001 ( mjd 51969 ) , as shown by observations with the rxte all - sky monitor ( asm ) ( announced by the mit asm team ) , the proportional counter array ( c.markwardt , private communication ) , and the wfcs . the asm light curve of the 2001 outburst is shown in fig . [ asmlc ] . the asm gives relatively poor coverage near the beginning of the outburst , so detailed information is not available on the initial rise . the high rate point at mjd 52000 is from a single asm dwell and may be due to an x - ray burst . after the initial outburst , the source flux decayed , and then rose again with an almost linear dependence of asm count rate with time over a period of roughly 20 days to a peak flux of 0.12 crab , nearly identical to the 1997 maximum flux . the source subsequently decayed over 10 days to a state of reduced flux , well above the level associated with quiescence in neutron star lmxbs @xcite . bursting activity was detected in the 2001 outburst with the wfcs , although not as extensively as during the 1997 outburst probably because the coverage was not as large as in 1997 . nevertheless , the wfcs detected 4 bursts in a single observation on 4 april 2001 ( mjd 52003 ) . these four bursts have peak fluxes and light curves similar to the bursts observed in 1997 . three other bursts were detected with the wfcs during the 2001 outburst , one 13 days earlier which is atypical ( natalucci et al . in preparation ) , one 162 days later , and one 179 days later . the four x - ray bursts found with the wfc triggered an rxte target - of - opportunity program which led to 4 observations , on 2001 april 6 , 9 , 12 , and 15 . all of our observations occurred during the second rise in flux , as indicated in fig . [ asmlc ] . data were obtained with the proportional counter array ( pca ) in a spectral mode ( standard 2 ) with 256 energy channels and 16 s time resolution , a low resolution timing mode ( standard 1 ) with no energy information and 0.125 s time resolution , and a high resolution timing mode ( event mode ) with 122 @xmath3s time resolution and 64 energy channels . in addition , burst catcher modes were used to acquire high time resolution data when the event rates exceeded the telemetry capacity of the event mode during x - ray bursts . ccc number & time & peak flux + 1 & apr 06 at 13:26:13 & @xmath4 + 2 & apr 12 at 14:20:30 & @xmath5 + 3 & apr 15 at 17:02:25 & @xmath6 + 4 & apr 15 at 18:37:08 & @xmath7 + we used the standard 1 data to search for x - ray bursts and found four bursts , see table [ bursttable ] and fig . we examined the spectral evolution of the bursts using burst catcher or event mode data with 64 channels of energy resolution . we extracted spectra for 0.25 s or 1 s intervals using all proportional counter units ( pcus ) on during each burst and all layers . the fluxes were corrected for deadtime effects . the maximum deadtime correction in the brightest burst was 7% . in order to eliminate the contribution of the persistent emission , we subtracted off a spectrum from 10 s of data preceding each burst , and fit the resulting spectra in the 3 - 20 kev band with an absorbed blackbody model with the column density fixed to the average value found from the persistent emission spectra described below . the first three bursts are similar with comparable peak fluxes . the decay ( @xmath2-folding ) times for the full band ( 260 kev ) pca count rate for bursts 2 and 3 were close to 3.0 s. burst 1 had a broader peak than bursts 2 and 3 , had a longer decay time of 5 s , and deviated significantly from an exponential decay . burst 4 was preceded by a small precursor event , beginning 9 s earlier with a peak count rate ( with persistent emission subtracted ) of @xmath8 of the main burst . substructure including at least two small peaks is present in the precursor event . the peak count rate of burst 4 was a factor of 4 lower than the other bursts and the corresponding temperature was lower . its decay time was 3.4 s , close to that of bursts 2 and 3 . the results of the spectral fits for the brightest burst , number 2 , are shown in fig . [ bur2spec ] . the burst shows an increase in radius and a simultaneous decrease in temperature near the peak , but the magnitude of the radius increase is not sufficient to classify the burst as an eddington - limited photospheric radius expansion burst . @xcite found that the average peak luminosity for photospheric radius expansion bursts from 5 bursters located in globular clusters , hence with known distances , was @xmath9 . using this luminosity as an upper limit on the peak luminosity in burst 2 , we place an upper limit on the distance to sax j1750.8 - 2900 of @xmath10 . this upper limit is consistent with the upper limit derived by @xcite of 7 kpc . using high time resolution data ( merged event lists from the event and burst catcher modes ) with no energy selection , we computed power spectra for overlapping 4 s intervals of data , with 0.125 s between the starts of successive intervals , and searched for excess power in the range 200 - 1200 hz . we found oscillations in the second burst with a maximum leahy normalized power of 49.3 at a frequency of 600.75 hz , occurring in the burst decay 5 s after the burst rise . allowing @xmath11 independent trials over the 20 s duration of the burst , the chance probability of occurrence is @xmath12 , equivalent to a @xmath13 detection . the dynamical power spectrum is shown in fig . [ bur2pow ] . in addition to the oscillations in the burst decay , there are oscillations present near 599.5 hz in the initial part of the burst rise . the oscillations begin at the burst onset ; no oscillations are present in the persistent emission before the burst onset . the frequency appears to change rapidly between burst rise and decay . to study the persistent emission , we used the standard-2 data for spectral information and a 122 @xmath3s time resolution event mode for timing information . we removed data around the x - ray bursts . we used only pcu 2 for the color and spectral studies as it was on during all of the observations and has a reasonably well understood response ( in particular , it does not suffer from the propane layer leak experienced by pcu 0 which was the only other pcu on during all of the observations ) . the background estimate was made using the bright source background files and the response was calculated using pcarmf v7.10 as supplied in ftools v5.1 . all pcus on during each observation were used for the timing analysis . we extracted an energy spectrum for each uninterrupted rxte observing window to investigate evolution of the energy spectrum . the energy range was limited to 3 - 20 kev and we included a 1% systematic error . we attempted to fit the spectra with a simple absorbed powerlaw model . this was unacceptable in all cases . addition of a gaussian emission line at an energy near the fe - k transition produced an acceptable fit for the first spectrum , but not for the others . the sum of a blackbody plus a gaussian with absorption from material with solar abundances and the sum of a multicolor disk blackbody plus a gaussian with absorption produced unacceptable fits . the sum of a comptonization model @xcite and a gaussian emission line with absorption produced acceptable fits , @xmath14 , to all of the spectra . we used this model to fit all of the persistent emission spectra . the limiting form of the comptonization spectrum at high temperatures over the fitted energy band is indistinguishable from a powerlaw given our limited statistics . therefore , only an upper limit on the comptonization temperature could be obtained for the first observation which had a spectrum consistent with the sum of a powerlaw and emission line with absorption as noted above . the comptonization temperature and optical depth show marked evolution over the observations , see fig . [ pers_spec ] . the temperature begins high , the lower limit on the temperature in the first observation is 8 kev , and decreases to 3 kev . simultaneously , the optical depth begins at 6 and increases to 11 for the later observations . the column density of the absorbing material shows some evidence for a decrease with time ; however , we caution that these spectra are not adequate to robustly constraint the column density . the average equivalent hydrogen column density is @xmath15 . the iron line parameters remain roughly constant across all the observations but are poorly constrained . all of the spectra were consistent with a centroid energy of 6.7 kev and this centroid energy was subsequently fixed for all of the fits . the width and normalization were allowed to vary . the best fit line widths were in the range 0.51.0 kev . the best fit equivalent widths ranged from 150 to 400 ev . the line width is consistent with previous results from neutron - star lmxbs @xcite . the equivalent width is higher than most previously reported values for neutron - star lmxbs ; the maximum equivalent width reported from asca is 170 ev @xcite . we produced a color - color diagram , see fig . [ ccd ] , using 256 s intervals extracted from background subtracted standard-2 data to study the source state . the april 6 and 9 observations form one cluster of points in the color - color diagram with high values of the hard color , while the april 12 and 15 data form a distinct cluster with lower values of the hard color . to further investigation the source state , we produced low frequency ( below 100 hz ) power spectra . we produced two spectra , one for each of the two clusters of points in the color - color diagram , see fig . [ lowfreq ] . the power spectrum for the april 6 and 9 observations shows strong band - limited noise , with an rms fraction of 12.8% in the 0.1100 hz range . this clearly identifies the source state as the ` island ' state @xcite . the power spectrum has the form of a broken powerlaw below 10 hz and a qpo is apparent above 10 hz . fitting the power spectrum with the sum of a broken powerlaw plus a lorentzian , we found excess noise above 20 hz and @xmath16 . we added an exponentially cutoff powerlaw to fit this excess noise and the fit improved to @xmath17 . the residuals of this fit show no systematic deviations and the fit is unlikely to be improved by addition of more continuum components . for the broken powerlaw , we find a break frequency of @xmath18 , an index below the break of @xmath19 and and index above the break of @xmath20 . the lorentzian centroid is @xmath21 . the break and qpo centroid frequencies are consistent with the correlation reported by @xcite . the power spectrum for april 12 and 15 shows weak timing noise . the rms fraction is 1.4% in the 0.1100 hz range . this indicates that the source was in the ` banana ' state . we fitted the power spectrum with the sum of a powerlaw and an exponentially cutoff powerlaw . this gave a reasonable fit with @xmath22 . the powerlaw index was @xmath23 . based on the timing and color information , we suggest that sax j1750.8 - 2900 is an atoll source . furthermore , we identify the source as being in the `` island '' state on april 6 and 9 , when the hard color had high values and strong timing noise was present , and in the `` banana '' state on april 12 and 15 , when the hard color had low values and the timing noise was weak . lcccccc apr 06 at 13:03:27 & 1344 & 0.46 & @xmath24 & @xmath25 & @xmath26 + apr 09 at 10:59:27 & 3024 & 0.45 & @xmath27 & @xmath28 & @xmath29 + apr 12 at 13:41:03 & 2336 & 0.37 & @xmath30 & @xmath31 & @xmath32 + apr 15 at 14:57:35 & 2960 & 0.35 & @xmath33 & @xmath34 & @xmath35 + apr 15 all data & 8608 & 0.35 & @xmath36 & @xmath37 & @xmath38 + we searched for high frequency qpos in each uninterrupted rxte observation window and in combinations of the various data segments . we calculated averages of 2 s power spectra for all pca events ( 260 kev ) and for events in the 4.7 - 20.8 kev energy band . we included events from all pcus on during each observation . we note that the true energy band for pcu0 likely differs somewhat from the nominal range due to the propane layer leak . we searched for peaks in the spectrum and fit any peak found with a lorentzian plus a constant equal to the calculated the poisson noise level . there were several qpo signals above 250 hz , see table [ qpotable ] . those found on april 9 and 12 are strong detections and are shown in fig . [ khzqpo ] . allowing for a number of trials equal to the search interval of 1200 hz divided by the qpo width @xcite and allowing several trial widths , we estimate a chance probability of occurrence of @xmath39 for the qpo on april 9 and @xmath40 for the qpo on april 12 . these qpos clearly establish sax j1750.8 - 2900 as a new member of the class of neutron - star low - mass x - ray binaries ( ns - lmxbs ) exhibiting khz qpos . the other signals have lower significance : @xmath41 for the qpo on april 6 and @xmath42 for the qpo on april 15 . the signal on april 15 is quite narrow and appears significant only in the 4.7 - 20.8 kev band power spectrum . in addition to the signals detected in individual uninterrupted observation segments , we also found a signal at 1253 hz in the sum of all of the april 15 data . this is the last entry in table [ qpotable ] . the detection has relatively low significance and must be considered tentative . however , if the detection is correct , then the frequency difference between the two qpos detected on april 15 is 317 hz . the formal error on the difference of the two frequency centroids is 9 hz . the 1253 hz peak may be broadened due to shifts in the centroid frequency over the integration , so the true uncertainty on the difference is somewhat larger . the frequency difference is consistent with half the frequency of the burst oscillations . the results presented here establish that sax j1750.8 - 2900 exhibits millisecond oscillations in its x - ray emission . the properties of the khz qpos in the persistent emission from sax j1750.8 - 2900 are similar to those of the other neutron - star lmxb khz qpo sources in terms of the observed frequency range and the oscillation amplitudes . additional measurements of sax j1750.8 - 2900 would be of interest to confirm our tentative detection of the second khz qpo branch and obtain a simultaneous measurement of the khz qpo frequency difference . the properties of the burst oscillations from sax j1750.8 - 2900 are also similar to those observed from other sources . the frequency shift seen in the single burst from sax j1750.8 - 2900 exhibiting oscillations is compatible with those seen from several other sources and smaller than the large shift ( by 1.32% ) seen from 4u 1916 - 053 @xcite . x - ray burst oscillation sources appear to form two distinct classes : `` fast oscillators '' showing burst oscillations near 600 hz with the precise frequency close to twice the frequency difference of the khz qpos seen in the persistent emission , and `` slow oscillators '' showing burst oscillations near 300 hz with the precise frequency near the difference of the khz qpo frequencies @xcite . our discovery of 600.75 hz oscillations from sax j1750.8 - 2900 establish that it is a member of the class of fast x - ray burst oscillators . our tentative detection of two contemporaneous khz qpos with a frequency difference near 300 hz is consistent with the properties of the other fast oscillators , but should be tested with additional observations . @xcite showed that the fast oscillators produce burst oscillations predominately , but not exclusively , in photospheric radius expansion bursts , while the slow oscillators have oscillations in bursts both with and without photospheric radius expansion . the burst from sax j1750.8 - 2900 in which we find oscillations shows weak , at best , evidence for photospheric radius expansion ; in particular , it does not meet the requirement of 20 km of radius expansion of @xcite . however , since we have a sample of only one burst with oscillations , our results are not inconsistent with those of @xcite . the fact that the behavior of the fast versus slow oscillators is different in regards to the occurrence of oscillations versus burst type ( radius expansion or not ) implies that the difference between the two classes is not an observational selection effect , i.e. due to the inclination of the neutron star spin axis relative to our line of sight , but rather a physical difference in the properties of the neutron stars @xcite . it would be of great interest to identify a model of the x - ray burst oscillations which explained both the large frequency shifts seen in some bursts @xcite and the dichotomy between the fast versus slow oscillators . we greatly appreciate the assistance by the duty scientists of the bepposax science operations center in the near to real - time wfc data analysis . we gratefully acknowledge the efforts of the rxte team , particularly jean swank and evan smith , in performing these target of opportunity observations . pk thanks mal ruderman for useful discussions and acknowledges support from nasa grants nag5 - 7405 , nag5 - 9097 , and nag5 - 9104 . jz acknowledges financial support from the netherlands organization for scientific research ( nwo ) .

we report the discovery of millisecond oscillations in the x - ray emission from the x - ray transient sax j1750.8 - 2900 . millisecond quasiperiodic oscillations ( khz qpos ) were present in the persistent emission with frequencies ranging from 543 hz to 1017 hz . oscillations at a frequency of 600.75 hz were present in the brightest x - ray burst observed . we derive an upper limit on the source distance of @xmath0 from this x - ray burst .

the bcs theory is considered as a conventional theory of superconductivity at a microscopic level , where bosonic excitations like phonons or spin fluctuations play the role of mediators in the formation of cooper pairs and hence superconductivity occurs in a metallic state . in recent years many varieties of superconductors have been discovered which are unconventional . some examples are 1 . cuprates , where anisotropic d - wave gaps occur with nodes ( vanishing gap ) at some symmetry points on the fermi surface @xcite , 2 . co - existence of superconductivity with anti - ferromagnetism ( cecusi@xmath3 ) or with ferromagnetism ( uge@xmath3 ) predominantly in heavy fermion systems @xcite , 3 . coexistence of superconductivity with charge / spin ordering ( nbse@xmath3 , cuprates ) @xcite , 4 . non - centrosymmetric heavy fermion systems ( cept@xmath4si , ceinsi@xmath4 ) @xcite , where lack of inversion symmetry gives rise to spin - orbit interaction with no definite parity in the ground state . in this case singlet and triplet pairings coexist , 5 . strong electronic correlations dominated so - called non - fermi liquid state , believed to be found in high @xmath5 oxides and in heavy fermion systems @xcite . in addition to these , there is another class of unconventional superconducting systems , the novel multiband superconductors . in these systems two or more energy bands are cut by the fermi energy giving rise to multiple energy gaps with different magnitudes in the different fermi sheets . recent measurements of tunnelling , point - contact spectroscopy , angle - resolved photoemission and specific heat provide clear evidence of multiple gap structures . examples of such systems are mgb@xmath3 , rni@xmath3b@xmath3c ( r= lu , y ) , 2h - nbse@xmath3 and many in the pnictide feas family . the bcs theory of superconductivity has been generalised to multiband systems . in one of our recent publications we have given a brief appraisal of the history of multiband bcs theory @xcite and have presented a theory of the time - reversal symmetry broken state in the bcs formalism . we are reminded that a phenomenological theory of superconductivity by ginzburg and landau [ gl ] was developed before the proposal of microscopic bcs theory . the gl approach is a successful theory of phase transitions with many practical applications . the basis of this theory rests on two approximations correct around the critical temperature ( @xmath5 ) ( i ) the order parameter , @xmath6 , is small near @xmath5 and ( ii ) @xmath7 , where @xmath8 . despite these limitations there is a myth that the gl theory applies not only around @xmath5 , rather it is useful for much lower temperatures . in early years soon after the appearance of bcs theory , gorkov @xcite established the equivalence of the bcs energy gap , @xmath9 , with the order parameter of the gl theory for single band superconductors with the above conditions ( i ) and ( ii ) . in multiband superconductors the equivalence has been investigated by many authors ( see a brief review in ref . @xcite ) . in both theories ( bcs and gl ) an additional interaction term appears due to interband interaction , which is recognised as the josephson term . this term is the lowest order coupling between the gaps ( in bcs ) and order parameter ( in gl ) in the different bands . the presence of josephson terms in multiband superconductors causes several problems in the gorkov type derivations . recently in a series of papers vagov and coworkers @xcite have made detailed analysis and established a generalisation of the standard gl theory ( which is correct to @xmath10 ) by retaining additional terms in the expansion up to order @xmath0 . in practice they have analysed the @xmath11 corrections to the order parameter for 1 , 2 and 3 bands . they call this formalism extended gl theory . this extended version with @xmath12 corrections seems to have improved the validity of the gl expansion to some lower temperatures away from @xmath5 in one- and two - band systems . in this paper we adopt the microscopic approach of gorkov generally for uniform multiband systems with isotropic ( spherical ) fermi surfaces . other types of fermi surfaces for dirty superconductors and with anisotropy can be done appropriately with more complications . we present here our detailed calculations of bcs gaps and gl order parameters for superconductors with one and two bands . in sec.2 we extend the gorkov technique to multiband superconductors in the absence of an external magnetic field , going beyond the standard / traditional model of gl . the coefficients for all terms in the series expansion of the self - consistent gap equation are given explicitly , and we show how to solve the resulting equations for the gap functions . in sec.3 results for the one band superconductor are presented showing clearly the departure of the standard gl order parameter ( with @xmath10 ) while comparing with the bcs result . higher order corrections are reported with impressive agreement with the bcs . we see that each additional term increases the range of @xmath13 for which the expansion is accurate . in sec.4 similar results for the two band superconductors are presented . these calculations are done for different interband couplings . in contrast with the single band results , additional terms in the two - band gl expansion only improve the agreement with the bcs result up to a certain value for @xmath13 . pushing beyond this point , the agreement becomes worse as additional terms are added . this disagreement is associated with the appearance of a second critical temperature in the weak coupling limit . in sec.5 we present the conclusions and summary of this work . the bcs theory is generalised to a multiband theory @xcite by allowing multiple fermion operators , which are identified by a band index , @xmath14 , and including a josephson interband term in the interaction . this term allows for cooper pairs to tunnel from band to band . with this generalisation , the effective multiband bcs hamiltonian in real space is given by @xmath15 where @xmath16 , @xmath17 is the vector potential , @xmath18 ( @xmath19 ) are fermionic annihilation ( creation ) operators , @xmath14 , @xmath20 are band indices , @xmath21 are spin indices , @xmath22 is the electron mass , @xmath23 is the chemical potential , @xmath24 are the interband coupling parameters , and @xmath25 is the superconducting gap . following the gorkov technique , the green function @xmath26 and anomalous green function @xmath27 can be written as a pair of coupled integral equations @xcite : @xmath28 where @xmath29 is the normal green function and @xmath30 , @xmath31 is the superconducting gap function in band @xmath14 , and the fermionic matsubara frequency @xmath32 , with @xmath33 . the normal green functions satisfy the equations @xmath34&{\ensuremath{\mathscr{g}_{\nu,\omega_n}^{(0)}(\mathbf{\mathbf{x}},\mathbf{\mathbf{x'}})}}\nonumber\\ & = \hbar\delta^3(\mathbf{x}-\mathbf{x'})\\ \left[-i\hbar\omega_n+\frac{\hbar^2}{2m_\nu}\left(\nabla-\frac{ie\mathbf{a}(\mathbf{x})}{\hbar c}\right)^2+\mu_\nu\right]&{\ensuremath{\widetilde{\mathscr{g}}_{\nu,\omega_n}^{(0)}(\mathbf{\mathbf{x}},\mathbf{\mathbf{x'}})}}\nonumber\\ & = \hbar\delta^3(\mathbf{x}-\mathbf{x'}).\end{aligned}\ ] ] using substitution , we can transform equations ( [ eq : normalgreenfunction ] ) and ( [ eq : anomalousgreenfunction ] ) into decoupled nonlinear integral equations , and by continued substitution we can write the anomalous green function as a series expansion in the gap and the normal green function @xmath35 the gap is defined in terms of the anomalous green function by @xmath36_{\nu\nu'}\delta^*_{\nu'}(\mathbf{x})=&\lim_{\eta\to 0^+}\sum_n e^{-i\omega_n\eta}\frac{1}{\beta\hbar}{\mathscr{f}_{\nu,\omega_n}^\dag(\mathbf{x},\mathbf{x})}. \label{eq : gapdefinition}\end{aligned}\ ] ] in this paper we are interested in finding the mean value for the gaps , so we will consider the case where the magnetic field is zero and the gap does not depend on @xmath37 , so the superconductor is uniform . by requiring the gap to satisfy equation [ eq : gapdefinition ] , we obtain the self - consistent gap equation in matrix form @xmath38 where @xmath39 is the interband coupling matrix with elements @xmath40 , @xmath41 is a column vector with elements @xmath42 , and @xmath43 is a column vector with elements given by @xmath44 @xmath45 with @xmath46 . equation [ eq : selfconsistentgap ] is a coupled equation involving the gaps from all bands , @xmath14 . these equations must be solved simultaneously . the normal green function can be solved in fourier space to find @xmath47 with @xmath48 . performing each of the real space integrals in equation [ eq : p ] produces a delta function , and these can be used to compute all but one of the @xmath49 space integrals , resulting in the simplified expression @xmath50 with @xmath51 is the density of states in band @xmath14 . when @xmath52 this integral diverges logarithmically , and so must be cut off at the debye energy , @xmath53 . in this case we find @xmath54 where @xmath55 is the euler - mascheroni constant and @xmath56 with @xmath5 to be defined later . the remaining terms with @xmath57 may be computed directly @xmath58 where @xmath59 is the riemann zeta function . putting this back together we find @xmath60 we then regroup terms to rewrite equation [ eq : selfconsistentgap ] in the form @xmath61 with @xmath62 , and @xmath63 is a diagonal matrix with elements @xmath51 on the diagonal . near the transition temperature , @xmath13 is a small parameter , so we will expand equation [ eq : newselfconsistant ] in powers of @xmath13 . to truncate this expansion , keeping only terms up to @xmath64 , we first make the scaling @xmath65 after scaling and then dividing through by @xmath10 we find @xmath66 then the gap is expanded in powers of @xmath13 , as is all the other dependence on @xmath13 in @xmath67 . the @xmath68 and @xmath69 expansions are given by @xmath70 we recover a set of equations for @xmath71 by collecting powers of @xmath13 in equation [ eq : lwgap ] and requiring that the equality holds for all @xmath13 . the leading order behaviour is a constant . collecting these constant terms leads to the lowest order equation @xmath72 this has a non - trivial solution if @xmath73 . we choose @xmath5 to be the largest value such that this equation is satisfied . we note that the @xmath5 of the combined system depends on the interband coupling . in the one band case , the well known solution for @xmath5 is @xmath74 while for the two - band case , the solution for @xmath5 is @xmath75 when @xmath76 , where @xmath77 is the critical temperature of the uncoupled first band , which is assumed to be the dominant band . we note that the critical temperature is enhanced over that of the dominant band due to the interband coupling , regardless of sign . now , since @xmath73 , there is at least one eigenvector of @xmath78 with a zero eigenvalue . we shall assume that this is non - degenerate , so that there is only one zero eigenvalue . we choose the base eigenvector to have the form @xmath79^t,\\ \rho_{i}=&\frac{c_{1,i}}{c_{1,1}},\\ c_{ijk\ldots , lmn\ldots}=&(-1)^{i+j+k+\ldots+l+m+n+\ldots}m_{ijk\ldots , lmn\ldots},\end{aligned}\ ] ] where @xmath80 is the cofactor of the matrix @xmath78 , and @xmath81 is the minor of @xmath78 , defined as the determinant of the matrix obtained by removing the rows @xmath82 and columns @xmath83 from @xmath78 . assuming all @xmath84 are finite and nonzero , we can then obtain a complete basis with the remaining vectors @xmath85^t.\end{aligned}\ ] ] the superconducting gaps can be written with this basis as @xmath86 putting this back into equation [ eq : tcequation ] and using the fact that @xmath87 and @xmath88 , @xmath89 , we find @xmath90 where @xmath91 is yet to be determined . the term linear in @xmath13 gives the equation @xmath92 this mixes @xmath93 with @xmath94 , however , as pointed out in ref , we can remove the @xmath94 dependence using the fact that @xmath95 . projecting this equation on to @xmath96 and using the solution for @xmath93 we find @xmath97 with @xmath98 and @xmath99 . this has the same form as the one band uniform g - l equation . kogan and schmalian @xcite pointed out that the gradient term is also the same as the one band g - l equation , and thus there is only one coherence length near @xmath5 , and the order parameters are proportional to each other . projecting equation [ eq : linearintau ] onto the other basis vectors , @xmath100 , results in a further set of equations for the higher components , @xmath101 . @xmath102 with @xmath103 , @xmath104 and @xmath105 . the indices @xmath106 and @xmath107 refer to the basis vectors , @xmath108 , not the band indices , @xmath14 . this process can be continued recursively to find the g - l approximation to any order . we provide the form for the terms @xmath109 and @xmath110 . @xmath111 @xmath112 all higher order terms can similarly be produced from the full definition of @xmath67 . and for moderate @xmath13 it converges quickly to the bcs solution . inset : a close up of the region near @xmath113 . there are singularities in the bcs function infinitesimally close to @xmath113 which prevent the extended g - l from converging at this point . ( b ) the magnitude of the lowest terms in the g - l expansion are shown on a log plot . the magnitude of the higher terms decays quickly except near the point @xmath113 where it remains finite . this shows that the expansion is converging on the region @xmath114.,scaledwidth=50.0% ] applying this procedure to a single band superconductor is fairly straight forward . the matrix @xmath78 becomes a number , and the equation for @xmath5 becomes trivial to solve . the basis vector @xmath115 so that @xmath116 in equation [ eq : deltapsibasis ] . this procedure has been performed for the one band case to high order , with the results shown in figure [ fig : extendedglonebandbigplot ] . the bcs solution is given by the bold black dots in the top plot . the thin red line that overshoots this is the conventional @xmath10 gl theory , while a selection of plots with higher order corrections up to @xmath0 with @xmath117 are also shown . the first correction , @xmath12 is seen as the dashed line just above the bcs solution @xcite , while higher order corrections are almost indistinguishable except near @xmath113 . including a larger number of corrections increases the range of convergence , and it is presumed that the infinite sum will converge for all @xmath114 . however for any large finite sum , the deviation near @xmath113 is expected to remain large . on the bottom plot of figure [ fig : extendedglonebandbigplot ] we plot the magnitude of each term in the sum . the error of any finite sum is approximately given by the magnitude of the next term in the sum , and so this plot can be viewed as an estimation of the error in any given finite sum . the magnitude of each term decreases in general except near @xmath113 , where , after the first few terms , it remains approximately constant . for the single band case , an exact form for each term in the expansion can be computed , though the number of terms needed increases rapidly . we report the result for the first three terms in the expansion . @xmath118 in two band gl , things progress in much the same way . however , there is now more a larger range of possibilities due to three parameters in the interband coupling matrix , @xmath24 , especially the role of the interband interaction , @xmath119 . we know from bcs theory that in the limit that the interband coupling goes to zero , the two gaps are independent and each has their own critical temperatures , which we label @xmath77 and @xmath120 respectively . when the interband coupling is small but nonzero , there is still a large change in the behaviour of the smaller gap near the temperature @xmath120 . however the critical temperature of the combined system is an enhancement of the dominant band s critical temperature . the exact lowest order solution can easily be calculated , with the result @xmath121 the higher order terms become increasingly complicated , however the results for specific parameters are calculated numerically to high order . in figure [ fig : bcsglplot ] we show plots of the bcs solution for a range of values for the interband coupling , @xmath119 . in a ) the first band is plotted , and it is seen that the interband coupling only has a weak effect on the behaviour of this band , while in b ) , the second gap shows a drastic change as @xmath119 increases , especially near @xmath120 , the critical temperature of the second band in the noninteracting limit . with the increase of the coupling strength , the large up - swell of the second band near this critical temperature gets washed out , so that at large coupling the plot looks reminiscent of a one band bcs plot . plots c ) and d ) depict the order parameters of band 1 and 2 respectively as calculated using the extended gl formalism derived earlier . for @xmath122 the behaviour shown in the gl plots is similar to that of the bcs plots above . however , for @xmath123 the behaviour of the gl plots is drastically different from the bcs plots , with the difference appearing sooner for smaller @xmath119 . the point where the solutions begin to disagree is very close to the location of @xmath120 , which in the small coupling limit is @xmath124 . while this finite summation approach does not prove that the series is divergent , it is clear that the sum has not converged in this range for the large number of terms computed . we expect that in general the sum will converge for all @xmath125 , but converge very slowly or diverge for @xmath126 . komendova et al . @xcite argue that there is a possibility of hidden criticality near @xmath120 which becomes critical in the limit that the coupling goes to zero . this feature is likely to be associated with the anomalous behavior of the gl gaps near this point , and is expected to prevent the series from converging below this point . surprisingly , while the bcs solution for the first band showed only a weak perturbation with the interband coupling , the non - convergent behaviour seen in the gl solution of the smaller band also affects the dominant band . this occurs for any small non - zero interband coupling , even though the solution converges for all @xmath13 if the interband coupling is zero . in figure [ fig : extendedglseries3bigplot ] the first two columns show the extended gl of band 1 and band 2 respectively as a function of @xmath127 for various @xmath119 . we can see that as the number of terms included in the expansion is increased , the gl solution departs from the bcs solution , shown as dots , in the region @xmath126 , and increasing the number of terms increases this difference . therefore , with this number of terms , the solution is not converging to the true solution in this range . the second two columns show the magnitude of each of the terms in the sum on a log plot . in these plots it is shown that there is approximately a pivot point above which the magnitude of the terms decrease , while below this point the magnitude of the terms increase . at the pivot point the magnitude of the terms remains approximately constant . the location of the pivot point is close to the point @xmath128 , especially for small interband coupling . the location of the point @xmath128 is shown as a vertical black line in the figure . as @xmath119 increases the location of the pivot point seems to move towards @xmath129 . however we know that @xmath120 is a constant . a possible reason for this behaviour of the pivot point is that as @xmath119 increases , @xmath120 does indeed remain constant , but @xmath5 increases , so that @xmath128 should move towards @xmath130 as @xmath119 increases . it is this increase in @xmath5 that makes the non - convergent point move towards zero as @xmath119 increases . in figure [ fig : extendedglseries5bigplot ] we have produced a similar plot to figure [ fig : extendedglseries3bigplot ] but where the gaps in the noninteracting limit have similar critical temperatures . with these parameters , the bcs solution shows that the dominant band is almost unperturbed by the interband interaction . at small interband interaction , the second band is weakly perturbed except near @xmath5 , however with increasing interband interaction , the second band quickly becomes indistinguishable from the first band . this is expected since the interband interaction causes the two bands to behave as a single band . since the properties of the two gaps are already similar in the uncoupled limit , only a reasonably small interband interaction is required before the two bands behave like a single band . when we look at the gl solution , we see that with these parameters and small interband coupling the region of validity of the solution is very tiny . even after the inclusion of a very large number of terms , the region where the gl solution has converged to the bcs solution is only in the range @xmath131 . when the interband coupling is increased , this range of convergence increases significantly . at large coupling , the solution converges over almost the complete temperature range . in this case the two gaps are almost identical . we see that in the two band case where the two gaps are close to degenerate and the interband coupling is very weak , the gl approximation is only valid in a very small temperature region near @xmath5 , and the theory should be applied with care . however , for the case where one band is very dominant , or where the interband coupling is very large , the gl theory performs very well , and converges quickly to the bcs result over a fairly large temperature range . in this paper we have reconstructed the relationship of the bcs theory with the gl theory with the limitations developed by gorkov in his ground - breaking work . the theory has been restricted to the case of a uniform system , but has been extended to allow multiple bands and large order in @xmath13 . this extends on the work of ref where the authors calculated a similar expansion keeping terms of order @xmath12 in the presence of a magnetic field . we have shown that in a one band superconductor the @xmath12 correction improves the magnitude of order parameter closer to the bcs value . higher order corrections for @xmath132 in @xmath0 improve the agreement with the bcs result except at @xmath129 , where the series for the gap appears to be nonconvergent . in the two band situation , the interband coupling plays a pivotal role in enhancing the smaller order parameter above the @xmath120 value in the bcs model . as the interband coupling increases the point of inflection around @xmath133 heals gradually . at large interband coupling both gaps look similar to a one band solution . the critical temperature of the system evolves smoothly out of the largest critical temperature , @xmath77 , and is enhanced by the interband coupling . in the gl model there are significant differences for both the gaps below @xmath126 . the large deviation persists for weaker interband couplings despite including larger @xmath0 corrections . this issue is significant when @xmath120 is close to the critical temperature @xmath5 . in this case the range of validity of the gl solution can be extremely small . the gl solution to the gaps below @xmath120 is unreliable , and therefore care must be taken when applying the gl model to multiband superconductors . when the interband coupling is larger or when one of the gaps is very dominant , the gl solution performs much better and including higher order terms can make the solution close to the bcs value over a large temperature range . similar to the one band case , the point @xmath129 is nonconvergent in the multiband solution regardless of interband coupling . in summary we have clearly demonstrated the importance of @xmath0 expansion for large @xmath134 for multiband gl superconductors . this point emphasises the weaker validity of the gl theory for lower temperatures , and especially for applications with small interband coupling . we are of the opinion that any use or misuse of gl theory has to be carefully examined considering its domain of applicability . the authors would like to thank the late john clem , vladimir kogan , alexei vagov , francois peeters , and nguyen van hieu for useful discussions . t. yoshida , m. hashimoto , i.m . vishik , z - x . shen , and a. fujimori , j. phys . . jpn . * 81 * , ( 2012 ) 011006 . g. knebel , d. aoki , j .- brison , l. howald , g. lapertot , j. panarin , s. raymond , and j. flouquet , physica staus solidi 14 june 2013 . bert , b. kalisky , c. bell , m. kim , y. hikita , h. y. hwang , and k. a. moler , nature physics * 7 * , ( 2011 ) 767771 . a. m. gabovich , a.i . voitenko , and m. ausloos , phys . rep . * 367 * , ( 2002 ) 583 . n. kimura and i. bonalde , non - centrosymmetric heavy - fermion superconductors , lecture notes in physics vol . 847 chapter 2 ed . e. bauer and m. sigrist ( 2012 ) . g. r. stewart , rev . * 73 * , ( 2001 ) 797 . b. j. wilson and m. p. das , j phys : condens . matter * 25 * ( 2013 ) 425702 . l p gorkov , sov phys jetp * 9 * ( 1959 ) 1364 . a. a. abrikosov , l. p. gorkov and i. e. dzyaloshinski , methods of quantum field theory in statistical physics , prentice hall , englewood cliffs , nj ( 1963 ) , see chapter 7 . e.h . brandt and m.p . das j supercond nov magn ( 2011 ) * 24 * 57 a. a. shanenko , m. v. miloevi , f. m. peeters and a. v. vagov , phys rev lett 106 ( 2011 ) 047005 . a. v. vagov , a. a. shanenko , m. v. miloevi , v. m. axt and f. m. peeters , prb * 85 * , 014502 ( 2012 ) a. v. vagov , a. a. shanenko , m. v. miloevi , v. m. axt and f. m. peeters , prb * 86 * , 144514 ( 2012 ) n. v. orlova , a. a. shanenko , m. v. miloevi , f. m. peeters , a. v. vagov and v. m. axt , , prb * 87 * , 134510 ( 2013 ) v. g. kogan and j schmalian , prb * 83 * 054515 ( 2011 ) l. komendova , y. chen , a.a . shanenko , m. v. miloevi , f.m . peeters phys rev lett * 108 * ( 2012 ) 207002 .

the recently discovered multiband superconductors have created a new class of novel superconductors . in these materials multiple superconducting gaps arise due to the formation of cooper pairs on different sheets of the fermi surfaces . an important feature of these superconductors is the interband couplings , which not only change the individual gap properties , but also create new collective modes . here we investigate the effect of the interband couplings in the ginzburg - landau theory . we produce a general @xmath0 expansion ( @xmath1 ) and show that this expansion has unexpected behaviour for @xmath2 . this point emphasises the weaker validity of the gl theory for lower temperatures and gives credence to the existence of hidden criticality near the critical temperature of the uncoupled subdominant band .

the tensor model was originally considered in @xcite to generalize the matrix model , which describes the two - dimensional simplicial quantum gravity , to higher dimensional cases . while the matrix model is a successful tool to analyze the two - dimensional simplicial gravity , the tensor model has not been successful in this direction , partly because of the absence of analytical methods to solve it and of physically appropriate interpretations of its partition function . in ref . @xcite , a new interpretation of the rank - three tensor model was proposed . namely , theory of a dynamical rank - three tensor may be regarded as that of dynamical fuzzy spaces . this proposal is based on the fact that a fuzzy space is described by an algebra of functions , which can be characterized by a rank - three tensor that defines multiplication , @xmath0 . this reinterpretation of the tensor model provides a new practical manner of extracting physics from the tensor model . in the original interpretation , it is necessary to compute the tensor model non - perturbatively , since the large volume limit of spaces corresponds to the large loop - number limit of the feynman diagrams of the tensor model . on the contrary , under the new interpretation , the semiclassical treatment of the tensor model is physically meaningful ; its classical solutions can be regarded as background fuzzy spaces , and small fluctuations around solutions as field fluctuations on fuzzy spaces . another key difference from the original proposal is that rank - three is enough as the rank of tensor to describe fuzzy spaces with arbitrary dimensions . this property drastically simplifies the structures of the tensor model , since various dimensional cases can be treated in a common framework . the rank - three tensor model has mainly been analyzed in numerical manners by the present author @xcite . in particular , for the tensor models that possess a certain gaussian type of classical solutions , it has numerically been shown that the properties of low - lying long - wavelength modes of small fluctuations around such gaussian backgrounds are in remarkable agreement with the general relativity in all the dimensional cases having been studied so far ( @xmath1 ) @xcite . namely , the general relativity was found to emerge in the tensor model as an effective long - wavelength description of the tensor model around a particular class of classical background solutions . this is also expected to be true in any other dimensions , since the framework and the procedure of analysis are common . this paper gives a summary of the results obtained so far concerning the emergence of the general relativity in the tensor model . there exist various versions of the rank - three tensor model @xcite . the simplest is the one that has a real symmetric rank - three tensor as its only dynamical variable , and has the invariance under the orthogonal group . in this paper , this simplest one is considered . the dynamical variable is a real - valued rank - three tensor @xmath2 , each index of which takes integers , @xmath3 . the number @xmath4 is the total number of linearly independent functions on a fuzzy space , or can more physically be interpreted as the number of `` points '' forming a fuzzy space . the variable @xmath2 is assumed to be totally symmetric , @xmath5 the algebra of products defined by @xmath6 is commutative but nonassociative in general . therefore the tensor model in this paper is a theory of dynamical commutative _ nonassociative _ fuzzy spaces . the basis of functions @xmath7 can be changed by linear transformations . a simple choice of equivalence of the basis functions is to assume that bases related by the orthogonal transformations represent equivalent fuzzy spaces . correspondingly , the tensor model must be invariant under the orthogonal transformation @xmath8 where @xmath9 is an arbitrary element of the orthogonal group @xmath10 . the definition of the system is given by a partition function , @xmath11 where @xmath12 is an action with the variable @xmath2 , and must be invariant under the orthogonal group transformation ( [ eq : trans ] ) . the integration measure @xmath13 must also be invariant under ( [ eq : trans ] ) , and is defined from the invariant metric in the space of @xmath2 given by @xmath14 so far , the emergence of the general relativity in the tensor model has only been shown around a particular class of backgrounds in the tensor model . these backgrounds have certain gaussian forms , and the algebras defined by them represent certain simple kinds of commutative _ nonassociative _ fuzzy flat spaces with arbitrary dimensions @xcite . in this section , to describe such genuine gaussian backgrounds , the indices of @xmath2 are assumed to take continuous values , while they will take finite discrete values in the actual analyses of the following sections . the gaussian backgrounds have the form , @xmath15 , \label{eq : cx}\ ] ] where @xmath16 and @xmath17 are positive numerical constants , @xmath18 are @xmath19-dimensional continuous coordinates , @xmath20 , and @xmath21 . the algebra of functions @xmath22 defines a commutative _ nonassociative _ fuzzy @xmath19-dimensional flat space considered in @xcite . because of the translational symmetry of ( [ eq : cx ] ) , it is generally more convenient to describe it in the momentum basis . by applying fourier transformation to the coordinate indices , one obtains the expression in the momentum basis because of the reality condition of the tensor in the coordinate basis , and one needs also an additional symmetric tensor @xmath23 for contracting the indices . these details are essentially important in the mode analysis of the following sections . ] as @xmath24,\ ] ] where @xmath25 and @xmath26 are positive numerical constants . one of the motivations for considering such particular solutions is the ( partial ) computability due to the gaussian forms . another is that the fuzzy spaces are physically well - behaved , because the fuzziness is well localized and the spaces are invariant under the poincare transformation , representing fuzzy @xmath19-dimensional flat spaces . moreover , as shown in the next section , there exists a natural correspondence between the metric field in the general relativity and the tensor around the gaussian backgrounds in the tensor model . in fact , there exist infinitely many actions that have such gaussian backgrounds as their classical solutions . two explicit examples have been studied so far @xcite . unfortunately , the explicit forms of the actions are very complicated and unnatural . this is a serious problem , which must be investigated in future study . however , what is interesting and remarkable in these actions in common is that each of them contains all the dimensional gaussian fuzzy flat spaces as its classical solutions , as illustrated in figure [ fig : potential ] . this means that all the dimensional spaces can be treated with one action in a unified manner . thus , for example , it is in principle possible to study transitions between spaces with distinct dimensions in the tensor model . in the physical interpretation of the tensor model , a classical solution should be regarded as a background space , and fluctuations of tensor around such a classical solution as field fluctuations on a background space . the main interest of the present study is whether such fluctuations can be identified with the general relativity or not . to check this , the procedure carried out so far in @xcite assumes a correspondence between the tensor model around the gaussian backgrounds and the metric tensor field in the general relativity . this correspondence enables one to compute the expectations about the tensor model from the general relativity . if the expectations successfully agree with the numerical analysis of the tensor model , one may conclude that the general relativity is emergent around the gaussian backgrounds in the tensor model . generalizing the gaussian backgrounds ( [ eq : cx ] ) in a coordinate invariant manner , one can derive a natural correspondence between the metric tensor field in the general relativity and the tensor around the gaussian backgrounds as @xmath27,\end{aligned}\ ] ] where @xmath28 , and @xmath29 denotes the geometric distance between @xmath30 and @xmath31 . the main assumption in this correspondence is that the low - lying long - wavelength fluctuations of the tensor around the gaussian backgrounds in the tensor model are exhausted by the metric field in the general relativity in the manner given in ( [ eq : correspondence ] ) . it should be noted that this correspondence could be modified in higher orders of the fuzziness @xmath32 . for example , there could exist corrections such as @xmath33 in ( [ eq : correspondence ] ) . although it is certainly possible to directly use the correspondence ( [ eq : correspondence ] ) in the comparison between the tensor model and the general relativity , it is much more convenient to use a tensor with a smaller rank . let me define @xmath34 small fluctuations @xmath35 around a classical solution @xmath36 induces fluctuations of @xmath37 as @xmath38 on the other hand , if one assumes a gaussian background and puts the assumed correspondence ( [ eq : correspondence ] ) into ( [ eq : tensordk ] ) , one obtains in the lowest order @xcite @xmath39 where the momentum basis is used , and @xmath40 is the fourier transform of @xmath41 , which describes the fluctuations of the metric tensor field around a flat background . in later sections , the analysis of eigenvectors gives @xmath35 of each fluctuation mode in the tensor model . then one can compute @xmath42 of each mode by putting this @xmath35 and the background @xmath36 into ( [ eq : tensordk ] ) . this @xmath42 obtained from the numerical analysis of the tensor model can be compared with the fluctuation modes of the metric tensor field in the general relativity through ( [ eq : metricdk ] ) . another important fact which can be derived from ( [ eq : correspondence ] ) is that the measure which must be used in the analysis of the general relativity is uniquely determined from the measure ( [ eq : cmeasure ] ) in the tensor model . by putting the correspondence ( [ eq : correspondence ] ) into ( [ eq : cmeasure ] ) , one obtains the dewitt supermetric @xcite , @xmath43,\ ] ] in the lowest order @xcite . in this section , i will study the small geometric fluctuations on @xmath19-dimensional flat tori in the general relativity to prepare for the comparison with the tensor model . the important point in the analysis is that not all of the fluctuations of the metric tensor field are the fluctuations of geometry , because of the gauge symmetry ( local translation symmetry ) in the general relativity . the relevant modes are only those which are normal to the gauge symmetry @xcite . the measure to be used to define this normality condition is the dewitt supermetric given in ( [ eq : supermetric ] ) , since the numerical analysis of the tensor model uses the corresponding measure ( [ eq : cmeasure ] ) as shown in the following section . for small fluctuations around a flat metric @xmath44 , the supermetric ( [ eq : supermetric ] ) is given by @xmath45.\ ] ] on the other hand , the infinitesimal gauge transformation on a flat background is given in the momentum basis by @xmath46 where @xmath47 is the fourier transform of local translation vector . at the vanishing momentum sector @xmath48 , as can be seen in ( [ eq : gaugemom ] ) , the gauge transformation is vacant , and all the components of the metric tensor are geometric degrees of freedom . by diagonalizing the supermetric ( [ eq : explicit ] ) , the modes can be shown to be classified into the following three orthogonal classes : + ( i ) 1 conformal mode : @xmath49 . + ( ii ) @xmath50 traceless diagonal modes : @xmath51 . + ( iii ) @xmath52 off - diagonal modes . at the nonvanishing momentum sector , one may take the momentum to be in the direction @xmath53 with obvious generalization to the other directions . then one can show that the modes normal to the gauge directions ( [ eq : gaugemom ] ) in the sense of the supermetric ( [ eq : explicit ] ) can be classified into the following three orthogonal classes : + ( i ) 1 diagonal mode : @xmath54 , @xmath55 for @xmath56 . + ( ii ) @xmath57 traceless diagonal modes : @xmath58 , @xmath59 . + ( iii ) @xmath60 off - diagonal modes : @xmath61 for @xmath62 , @xmath63 . @xmath42 for the geometric fluctuations of each mode in the general relativity can be computed by putting these results into ( [ eq : metricdk ] ) , and can be compared with the numerical analysis of the tensor model . in this section , i will give a brief summary of the numerical analyses having been done so far on the small fluctuations around the gaussian backgrounds in the tensor model @xcite . to unambiguously determine the spectra , the fluctuations must be normalized in a manner respecting the symmetries of the tensor model . the tensor @xmath2 is symmetric under the exchanges of the indices as in ( [ eq : csym ] ) , and the tensor model has the orthogonal group symmetry ( [ eq : trans ] ) . thus the independent components of fluctuations are normalized through ( [ eq : cmeasure ] ) as @xmath64\ , d c_{abc}\ , d c^{abc}=\sum_{(abc ) } d \tilde c_{(abc ) } \ , d \tilde c^{(abc)},\ ] ] where @xmath65 denotes an order - independent set of three indices , @xmath66}\,c_{abc}$ ] are the independent normalized components , and @xmath67 $ ] is the multiplicity defined by @xmath68=\left\ { \begin{array}{ll } 1 & \hbox{for } a = b = c,\\ 3 & \hbox{for } a = b\neq c,\ b = c\neq a,\ c = a\neq b,\\ 6 & \hbox{all different}. \end{array } \right.\ ] ] then the coefficient matrix for the normalized fluctuations in the quadratic order around a background @xmath69 is given by @xmath70 the numerical analysis is carried out to obtain the eigenvalues and eigenvectors of the matrix ( [ eq : m ] ) , and then the results are compared with the general relativity through the procedure explained in the previous sections . the genuine gaussian backgrounds presented in section [ sec : gauss ] can not be used as a background @xmath69 , because of their infinite number of degrees of freedom . thus the actual numerical analyses have been done around the backgrounds of @xmath19-dimensional fuzzy flat tori . to describe such backgrounds of fuzzy flat tori in terms of @xmath2 , the indices are assumed to take integer momenta bounded by a cut off , and the momentum conservation , @xmath71 , is assumed on account of the translational symmetry of such flat tori . in fact , under these assumptions , one can numerically find classical solutions that resemble the genuine gaussian backgrounds @xcite . using these numerical classical solutions as backgrounds , the coefficient matrices ( [ eq : m ] ) have been analyzed for dimensions @xmath72 in @xcite , and also for @xmath1 with an approximate method explained below @xcite . the numerical analyses have shown that the spectra of the modes with long - wavelengths can roughly be classified into the following three classes . + ( i ) the `` heavy '' modes with spectra of order 1 or larger . + ( ii ) in @xmath73 , there exist low - lying modes with non - vanishing spectra of order much smaller than 1 . + ( iii ) the zero modes with vanishing spectra . + it is observed that the two classes ( i ) and ( ii ) are rather clearly separated so that they are located hierarchically . the spectra in the class ( ii ) have been shown to form trajectories of the fourth power of momenta . moreover , the spectral patterns and the mode profiles have been shown to be in good agreement with the geometric degrees of freedom in the general relativity discussed in section [ sec : gr ] , by comparing ( [ eq : tensordk ] ) computed numerically from the tensor model and ( [ eq : metricdk ] ) from the general relativity . this shows that the general relativity is emergent around such backgrounds , and that the lowest effective actions are composed of curvature quadratic terms . as for the class ( iii ) , by counting their numbers , these modes have been identified with the modes of the @xmath10 symmetry transformations in the tensor model spontaneously broken to the remaining symmetry @xmath74 of the torus backgrounds . therefore these zero spectra are just the gauge modes of the tensor model . the relation between these zero modes and the local gauge symmetry ( local translation symmetry ) of the general relativity has been studied in @xcite , which will be summarized in the following section . in the approximate method used in @xcite , the backgrounds @xmath69 in ( [ eq : m ] ) are not taken to be the classical solutions , but to be approximate ones , the gaussian backgrounds ( [ eq : cp ] ) with the modifications that the continuum @xmath75-functions are replaced with kronecker deltas , and that @xmath76 is taken in the range @xmath77 for the approximation to be good . in this approximation , it is easier to numerically analyze the cases with larger @xmath78 , and the agreement between the spectra in the class ( ii ) and the geometric degrees of freedom in the general relativity has been observed more clearly than without approximations . part of the results for @xmath79 are shown in figures [ fig : dim2l10spec ] and [ fig : dim2l10 ] . especially , figure [ fig : dim2l10 ] shows very clearly the agreement of the mode profiles between the tensor model and the general relativity . fuzzy flat torus for cut - off @xmath80 and @xmath81 . the horizontal axis is the size of momentum @xmath82 of fluctuation modes , and the vertical axis is the spectral value . the solid line is @xmath83 . ] for the low - lying mode at @xmath84 sector for @xmath79 . the left and right figures are ( [ eq : tensordk ] ) from the numerical analysis of the tensor model and ( [ eq : metricdk ] ) for the diagonal mode in the general relativity , respectively . the axes are @xmath85.,title="fig:",width=226 ] for the low - lying mode at @xmath84 sector for @xmath79 . the left and right figures are ( [ eq : tensordk ] ) from the numerical analysis of the tensor model and ( [ eq : metricdk ] ) for the diagonal mode in the general relativity , respectively . the axes are @xmath85.,title="fig:",width=226 ] the connection between the spontaneously broken @xmath10 symmetry and the local gauge symmetry ( local translation symmetry ) in the general relativity has been discussed in @xcite . in the paper , the brst gauge fixing procedure has been applied to the @xmath86 symmetry of the tensor model , and the spectra of the ghost quadratic term have been studied numerically . then the appearance of emergent massless ghost fields has been observed , and they have been identified with the reparametrization ghost fields coming from the brst gauge fixing of the general relativity . let me start with the gauge fixing in the tensor model . the off - shell nilpotent brst transformation in the tensor model is defined by @xmath87 where @xmath88 and @xmath89 are the ghosts ( anti - ghosts ) , and the bosonic auxiliary variables @xcite , respectively . here @xmath90 is the infinitesimal orthogonal transformation defined by @xmath91 where @xmath92 are the elements of the lie algebra @xmath93 in the vector representation , and @xmath94 is the structure constant defined by @xmath95=f^{ij}{}_k t^k$ ] . now let me define a new dynamical variable @xmath25 by shifting the original variable @xmath96 by a background @xmath69 , @xmath97 then a natural gauge fixing plus faddeev - popov action is given by @xmath98 where @xmath99 is the inner product associated with the measure ( [ eq : cmeasure ] ) . the reason why this is natural is that the gauge fixing conditions ( @xmath100 ) only allow @xmath25 to be normal to the symmetry directions around the background @xmath69 . this action contains the ghost quadratic term as @xmath101 figure [ fig : ghostspec ] shows the spectra of this ghost quadratic term and the ratio of the two trajectories obtained numerically for a gaussian background of @xmath79 flat torus . , @xmath102 , @xmath103 . the right figure shows the ratios of the two trajectories . the horizontal axis is the momentum size @xmath104.,title="fig:",width=302 ] , @xmath102 , @xmath103 . the right figure shows the ratios of the two trajectories . the horizontal axis is the momentum size @xmath104.,title="fig:",width=302 ] on the other hand , the off - shell nilpotent brst gauge transformation in the general relativity is given by @xmath105 corresponding to ( [ eq : shiftca ] ) , let me define a new dynamical field @xmath106 by shifting the metric by a flat background , @xmath107 . then , an action naturally corresponding to ( [ eq : sgftensor ] ) is given by @xmath108 where @xmath109 is the inner product associated with the dewitt supermetric ( [ eq : supermetric ] ) . in fact , the gauge fixing conditions in only allows @xmath106 to be normal to the infinitesimal gauge transformations of the flat background . this action contains the ghost kinetic term as @xmath110+\cdots.\ ] ] this kinetic term contains the longitudinal ( @xmath111 ) and normal ( @xmath112 ) modes with spectra @xmath113 and @xmath114 , respectively . in fact , the ratio of spectra @xmath115 agrees well with the numerical analysis of the tensor model as plotted in figure [ fig : ghostspec ] . the mode profiles have also been checked as shown in figure [ fig : ghostprofile ] for @xmath79 . the comparison of ghosts between the numerical analysis of the tensor model and the general relativity has also been done for @xmath116 , and clear agreement has been obtained . , through @xmath42 in ( [ eq : tensordk ] ) and ( [ eq : metricdk ] ) . the left couple of figures show the profiles of the modes in the lower and upper trajectories in the tensor model . the right couple of figures show the profiles of the normal ( @xmath117 ) and the longitudinal ( @xmath118 ) modes in the general relativity . , title="fig:",width=147 ] , through @xmath42 in ( [ eq : tensordk ] ) and ( [ eq : metricdk ] ) . the left couple of figures show the profiles of the modes in the lower and upper trajectories in the tensor model . the right couple of figures show the profiles of the normal ( @xmath117 ) and the longitudinal ( @xmath118 ) modes in the general relativity . , title="fig:",width=147 ] , through @xmath42 in ( [ eq : tensordk ] ) and ( [ eq : metricdk ] ) . the left couple of figures show the profiles of the modes in the lower and upper trajectories in the tensor model . the right couple of figures show the profiles of the normal ( @xmath117 ) and the longitudinal ( @xmath118 ) modes in the general relativity . , title="fig:",width=147 ] , through @xmath42 in ( [ eq : tensordk ] ) and ( [ eq : metricdk ] ) . the left couple of figures show the profiles of the modes in the lower and upper trajectories in the tensor model . the right couple of figures show the profiles of the normal ( @xmath117 ) and the longitudinal ( @xmath118 ) modes in the general relativity . , title="fig:",width=147 ] in a series of papers , i have studied the tensor model with actions which possess the gaussian backgrounds as their classical solutions . these backgrounds represent fuzzy flat spaces with arbitrary dimensions , and the small fluctuations around them have been compared with the general relativity on flat backgrounds through numerical analyses . the numerical analyses for dimensions @xmath1 have shown that the long - wavelength low - lying fluctuation spectra are in one - to - one correspondence with the geometric fluctuations in the general relativity . the analyses have also shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds corresponds to the local gauge symmetry ( local translational symmetry ) of the general relativity . these results should be valid in all dimensions , because of the dimensional independence of the framework and of the way of analysis . thus , the tensor model provides an interesting model of simultaneous emergence of space , the general relativity and its gauge symmetry of translation in general dimensions . there seem to exist various questions about the results . for example , the agreement between the modes in the tensor model and in the general relativity should be checked also for higher orders of fluctuations , although such higher order agreement is expected , because the general relativity ( possibly with modified actions ) is the only theory of symmetric rank - two tensor field with the gauge symmetry . this will require more efficient numerical facility and/or technical developments . another important question is the range of generality of the results , which have only been shown so far around the gaussian backgrounds in a few actions of examples . for the tensor model to be really interesting , such emergence should be shown to be common phenomena in more general settings . j. ambjorn , b. durhuus and t. jonsson , `` three - dimensional simplicial quantum gravity and generalized matrix models , '' mod . a * 6 * , 1133 ( 1991 ) . n. sasakura , `` tensor model for gravity and orientability of manifold , '' mod . lett . a * 6 * , 2613 ( 1991 ) . n. godfrey and m. gross , `` simplicial quantum gravity in more than two - dimensions , '' phys . d * 43 * , 1749 ( 1991 ) . n. sasakura , `` an invariant approach to dynamical fuzzy spaces with a three - index variable , '' mod . phys . a * 21 * , 1017 ( 2006 ) [ arxiv : hep - th/0506192 ] . y. sasai and n. sasakura , `` one - loop unitarity of scalar field theories on poincare invariant commutative nonassociative spacetimes , '' jhep * 0609 * , 046 ( 2006 ) [ arxiv : hep - th/0604194 ] . a. connes , `` noncommutative geometry , '' _ academic press ( 1994 ) 661 p_. + j. madore , `` an introduction to noncommutative differential geometry and its physical applications , '' lond . math . note ser . * 257 * , 1 ( 2000 ) . + a. p. balachandran , s. kurkcuoglu and s. vaidya , `` lectures on fuzzy and fuzzy susy physics , '' arxiv : hep - th/0511114 . + r. j. szabo , `` quantum field theory on noncommutative spaces , '' phys . * 378 * , 207 ( 2003 ) [ arxiv : hep - th/0109162 ] . n. sasakura , `` an invariant approach to dynamical fuzzy spaces with a three - index variable - euclidean models , '' in the proceedings of 4th international symposium on quantum theory and symmetries ( qts-4 ) , varna , bulgaria , 15 - 21 aug 2005 [ arxiv : hep - th/0511154 ] . n. sasakura , `` tensor model and dynamical generation of commutative nonassociative fuzzy spaces , '' class . * 23 * , 5397 ( 2006 ) [ arxiv : hep - th/0606066 ] . n. sasakura , `` the fluctuation spectra around a gaussian classical solution of a tensor model and the general relativity , '' int . j. mod . a * 23 * , 693 ( 2008 ) [ arxiv:0706.1618 [ hep - th ] ] . n. sasakura , `` the lowest modes around gaussian solutions of tensor models and the general relativity , '' int . j. mod . a * 23 * , 3863 ( 2008 ) [ arxiv:0710.0696 [ hep - th ] ] . n. sasakura , `` gauge fixing in the tensor model and emergence of local gauge symmetries , '' prog . . phys . * 122 * , 309 ( 2009 ) [ arxiv:0904.0046 [ hep - th ] ] . b. s. dewitt , `` quantization of fields with infinite - dimensional invariance groups . generalized schwinger - feynman theory , '' j. math . phys . * 3 * , 1073 ( 1962 ) . r. ferrari and l. e. picasso , `` spontaneous breakdown in quantum electrodynamics , '' nucl . b * 31 * , 316 ( 1971 ) . r. a. brandt and w. c. ng , `` gauge invariance and mass , '' phys . d * 10 * , 4198 ( 1974 ) . a. b. borisov and v. i. ogievetsky , `` theory of dynamical affine and conformal symmetries as gravity theory of the gravitational field , '' theor . math . * 21 * , 1179 ( 1975 ) [ teor . mat . * 21 * , 329 ( 1974 ) ] . t. kugo and s. uehara , `` general procedure of gauge fixing based on brs invariance principle , '' nucl . b * 197 * , 378 ( 1982 ) .

this paper gives a summary of the author s works concerning the emergent general relativity in a particular class of tensor models , which possess gaussian classical solutions . in general , a classical solution in a tensor model may be physically regarded as a background space , and small fluctuations about the solution as emergent fields on the space . the numerical analyses of the tensor models possessing gaussian classical background solutions have shown that the low - lying long - wavelength fluctuations around the backgrounds are in one - to - one correspondence with the geometric fluctuations on flat spaces in the general relativity . it has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds can be identified with the local translation symmetry of the general relativity . thus the tensor model provides an interesting model of simultaneous emergence of space , the general relativity , and its local gauge symmetry of translation . = 17.5pt plus 0.2pt minus 0.1pt # 1([#1 ] )

according to the instanton liquid model ( ilm ) instantons are largely responsible for the spontaneous breaking of chiral symmetry and the low energy properties of light hadrons @xcite . this has been demonstrated using phenomenological instanton liquid models . the two main tasks involved in the construction are * producing an equilibrium instanton ensemble assuming some interaction between the pseudoparticles . this is typically done with a variational calculation @xcite . * computing hadronic observables using the above instanton ensemble as the gauge background . this is a formidable task and the computation necessarily involves several approximations . it would thus be very desirable to check the physical picture emerging from the ilm , starting from first principles . in the last few years there has been a considerable effort in this direction , using lattice simulations of qcd as a tool . the properties of the instanton ensemble have been extracted from monte carlo generated lattice configurations using several different `` smoothing '' techniques . the most important parameters , the topological susceptibility , the instanton density , and the average instanton size were found to be in qualitative agreement with the ilm . on the other hand although there is a considerable amount of indirect evidence @xcite so far the lattice has not given any direct results concerning the second part of the above programme . in ref . @xcite cooled lattice configurations were used to show that low eigenmodes of the dirac operator saturate the point - to - point hadronic correlators . more recently a strong correlation was found between the chiral density of low modes of the staggered dirac operator and the topological charge density @xcite . these results suggest that instantons probably really play an important role in qcd . all this work however has involved monte carlo generated or cooled lattices that contained other gauge field fluctuations in addition to the instantons . therefore , based solely on these studies it is impossible to separate the part of the effects due to the instantons and the part ( if any ) that is due to the rest of the gauge field fluctuations . to my knowledge , the only lattice study so far that makes such a separation possible has appeared in ref . @xcite . there , the instanton content of smoothed gauge configurations was reconstructed . hadron spectroscopy done on these artificial instanton backgrounds was compared to the spectroscopy on the corresponding smoothed configurations . while the smoothed ensemble yielded physically reasonable results for the pion and the rho mass , the instantons failed to reproduce this . in fact wilson spectroscopy on the instanton ensemble was hardly different from free field theory ( trivial gauge field background ) . in view of the positive results cited above , this was highly unexpected , especially that at first sight the only difference between the compared two ensembles appeared to be the presence versus the lack of confinement on the smoothed and the artificial instanton ensembles respectively . according to the ilm , confinement is not essential for the low energy properties of light hadrons @xcite . the findings of ref . @xcite indeed suggest that the physics in the chiral limit is rather insensitive to the string tension . with less than half of the original string tension , the low fermion modes on the cooled configurations still produced physically sensible hadron correlators . in the present paper i address the question of how to reconcile the insensitivity of light hadrons to the string tension ( also expected from the ilm ) with the failure of the instantons alone to reproduce the correct physics ( found in @xcite ) . i will exclusively consider the su(2 ) gauge group . along the way i will also obtain some useful insight into the structure of the low end of the spectrum of lattice dirac operators and in particular i will show the importance of using a ( close to ) chiral lattice dirac operator . this is essential for a faithful reproduction of continuum - like quark ( quasi)zero modes . the plan of the paper is as follows . in section [ se : zmz ] i briefly discuss the instanton liquid picture of physics in the chiral limit , and in particular the role of the zero modes and the so called `` zero mode zone '' . in section [ se : cs ] i point out the role of the explicit chiral symmetry breaking of the lattice dirac operator and the importance of minimising it . i show that the optimised clover dirac operator introduced in @xcite performs much better in this respect than the wilson operator . i explicitly study the zero mode zone in the simplest nontrivial case , that of an instanton antiinstanton pair and find that the mixing of zero modes closely follows the pattern expected in the continuum . in section [ se : spectrum ] i find that the optimised clover operator produces spectroscopy results markedly different from the wilson operator but it still falls short of reproducing the correct physics on the instanton backgrounds . instead , it yields a peculiar mixture of qcd and free field theory . i demonstrate that this is due to the presence of a large number of low lying free field modes in the spectrum of the dirac operator with the instanton backgrounds that produce artificially light hadrons . the free modes are completely absent from the smoothed configurations . they can be also eliminated from the instanton configurations by superimposing an ensemble of zero momentum gauge field fluctuations ( torons ) over the instantons . these gauge field fluctuations survive even deep in the perturbative regime . i argue that they are probably needed to compensate for the anomalously large small - volume effects present in the instanton ensemble due to the lack of confinement . by comparing the density of eigenvalues around zero on the physical ( smoothed ) , the toron , and toron plus instanton ensembles , i obtain direct evidence that the instantons are responsible for chiral symmetry breaking . after the elimination of the free quark modes , the instanton backgrounds are shown to reproduce the correct physical pion and rho correlators . section [ se : con ] contains my conclusions along with some speculations on still unanswered questions . finally , in the appendix , for reference i collected some general properties of wilson type lattice dirac operators that i used in the main text . some of these can be also found in the literature . in this section i describe how instantons can produce low lying modes of the dirac operator and why this is important for the physics in the chiral limit . for simplicity i start with the continuum theory and in the second part of this section i discuss the complications that appear on the lattice . let @xmath0 be the continuum massless dirac operator which implicitly depends on a gauge field background . the basic building block of the physics in the chiral limit is the @xmath1 limit of the massive quark propagator , @xmath2 . from the spectral decomposition in terms of eigenmodes of @xmath0 @xmath3 it is easily seen that in the chiral limit the most important modes are the ones with eigenvalues @xmath4 close to 0 . the crucial assumption of the ilm is that it is the instantons that are responsible for generating the bulk of these lowest eigenmodes . this can be qualitatively understood as follows . according to the atiyah - singer index theorem , in the background of an instanton ( antiinstanton ) @xmath0 has at least one negative ( positive ) chirality zero mode . in the presence of an infinitely separated instanton antiinstanton pair we still expect to find opposite chirality zero modes localised on the two objects . if the members of the pair are brought closer to one another , the two degenerate zero eigenvalues will in general split into two complex ones still close to the origin . a very simplistic but qualitatively correct description of this can be given as follows . let @xmath5 and @xmath6 be the zero modes of the dirac operators @xmath7 and @xmath8 respectively . since we consider @xmath0 in different gauge backgrounds , we explicitly indicate this ; @xmath9 and @xmath10 refer to an instanton and an antiinstanton . we now want to describe the spectrum of @xmath11 where @xmath12 is some superposition of the gauge fields @xmath9 and @xmath10 . since @xmath5 and @xmath6 are of opposite chirality , @xmath13 . let us arbitrarily complete the set of these two vectors into an orthonormal basis and construct the matrix of @xmath11 in this basis . the elements of this matrix in the subspace spanned by @xmath5 and @xmath6 are @xmath14 where @xmath15 the diagonal matrix elements vanish because @xmath0 maps left handed vectors into right handed ones and vice versa . by a suitable choice of the phases , the off - diagonal elements can always be made real and due to the anti - hermiticity of @xmath0 they are of equal magnitude and have opposite signs . if we now assume that @xmath16 and @xmath17 lie approximately in this two - dimensional subspace , i.e. the matrix of @xmath11 contains the above @xmath18 block - diagonal part , then @xmath11 can be easily seen to have two complex conjugate eigenvalues @xmath19 with the corresponding ( unnormalised ) eigenvectors being @xmath20 . this mechanism can be generalised to gauge fields which are superpositions of several instantons and antiinstantons . the basis in this case consists of the zero modes of the individual ( anti)instantons . the only additional approximation involved is the assumption that the zero modes corresponding to different ( anti)instantons are orthogonal . if the pseudoparticles are well separated , this is approximately true . one can thus consider the matrix of @xmath0 restricted to the subspace of zero modes called the `` zero mode zone '' in a general instanton background . if the off diagonal elements of this matrix are small then it will have complex eigenvalues close to the origin , in addition to a number of exact zero eigenvalues corresponding to the total topological charge . moreover , the complex eigenmodes will be approximately linear combinations of several instanton and antiinstanton zero modes and thus become highly delocalised , making it possible to propagate quarks to large distances . the most important quantities in this construction are the off diagonal instanton antiinstanton matrix elements , the @xmath21 s . in the simplest case of only one pair , @xmath21 is known to depend on the distance and the relative orientation of the members of the pair as @xmath22 where @xmath23 describes the relative orientation in group space , @xmath24 and @xmath25 is the relative position of the instanton and the antiinstanton @xcite . @xmath26 is a function of the instanton and antiinstanton scale parameters and the distance between them . at small distances it depends on the ansatz used to combine the field of the instanton and the antiinstanton , at large separation it is expected to fall off as @xmath27 . we have seen how instanton zero modes can give rise to the low - lying eigenmodes that are the most important ones in the chiral limit . close to the chiral limit the number of modes giving a substantial contribution to the quark propagator becomes smaller and smaller . it is thus crucial to reproduce these low eigenmodes and eigenvalues as faithfully as possible on the lattice if we want to test the ilm . the most important obstruction to this is that due to the nielsen - ninomiya theorem it is impossible to construct a local lattice dirac operator describing one fermion species with exact chiral symmetry @xcite . in the present paper i shall limit the discussion to wilson type lattice dirac operators that describe one fermion species but explicitly break chiral symmetry . in the absence of chiral symmetry the fermion zero modes corresponding to the topology of the gauge field are not protected ; any small fluctuation of the gauge field can shift the zero eigenvalues . in order to understand how this happens , a useful starting point is the following property shared by the lattice and the continuum dirac operator . let @xmath28 be an eigenvector of @xmath0 corresponding to the eigenvalue @xmath29 . if @xmath0 has no degeneracies then @xmath30 if and only if @xmath29 is real . this is a simple consequence of the so - called `` @xmath31 hermiticity '' , @xmath32 , a common property of the continuum and the lattice dirac operator . a proof of the above statement is given in the appendix . in the continuum , @xmath0 is anti - hermitian and has eigenvalues on the imaginary axis . thus the only real eigenvalue it can have is 0 . chirality makes a sharp distinction between zero and non - zero modes in the continuum . zero modes have a chirality of @xmath33 while all the non - zero eigenmodes have zero chirality . since the eigenmodes and their chiralities depend continuously on the gauge field , zero modes are protected , they can not be shifted away from zero by continuous deformations of the gauge field . on the other hand , the lattice dirac operator is not anti - hermitian , and its eigenvalues are not constrained to be on the imaginary axis . while lattice zero modes are still protected from being shifted off the real axis by smooth deformations of the gauge field ( this would cause their chirality to jump to zero ) , there is nothing preventing them from moving continuously along the real axis . indeed , a lattice dirac operator will in general not have any exact zero modes unless the gauge field is fine tuned . it is now clear that the lattice analogues of the continuum zero modes are real modes of the dirac operator . these have non - vanishing chiralities but in general their magnitude is smaller than 1 . the crucial role chiral symmetry plays in this discussion can be understood by noting that chiral symmetry of @xmath0 ( i.e. @xmath34 ) together with @xmath31 hermiticity would imply that @xmath0 is anti - hermitian and has protected zero modes . the smaller the explicit chiral symmetry breaking of @xmath0 is , the more its low lying spectrum will resemble that of an anti - hermitian operator . recently it has been shown that the closest a lattice dirac operator can come to being chirally symmetric is by obeying the ginsparg - wilson relation and thus having an ultralocal chirality breaking in the quark propagator @xcite . in this case the spectrum of @xmath0 lies on a circle in the complex plane passing through the origin . indeed the low end of this spectrum is almost on the imaginary axis and there are protected zero modes . i shall now discuss how the continuum description of the zero mode zone presented in the previous subsection has to be modified on the lattice . assume that there is an instanton on the lattice . in the limit when the instanton size goes to infinity ( in units of the lattice spacing ) , the instanton `` does not see '' the lattice at all , and the continuum description applies . the lattice dirac operator , @xmath7 has an exact zero eigenvalue with a corresponding @xmath35 chirality eigenmode @xmath5 . if the instanton is not infinitely large , the real eigenvalue gets shifted away from zero , its magnitude depends on the instanton size and the chirality breaking of @xmath7 @xcite . the corresponding eigenmode also becomes only an approximate eigenvector of @xmath31 and thus @xmath36 . let us now consider the dirac operator in the presence of an instanton antiinstanton pair . following the continuum discussion in the previous subsection , its matrix elements between the ( lattice approximate ) `` zero modes '' @xmath5 and @xmath6 can be parametrised as @xmath37 where @xmath38 is real , @xmath39 and @xmath40 are non - negative real numbers . in the su(2 ) case the diagonal elements are automatically real and the off - diagonal ones can be also made real by a suitable choice of the phases of the eigenvectors . this will be proved in the appendix . due to the @xmath31 hermiticity of @xmath0 , since @xmath5 and @xmath6 are approximate eigenvectors of @xmath31 , the off - diagonal terms have opposite signs and about the same magnitude . the eigenvalues of this matrix are @xmath41 with the corresponding ( unnormalised ) eigenvectors being @xmath42 the most important feature of these eigenvectors is that as in the continuum they are mixtures of the instanton and antiinstanton zero modes with roughly the same magnitude ( the ratio being @xmath43 ) and with a relative phase @xmath44 , up to a correction proportional to @xmath38 . this mechanism produces highly delocalised eigenmodes in the zero mode zone and thus facilitates the propagation of quarks by jumping from instanton to antiinstanton . maintaining this feature on the lattice is thus crucial for the description of the zero mode zone . the requirement for the proper mixing of zero modes is the inequality @xmath45 for producing qualitatively continuum - like eigenmodes , the difference of the diagonal elements has to be much smaller than the magnitude of the off - diagonal mixing matrix elements of @xmath11 . otherwise the two eigenvalues will be real and the corresponding eigenmodes rapidly become localised on the instanton and the antiinstanton respectively . recall that in the continuum , the mixing matrix elements @xmath46 are proportional to @xmath47 , where @xmath48 is the invariant angle of the instanton antiinstanton relative orientation ( see eq . ( [ eq : relor ] ) ) . @xmath38 ( and @xmath49 ) is expected to be of the order of the real eigenvalues of @xmath7 and @xmath8 , both small but nonzero numbers . this means that inequality ( [ eq : ineq ] ) will be violated for @xmath48 sufficiently close to @xmath50 . we can minimise the range in relative orientation , where this happens by making @xmath38 small , i.e. having the would - be zero modes close to zero . in this section i study how the general features of the zero mode zone described in the previous section are realised by two commonly used lattice dirac operators , the wilson and the clover operator . the gauge background i use for this test contains an instanton of size @xmath51 and an antiinstanton of size @xmath52 , separated by a distance @xmath53 in the time direction . the lattice size is @xmath54 and i always use antiperiodic boundary conditions in the time direction , periodic in all other directions . these parameters are typical in the spectroscopy calculations that i shall present in the next section . let me start with the wilson operator . in the presence of only the ( anti)instanton with the above parameters , @xmath55 has a real eigenvalue at ( 0.17 ) 0.51 with chirality ( 0.39 ) -0.53 . it is clear that for these small instantons the wilson operator does not even come close to satisfying the conditions of the previous section : the `` zero modes '' are far away from zero with chiralities of magnitude much smaller than 1 . the simplified description of the instanton - antiinstanton configuration in terms of the @xmath18 matrix is then inadequate . if the instanton and antiinstanton are close to being parallel @xmath56 then the real part of the ( anti)instanton related eigenvalues is around 0.3 . moreover already at @xmath57 the two complex eigenvalues become real and the corresponding modes are rather localised at the instanton and the antiinstanton respectively . a much better description of the zero mode zone is provided by the optimised clover action proposed in ref . ( see also @xcite for more discussion and a recent scaling test of this action . ) the clover coefficient was chosen to minimise the range in which the real eigenvalues occur in the physical branch of the spectrum on locally smooth gauge backgrounds . as explained in the previous section , this is exactly what is needed for a good description of the mixing of zero modes . the optimal value for @xmath58 was found to be around 1.2 on a set of ape smeared @xmath59 quenched su(3 ) configurations . physically these are quite similar to the su(2 ) configurations appearing in the present study , therefore in what follows i shall always choose @xmath60 . the real modes occurring in the single ( anti)instanton background already reveal that the clover action is much closer to the continuum than the wilson action . the eigenvalues are located at -0.014 and -0.002 and the chiralities of the corresponding eigenmodes are 0.993 and -0.995 respectively . in the instanton plus antiinstanton background the lowest two modes are always complex for almost any relative orientation of the pair , even at @xmath61 . we can hope that the simple description in terms of the mixing matrix works well in this case . indeed , the magnitude of the diagonal elements is small , neither @xmath38 nor @xmath29 goes above 0.02 in magnitude for any relative orientation . in fig . [ fig : t12 ] i plotted the two off - diagonal matrix elements , @xmath62 versus @xmath48 along with the one - parameter fits of the form const.@xmath63 . as expected , the mixing matrix elements are very well described by the continuum ansatz of eq.([eq : relor ] ) . a comparison of the eigenvalues of the @xmath18 mixing matrix and the explicitly computed eigenvalues of the lattice dirac operator in the corresponding gauge backgrounds shows agreement to within @xmath64 in the whole range of @xmath65 . we can conclude that the @xmath60 clover operator should give a satisfactory treatment of the zero mode zone as long as the instanton size does not drop below around @xmath51 . this is of course true only if the gauge configuration is locally smooth . otherwise the fermion mass acquires an additive renormalisation which completely destroys the above features important for the description of the zero mode zone . having a fermion action that is expected to describe the zero modes and their mixing reasonably well , we can discuss the main topic of the paper , namely , what part of qcd is reproduced by instantons alone . in this section i present quenched hadron spectroscopy results obtained with the @xmath60 clover fermion action . the gauge field backgrounds on which the spectroscopy is done are the ones used in ref . the starting point is a set of 28 @xmath54 su(2 ) configurations generated with a fixed point action . the lattice spacing is @xmath66fm , as fixed by the sommer parameter of the heavy quark potential . these monte carlo generated configurations are not yet suitable for our purposes since they are not locally `` smooth '' . both the instantons and the details of the zero mode zone are obscured by short distance fluctuations . cycling , a smoothing technique based on the renormalisation group , has been shown to preserve the main physical features of the gauge configurations . after 9 cycling steps the string tension @xcite and @xmath67 versus @xmath68 are essentially unchanged . moreover , these lattices are smooth enough that their instanton content can be unambiguously identified . in fact , about @xmath69 of the action is accounted for by instantons ( assuming there is no interaction between them ) . we can create artificial lattice configurations that have the same instanton content as the smoothed ones . the instanton sizes and locations are reproduced but the relative orientation in group space is distributed according to the su(2 ) haar measure . recently a study of the relative orientation appeared in @xcite , however reproducing the relative orientations would make our procedure much more cumbersome . i compare the spectroscopy done on the smoothed ( cycled ) lattices and the corresponding artificial instanton configurations with the same instanton content . these ensembles are both locally smooth enough that the optimised clover action should give a satisfactory description of their zero mode zone . in fig . [ fig : pirho ] the vector meson mass is plotted as a function of the pseudoscalar mass for both ensembles . the correlators were always fitted with the assumption that there is only one lightest particle dominating both the pseudoscalar and the vector channels . i also show the @xmath70 free field line . in fact , the wilson fermion action produces exactly degenerate free field like pions and rhos on the instanton backgrounds . compared to the wilson action , the clover action shows a marked improvement , but it still fails to reproduce the @xmath67 vs. @xmath68 observed on the smoothed lattices . [ fig : pi ] shows the pion mass sqared vs. the bare quark mass for both ensembles . on the smoothed ensemble @xmath71 , as expected from pcac . this is clearly not the case on the instanton ensemble , where a best fit to the form @xmath72 gives @xmath73 which is between @xmath74 and the free field value , @xmath75 . we also note that since both ensembles are locally very smooth , the additive mass renormalisations are very close to zero . therefore , it is also meaningful to compare hadron masses obtained at the same bare quark mass on the two gauge ensembles . this comparison reveals that the instanton ensemble typically produces much lighter hadrons than the smoothed ensemble . in spite of its substantially improved chiral properties , even the clover dirac operator fails to reproduce the correct physics of qcd on the instanton backgrounds . instead , it yields a peculiar mixture of qcd and free field theory . is there a problem with the instanton liquid model or the physics of these instanton lattices is still not close enough to the continuum ? to answer this question we look at the low lying eigenmodes of the dirac operator in more detail . in fig . [ fig : dspec ] i plot the lowest 30 eigenvalues in the complex plane for the instanton and the smoothed ensemble . the eigenvalues of all 28 configurations are superimposed . the boundary condition is antiperiodic in the time direction , periodic in all other directions . in both cases most of the eigenvalues lie very close to a circle passing through the origin . this is a sign of the approximate chiral symmetry of the action . the most striking difference between the two ensembles is the appearance of a gap at around im@xmath29=0.5 on the instanton ensemble . it is also instructive to compare the density of eigenvalues . since the low eigenvalues are almost imaginary i project them on the imaginary axis and plot the densities as a function of the imaginary part of the eigenvalues . this corresponds to the density of eigenvalues around zero in the continuum . in fig . [ fig : ev_hist1 ] we can compare the densities corresponding to the two ensembles . the real modes have been removed , these would show up as spikes at the origin . the densities at zero seem to agree quite well . according to the banks - casher relation the chiral condensate is proportional to the density of modes around zero @xcite ( excluding exact zero modes , which do not contribute in the infinite volume limit ) . this shows that on both the instanton and the smoothed ensemble chiral symmetry is broken and the value of the chiral condensate is approximately the same . apart from the vicinity of the origin , however , the two distributions are quite different . on the instanton ensemble there is a substantial `` piling up '' of modes above im(@xmath76 and then a `` thinning '' of modes at 0.5 . what is the reason of this huge difference between the two distributions ? the answer can be easily given by looking at the eigenvectors of the dirac operator . it turns out that the quark density of the eigenvectors on the instanton ensemble corresponding to the peak of the distribution are much more delocalised than all the other eigenvectors occurring in either ensemble . in fact , these modes are essentially free quark modes . this can be seen as follows . on a trivial gauge field configuration ( all links are 1 ) with periodic bondary conditions in all directions , there are @xmath77 trivial ( constant ) zero modes , this is the number of c - number degrees of freedom corresponding to one fermion flavour . in the presence of antiperiodic boundary condition in the time direction , the lowest eigenmodes are shifted away from zero , to @xmath78 since in our case @xmath79 . both eigenvalues are @xmath77-fold degenerate . the mixing of the free field modes into a particular eigenmode @xmath28 can be characterised by @xmath80 , where @xmath81 is the projection of the normalised eigenmode @xmath28 onto the eigenspace corresponding to the eigenvalue @xmath82 . this quantity is not gauge invariant therefore i work in landau gauge . any generic fermion mode on a non - trivial but locally smooth configuration will have a nonzero projection on the @xmath82 eigenspaces . the question is whether this is just an accidental mixing or the given ensemble really contains close to free field modes . to decide this , a good quantity to look at is @xmath83 . if the mixing is accidental , we expect @xmath84 to fluctuate around 1 , independently of the corresponding eigenvalue @xmath29 . on the other hand , if there are free field like modes on a given configuration then @xmath84 will increase substantially when @xmath29 approaches the free field eigenvalue @xmath85 . in fig . [ fig : proj ] i plotted @xmath84 versus the distance of the given eigenvalue from the free field mode @xmath85 . the figure shows a random selection of eigenvalues with imaginary parts between 0.0 and 0.2 . in the smoothed ensemble @xmath84 fluctuates around 1 everywhere , there is no trace of the free field modes . on the other hand , in the instanton ensemble , the modes close to @xmath85 have a substantially larger projection on the @xmath85 eigenspace than on the @xmath86 subspace . in fact , these modes , close to @xmath85 have @xmath87 , so they are essentially free field modes . a detailed study of the chiral density @xmath88 reveals that all the modes with @xmath89 look like mixtures of instanton zero modes with the density concentrated in several lumps . this is to be contrasted with the chiral density of the modes with @xmath90 that spreads almost homogenously over the whole lattice , as expected of the lowest free field eigenmodes . now we can understand why the instanton ensemble gave substantially smaller hadron masses than the smoothed ensemble . the reason is that the former had a large number of free quark modes . even if the zero mode zones of the two ensembles are similar , the free field modes provide a very efficient way to propagate quarks to large distances . this is why the instanton ensemble shows a peculiar mixture of qcd and free field characteristics . in the smoothed ensemble confinement completely eliminates the freely propagating quarks . according to the instanton liquid model , instantons alone , without confinement , can reproduce most of the properties of the light hadrons . our result shows that in the presence of instantons only , without confinement , free quark modes provide a very effective way to propagate quarks and they contaminate the hadron correlators producing unusually light hadrons . one possible way to suppress the free quark modes is to add to the instantons some locally very smooth background that contains only long wavelength fluctuations . there has to be enough long distance structure to suppress the free propagation of quarks but on the other hand they have to be locally smooth enough not to distort the instantons . cycling is a very efficient way to create such backgrounds . to this end i started with an ensemble of wilson @xmath91 ( @xmath92fm ) @xmath93 su(2 ) gauge configurations , performed 4 cycling steps and finally inverse blocked them to size @xmath54 . this produced a set of lattices extremely smooth on the few lattice spacing scale but containing the longest wavelength fluctuations characteristic to a @xmath94@xmath95 lattice . their spatial size was a bit smaller , their temporal size a bit larger than the confinement scale . moreover these configurations had a small enough physical size that they almost never contained instantons . after superimposing these lattices on the instanton configurations by simply multiplying the corresponding links ( in landau gauge ) , i checked that the original topological charge of the instanton configurations changed very little . to show how the addition of the smooth background affects the hadron correlators , in fig . [ fig : pirho_b2.4 ] i plotted typical pseudoscalar and vector correlators ( at @xmath96 ) . the change is dramatic . while on the instanton configurations the pion and the rho are very light ( crosses ) , the addition of the smooth background brings the correlators ( bursts ) very close to the physical ones obtained on the smoothed lattices ( boxes ) . the good agreement of the physical correlators and the instanton plus smooth background correlators persists in the whole range of masses where i tested it ( @xmath97 ) . it would be tempting to interpret this as a consequence of confinement ; the superimposed smooth backgrounds still contained the longest scale fluctuations of the confining lattices and these removed the free field modes that contaminated the spectroscopy before . but is confinement really needed for that ? this can be easily tested by replacing the superimposed backgrounds with ones of much smaller physical size . the lattices that i used for this purpose were generated in exactly the same way as before , except that the starting @xmath98 lattices were produced at wilson @xmath99 . these lattices have a tiny physical size , they are expected to be completely perturbative . in fig . [ fig : pirho_b3.0 ] i plotted again the pion and rho correlators at @xmath96 . quite surprisingly , the correlators have absolutely no dependence on the @xmath100 at which the background was created , they are the same on the instantons plus confining and instantons plus perturbative configurations . apparently , there is still something nontrivial on these physically very tiny perturbative lattices that has the same effect on the correlators as a confining background . what could this be ? the answer can be given by noting that the @xmath101 limit of yang - mills theory on the torus is not completely trivial because there is not only one gauge field configuration with zero action . the set of flat ( minimal action ) gauge field configurations can be characterised by the su(2 ) conjugacy classes represented by four mutually commuting gauge group elements corresponding to the ( constant ) polyakov loops in the four perpendicular directions . these are essentially zero momentum gauge field configurations , with a constant gauge potential ( link ) , sometimes referred to as torons . their contribution to physical quantities is a finite size effect , and has been estimated in ref . @xcite . to explicitly check that it is really the torons that affect the correlators so strongly , i extracted the `` toron content '' of the perturbative background fields . this was done in the following way . after landau gauge fixing i averaged the polyakov loops in all four directions . since the polyakov loops along any given direction were almost constant ( up to small perturbative fluctuations ) , the averages were close to being su(2 ) elements . i projected the averages back onto su(2 ) and then distributed the average polyakov loop evenly among the links in the given direction . this resulted in configurations with constant gauge fields ( links in any direction ) that carried the average polyakov loops of the original perturbative configurations . i shall refer to these lattices as `` toron '' configurations . we can now compare the correlators in the instantons plus toron background and the instantons plus the full perturbative background . the result again in the typical case of @xmath96 is shown in fig . [ fig : toron ] . the instantons and the torons together fully reproduce the correlators , demonstrating that the most important fluctuations on the perturbative lattices are indeed the torons . it is also quite instructive to compare the distribution of the lowest eigenvalues of the dirac operator on the different types of gauge configurations . [ fig : bc_hist ] shows the densities on the smoothed lattices ( solid lines ) , the perturbative ones ( dotted line ) and the lattices containing both the instantons and the perturbative backgrounds ( dashed line ) . the three distributions agree quite well away from zero . however around zero , the perturbative ensemble produces only a very low density ( compatible with zero ) , while the two other ensembles yield qualitatively similar densities , both nonzero . this demonstrates very clearly that it is the instantons that are responsible for creating a nonzero eigenvalue density around zero which through the banks - casher relation @xcite implies the spontaneous breaking of chiral symmetry . we also note that the peak corresponding to the free field modes ( see fig . [ fig : ev_hist1 ] ) has been completely eliminated by the perturbative backgrounds . this again can be attributed to the torons , as can be checked by looking at the eigenvalue distribution on the toron lattices ( not shown ) which is very similar to the dotted line in fig . [ fig : bc_hist ] . in the present paper i addressed the question of how much of qcd in the chiral limit is reproduced by restricting the fluctuations of the gauge field to instantons only . i reconstructed the instanton content of smoothed monte carlo lattice configurations and compared hadron spectroscopy on this instanton ensemble to the spectroscopy on the original `` physical '' smoothed configurations . the fermion action used for the spectroscopy was a `` chirally optimised '' clover action . i explicitly studied the fermion zero modes in the presence of an instanton and also the mixing of the zero modes corresponding to an instanton and an antiinstanton . i found that in these simple cases the optimised action described the continuum features of the zero mode zone rather well . in spite of this , the optimised action still failed to reproduce the physical hadron spectrum . it yielded anomalously light hadrons with a mixture of qcd and free - field like features . a closer look at the spectrum of the dirac operator revealed that on the instanton ensemble it had a large number of free quark modes . these were absent on the physical smoothed configurations . they provided a very efficient way of propagating quarks to large distances thereby substantially reduced the hadron masses and contaminated the spectroscopy . superimposing an ensemble of very smooth , essentially perturbative gauge field fluctuations on the instantons , was enough to eliminate the free quark modes and to restore the physical hadron correlators . by comparing the density of low eigenvalues of the dirac operator i also obtained direct evidence that it is the instantons that are responsible for creating a nonzero density of modes around zero and by the banks - casher relation also for chiral symmetry breaking . the important fluctuations on the superimposed perturbative lattices , needed to recover the physical hadron correlators , turned out to be zero momentum ( constant link or gauge potential ) configurations , torons . i would like to emphasise here that this toron ensemble was a perturbative one , produced at large @xmath100 i.e. small physical lattice size . the meson correlators appeared to be largely independent of the @xmath100 at which the superimposed background was created , as long as it was above the finite temperature phase transition on the given lattice size . it seems rather surprising that such `` mild '' gauge field fluctuations as the torons , can have so profound an impact on physical quantities . after all , torons exist only on the torus , they are absent in an infinite space - time and thus constitute only finite size effects . on the other hand we know that the instantons by themselves do not confine @xcite . this means that the instanton ensemble does not produce a mass gap and it is expected to exhibit much stronger finite size effects than qcd , from which the instantons have been extracted . in fact , the free field modes that are eliminated by the torons are also absent in the infinite volume limit , since they are not normalisable . in free field theory the density of modes around zero is proportional to the linear size of the box @xmath102 , whereas the density of instanton related modes grows proportionally to the volume . it is thus not inconceivable that in increasingly bigger volumes , although the torons have less and less influence , at the same time they become less and less needed since the relative importance of the free field modes also dies out . the torons might be needed only for compensating the unnaturally large finite size effects due to the absence of confinement in the instanton ensemble . what is the picture emerging from this lattice study concerning the instanton liquid model ? the qcd vacuum contains an ensemble of instantons that is generated by the non - perturbative gauge field dynamics . the instantons by themselves break chiral symmetry but hadron correlators in the instanton backgrounds are strongly contaminated by freely propagating quarks . this yields anomalously light mesons and a small splitting between the pseudoscalar and the vector channel . the free quarks can be eliminated and the physical hadron correlators can be restored by superimposing a perturbative ensemble of zero momentum gauge configurations ( torons ) on the instantons . this suggests that the contamination by free quarks is most likely a finite volume effect which , in qcd , is completely suppressed by confinement . more evidence in favour of this scenario could be gathered by studying the volume dependence of these effects on larger volumes . another important question , not answered by the present study is why it is exactly the perturbative toron ensemble that is needed to restore the physical meson correlators . it would be also very desirable to extend this investigation to a larger class of observables , in particular baryon correlators , for which the su(3 ) case need be considered . finally , for more precise tests a better chiral action , such as the overlap @xcite could be used . in the appendix , for reference , i collected some properties of wilson type lattice dirac operators that were used in the main body of the paper . some of these results can be found in the literature , in particular in refs . @xcite . i assume that the dirac operator satisfies @xmath31 hermiticity , i.e. @xmath103 let @xmath0 be a @xmath31-hermitian dirac operator with no degeneracy in its spectrum , i.e. having as many different eigenvalues as the dimension of the space it acts on . let @xmath29 , and @xmath49 be two eigenvalues of @xmath0 with the corresponding eigenvectors being @xmath28 and @xmath104 . then @xmath105 if and only if @xmath106 . _ proof : _ a simple consequence of @xmath31 hermiticity is that @xmath107 which implies @xmath108 it immediately follows that if @xmath109 then @xmath106 . to prove the converse we first note that since @xmath0 has as many different eigenvalues as the dimension of the space it acts on , its eigenvectors form a basis . we have already seen that the only eigenvector on which @xmath110 can have a nonzero projection , is @xmath111 . if this projection were also zero , then @xmath110 would be orthogonal to all the vectors in a complete set and consequently it would be zero . this is impossible since @xmath31 has a trivial kernel . for proving that @xmath112 it was essential that the eigenvalues of @xmath0 be non - degenerate and its eigenvectors form a complete set . generally this is expected to be the case unless the gauge field is fine tuned . there is however an important exception . when topology as seen by the fermions changes through a smooth deformation of the gauge fields , two opposite chirality real eigenvalues collide and leave the real axis as a complex conjugate pair . it can be shown that when the two eigenvalues coincide , the corresponding eigenspace is one - dimensional , the eigenvector has zero chirality , @xmath0 does not possess a complete set of eigenvectors and therefore has no spectral decomposition . a simple consequence of statement 1 is that if @xmath0 has a complete non - degenerate spectrum , then the chirality of an eigenvector , @xmath88 is nonzero if and only if the corresponding eigenvalue , @xmath29 , is real . in the remainder of the appendix i shall prove the following property of wilson type su(2 ) lattice dirac operators . let @xmath113 and @xmath114 be real eigenvectors of two ( not necessarily the same ) su(2 ) lattice dirac operators such that the corresponding eigenvalues and their complex conjugates are non - degenerate . it is then possible to choose the phases of @xmath113 and @xmath114 such @xmath115 , @xmath116 , @xmath117 , and @xmath118 are all real for any ( third ) lattice dirac operator @xmath0 . _ proof : _ besides @xmath31 hermiticity , this property depends on an additional symmetry of the dirac operator relating @xmath0 to its complex conjugate @xcite . this is specific to the su(2 ) case . let @xmath119 be the charge conjugation operator and @xmath120 , where @xmath121 is a pauli matrix acting in colour space . using that @xmath122 ( where the last equality follows because we work in a representation with all @xmath123 s hermitian ) , and that for any @xmath124 @xmath125 , it is not hard to prove that @xmath126 now if @xmath28 is an eigenvector of @xmath0 with eigenvalue @xmath29 then @xmath127 which means that @xmath128 is an eigenvector of @xmath0 with eigenvalue @xmath129 . due to the non - degeneracy of the eigenvalues and that @xmath130 , it follows that @xmath131 in the special case when @xmath29 is real , the phase choice @xmath132 eliminates the extra phase factor on the r.h.s . ( [ eq : phase ] ) . now writing @xmath133 , and making use of eqs . ( [ eq : dstar ] ) and ( [ eq : phase ] ) with the above phase choice , the non - diagonal matrix elements can be easily seen to be real . the same argument also shows that the diagonal matrix elements are automatically real , regardless of the phase choice . this completes the proof . we would like to note that while any wilson type dirac operator has a spectrum symmetric with respect to the real axis ( this follows from @xmath31 hermiticity ) , in general there is no simple relation between the eigenvectors corresponding to complex conjugate eigenvalues . the only exception is the su(2 ) case , when they are related by eq . ( [ eq : phase ] ) . i thank pierre van baal for discussions , tom degrand and anna hasenfratz for correspondence . the computations were performed using a computer code based on the milc @xcite code and the arpack @xcite package . i thank the milc collaboration and the authors of the arpack package for making their code publicly available . this work has been supported by fom . de forcrand , m. garca prez , and i .- o . stamatescu , nucl . phys . * b499 * ( 1997 ) 409 ; c. michael , and p.s . spencer , phys . * d52 * ( 1995 ) 4691 ; d.a . smith , and m.j . teper , phys . * d58 * ( 1998 ) 014505 ; b. alles , m. campostrini , a. di giacomo , y. gunduc , e. vicari , phys . * d48 * ( 1993 ) 2284 ; a. hasenfratz , and c. nieter , phys . * b439 * ( 1998 ) 366 . edwards , u.m . heller , j. kiskis , and r. narayanan , phys . * 82 * ( 1999 ) 4188 ; r.g . edwards , u.m . heller , j. kiskis , and r. narayanan , chiral condensate in the deconfined phase of quenched gauge theories , hep - lat/9919941 .

i address the question of how much of qcd in the chiral limit is reproduced by instantons . after reconstructing the instanton content of smoothed monte carlo lattice configurations , i compare hadron spectroscopy on this instanton ensemble to the spectroscopy on the original `` physical '' smoothed configurations using a chirally optimised clover fermion action . by studying the zero mode zone in simple instances i find that the optimised action gives a satisfactory description of it . through the banks - casher formula , instantons by themselves are shown to break chiral symmetry but hadron correlators on the instanton backgrounds are strongly influenced by free quark propagation . this results in unnaturally light hadrons and a small splitting between the vector and the pseudoscalar meson channels . superimposing a perturbative ensemble of zero momentum gauge field fluctuations ( torons ) on the instantons is found to be enough to eliminate the free quarks and restore the physical hadron correlators . i argue that the torons that are present only in finite volumes , are probably needed to compensate the unnaturally large finite size effects due to the lack of confinement in the instanton ensemble . = 1ex plus0.5ex minus0.2ex * instantons and chiral symmetry on the lattice * + * tams g. kovcs * + _ instituut - lorentz for theoretical physics , p.o.box 9506 + 2300 ra , leiden , the netherlands _ + e - mail : kovacs@lorentz.leidenuniv.nl +

the phenomenon of shear localization during high strain - rate deformations of metals @xcite is a striking instance of material instability in mechanics , that has attracted considerable attention in the mechanics literature ( _ e.g. _ @xcite ) as well as a number of mathematical studies ( @xcite and references therein ) . material instability is typically associated with ill - posedness of an underlying initial value problem , what has coined the term _ hadamard instability _ for its description in the mechanics literature . it should however be emphasized that while hadamard instability indicates the catastrophic growth of oscillations around a mean state , what is observed in localization is the orderly albeit extremely fast development of coherent structures : the shear bands . the latter is a nonlinear phenomenon and a linearized analysis can only serve as a precursor to such behavior . we work with the simplest model capturing the mechanism of shear band formation , a simple shear motion of a non - newtonian fluid with temperature decreasing viscosity @xmath0 , @xmath1 where @xmath2 describes the strain - rate sensitivity which typically in these problems satisfies @xmath3 . mathematical studies of , for power law viscosities @xmath4 , @xmath5 , show that the uniform shear is asymptotically stable for @xmath6 @xcite , but indicate unstable response in the complementary region @xmath7 , @xcite . the mathematical analysis has not so far characterized the precise behavior in the instability region ; however , detailed and resolved numerical studies ( _ e.g. _ @xcite ) show that instability is followed by formation of shear bands . the difficulty is that in order to analyze the unstable regime one needs to account for the combined effect of parabolic regularizations in conjunction to ill - posed initial value problems . an asymptotic theory for calculating an effective equation for the compound effect is proposed in @xcite . it is based on the theory of relaxation systems and the chapman - enskog expansion and produces an effective asymptotic equation . for a power law the effective equation changes type from forward to backward parabolic , across the threshold @xmath8 of the parameter space . the criteria of @xcite are based on asymptotic analysis and as such are formal in nature . our objective , here , is to develop a mathematical theory for the onset of localisation and the emergence of a nonlinear localized state , thus providing a rigorous justification to the behavior conjectured by the effective equation in @xcite . we employ the constitutive hypothesis of an _ exponential law _ for the viscosity coefficient , @xmath9 the model for the exponential law admits a special class of solutions describing uniform shearing @xmath10 the graph @xmath11 describing the average stress vs. average strain response is monotonically decreasing , and thus the deformation of uniform shear is characterized by strain softening response . we set the following goals : 1 . to study the linearized problem around the uniform shearing solution and to assess the effects of the various parameters of the problem 2 . to inquire whether there are special solutions of the nonlinear problem that indicate the emergence of localized deformations and formation of shear bands . since the base solution around which we linearize is time - dependent the linearized analysis necessarily involves analysis of non - autonomous systems and also perturbations have to be compared against the base solution . we perform a linearization using ideas of relative perturbations @xmath12 leading to linearized system @xmath13 with @xmath14 computed by @xmath15 . this problem turns out to have the following behavior : 1 . [ en : introa]for the case without thermal diffusion @xmath16 : + when @xmath17 the high frequency modes grow exponentially fast with the exponent of the order of the frequency what is characteristic of ill - posedness and _ hadamard instability_. + when @xmath2 the modes are still unstable but the rate of growth is bounded independently of the frequency . in fact , the behavior in that range is characteristic of _ turing instability_. 2 . the effect of thermal diffusion ( @xmath18 ) for the non - autonomous linearized model can also be assessed : perturbations do grow initially in the early stages of deformation , but over time the effect of the diffusion intensifies and eventually the system gets stabilized . for a quantitative description of the above qualitative statement we refer to lemmas [ lemdecay ] and [ lembound ] . in addition , this behavior can also occur for the full solution of the nonlinear problem subject to simple shear boundary conditions as indicated in numerical simulations presented in figure [ mtstb ] . this type of metastable response was already conjectured ( for a somewhat different model ) in numerical experiments of @xcite . the linearized analysis presented here offers a theoretical explanation of this interesting behavior . in the second part of this article we consider the adiabatic variant of the model with exponential constitutive law . the system is expressed for @xmath19 in the form @xmath20 and is now considered on the whole space @xmath21 . for the system , we construct a class of localizing solutions @xmath22 \sigma ( x , t ) & = \sigma_s ( t ) \frac{1}{\phi_\lambda ( t ) } \ , \sigma_\lambda \big ( \sqrt{\lambda } x \phi_\lambda ( t ) \big ) , \\[2pt ] \theta(x , t ) & = \theta_s(t ) + \lambda \tfrac{n+1}{\alpha } ( \theta_s(t ) - \theta_0 ) + \theta_\lambda \big ( \sqrt{\lambda } x \phi_\lambda ( t ) \big ) , \end{aligned}\ ] ] where @xmath23 depends on the parameter @xmath24 , and is determined via the solution @xmath25 of the singular initial value problem , with @xmath26 . the existence of the solution @xmath25 is established in theorem [ connection ] where most importantly its behavior at the origin and at infinity is made precise . the emerging solution depends on three parameters : @xmath27 linked to the uniform shear in , a parameter @xmath28 which can be viewed as a length scale of initial data , and a parameter @xmath29 determining the size of the initial profile . to our knowledge this is the first instance that the compound effect of hadamard instability and parabolic regularizations is mathematically analyzed in a nonlinear context . the reader should contrast the nonlinear response captured by the exact solution to the response of the associated linearized problem . the solutions indicate that the competition among hadamard instability and momentum diffusion in a nonlinear context can lead to a nonlinear coherent structure that localizes in a narrow band ( see theorem [ shearband ] and remark [ rmkbeh ] ) . by contrast , for a linear model , the combined effect of hadamard instability and parabolicity leads to unstable oscillatory modes growing in amplitude ( see ) . this justifies the scenario that nonlinearity can kill the oscillations caused by hadamard instability so that the non - uniformity of deformation produced by the instability merges into a single localized zone . it would be interesting to explore this type of response for other models related to material instability in solid mechanics . the exposition is organized as follows : section [ secdescription ] presents a description of the problem and the formulation via relative perturbations . the linearized analysis is presented in section [ stabar ] , first the part that can be done via spectral analysis for autonomous systems , and then the part concerning the effect of thermal diffusion , which is based on energy estimates for a non - autonomous linear system . detailed asymptotics for the eigenvalues and the connection with hadamard and turing instabilities are presented there . the presentation of the localized solutions is split in sections [ seclocal]-[seclocalii ] : section [ seclocal ] presents the invariance properties , the _ ansatz _ of localized solutions , and also introduces the relevant auxiliary problem , . the construction of solutions to the singular initial - value problem , and the proof of their properties is the centerpiece of our analysis and forms the objective of section [ secss ] . the construction is based on a series of steps , involving desingularization , a nonlinear transformation , and eventually the existence of a heteroclinic orbit via the poincar - bendixson theorem . the construction of the localized solutions is then effected in section [ seclocalii ] . in this work we study the onset of localization and formation of shear bands for the model describing shear motions of a non - newtonian fluid with viscosity @xmath30 decreasing with temperature . for concreteness we take a viscosity satisfying the exponential law and are interested in the range where the rate sensitivity @xmath31 is positive and small . the model then becomes @xmath32 and describes shear motions between two parallel plates located at @xmath33 and @xmath34 with @xmath35 the velocity in the shear direction , @xmath36 the shear stress , @xmath37 the temperature , and @xmath38 depicting the strain rate . the boundary conditions are taken @xmath39 equation reflects prescribed velocities at the boundaries , while manifests that the bounding plates are adiabatic . note that implies that the stress satisfies @xmath40 and that @xmath41 the initial conditions are @xmath42 throughout the study we will consider two cases : when @xmath43 the process is adiabatic and the resulting model will be the main object of study , as it is the simplest model proposed for studying the phenomenon of formation of shear bands . the case @xmath18 will also be studied in section [ stabar ] as a paradigm to assess the effect of thermal diffusion . the problem , , admits a class of special solutions describing uniform shear @xmath44 for the special case of the exponential law , @xmath45 and @xmath46 read @xmath47 these are universal solutions , _ i.e. _ they hold for all values of the parameters @xmath31 , @xmath48 , @xmath49 in the model including the limiting elliptic initial value problem when @xmath50 , @xmath43 . the graph @xmath51 is viewed as describing the stress vs. average strain response . as the graph is decreasing , it means that there is always softening response in the course of the deformation . the system has served as a paradigm for the mathematical study of the phenomenon of shear bands ( _ e.g. _ @xcite ) occurring during the high strain - rate deformations of metals . formation of shear bands is a well known material instability that usually leads to rupture , and has received considerable attention in the mechanics literature ( _ e.g. _ @xcite ) . the reader is referred to @xcite for a description of the problem intended for a mathematically oriented reader that is briefly also outlined below . it was recognized by zener and hollomon @xcite ( see also @xcite ) that the effect of the deformation speed is twofold : ( a ) to change the deformation conditions from isothermal to nearly adiabatic ; under nearly adiabatic conditions the combined effect of thermal softening and strain hardening tends to produce net softening response . ( b ) strain rate has an effect _ per se _ and induces momentum diffusion that needs to be included in the constitutive modeling . a theory of thermoviscoplasticity is commonly used to model shear bands and we refer to @xcite for various accounts of the modeling aspects . the basic mechanism for localization is encoded in the behavior of the following simple system of differential equations @xmath52 ( which is the adiabatic case @xmath43 of ) . this model is appropriate for a non - newtonian fluid with temperature dependent viscosity , and is also a simplification of the models proposed in @xcite to capture the mechanism for shear band formation in high strain - rate plastic deformations of metals . the argument - adapted to the language of a temperature dependent fluid - goes as follows : as the fluid gets sheared the dissipation produces heat and elevates the temperature of the fluid . the viscosity decreases with the elevation of the temperature and the fluid becomes softer and easier to shear . one eventuality is that the shear deformation distributes uniformly across the material in the same manner as it does for the usual newtonian fluids . this arrangement is described by the uniform shearing solution . a second mode of response is suggested when comparing the behavior of the fluid in two spots : one hot and one cold . since the viscosity is weaker in the hot spot and stronger in the cold spots , the amount of shear generated in the hot spot will be larger than the shear generated in the cold spot and in turn the temperature difference will be intensified . it is then conceivable that the deformation localizes into a narrow band where the material is considerably hotter ( and weaker ) than the surrounding environment . the two possibilities are depicted in fig . [ shearflow ] . the outcome of the competition of thermal softening and momentum diffusion is captured by the behavior of solutions to . a series of works analyze the system mostly for power law viscosities @xmath53 these studies show stable response @xcite when @xmath54 , and indicate unstable response @xcite but without characterizing the behavior in the complementary region @xmath55 . more precisely , in the unstable regime tzavaras @xcite shows that development of shear bands can be induced by stress boundary conditions leading to finite - time blow - up , and that their occurrence is associated with a collapse of stress diffusion across the band . on the other hand , bertsch , peletier and verduyn - lunel @xcite show that if the problem has global solutions and there is a sufficiently large initial temperature perturbation , then a shear band appears as the asymptotic in time state . careful and detailed numerical investigations @xcite indicate that instability is widespread in the parameter regime @xmath55 and is typically followed by the development of shear bands . the behavior in the instability domain " is at present poorly understood , especially regarding the precise mechanism of formation of the shear band . the reason is the response in this regime results from the competition between an ill - posed problem and a parabolic regularization , and little mathematical theory is available to study such effects . nevertheless , an asymptotic theory similar to the chapman - enskog expansion is used by katsaounis - tzavaras @xcite to calculate an effective equation for the compound effect of thermal softening and strain - rate sensitivity . the effective equation changes type from forward to backward parabolic across the instability threshold , and thus provides an asymptotic criterion for the onset of localization . in the present work , we will develop a mathematical framework for studying the competition and for examining the conjecture of @xcite . our analysis is facilitated by the use of the exponential law inducing special scaling properties on the resulting system of partial differential equations . in the mechanics literature shear localization is typically associated with hadamard - instability . this is easily put on display for , by considering the limiting case @xmath50 and @xmath43 . in the parlance of mechanics , this corresponds to rate - insensitive deformations and absence of heat conduction , and leads to the system of conservation laws @xmath56 for @xmath57 the wave speeds are both imaginary , @xmath58 , and the linearized initial value problem for is ill - posed . two questions then emerge : 1 . to assess the combined effect of the competition between hadamard instability and momentum diffusion and/or heat conduction ( as manifested by small @xmath2 and @xmath18 respectively ) . while hadamard instability indicates the catastrophic growth of oscillations around a mean state , it does not by itself explain the formation of coherent structures typically observed in localization . the latter is an outcome of the nonlinear effects and has to be assessed at the level of the nonlinear problem . the first objective is to develop quantitative criteria for the stability of the uniform shearing solutions . since the basic solutions are time dependent an issue arises how to define their stability . following @xcite , we call the uniform shearing solution _ asymptotically stable _ if perturbations of the solution decay faster than the basic solution , and we call it _ unstable _ if they grow faster than the uniform shearing solution . it is expedient to introduce a relative perturbation formulation of the problem : for the _ exponential law _ , we introduce the state @xmath59 describing the relative perturbation and defined through the transformation @xmath60 where @xmath61 is given by and satisfies in particular that @xmath62 using and we derive the system satisfied by the relative perturbation @xmath59 in the form @xmath63 \theta_t - \kappa \theta_{xx } & = \sigma_s ( t ) \ , \big ( \sigma \ , u -1 \big ) , \\[2pt ] \sigma & = e^{-\alpha \theta } u^n . \end{aligned}\ ] ] note that the uniform shearing solution is mapped to the state @xmath64 , @xmath65 and @xmath66 , and the latter is an equilibrium for the system . the problem then becomes to study the stability of the equilibrium @xmath67 for the non - autonomous system . in this section we present an analysis of the linearized system . since the uniform shearing solution is time - dependent the linearized analysis will seek to classify the precise growth in - time of perturbations . we compute the linearized problem , using the ansatz @xmath68 where @xmath69 is a small parameter measuring the size of the perturbation and @xmath70 . expansion of the equation @xmath71 gives to the leading order of @xmath69 @xmath72 while linearizing the other two equations gives @xmath73 collecting the equations together and neglecting the terms of @xmath74 we obtain the linearized system : @xmath75 { \bar \theta}_t - \kappa { \bar \theta}_{xx } & = \sigma_s ( t ) \ , \big ( ( n+1 ) { \bar u}- \alpha { \bar \theta}\big ) , \\[2pt ] { \bar \sigma}&= n { \bar u}- \alpha { \bar \theta}. \end{aligned } \right .\ ] ] the boundary conditions and and the transformations and , imply that the induced boundary conditions for the linearized system are @xmath76 in view of the transformations and , the relation between the original solution @xmath77 and the solution of the linear problem is as follows @xmath78 our analysis of the linearized equation is facilitated by introducing the transformation of variables @xmath79 where @xmath80 is a rescaling of time , defined by @xmath81 the map @xmath80 is invertible and its inverse is given by @xmath82 one computes that @xmath83 satisfy the linear system @xmath84 \frac{{\partial}{\hat \theta}}{{\partial}\tau } - \kappa c_0 e^{\alpha \tau } { \hat \theta}_{xx } & = ( n+1 ) { \hat u}- \alpha { \hat \theta } , \\[2pt ] { \hat \sigma}&= n { \hat u}- \alpha { \hat \theta } , \end{aligned } \right .\ ] ] with boundary conditions @xmath85 for further reference , we record that the original variables @xmath86 and the solution @xmath87 of the rescaled problem - are related through the formulas @xmath88 we next consider the system @xmath89 \frac{{\partial}{\hat \theta}}{{\partial}\tau } - k { \hat \theta}_{xx } & = ( n+1 ) { \hat u}- \alpha { \hat \theta } , \end{aligned } \right .\ ] ] with boundary conditions . note that in the thermal diffusion coefficient @xmath90 is time dependent , while in this coefficient is frozen in time . we analyze the solutions of via an eigenvalue analysis . this provides the following information : 1 . for @xmath91 it provides the stability or instability properties of the linearized system . 2 . for @xmath92 the eigenvalue analysis will only offer an indication of the effect of thermal diffusion on the linearized analysis . this will be complemented by an energy estimate analysis of the exact system in the following section . the solutions of - are expressed as a cosine fourier series @xmath93 where the fourier coefficients @xmath94 satisfy the ordinary differential equations @xmath95 the eigenvalues of the matrix @xmath96 are the roots of the binomial equation @xmath97 * the mode @xmath98*. for @xmath99 the equation becomes @xmath100 and has two eigenvalues @xmath101 and @xmath102 . this corresponds to marginal stability for the linearized system . however , any instability associated with the eigenvalue @xmath101 is eliminated by , which imply for the perturbation @xmath103 in the condition @xmath104 . indeed , the system for @xmath105 takes the form @xmath106 this implies , using , , and the boundary condition , that @xmath107 and clearly @xmath108 decays . thus , the @xmath109-th mode decays . * the modes @xmath110*. for these modes the eigenvalues are @xmath111 where the discriminant @xmath112 since @xmath113 we conclude that @xmath114 in summary : 1 . if @xmath115 then all modes @xmath116 are unstable . 2 . if @xmath92 the first few modes might be unstable , that is the modes @xmath117 for which @xmath118 , but the high frequency modes are stable . 3 . the higher @xmath119 the fewer the number of unstable modes . for @xmath119 larger than a certain threshold ( here equal to @xmath120 ) all modes are asymptotically stable . we analyze the behavior of the linearized modes as a function of @xmath117 and discuss the nature of the instability in the various special cases of interest . the analysis reveals the role of the parameters @xmath121 regarding the nature of the instability . in this regime @xmath122 one eigenvalue is negative and the other strictly positive and increasing with @xmath117 . using the taylor series development @xmath123 we obtain the following asymptotic developments for the eigenvalues @xmath124 referring to the formulas , and , we see that the @xmath117-th mode of the perturbation grows to the leading order like @xmath125 clearly the perturbation grows much faster than the uniform shearing solution @xmath126 given by where the growth is logarithmic , and the behavior is characteristic of _ hadamard instability _ , that is the high frequency oscillations grow at a catastrophic rate . in this case the eigenvalues of the @xmath117-th mode are given by the formulas @xmath127 again we have one negative and one positive eigenvalue and thus unstable response in this regime as well . using once again the taylor expansion we obtain the asymptotic expansions @xmath128 for the positive and negative eigenvalues respectively . the positive eigenvalue @xmath129 satisfies the following properties : @xmath130 set @xmath131 and recall that @xmath132 satisfies the identity @xmath133 differentiating , we derive @xmath134 next , using the elementary bound @xmath135 for @xmath136 we obtain @xmath137 _ + ( x ) & = - + ( n x + ) < < . @xmath137 this shows that @xmath138 and provides and . then follows from . the following remarks are in order : observe first that , by virtue of , and , the @xmath117-th mode of the perturbation grows like @xmath139 clearly the perturbation grows much faster than the uniform shearing solution @xmath126 , but the rate of growth is bounded in @xmath117 . on the other hand , the growth becomes very fast as the rate sensitivity @xmath140 . the behavior in this regime is characterized by _ turing instability_. recall , that turing instability @xcite corresponds to the following scenario that occurs in some instances of dynamical systems . there are two dynamical systems @xmath141 where the spectrum of both matrices @xmath142 and @xmath143 lies in the left - hand plane , but the trotter product of the two systems @xmath144 exhibits unstable response . indeed , the dynamical system with @xmath115 may be visualized as the trotter product of the two systems @xmath145 and @xmath146 each of and has spectrum in the left half space but the eigenvalues of have unstable response . in this regime the eigenvalues are given by and they are real and have the properties @xmath147 therefore , the high modes are stable , but the low modes might be unstable if @xmath119 is sufficiently small . the most unstable " mode is the first mode . one can again carry out the large @xmath117 asymptotics of the eigenvalues . the result , in the case that @xmath148 , reads @xmath149 a similar formula holds when @xmath150 . consider now the linearized system with boundary condition @xmath151 . since we are interested in perturbations of the uniform shearing solution we will consider initial data that satisfy @xmath152 and thus focus on solutions with @xmath153 the objective is now to analyze the effect of the time - dependent diffusion on the dynamics of the system . the linearized analysis of the system with frozen coefficients suggests to expect that diffusion stabilizes the system and its effect is intensified with the passage of time . indeed , here we provide a rigorous proof of this statement via energy estimates . [ lemdecay ] there exist @xmath154 and @xmath155 such that @xmath156 and @xmath157 we multiply @xmath158 by @xmath159 and @xmath160 by @xmath161 and obtain the identities @xmath162 using cauchy - schwarz inequality , gives @xmath163 in view of , we may use the poincare inequality @xmath164 combining the above , we note that for @xmath142 a positive constant we have @xmath165 next we select @xmath142 such that @xmath166 , and deduce that for @xmath142 thus selected there is a choice @xmath167 such that @xmath168 select now @xmath169 such that @xmath170 for @xmath171 ( recall from that @xmath172 when @xmath173 ) . on the interval @xmath171 , we have the differential inequality @xmath174 the latter implies and yields , upon using gronwall s inequality and , @xmath175 and concludes the proof the next lemma provides control of the intermediate times . [ lembound ] there exist @xmath176 and a constant @xmath177 such that @xmath178 using again and we have for any @xmath179 @xmath180 select now @xmath143 large so that @xmath181 for all @xmath182 , and then we obtain @xmath183 from where follows via gronwall s inequality . we conclude now the analysis of stability . suppose that for some @xmath184 the initial data satisfy @xmath185 and are controlled in @xmath186 by @xmath187 then , imply @xmath188 together they imply that the state @xmath189 is stable in @xmath186 , and in fact on account of it is asymptotically stable . we state the result in the following : for the initial - boundary value problem , with @xmath18 , if the initial perturbation @xmath190 satisfies @xmath191 then the equilibrium @xmath189 is asymptotically stable in @xmath186 . the proof of the above lemmas and the analysis of the linearized problem with frozen coefficients indicate that the perturbation of the base solution may drift far from the origin until eventually it is recaptured as the effect of the diffusion intensifies with time . this phenomenon of _ metastability _ also appears in numerical experiments for the nonlinear problem ; an instance of these experiments appears in figure [ mtstb ] . we solve numerically the rescaled system for @xmath192 . the initial conditions are perturbations of the uniform shearing solutions and they are : @xmath193 and @xmath194 , where @xmath195 is a small gaussian perturbation centered at the middle of the interval . the numerical method is based on adaptivity in space and time . an adaptive finite element method is used for the spatial discretisation while a runge - kutta method with variable time step is used for the discretisation in time . further details concerning the numerical scheme and the adaptivity criteria can be found in @xcite . in figure [ mtstb ] the evolution in time ( in natural logarithmic scale ) is presented for the original variables @xmath35 and @xmath37 in ; @xmath37 is sketched in natural logarithmic scale . we notice that , after a transient period of localization , thermal diffusion dominates the process , inhomogeneities of the state variables dissipate , and eventually the solution approaches a uniform shearing solution . ( left ) , @xmath196 ( right).,title="fig:",width=302,height=226 ] ( left ) , @xmath196 ( right).,title="fig:",width=302,height=226 ] we consider next the problem on the whole real line and put aside the boundary conditions . the objective is to construct special solutions that capture a process of localization . this is done over three sections . the present one is preliminary in nature and discusses various properties that make the construction feasible and the setting of the problem . the main step is performed in section [ secss ] where a special class of self - similar solutions are constructed . the localized solutions are then presented in section [ seclocalii ] . we consider on @xmath197 and recall the notation @xmath198 . the uniform shearing solutions are defined by solving . introduce the transformation @xmath199 which involves a passage to relative perturbations ( as in section [ secrelpert ] ) and a change of time scale from the original to a rescaled time @xmath80 . the scaling @xmath80 is precisely the one used in the linearized analysis of sections [ secchatime ] and [ seclinfroz ] ; it is defined by @xmath200 it satisfies @xmath201 where @xmath202 is as in , @xmath203 . the scaling @xmath204 satisfies @xmath205 as @xmath206 , and @xmath80 is invertible with inverse map given by @xmath82 the functions @xmath207 satisfy the system of partial differential equations @xmath208 { \hat \theta}_\tau - \kappa c_0 e^{\alpha \tau } { \hat \theta}_{xx } & = \big ( { \hat \sigma}\ , { \hat u}-1 \big ) , \\[2pt ] { \hat \sigma}&= e^{-\alpha { \hat \theta } } { \hat u}^n . \end{aligned}\ ] ] we emphasize that * for @xmath18 the system is non - autonomous , which is natural in view of the nature of the transformation . * notably , for @xmath43 the system becomes autonomous . next , we consider the adiabatic case @xmath43 where takes the form @xmath209 { \hat \theta}_\tau & = \big ( { \hat \sigma}\ , { \hat u}-1 \big ) , \\[2pt ] { \hat \sigma}&= e^{-\alpha { \hat \theta } } { \hat u}^n \ , . \end{aligned}\ ] ] to understand the long time response of we perform a formal asymptotic analysis following ideas from @xcite . fix an observational time scale @xmath169 and introduce the change of variables @xmath210 one easily checks that @xmath211 satisfies the system @xmath212 \tfrac{1}{t } \ , { \tilde \theta}_s & = \big ( { \tilde \sigma}\ , { \tilde u}-1 \big ) , \\[2pt ] { \tilde \sigma}&= e^{-\alpha { \tilde \theta } } { \tilde u}^n . \end{aligned}\ ] ] we are interested in calculating the equation describing the effective response of for @xmath169 sufficiently large . this asymptotic analysis involves a parabolic equation @xmath158 , which can be thought as a moment equation , and equations @xmath160 and @xmath71 , which may be thought as describing a relaxation process towards the equilibrium curve described by @xmath213 note that the equilibria are a one - dimensional curve parametrized by @xmath214 . to calculate the effective equation we perform a procedure analogous to the chapman - enskog expansion of kinetic theory . we refer to ( * ? ? ? * sec 5 ) for the details of the asymptotic analysis . the procedure produces the following effective equation @xmath215 which captures the effective response up to order @xmath216 . the leading term is backward parabolic and the first order correction is fourth order . one checks that the effect of the fourth order term is to stabilize the equation , in the sense that the linearized equation around the equilibrium @xmath217 is stable ( see @xcite for the details ) . it was further shown in @xcite that admits special solutions that exhibit localization , which is not surprising as the leading response in is backward parabolic . localized solutions of a similar nature will be constructed here directly for the system . this is a more unexpected " behavior , because is at least _ pro - forma _ a hyperbolic - parabolic " system , and the explanation for this behavior lies precisely in the asymptotic relation between and . it is easy to check that for @xmath43 the system is invariant under the one - parameter scaling transformation @xmath218 for any @xmath219 . it is also easy to check that this is the only scaling transformation under which the problem is invariant . this transformation has the distinctive feature that the scaling of space is independent of the scaling of time . finally , the system is not scaling invariant when the thermal diffusion @xmath49 is positive . the scaling invariance motivates a change of variables and a related _ ansatz _ of solutions . introduce a variable in time scaling @xmath220 , the variable @xmath221 , and the change of variables @xmath222 we denote by @xmath223 and by @xmath224 . introducing to , we obtain that the function @xmath225 satisfies the system of partial differential equations @xmath226 { \bar \theta}_\tau - \frac{\dot r}{r } \left ( \tfrac{n+1}{\alpha } + \xi { \bar \theta}_{\xi } \right ) , - \kappa c_0 \frac{e^{\alpha \tau}}{r^2 ( \tau ) } { \bar \theta}_{\xi \xi } & = \big ( { \bar \sigma}\ , { \bar u}-1 \big ) , \\[2pt ] { \bar \sigma}&= e^{-\alpha { \bar \theta } } { \bar u}^n . \end{aligned}\ ] ] the resulting system depends on the choice of the scaling function @xmath227 . we will be interested in solutions that are steady states in the new variables . of course such solutions are dynamically evolving in the original variables . there are two cases that produce simplified results of interest : 1 . consider the case @xmath18 , and select @xmath228 . then the system is consistent with the ansatz of steady state solutions @xmath229 and the latter are constructed by solving the system of ordinary differential equations @xmath230 -\frac{\alpha}{2 } \left ( \tfrac{n+1}{\alpha } + \xi { \bar \theta } ' \right ) - \kappa c_0 { \bar \theta } '' & = { \bar \sigma}\ , { \bar u}-1 , \\[2pt ] { \bar \sigma}&= e^{-\alpha { \bar \theta } } { \bar u}^n . \end{aligned}\ ] ] + since @xmath231 as @xmath232 increases , solutions of will produce defocusing solutions for . a study of is expected to give information about the convergence to the uniform shearing state in the non adiabatic setting . this is not the focus of the present work and we will not deal further with the system . 2 . instead we consider the case @xmath43 . we may now select any exponential function for @xmath227 but are most interested in @xmath233 with @xmath24 as this produces focusing solutions . with this selection , @xmath229 is a steady solution for with @xmath43 provided it satisfies @xmath234 \lambda \left ( \tfrac{n+1}{\alpha } + \xi { \bar \theta } ' \right ) & = { \bar \sigma}\ , { \bar u}-1 , \\[2pt ] { \bar \sigma}&= e^{-\alpha { \bar \theta } } { \bar u}^n , \end{aligned}\ ] ] where @xmath24 is a parameter . note that : * the form of the transformation is motivated by the scaling invariance . * the scaling connects the two variables and produces interesting systems even in the case @xmath49 positive when the system is not scale invariant . we are ready to state and prove the main result of the paper . this concerns the adiabatic case @xmath378 of system , taken now on the whole real line @xmath379 and expressed in terms of the variable @xmath19 in the form : @xmath380 our analysis is based on the transformations of section [ seclocal ] and the self - similar solution constructed in section [ secss ] . [ shearband ] let @xmath381 and @xmath247 be given . for any @xmath24 , the system admits a special class of solutions describing localization of the form : @xmath382 \label{locsigma } \sigma ( x , t ) & = \sigma_s ( t ) \left ( \tfrac{\alpha}{c_0 } t + 1 \right ) ^ { - \tfrac{\lambda}{\alpha } } \sigma_\lambda \left ( \sqrt{\lambda } x \left ( \tfrac{\alpha}{c_0 } t + 1 \right ) ^{\tfrac{\lambda}{\alpha } } \right ) , \\[2pt ] \label{loctheta } \theta(x , t ) & = \left ( 1 + \lambda \tfrac{n+1}{\alpha } \right ) \theta_s ( t ) - \lambda \tfrac{n+1}{\alpha } \theta_0 + \theta_\lambda \left ( \sqrt{\lambda } x \left ( \tfrac{\alpha}{c_0 } t + 1 \right ) ^{\tfrac{\lambda}{\alpha } } \right ) , \end{aligned}\ ] ] where @xmath46 , @xmath45 are the uniform shear solutions in , @xmath203 , while @xmath383 is an even function defined on @xmath384 , which solves the initial value problem , with @xmath385 for @xmath386 . [ rmkbeh ] 1 . 2 . because @xmath387 goes to infinity in time , the level curves of @xmath388 accumulate along the @xmath389 axis in the @xmath390 plane . the functions , , thus describe a localizing solution where the flow tends to concentrate around @xmath33 as the time proceeds . the solution , , is defined on @xmath391 and emanates from initial data @xmath392 \sigma ( x,0 ) & = \sigma_s ( 0 ) \sigma_\lambda \big ( \sqrt{\lambda } x \big ) , \\[2pt ] \theta(x,0 ) & = \theta_s ( 0 ) + \theta_\lambda \big ( \sqrt{\lambda } x \big ) \ , . \end{aligned}\ ] ] the initial data depend on two parameters : the parameter @xmath28 which can be thought as a length scale in initial data , and @xmath29 in which describes the amplitude of the initial perturbation . this parametric family of solutions is valid for any @xmath24 . if @xmath28 is large then the length scale of the initial perturbation is short while the growth in time and the rate of localization for the solution become fast . we introduce the changes of variables and the stationary variant of , which once combined together reads @xmath393 we select @xmath394 with @xmath24 . according to the analysis in section [ secinvariance ] the function @xmath395 is sought as a solution of defined on the interval @xmath396 . the desired @xmath395 is constructed as follows . first , observe the relation between the system and the system . let @xmath29 be fixed , and let @xmath397 be the solution of with @xmath26 and , defined for @xmath398 . this solution is well defined by theorem [ connection ] . using a direct computation , the smoothness properties in ( iii ) of theorem [ connection ] , and we conclude that 1 . the function @xmath399 defined by @xmath400 satisfies on @xmath401 and the condition @xmath402 . if we extend @xmath399 on @xmath403 by setting @xmath404 for @xmath405 , the resulting function @xmath406 , @xmath407 is a solution of defined on the whole real line that is even . noting that @xmath408 we combine with and to arrive at , , . in figure [ uqsol ] the profiles of @xmath409 and @xmath410 in and , respectively , are drawn at various instances of time from @xmath411 to @xmath412 . to achieve these profiles , we construct numerically the heteroclinic orbit for whose existence is justified in proposition [ thode ] . the construction is simple , if one exploits the special properties of system , and proceeds as follows : initial conditions @xmath413 are selected close to the equilibrium @xmath303 inside the region @xmath311 and in the direction of the stable manifold of @xmath284 for the linearized problem . then is solved backward in time , using an ode solver , and since @xmath414 is an unstable node ( for the forward problem ) the resulting orbit is a good approximation of the heteroclinic . for the integration of we use an explicit runge kutta predictor - corrector method of order ( 4,5 ) with adaptive time stepping . then @xmath29 is selected by choosing an appropriate parametrisation @xmath344 following the construction in . , and @xmath410 , .,title="fig:",width=302,height=226 ] , and @xmath410 , .,title="fig:",width=302,height=226 ] for the numerical runs the constitutive parameters are @xmath415 , @xmath416 . the parameters related to the solution are @xmath417 , @xmath418 while for the selected reparametrization of the heteroclinic orbit the value of @xmath419 . once the heteroclinic has been constructed , the profiles @xmath420 are computed via the changes of variables and . finally , @xmath409 and @xmath410 are computed by , respectively with @xmath417 , and are presented in fig [ uqsol ] . in this section we consider the system of ordinary differential equations @xmath235 \nu \left ( \tfrac{n+1}{\alpha } + \xi \theta ' \right ) & = \sigma \ , u -1 , \\[2pt ] \sigma & = e^{-\alpha \theta } u^n \ , , \end{aligned}\ ] ] on the domain @xmath236 with @xmath237 , subject to the initial condition @xmath238 this is an auxiliary problem that will be studied in detail in this section . we emphasize that it does not coincide with the integrated form of , however it will be used together with additional properties in order to construct solutions of and localizing solutions for in the following section . in this section , we are interested in constructing solutions @xmath239 for the singular initial value problem , for any @xmath237 . the following elementary remarks are in order : 1 . when @xmath240 , simplifies to @xmath241 \sigma ( 0 ) & = \sigma_0 \end{aligned } \right . \left \ { \begin{aligned } \sigma \ , u & = 1 \\[2pt ] \theta & = \frac{n+1}{\alpha } \log u \end{aligned } \right . .\ ] ] the solution of is computed explicitly by @xmath242 2 . when @xmath243 , the system is invariant under the family of scaling transformations @xmath244 for @xmath245 . there is a special solution of that is self - similar with respect to the scaling invariance and reads @xmath246 this solution does not however satisfy the initial condition . we next construct a function @xmath239 which satisfies , for all values of @xmath237 and @xmath247 . we will show that the constructed solution will have the inner behavior of and the outer behavior of . [ connection ] let @xmath247 be fixed . let us note @xmath248 . given @xmath237 there exists a solution @xmath249 of , defined for @xmath250 and satisfying the properties : 1 . it achieves the initial data @xmath251 where @xmath29 as in and @xmath252 , @xmath253 are defined via @xmath254 2 . as @xmath255 it has the limiting behavior @xmath256 \theta ( \xi ) & = - \tfrac{n+1}{\alpha } \log \xi + o(1 ) . \end{aligned}\ ] ] 3 . the solution has the regularity @xmath257 , @xmath258 , @xmath259 . it satisfies @xmath260 and has the taylor expansion as @xmath261 , @xmath262 where as usual @xmath263 ( but the rate may degenerate as @xmath264 tends to @xmath109 ) . _ step 1 . _ motivated by the special solution we introduce a change of variables in two steps : @xmath265 \theta ( \xi ) & = - \tfrac{n+1}{\alpha } \log \xi + { \bar \theta } ( \xi ) & & = - \tfrac{n+1}{\alpha } \log \xi + \theta ( \log \xi ) , \nonumber\end{aligned}\ ] ] indeed , we easily check that @xmath266 satisfies the system of equations @xmath267 \nu \xi \ , { \bar \theta } ' & = { \bar \sigma } \ , { \bar u } -1 , \\[2pt ] { \bar \sigma } & = e^{-\alpha { \bar \theta } } { \bar u}^n \ , . \end{aligned}\ ] ] in the second step , we introduce in the change of independent variable @xmath268 and derive that @xmath269 satisfy @xmath270 \nu \frac{d\theta}{d\eta } & = \sigma \ , u -1 , \\[2pt ] \sigma & = e^{-\alpha \theta } u^n \ , . \end{aligned}\ ] ] the system is far simpler than the original in that it is autonomous and non singular . we next consider and perform a further simplification , via the change of dependent variables @xmath271 note that @xmath272 . we now compute @xmath273 and @xmath274 in summary , we conclude that @xmath275 satisfies the autonomous system of ordinary differential equations @xmath276 where @xmath277 there are two advantages in relative to : 1 . is expressed via only two independent variables @xmath278 . the equilibrium at infinity in is pulled at the axis for via the transformation . we perform a phase space analysis of and establish the existence of a heteroclinic connection . we show : [ thode ] consider the autonomous planar system of differential equations with @xmath279 defined by . for any @xmath237 the system has the following properties : * @xmath280 has two equilibria @xmath281 , @xmath282 in the first quadrant . both @xmath283 and @xmath284 are hyperbolic points , @xmath283 is a repelling node while @xmath284 is a saddle point . * there exists a heteroclinic orbit connecting @xmath283 to @xmath284 . _ proof of proposition [ thode ] . _ 1 . the system has the special solution @xmath285 this solution splits the phase portrait into the left and right plane and acts as a barrier from crossing from the one side to the other . the equilibria of are computed by solving @xmath286 and they are @xmath287 and @xmath288 and @xmath289 . focusing on the behavior in the first quadrant we neglect the equilibrium @xmath290 . a computation shows @xmath291 -2 \tfrac{n+1}{n } a & \tfrac{\alpha}{n\nu } c_\nu \end{pmatrix } .\ ] ] at the equilibrium @xmath283 we have @xmath292 and we have eigenvalues @xmath293 with eigenvector @xmath294 , and @xmath295 with eigenvector @xmath296 . in view of , we have @xmath297 . the local behavior of solutions of around the equilibrium @xmath283 is that of a repelling node . standard phase plane theory for second order systems ( _ e.g. _ @xcite ) states that there is a small neighbourhood of the equilibrium @xmath283 such that for any given solution @xmath298 of there are constants @xmath299 , @xmath300 such that @xmath301 as @xmath302 . 4 . at the point @xmath303 equation gives @xmath304 the eigenvalues @xmath305 , computed by @xmath306 are @xmath307 with eigenvector @xmath308 and @xmath309 with eigenvector @xmath310 . @xmath284 is a saddle point . 5 . consider the region @xmath311 bounded by the curves * @xmath312 is the @xmath313-axis defined by the equation @xmath314 , * @xmath315 is the horizontal line @xmath316 , * @xmath317 is the parabola @xmath318 . + the region @xmath311 is expressed as @xmath319 see figure [ fig : phase2 ] . + + we will show that @xmath311 is negatively invariant . + indeed , the flow of is parallel to @xmath312 as indicated in ( b ) . on the horizontal line @xmath320 the system gives @xmath321 on the curve @xmath322 the system gives @xmath323 therefore @xmath311 is negatively invariant . 6 . for completeness we also show that the vector @xmath324 associated to the eigenvalue @xmath325 at @xmath284 points towards the interior of the region @xmath311 . recall that @xmath284 is the upper right corner of the region @xmath311 and that * the tangent vector to @xmath315 is @xmath326 , * the tangent vector to @xmath317 at @xmath303 is @xmath327 , * the eigenvector @xmath328 . + we will show that @xmath329 this then implies that the vector @xmath330 when placed at the corner @xmath284 points strictly towards the interior of @xmath311 . to show recall that @xmath307 and use and to rewrite @xmath331 where @xmath332 this shows @xmath333 and concludes . it is easy to check that there can be no closed limit cycle in @xmath311 because in that region @xmath334 the poincar - bendixson theorem implies that there exists a heteroclinic orbit joining @xmath335 and @xmath336 which emanates along the direction of the stable manifold at @xmath336 and progresses backwards in time towards the node at @xmath335 . we turn now to defining the solution of , . let @xmath337 stand for a parametrization of the heteroclinic orbit . in view of and the fact that @xmath338 , there exists a constant @xmath299 such that @xmath339 due to the fact that the vertical axis is a trajectory of its own and thus distinct from the trajectory we are considering , and due to the form of the phase portrait we can assert that @xmath340 . for any @xmath341 the function @xmath342 offers an alternative parametrization of the same heteroclinic orbit . applying to this parametrization yields @xmath343 choose @xmath344 such that @xmath345 and let @xmath298 denote the corresponding parametrization of the heteroclinic : @xmath346 the selected parametrization satisfies as @xmath302 @xmath347 where as usual @xmath348 as @xmath302 . recalling the changes of variables and we define @xmath349 the function clearly satisfies the system and , since @xmath350 as @xmath351 , it satisfies @xmath352 \theta ( \xi ) & = - \tfrac{n+1}{\alpha } \log \xi + o(1 ) , \end{aligned}\ ] ] as @xmath353 goes to infinity , which shows . in the last step , we study the behavior near the ( apparent ) singular point at @xmath354 and prove that the solution is in fact regular there and satisfies , and . clearly and @xmath160 imply that @xmath355 as @xmath261 . let @xmath252 and @xmath253 be defined by . since @xmath356 , gives @xmath357 now , , , imply @xmath358 and together with and provide . observe now that @xmath359 using , and , we obtain @xmath360 \\ & = \tfrac{1}{n } u^3 b - u \tfrac{\alpha}{\nu n } \frac{c_\nu}{b } \big ( b - \frac{1}{c_\nu } \big ) \\ & = \tfrac{1}{n } \tfrac{u_0 ^ 3}{c_\nu } e^{3 \eta}(1 + o(1 ) ) - \tfrac{\alpha}{\nu n } u_0 o(1 ) e^{2\eta } \qquad \mbox{as $ \eta \to -\infty$ } \end{aligned}\ ] ] and thus @xmath361 since @xmath362 \cap c^1(0,1)$ ] the mean value theorem implies @xmath363 and implies @xmath364 exists and @xmath365 . we conclude that @xmath366 has the taylor expansion @xmath158 . in turn , @xmath367 , @xmath368 , @xmath369 and @xmath370 achieves the taylor expansion @xmath160 . turning to @xmath371 , observe that by and @xmath71 @xmath372 while @xmath373 in addition , @xmath374 combined with the formula @xmath375 gives @xmath376 this shows @xmath377 has the taylor expansion @xmath71 at the origin . we considered the system as a paradigm to study the onset of localization and formation of shear bands and to examine the role of thermal softening , strain - rate hardening and thermal diffusion in their development . the stability of the uniform shear solution leads , in general , to the analysis of a non - autonomous system . the present model has the remarkable property that the system of relative perturbations is autonomous in the adiabatic case ( @xmath421 ) . a study of linearized stability for the case with no thermal diffusion @xmath378 shows that : ( i ) for @xmath17 the high frequency modes grow exponentially fast , with the exponent proportional to the frequency , what is characteristic of ill - posedness and hadamard instability . ( ii ) when @xmath2 the modes are still unstable but the rate of growth is bounded independently of the frequency . the behavior in that range is that characteristic of turing instability . the effect of thermal diffusion ( @xmath18 ) for the non - autonomous linearized model was also assessed ; it was shown that perturbations grow in the early stages of deformation , but over time the compound effect of diffusion stabilizes the response of the system . then we turned to the nonlinear system and constructed a class of self - similar solutions on the real line , leading to the explicit solutions , , . their construction is based on the self - similar profiles of , , in turn determined by a suitable heteroclinic connection . they depend on three parameters : @xmath27 linked to the uniform shear in , a parameter @xmath28 which can be viewed as a length scale of initial data , and @xmath29 determining the size of the initial profile . the solutions presented in fig [ uqsol ] evidently exhibit localisation : as time increases a coherent structure forms with the deformation localizing in a narrow band . in this self - similar solution there is no trace of the oscillations . a comparison with the modes of the linearized problem indicates that the nonlinearity suppresses the oscillations and a seamless coherent structure emerges as the net outcome of hadamard instability , rate dependence , and the nonlinear structure of the problem . , _ localization and shear bands in high strain - rate plasticity_. in nonlinear conservation laws and applications , a. bressan , g .- q . chen , m. lewiscka and d. wang , eds ; i m a vol . math . appl . , 153 , springer , new york , 2011 , pp 365 - 378 .

shear localization occurs in various instances of material instability in solid mechanics and is typically associated with hadamard - instability for an underlying model . while hadamard instability indicates the catastrophic growth of oscillations around a mean state , it does not by itself explain the formation of coherent structures typically observed in localization . the latter is a nonlinear effect and its analysis is the main objective of this article . we consider a model that captures the main mechanisms observed in high strain - rate deformation of metals , and describes shear motions of temperature dependent non - newtonian fluids . for a special dependence of the viscosity on the temperature , we carry out a linearized stability analysis around a base state of uniform shearing solutions , and quantitatively assess the effects of the various mechanisms affecting the problem : thermal softening , momentum diffusion and thermal diffusion . then , we turn to the nonlinear model , and construct localized states - in the form of similarity solutions - that emerge as coherent structures in the localization process . this justifies a scenario for localization that is proposed on the basis of asymptotic analysis in @xcite .