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we have described the theory of a previously unstudied form of the josephson interference effect that can occur in nanoscale @xmath0 junctions due to the coupling of the orbital angular momentum of transverse electronic subbands with an axial magnetic flux . we found in section [ sec : effect_of_length ] the regimes in which this interference effect dominates the @xmath3 vs @xmath4 characteristics of the junction . an idealized model of an @xmath0 junction was used , with several simplifying assumptions , in order to elucidate the mechanism of the effect . we discuss generalizations of the model below , in particular those modifications that may be necessary to directly model experimental devices . + _ fwvm and barriers at the interfaces _ no barriers were assumed at the @xmath2-@xmath1 interfaces , and fwvm was neglected . the effective mass for electrons @xmath20 was assumed to be uniform throughout the junction . these assumptions allowed kulik s method of matching the wavefunctions at the interfaces to be used to calculate the bound state and the continuum currents . we stress that the basic mechanism of the orbital interference effect , i.e. the modification of the @xmath1-section wavenumbers in the presence of the axial field , is independent of fwvm and interfacial barriers . therefore , the main features of the @xmath3 oscillations ( periodicity , amplitude ) should only be modified by fwvm and barriers as higher order corrections . the exact shape of the junction cpr and the @xmath3 vs @xmath4 curves , however , depends on the details of the interfaces . accurate modeling of experiments must take this into account , based on the material and interfacial properties specific to a particular experimental implementation . + as was discovered in studies of andreev reflection at nb - inas interfaces @xcite , fwvm modifies the cpr of superconductor / semiconductor / superconductor junctions . in the general case , where fwvm and barriers are included at the @xmath2-@xmath1 interfaces , the bound state energies and the continuum current must be calculated from the transmission matrix formalism @xcite , in which the transmission matrix includes a normal ( specular ) reflection coefficient as well as an andreev ( retro ) reflection coefficient . the values of these coefficients depend on the material details of the junction . the cprs of junctions with fwvm @xcite and barriers @xcite have been previously calculated ( numerically ) at zero magnetic field . the effect of both mechanisms is to make the cpr more closely resemble a sinusoidal curve . since the @xmath1-section wavenumbers have the same coupling to the axial field shown in eq . [ eq : wavenumbers ] regardless of fwvm or barriers , we expect the phase - shift mechanism leading to interference to remain . however , the shape of the oscillations in @xmath3 should appear more sinusoidal , following the shape of the cpr . + normal reflection of the quasiparticle wavefunctions from the @xmath2-@xmath1 barriers will result in a higher order correction to the periodicity of the interference effect . this is due to an increased average phase pickup as the quasiparticle spends more time in the junction . on the other hand , this should also lead to some randomization of the phase . the former would lead to shorter period oscillations ( i.e the effect occurs at lower fields ) , while the latter will partially smear out the interference effect , reducing the amplitude of oscillations . similar effects are expected from elastic backscattering occurring in the @xmath1-section in non - ballistic junctions . are beyond the scope of this paper , and are left to future work . + _ general form of @xmath42 _ a spatially uniform pairing potential was assumed in the @xmath2-sections at all magnetic fields ( eq . [ eq : delta ] ) . this was justified by assuming cylindrical @xmath2-sections , and restricting the cylinder diameter to be smaller than the superconducting coherence length in the @xmath2-sections . however , experimental fabrication of nanoscale @xmath0 junctions is usually done by evaporating or sputtering metallic ( e.g. al or nb ) thin film contacts onto a semiconducting nanowire . in this case , the geometry of the @xmath2-section is not cylindrical but @xmath300 shaped . the lack of cylindrical symmetry necessitates , in principle , a 3-dimensional numerical calculation of @xmath301 using self - consistent methods . however , as long as the @xmath1-section can be assumed to be cylindrically symmetric , our model should closely approximate the experimental situation . this is because the interference effect depends mainly on the eigensolutions in the @xmath1-section , particularly the orbital angular momentum states and their coupling to the flux . the details of the eigensolutions in the @xmath2-section do not play a direct role , other than asserting the form of the wavefunction ansatz ( eq . [ eq : psi_plus ] ) is valid . similarly , the axial field can induce a non - uniformity in @xmath42 . if the thickness of the metallic film becomes larger than the @xmath2-section coherence length , fluxoid quantization can result in a @xmath302-dependent phase for @xmath42 in the presence of the field . inserting a @xmath302-dependent @xmath42 in the bdg equations ( eq . [ eq : bdg ] ) will affect the bound state solutions , likely requiring a 3-dimensional numerical solution . however , we still expect the interference effect to depend mainly on the states in the @xmath1-section . + _ general radial wavefunctions _ the shell - conduction model was used in order to simplify computations , and to help gain intuitive insight into the problem ; it is not strictly necessary for the main arguments of the paper . indeed , we found in section [ sec : semiclassical ] that the semiclassical phase shift @xmath165 is only weakly dependent on the radius @xmath40 , so the interference effect should be present for general radial wavefunctions . in future work , the radial wavefunctions in the @xmath1-section , @xmath78 , and the corresponding single particle energies in the presence of the field will be numerically calculated , yielding the appropriate wavenumbers @xmath111 for electron- and hole - like solutions . we expect the term @xmath237 appearing in eqs . [ eq : quantization - rule ] , [ eq : continuum - current ] will continue to result phase shifts similar to those seen in the present model . + _ zeeman and spin - orbit effects _ in order to study the the orbital josephson interference effect in isolation , zeeman and spin - orbit effects were neglected in our analysis . it is useful to ask under what circumstances should the orbital effects or the zeeman + spin - orbit effects dominate ? the critical current of a short , insb @xmath0 junction , including spin - orbit and zeeman effects , was studied in ref.@xcite . the bound state energies were solved , and a phase - shift was observed in the energy - versus - phase curves due to the zeeman effect . similar to the mechanism described in our analysis , the zeeman effect modifies the @xmath1-section wavenumbers , but based on the spin state rather than the orbital state . this results in an oscillation of @xmath3 , with the first minimum occurring at @xmath303 , where @xmath304 is the fermi velocity , @xmath305 is the effective land g - factor , and @xmath306 is the bohr magneton . for a 200 nm insb junction with @xmath307 , this evaluates to @xmath308 t for @xmath304 corresponding to @xmath309 mev , but for other materials @xmath310 is typically much larger . considering inas with a moderately large @xmath311 gives @xmath312 t ( again for @xmath313 nm and @xmath309 mev ) . since we used the effective mass for inas in our calculations , we can directly compare with the results of figure [ fig : fig7_res]c for a 200 nm long junction . the orbital effect should dominate in this case , as the first minimum of @xmath3 is at a flux corresponding to @xmath314 t. the consequence of the inclusion of the spin - orbit coupling is a smaller correction : the so called anomalous josephson effect , in which the current is no longer an even function of the superconducting phase : @xmath315 . + we conclude that for insb devices , the zeeman effect could easily dominate . this is especially true if either of the following conditions hold : ( i ) if only @xmath316 subbands are occupied ( i.e. small chemical potential ) , or ( ii ) in a perpendicular field , where the zeeman effect is present but the orbital effect is suppressed . on the other hand , for most low spin - orbit semiconductor materials , @xmath317 and we would expect the orbital subband effect to dominate in an axial field experiment , unless only @xmath287 subbands are occupied . + _ magnetic depairing _ field - induced depairing suppresses both superconductivity in the @xmath2-sections and proximity superconductivity in the @xmath1-section @xcite . for a type - ii superconductor with a relatively large gap such as nb , we can assume depairing in the @xmath1-section should dominate . for diffusive junctions in the narrow junction limit , the usadel equations predict a monotonic gaussian decay @xmath318 for a perpendicular magnetic flux @xmath4 through the @xmath1-section @xcite , where @xmath319 is a numerical constant . a similar effect should apply to the axial field case , except with a slower magnetic field decay due to a smaller cross - sectional area ( smaller flux ) . this is expected to produce a monotonic decay envelope superimposed on the critical current oscillations , and should be taken into account when modeling experimental data . + _ summary _ the idealized model studied here serves to demonstrate a novel form of josephson interference due to orbital angular momentum states , with the unusual property of flux - aperiodic oscillations . extensions to the model discussed above , most importantly fwvm and barriers at the @xmath2-@xmath1 interfaces , will be useful for describing experimental implementations . we thank a. j. leggett and s. frolov for helpful discussions . this work was supported by nserc and the ontario ministry for research and innovation .

numerical calculations of the critical current versus axial field reveal flux - aperiodic oscillations that we identify as a novel form of josephson interference due to this orbital subband effect . this behavior is studied as a function of junction length and chemical potential . finally , we discuss extensions to the model that may be useful for describing realistic devices . [ qcalc_standalone ] the josephson effect is characterized by a current - phase relationship ( cpr ) linking macroscopic current flow to the phase gradient of the superconducting order parameter . efforts to realize majorana fermion quasiparticles in 1d semiconductors with strong spin - orbit interaction and proximity coupling to a superconductor have further raised interest in this type of junction . this is a regime in which the general theorem of byers and yang does not apply . for certain junction parameters , the interference effect can dominate the vs characteristics . the aim of this paper is to theoretically describe this type of josephson interference in a fully quantum mechanical way . in particular , we are interested in understanding the effect in isolation from the additional complications of real devices , such as non - cylindrical contact geometry , interfacial potential barriers , etc .

a semiconductor nanowire based superconductor - normal - superconductor ( ) junction is modeled theoretically . a magnetic field is applied along the nanowire axis , parallel to the current . the bogoliubov - de gennes equations for andreev bound states are solved while considering the electronic subbands due to radial confinement in the-section . the energy - versus - phase curves of the andreev bound states shift in phase as the-section quasiparticles with orbital angular momentum couple to the axial field . a similar phase shift is observed in the continuum current of the junction . the quantum mechanical result is shown to reduce to an intuitive , semi - classical model when the andreev approximation holds . numerical calculations of the critical current versus axial field reveal flux - aperiodic oscillations that we identify as a novel form of josephson interference due to this orbital subband effect . this behavior is studied as a function of junction length and chemical potential . finally , we discuss extensions to the model that may be useful for describing realistic devices . [ qcalc_standalone ] the josephson effect is characterized by a current - phase relationship ( cpr ) linking macroscopic current flow to the phase gradient of the superconducting order parameter . the precise form of the cpr for a superconducting weak link depends on intrinsic factors such as junction geometry , material properties , coherence lengths , etc . , in addition to extrinsic variables like temperature and magnetic field . in superconductor - normal - superconductor ( ) junctions in which the-section is long enough to suppress direct tunnelling of cooper pairs , but shorter than the-section phase coherence length , a supercurrent may be carried by quasiparticles undergoing andreev reflection at the- interfaces . planar junctions of width large compared to the-section superconducting coherence length have been studied in great detail ( width refers to the dimension perpendicular to the current ) . these have revealed , for example , fraunhofer oscillations of the critical current with respect to an externally applied out - of - plane magnetic field . for junction widths comparable to the-section coherence length , i.e. the narrow junction limit , this becomes a quasi - gaussian , monotonic decay of the critical current . recently , attention has been given to nanoscale , quasi one - dimensional ( 1d ) junctions , such as those readily engineered by contacting semiconductor nanowires with superconducting leads . gating the semiconducting-section allows for modulating the supercurrent by controlling the chemical potential . the oscillations of the magnetoresistance of a nanowire junction in the voltage - biased state ( i.e. no dc supercurrent ) versus an axial magnetic field have been studied . efforts to realize majorana fermion quasiparticles in 1d semiconductors with strong spin - orbit interaction and proximity coupling to a superconductor have further raised interest in this type of junction . theoretical results have indicated that the behaviour of the critical current in such a junction versus magnetic field and chemical potential can be used to identify topological phases . + previous theoretical descriptions of quasi-1d junctions have not fully considered the effects of nanoscale confinement on the cpr , in particular the implications of orbital angular momentum coupling to an external magnetic flux . here we provide a quantum mechanical description of an idealized junction with a flux applied along the nanowire axis ( parallel to the current ) . for a planar junction , no significant modification of the cpr with an axial flux is expected , as azimuthal motion of the carriers is absent . however , for a cylindrical geometry , azimuthal motion leads to a non - trivial effect which we identify as a previously unstudied form of josephson interference . this is due to the coupling between andreev quasiparticles ( bound states and continuum states ) with orbital angular momentum and the axial flux , which results in phase shifts of the energy - versus - phase for these current carrying states . the total current summed over all channels ( occupied orbitals ) can display interference . in contrast to fraunhofer interference in wide planar junctions , the flux is aligned _ with _ the current and the oscillations are _ not _ periodic in the flux quantum . this effect is only present in nanoscale junctions with lateral dimensions ( i.e. diameter ) smaller than the london penetration depth . this is a regime in which the general theorem of byers and yang does not apply . it is shown that the supercurrent from continuum states also contributes to this interference . for certain junction parameters , the interference effect can dominate the vs characteristics . semiclassically , the effect is intuitively understood by the pickup of a magnetic phase by andreev pairs with an azimuthal velocity component as they cross the junction ballistically . the aim of this paper is to theoretically describe this type of josephson interference in a fully quantum mechanical way . in particular , we are interested in understanding the effect in isolation from the additional complications of real devices , such as non - cylindrical contact geometry , interfacial potential barriers , etc . we consider in the discussion section how to modify the present model to better describe realistic devices . here , we consider the case where the diameter is smaller than the superconducting coherence length in the-section , so that the phase of the order parameter is uniform around the-section circumference in any magnetic field up to the critical field of the leads , . spin - orbit and zeeman effects in the-section ( e.g. relevant to iii - v semiconductor nanowires ) are neglected , and we assume no barriers at the- interfaces . furthermore , we neglect magnetic depairing effects .

astro-ph9705057

i

when galaxies formed and how they evolved are among the central issues of cosmology today . the traditional clue to this problem has been provided by the number count of galaxies as a function of apparent magnitude . early observations in the blue band showed that the number of galaxies per unit angular area increases faster than is expected in simple models for fainter magnitude ( koo & kron 1992 , for a review ) . it has become clear , however , that the number count alone does not allow an unambiguous interpretation . over the last decade , much effort has been expended to obtain redshift of faint galaxies ( broadhurst , ellis & shanks 1988 , hereafter bes ; colless et al . 1990 ; 1993 ; lilly , cowie & gardner 1991 ; koo & kron 1992 ; songaila et al . 1994 ; glazebrook et al . 1995a ; lilly et al . 1995 ; ellis et al . these redshift surveys have revealed , when combined with the count data , that the distribution of galaxies in the universe is much more puzzling than it looked : the excessive number of galaxies observed in the number count seemed to be most simply explained by assuming luminosity evolution of galaxies ( e.g. , koo & kron 1992 ) , but the redshift surveys have indicated that the shape of the redshift distribution appears almost as is predicted with a no - evolution model , with the normalization , however , being larger by a factor of two . simple models with all galaxies undergoing substantial luminosity evolution , which account for the steep slope of the number count , predict too many galaxies having high redshift to be consistent with the observations ( e.g. , ellis 1990 ; lilly 1993 ) ( though this seems to have been somewhat overstated ) . there have been a number of attempts to solve this problem : some authors speculated that the excess count is due to a new population of galaxies which have undergone star burst at rather low redshift ( bes ; cowie , songaila and hu 1991 ) . babul & rees ( 1992 ) and babul & ferguson ( 1996 ) interpreted this new population as dwarf galaxies from collapsed lyman @xmath12 clouds , eventually faded into galaxies making up an invisible population . some other authors have assumed a heuristic morphology - dependent luminosity evolution so that subluminous galaxies evolve fast to explain high counts while keeping the shape of redshift distribution as is in the no - evolution model , rather than introducing a new population ( bes ; lilly 1993 ; phillips & driver 1995 ) . another class of explanation invokes mergers of galaxies with a very high merging rate ( guiderdoni & rocca - volmerange 1991 ; broadhurst , ellis & glazebrook 1992 ; carlberg 1992 ) . the required rate to explain the steep slope of the @xmath4 band count is so high that a typical galaxy gains 50% of mass over the past 5 gyr . more recently , gronwall & koo ( 1995 ) suggested that the presence of an abundant non - evolving very blue dwarf population accounts for both counts and redshift data , although the meaning of this blue population is not clear to us . observational data have now accumulated for both counts and redshifts over a wide range of colours from @xmath4 to near infrared @xmath6 band , and the deepest redshift reaches beyond @xmath13 ( glazebrook et al . 1995a ; songaila et al . 1994 ; lilly et al . a large number of redshifts contained in the survey have enabled a direct construction of the @xmath4 band luminosity function as a function of redshift ( lilly et al . 1995 ; ellis et al . 1996 ; see also eales 1993 ) . this prompts us to think that we already have enough data to understand the evolution of galaxies below @xmath14 . in addition , more varieties of clues have become available to understand the nature of faint galaxies . sharp images of galaxies obtained with the hst have enabled a study of morphology to @xmath5=24 mag , giving us a direct probe concerning how the faint galaxies look like ( driver , windhorst & griffiths 1995 ; glazebrook et al . 1995b ; abraham et al . 1996 ; driver et al . 1996 ) . the work has shown that the fraction of irregular galaxies increases sharply with magnitude beyond @xmath15 mag . the same observations indicate that giant galaxies evolve only slowly , although they do indicate some evolution in colour , brightness and the detail of morphology . another piece of evidence that giant galaxies evolve only slowly , if at all , comes from a mg ii quasar - absorption - line selected sample ( steidel , dickinson & persson 1994 ) . the luminosity of galaxies that yield mg ii absorption lines for a quasar in their vicinity is remarkably constant both in the rest - frame @xmath4 and @xmath6 bands from @xmath16 to @xmath171 . there are also a number of observations that tell us about the content of galaxies . the information that would directly constrain the model for evolution of galaxies is the estimate of the global star formation rate at @xmath18 ( gallego et al . 1995 ) and high @xmath19 ( cowie , hu & songaila 1995 ; see also madau et al . 1996 for a more recent work ) from the strength of h@xmath12 or [ o ii]@xmath20 emission lines . this quantifies the inference from early observations concerning the increase of star formation activity in the past ( bes ) . yet another useful information is given by the measurement of the neutral hydrogen abundance in the universe as a function of redshift ( lanzetta , wolfe & turnshek 1995 ; storrie - lombardi , mcmahon & irwin 1996 ) . the data show clear depletion of hi gas with time , suggesting that this gas has been used for star formation . this information can be used to constrain the total volume emissivity of galaxies ( see , pei & fall 1995 ) , on which direct information is also available from galaxy observations ( lilly et al . 1996 ) . from accumulating observations we now have a reasonable picture for the evolution of galaxies ( fukugita , hogan & peebles 1996 ) : giant galaxies were already mature at @xmath21 and evolve only slowly lower than this redshift , whereas subluminous galaxies evolve rapidly in the same redshift range . the latter are probably young galaxies formed close to @xmath21 . it has also been speculated that most of elliptical galaxies and bulges of the spiral galaxies , which altogether we call spheroids , formed before @xmath22 , and spiral disks were assembled in the redshift range from @xmath23 to 1 . in this paper we present a simple but quantitative model for the history of galaxies , which is broadly consistent with the observations . the model employs a population synthesis model of stars . in the traditional attempts , galaxies are assumed to have fully assembled at an early epoch of the universe ( searle , sargent & bagnuolo 1973 ; sandage 1986 ; yoshii & takahara 1988 ) , and the star formation rates are adjusted to reproduce galaxy colours today . in our attempt it is crucial to properly build in the epoch of the formation of galaxies in the stellar population synthesis model . for this purpose we first show that one can estimate the age of the galaxies as seen today from a simultaneous consideration of the colour and gas content of galaxies with the aid of moderate assumptions on the star formation history , and infer the age of irregular galaxies and also of disks of spiral galaxies . for elliptical galaxies , and also bulges of spiral galaxies from family resemblance ( both are called spheroids ) , we adopt the conventional view that they formed at high @xmath19 as argued by eggen , lynden - bell and sandage ( 1962 ) . we assume that the number of galaxies is conserved since they were born , and it is constrained by the local type - dependent luminosity function . we calculate the evolution of bulges and disks separately and assume delayed formation of disks and irregular galaxies . the spheroids are supposed to follow the passive evolution to now . galaxies are grouped into four morphology types : e / s0 , sab , sc , and sdm - irr which is simply referred to as irr in this paper ; low luminosity dwarf galaxies do not play an essential role in our argument . the mix of morphological types of galaxies is not constant , but depends on absolute luminosity . the predictions of this model are compared with the observation of the galaxy number count and the redshift distribution in @xmath4 , @xmath5 and @xmath6 bands . we do not discuss the @xmath24 band count , since we can not model reliable @xmath6- and @xmath25-corrections for this band , and @xmath24 band data are too sensitive to the detail of galaxy activity . we discuss the evolution of gas and star content of galaxies predicted in the model . through these comparisons with the observations we may conclude that the present model is reasonably well constrained for @xmath26 . the prediction for @xmath27 , however , is basically an extrapolation , and ill - constrained . a number of checkpoints are also discussed to prove or disprove the validity of the model and , more importantly , underlying assumptions taken in this paper . we do not consider the no - evolution model seriously , since it is clearly unphysical . we know that elliptical galaxies _ must _ evolve even in the absence of new star formation after the burst , since the stars in the main sequence branch continuously evolve to giants , and the total luminosity should decrease with time ( tinsley 1972 ) . this basic feature is a solid prediction of the stellar population synthesis model , although quantitative details depend on models , in particular on assumed initial mass functions . for spiral galaxies , there are more uncertainties : our model corresponds to the case where luminosity changes very little with time : the decrease of light due to evolution of stars is compensated by newly born stars formed from the gas replenished into the disk . we note that the present model is quite different from `` cdm cosmologies '' , where morphologies of galaxies change from time to time due to mergers down to low redshift ( kauffmann , guiderdoni & white 1994 ; cole et al . that is , elliptical galaxies are not quite old , but formed by collision of , for instance , two spiral galaxies at rather low redshift ( toomre 1977 ) . one of the original reasons for interest in faint galaxy counts is that it may offer a chance to test world geometry ( sandage 1961 ) . a number of attempts have been made aiming at this goal ( e.g. , yoshii & takahara 1988 ; fukugita et al . 1990 ; gardner , cowie & wainscoat 1993 ; gronwall & koo 1995 ) . the reliability of the results , however , is strongly affected by the uncertainties of the model . in this paper we discuss where the predictions are reasonably robust . in particular we discuss the possibility to test the presence or absence of the cosmological constant . in the next section we discuss how to estimate the ages of galaxies from their colours and gas content . the details of the model and input assumptions are discussed in section 3 . in section 4 we present the predictions of the model and compare them with the observations in @xmath6 , @xmath4 and @xmath5 bands , and with the constraint on evolution from a mg ii quasar absorption line selected sample . the evolution of baryons in stars and in gas is also discussed . section 5 is devoted to discussion and summary of the problems .

a simple quantitative model is presented for the history of galaxies to explain galaxy number counts , redshift distributions and some other related observations . we first infer that irregular galaxies and the disks of spiral galaxies are young , probably formed at from a simultaneous consideration of colours and gas content under a moderate assumption on the star formation history . assuming that elliptical galaxies and bulges of spiral galaxies , both called spheroids in the discussion , had formed early in the universe , thus estimated into account , we construct a model for the history of galaxies employing a stellar population synthesis model . we assume that the number of galaxies does not change except that some of them ( irregulars ) were newly born , and use a morphology - dependent local luminosity function to constrain the number of galaxies . we represent the galaxies by e / s0 , sab , sc and irr ; low luminosity dwarfs or any objects unobservable today do not play a role in our considerations . in our model , spheroids follow passive evolution and the luminosity of spiral galaxies evolves only very slowly for a wide redshift interval due to a counterbalance between fading stars and new star formation from the gas replenished from intergalactic space . irregular galaxies evolve moderately fast for . the predictions of the model are compared with the observation of galaxy number counts and redshift distributions for the , and colour bands . we argue that models are disfavoured , unless the basic assumptions of the present model are abandoned . the band observations reach quite high redshift : for instance observations at=23 mag may explore the formation epoch , which could be as high as . on the other hand , galaxies observed in the band are dominated by disks and irregulars , spheroids making a very small contribution . consistency is also shown with the constraint on the luminosity evolution from a mg ii quasar - absorption - line selected sample . the amount of baryons in disk stars is increasing , as they form to , which just offsets the decrease of neutral gas towards the present epoch , as inferred from quasar absorption line surveys . 814i_f814w

a simple quantitative model is presented for the history of galaxies to explain galaxy number counts , redshift distributions and some other related observations . we first infer that irregular galaxies and the disks of spiral galaxies are young , probably formed at from a simultaneous consideration of colours and gas content under a moderate assumption on the star formation history . assuming that elliptical galaxies and bulges of spiral galaxies , both called spheroids in the discussion , had formed early in the universe , the resulting scenario is that spiral galaxies formed as intergalactic gas accreting onto pre - existing bulges mostly at ; irregular galaxies as seen today formed by aggregation of clouds at . taking the formation epochs thus estimated into account , we construct a model for the history of galaxies employing a stellar population synthesis model . we assume that the number of galaxies does not change except that some of them ( irregulars ) were newly born , and use a morphology - dependent local luminosity function to constrain the number of galaxies . we represent the galaxies by e / s0 , sab , sc and irr ; low luminosity dwarfs or any objects unobservable today do not play a role in our considerations . in our model , spheroids follow passive evolution and the luminosity of spiral galaxies evolves only very slowly for a wide redshift interval due to a counterbalance between fading stars and new star formation from the gas replenished from intergalactic space . irregular galaxies evolve moderately fast for . the predictions of the model are compared with the observation of galaxy number counts and redshift distributions for the , and colour bands . we show that band observations are largely controlled by spheroids , which make them particularly suitable to study cosmology . we argue that models are disfavoured , unless the basic assumptions of the present model are abandoned . the band observations reach quite high redshift : for instance observations at=23 mag may explore the formation epoch , which could be as high as . on the other hand , galaxies observed in the band are dominated by disks and irregulars , spheroids making a very small contribution . it is shown that young irregular galaxies cause the steep slope of the counts . the fraction of irregular galaxies increases with decreasing brightness : at mag , they contribute as much as spiral galaxies . thus , `` the faint blue galaxy problem '' is solved by invoking young galaxies . this interpretation is corroborated by a comparison of our prediction with the morphologically - classified galaxy counts in the band . we do not invoke sporadic star bursting : star formation takes place steadily as does today , but galaxies ( especially irregulars ) are gaseous at higher redshift , and hence star formation is much more active than today . consistency is also shown with the constraint on the luminosity evolution from a mg ii quasar - absorption - line selected sample . we estimate that 2/3 of the baryons in stars are stored in spheroids and 1/3 in disks , only% being in irregular galaxies . the amount of baryons in disk stars is increasing , as they form to , which just offsets the decrease of neutral gas towards the present epoch , as inferred from quasar absorption line surveys . 814i_f814w

0705.2707

i

since tubules were fabricated successfully from chiral lipid molecules , chiral lipid structures have attracted much experimental attention @xcite . most of chiral structures in experiments are self - assembled from dc@xmath3 which is a typical chiral molecule . et al . _ have observed that the spherical vesicles in solution have very weak circular dichroism signal while tubules have strong one @xcite . et al . _ have investigated the chiral lipid tubules formed from various proportions of left- and right - handed dc@xmath3 molecules and found that they are of similar radii @xcite , which reveals that the radii of chiral tubules do not depend on the strength of the molecular chirality . fang s group has carefully resolved the molecular tilting order in the tubules and concluded that the projected direction of the molecules on the tubular surfaces departs 45@xmath2 from the equator of the tubules at the uniform tilting state @xcite . helical ripples in lipid tubules are also observed by the same group with atomic force microscopy @xcite . their pitch angles are found to be concentrated on about 5@xmath2 and 28@xmath2 @xcite . cholesterol is another kind of chiral molecules used in the experiments where helical stripes with pitch angles @xmath4 and @xmath5 are usually observed @xcite . additionally , oda _ et al . _ have reported twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions @xcite . it is found that the twisted ribbons can be tuned by the introduction of opposite - handed chiral counterions in various proportions @xcite . from the experimental data @xcite , we see that the ratio between the width and pitch of the ribbons is proportional to the relative concentration difference of left- and right - handed enantiomers in the low relative concentration difference region . can we interpret all or at least most of above experimental results within a unified theory ? there are several theoretical discussions on chiral lipid membranes ( clms ) in the previous literature , where the chiral molecules are assumed to be in a smectic c@xmath6 phase at which the direction of the molecules is tilted from the normal of the membranes at a constant angle . the possible free energy of clms is discussed by helfrich and prost from symmetry arguments @xcite . their theory has been further developed and applied in many studies @xcite . nelson and powers have investigated the thermal fluctuations of clms @xcite . et al . _ have discussed tubules with helically modulated tilting state and helical ripples in tubules @xcite . komura and ou - yang have given an explanation to the high - pitch helical stripes of cholesterol molecules @xcite . due to the complicated form of the free energy used in these theories @xcite , it is almost impossible to obtain the general euler - lagrange equations corresponding to the free energy . thus one can not determine whether a configuration , such as a twisted ribbon , a tubule with helically modulated tilting state or a helical stripe , is a true equilibrium structure or not . this difficulty was always ignored in previous discussion of both tubules with helically modulated tilting state and helical stripes @xcite . can we confront this difficulty and construct a more concise theory of clms consistent with the experiments , in which we can unambiguously say which configuration is a genuine equilibrium structure ? we will address these questions in this paper , which is organized as follows : in sec . [ endensity ] , we introduce a concise free energy density of clms which includes the contributions of the bending and the surface tension of the membranes , as well as the chirality and orientational variation of the tilting molecules . in sec . [ closedclms ] , we present the euler - lagrange equations for clms without free edges and use them to explain experimental data @xcite . we predict a torus with the ratio between its two generated radii larger than @xmath1 , which has not yet been observed in the experiments on self - assembled clms . in sec . [ openclms ] , we present the euler - lagrange equations and boundary conditions for clms with free edges and use them to discuss experimental data @xcite . [ summary ] is a brief summary and discussion . in the appendixes , we briefly derive the euler - lagrange equations for clms without free edges and the euler - lagrange equations as well as boundary conditions for clms with free edges through the variational method @xcite developed by one of the present authors . several mathematical details are also put in the appendixes .

a theory of chiral lipid membranes is proposed on the basis of a concise free energy density which includes the contributions of the bending and the surface tension of membranes , as well as the chirality and orientational variation of tilting molecules . this theory is consistent with the previous experiments [ j.m . et al . _ , et al . _ , natl . acad . sci . usa * 102 * , 7438 ( 2005 ) ] on self - assembled chiral lipid membranes of dcpc . a torus with the ratio between its two generated radii larger than is predicted from the euler - lagrange equations . et al . _ , additionally , the present theory can explain twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions .

a theory of chiral lipid membranes is proposed on the basis of a concise free energy density which includes the contributions of the bending and the surface tension of membranes , as well as the chirality and orientational variation of tilting molecules . this theory is consistent with the previous experiments [ j.m . schnur _ et al . _ , science * 264 * , 945 ( 1994 ) ; m.s . spector _ et al . _ , langmuir * 14 * , 3493 ( 1998 ) ; y. zhao , _ et al . _ , proc . natl . acad . sci . usa * 102 * , 7438 ( 2005 ) ] on self - assembled chiral lipid membranes of dcpc . a torus with the ratio between its two generated radii larger than is predicted from the euler - lagrange equations . it is found that tubules with helically modulated tilting state are not admitted by the euler - lagrange equations , and that they are less energetically favorable than helical ripples in tubules . the pitch angles of helical ripples are theoretically estimated to be about 0 and 35 , which are close to the most frequent values 5 and 28 observed in the experiment [ n. mahajan _ et al . _ , langmuir * 22 * , 1973 ( 2006 ) ] . additionally , the present theory can explain twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions . the ratio between the width and pitch of twisted ribbons is predicted to be proportional to the relative concentration difference of left- and right - handed enantiomers in the low relative concentration difference region , which is in good agreement with the experiment [ r. oda _ et al . _ , nature ( london ) * 399 * , 566 ( 1999 ) ] .

0705.2707

c

in this paper , we have focused on a concise free energy from which the euler - lagrange equations of clms without free edges and euler - lagrange equations as well as boundary conditions of clms with free edges can be derived analytically . their implications for extant and future experiments can be summarized as follows . \(i ) our theory predicts the same probability for left- and right - handed spherical vesicles existing in solution . thus no evident circular dichroism signal should be observed , which is consistent with the experiment @xcite . \(ii ) the radius of a tubule at uniform tilting state satisfies eq.([tubradius ] ) . it reveals that the radius is independent of molecular chirality , which is consistent with the experiment @xcite where the tubular radius is insensitive to the proportion of left- and right - handed dc@xmath3 in the tubule . we find that the projected direction of the molecules on the tubular surface departs 45@xmath2 from the equator of the tubule with the uniform tilting state , which is in good agreement with the experiment @xcite . \(iii ) tubules with helically modulated tilting state are not admitted by the euler - lagrange equation ( [ el2 ] ) for non - vanishing @xmath20 . the orientational variation of the tilting lipid molecules breaks the force balance along the normal direction of the tubule . thus a tubule with helically modulated tilting state is not an equilibrium structure within our theoretical framework . \(iv ) helical ripples in tubules are equilibrium structures . the pitch angle of helical ripples is estimated as about 0@xmath2 and 35@xmath2 for @xmath111 , which are close to the experimental values 5@xmath2 and 28@xmath2 observed by fang s group @xcite . \(v ) tori with a ratio of generated radii larger than @xmath1 are predicted by our theory , eq.([torusrad ] ) , which have not yet been observed in the experiments . \(vi ) helical stripes with free edges at either uniform tilting state or helical modulated tilting state are not possible equilibrium configurations . \(vii ) twisted ribbons satisfy euler - lagrange equations ( [ el3 ] ) , ( [ el4 ] ) and boundary conditions ( [ bc1])([bc4 ] ) . the ratio between the width and pitch of the ribbons is proportional to the relative concentration difference of left- and right - handed enantiomers in the low relative concentration difference region , which is in good agreement with the experiment @xcite . finally , we have to list a few open problems which should be addressed in the future work . \(i ) within the present theory , we can not give a simple explanation for the experiments on clms of pure cholesterol @xcite where helical stripes with pitch angles @xmath4 and @xmath5 are usually observed . it has recently been found that the cholesterol helical stripes have good crystal structure @xcite , which is out of the range of our theory . there might be two possible ramifications based on the present theory : one would be to include anisotropic bending effects @xcite in the free energy density ( [ energy2 ] ) ; another one would be to consider a line tension @xmath166 depending on the angle between the directions of the tilting and the free edges @xcite in the free energy ( [ openfe ] ) . \(ii ) we can not yet interpret the twisted ribbon - to - tubule transition with increasing the relative concentration difference of the left- and right - handed enantiomers reported in the recent experiment by oda s group @xcite . a possible reason is that the parameters except @xmath20 in our theory are independent of the relative concentration difference @xcite . additionally , the ribbons and tubules observed in this experiment are usually multi - bilayer structures @xcite . then a decoupling effect @xcite between neighbor bilayers might occur . an extended theory including these two factors might be required to investigate the mechanism responsible for the transition . \(iii ) we have ignored the effect of singular points in clms . although the features of singular points in 2d planar films or achiral nematic spherical , torus vesicles and other manifolds have been fully investigated in previous literature @xcite , it remains a challenge to study the properties of singular points in clms @xcite .

the pitch angles of helical ripples are theoretically estimated to be about 0 and 35 , which are close to the most frequent values 5 and 28 observed in the experiment [ n. mahajan _ the ratio between the width and pitch of twisted ribbons is predicted to be proportional to the relative concentration difference of left- and right - handed enantiomers in the low relative concentration difference region , which is in good agreement with the experiment [ r. oda _ et al .

a theory of chiral lipid membranes is proposed on the basis of a concise free energy density which includes the contributions of the bending and the surface tension of membranes , as well as the chirality and orientational variation of tilting molecules . this theory is consistent with the previous experiments [ j.m . schnur _ et al . _ , science * 264 * , 945 ( 1994 ) ; m.s . spector _ et al . _ , langmuir * 14 * , 3493 ( 1998 ) ; y. zhao , _ et al . _ , proc . natl . acad . sci . usa * 102 * , 7438 ( 2005 ) ] on self - assembled chiral lipid membranes of dcpc . a torus with the ratio between its two generated radii larger than is predicted from the euler - lagrange equations . it is found that tubules with helically modulated tilting state are not admitted by the euler - lagrange equations , and that they are less energetically favorable than helical ripples in tubules . the pitch angles of helical ripples are theoretically estimated to be about 0 and 35 , which are close to the most frequent values 5 and 28 observed in the experiment [ n. mahajan _ et al . _ , langmuir * 22 * , 1973 ( 2006 ) ] . additionally , the present theory can explain twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions . the ratio between the width and pitch of twisted ribbons is predicted to be proportional to the relative concentration difference of left- and right - handed enantiomers in the low relative concentration difference region , which is in good agreement with the experiment [ r. oda _ et al . _ , nature ( london ) * 399 * , 566 ( 1999 ) ] .

1303.0944

i

a possible approach for dealing with the intractability of a given decision or optimization problem is to identify restrictions on input instances under which the problem can still be solved efficiently . one generic framework for describing a kind of such restrictions in case of graph problems is the following : given a graph @xmath0 , does @xmath0 admit non - negative integer weights on its vertices ( or edges , depending on the problem ) and a set @xmath1 of integers such that a subset @xmath2 of its vertices ( or edges ) has property @xmath3 if and only if the sum of the weights of elements of @xmath2 belongs to @xmath1 ? property @xmath3 can denote any of the desired substructures we are looking for , such as matchings , cliques , stable sets , dominating sets , etc . if weights as above exist and are given with the graph , and the set @xmath1 is given by a membership oracle , then a dynamic programming algorithm can be employed to find a subset with property @xmath3 of either maximum or minimum cost ( according to a given cost function on the vertices / edges ) in @xmath4 time and with @xmath5 calls of the membership oracle , where @xmath6 is the number of vertices ( or edges ) of @xmath0 and @xmath5 is a given upper bound for @xmath1 @xcite . the advantages of the above framework depend both on the choice of property @xmath3 and on the constraints ( if any ) imposed on the structure of the set @xmath1 . for example , if @xmath3 denotes the property of being a stable ( independent ) set , and set @xmath1 is restricted to be an interval unbounded from below equivalently , @xmath1 is of the form of the form @xmath7 $ ] for some non - negative integer @xmath8 , we obtain the class of _ threshold graphs _ @xcite , which is very well understood and admits several characterizations and linear time algorithms for recognition and for several optimization problems @xcite . if @xmath3 denotes the property of being a dominating set and @xmath1 is an interval unbounded from above , that is , it is of the form @xmath9 for some non - negative integer @xmath8 , we obtain the class of _ domishold graphs _ @xcite , which enjoy similar properties as the threshold graphs . on the other hand , if @xmath3 is the property of being a _ maximal _ stable set and @xmath1 is restricted to consist of a single number , we obtain the class of _ equistable graphs _ @xcite , for which the recognition complexity is open ( see , e.g.,@xcite ) , no structural characterization is known , and several @xmath10-hard optimization problems remain intractable on this class . for instance , it is shown in @xcite that the maximum independent set and the minimum independent dominating set problems are apx - hard in the class of equistable graphs . as the above examples show , the resulting class of graphs can be either _ hereditary _ ( that is , closed under vertex deletion)as in the case of threshold or domishold graphs , or not as in the case of equistable graphs . when the resulting graph class is not hereditary , it is natural to consider the hereditary version of the property , in which the requirement ( the existence of weights and the set @xmath1 ) is extended to all induced subgraphs of the given graph . in this paper , we introduce and study the case when @xmath3 is the property of being a total dominating set and @xmath1 is an interval unbounded from above . given a graph @xmath11 , a _ total dominating set _ ( a td set , for short ) is a subset @xmath12 of the vertices of @xmath0 such that every vertex of @xmath0 has a neighbor in @xmath12 . this notion has been extensively studied in the literature , see , e.g. the monographs @xcite and the survey paper @xcite . a graph @xmath13 is said to be _ total domishold _ ( td for short ) if there exists a pair @xmath14 where @xmath15 is a weight function and @xmath16 is a threshold such that for every subset @xmath17 , @xmath18 if and only if @xmath19 is a total dominating set in @xmath0 . a pair @xmath14 as above will be referred to as a _ total domishold structure _ of @xmath0 . notice that the above definition allows @xmath0 to have isolated vertices . every graph with an isolated vertex is total domishold , even though it does not have any td sets . the complete graph of order @xmath6 is total domishold . indeed , a subset @xmath20 is a total dominating set of @xmath21 if and only if @xmath19 is of size at least two , and consequently the pair @xmath22 where @xmath23 for all @xmath24 is a total domishold structure of @xmath21 . on the other hand , the @xmath25-cycle @xmath26 is not a total domishold graph ( cf . theorem [ prop : forbidden - induced - subgraphs ] in section [ sec : htd - partial - characterizations ] ) . it is easy to see that adding a new vertex to the @xmath25-cycle and connecting it to exactly one vertex of the cycle results in a total domishold graph . therefore , contrary to the classes of threshold and domishold graphs , the class of total domishold graphs is not hereditary . this motivates the following definition : a graph @xmath0 is said to be _ hereditary total domishold _ ( htd for short ) if every induced subgraph of it is total domishold . we initiate the study of td and htd graphs . we identify several operations preserving the class of td graphs , which , together with results from the literature @xcite , imply that the class of htd graphs properly contain the classes of threshold graphs and the complements of domishold graphs ( section [ sec : properties ] ) . we exhibit close relationships between total domishold graphs , threshold boolean functions and threshold hypergraphs ( in section [ sec : td - graphs - bfs - hypergraphs ] ) , and use them to develop a polynomial time recognition algorithm for total domishold graphs , a polynomial time algorithm for the total dominating set problem in the class of total domishold graphs , and a @xmath27-approximation algorithm for the dominating set problem in the class of total domishold graphs ( in section [ sec : algorithmic - aspects ] ) . as our main result , we completely characterize the htd graphs ( in section [ sec : htd - partial - characterizations ] ) , both in terms of forbidden induced subgraphs , and in terms of properties of certain boolean functions derived from induced subgraphs of the graph . we conclude the paper with some open problems in section [ sec : conclusion ] .

a total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set . we introduce and study graphs that admit non - negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold . we show that these graphs , which we call total domishold graphs , form a non - hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs , and are closely related to threshold boolean functions and threshold hypergraphs . we present a polynomial time recognition algorithm of total domishold graphs , and characterize graphs in which the above property holds in a hereditary sense . ( 2010 ) : 05c69

a total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set . we introduce and study graphs that admit non - negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold . we show that these graphs , which we call total domishold graphs , form a non - hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs , and are closely related to threshold boolean functions and threshold hypergraphs . we present a polynomial time recognition algorithm of total domishold graphs , and characterize graphs in which the above property holds in a hereditary sense . our characterization is obtained by studying a new family of hypergraphs , defined similarly as the sperner hypergraphs , which may be of independent interest . * keywords : * total domination ; total domishold graph ; split graph ; dually sperner hypergraph ; threshold hypergraph ; threshold boolean function ; forbidden induced subgraph characterization . ( 2010 ) : 05c69

1612.03355

i

no - slip billiard systems have received so far very little attention despite some interesting features that distinguish them from the much more widely studied standard billiards . these non - standard types of billiards are discrete time systems in dimension @xmath2 ( after taking the natural poincar section of a flow ) representing a rotating disc with unit kinetic energy that moves freely in a billiard domain with piecewise smooth boundary . although not hamiltonian , these systems are nevertheless time reversible and leave invariant the canonical billiard measure . they also exhibit dynamical behavior that is in sharp contrast with standard billiards . for example , they very often contain period-@xmath1 orbits having small elliptic islands . these regions exist amid chaos that appears , in numerical experiments , to result from the usual mechanisms of dispersing and focusing . we are aware of only two articles on this subject prior to our @xcite : one by broomhead and gutkin @xcite showing that no - slip billiard orbits in an infinite strip are bounded , and another by wojtkowski , characterizing linear stability for a special type of period-@xmath1 orbit . in this paper we extend their results as will be detailed shortly , and develop the basic theory of no - slip billiards in a more systematic way . in this section we explain the organization of the paper and highlight our main new results . section [ preliminaries ] gives preliminary information and sets notation and terminology concerning rigid collisions . it specializes the general results from @xcite ( stated in that paper in arbitrary dimension for bodies of general shapes and mass distributions ) to discs in the plane with rotationally symmetric mass distributions . the main fact is briefly summarized in proposition [ classification ] . although the classification into specular and no - slip collisions is the same as in @xcite , our approach is more differential geometric in style and may have some conceptual advantages . for example , we derive this classification ( in @xcite ) from an orthogonal decomposition of the tangent bundle @xmath3 restricted to the boundary @xmath4 ( orthogonal relative to the kinetic energy riemannian metric in the system s configuration manifold @xmath5 ) into physically meaningful subbundles . this orthogonal decomposition is explained here only for discs in the plane . by a planar _ no - slip billiard _ system we mean a mechanical system in @xmath6 in which one of the colliding bodies , which may have arbitrary shape , is fixed in place , whereas the second , moving body is a disc with rotationally symmetric mass distribution ; post - collision velocities ( translational and rotational ) are determined from pre - collision velocities via the no - slip collision map and between consecutive collisions the bodies undergo free motion . contrary to the standard case , the moving particle s mass distribution influences the collision properties ( via an angle parameter which is denoted @xmath7 throughout the paper ) . the main definitions and notations concerning no - slip billiard systems , in particular the notions of _ reduced phase space _ , _ velocity phase space _ , and the _ product , eigen- _ and _ wavefront _ frames , are introduced in section [ noslip billiards definitions ] . a special feature of no - slip billiards around which much of the present study is based , is the ubiquitous occurrence of period-@xmath1 trajectories . the general description of these trajectories is given in section [ periodic section ] . in section [ sec : differential ] we obtain the differential of the no - slip billiard map and show the form it takes for period-@xmath1 trajectories . for general collision systems as considered in @xcite ( satisfying energy and momenta conservation , time reversibility , involving impulsive forces that can only act at the contact point between colliding bodies ) , the issue of characterizing smooth invariant measures still needs much further investigation , although it is shown there that the canonical ( liouville ) measure is invariant if a certain field of subspaces defined in terms of the collision maps at each collision configuration @xmath8 is parallel with respect to the kinetic energy metric . this is the case for planar no - slip billiards , so that the standard billiard measure is still invariant . ( note , however , that the configuration manifold is now @xmath9-dimensional . ) a detailed proof of this fact , in addition to comments on time reversibility are given in section [ measure reversible ] . dynamics proper begins with sections [ wedge section ] and [ higher order polygon ] , which are concerned with no - slip billiard systems in wedge - shaped regions and polygons . here we generalize the main result from @xcite . in that paper it is shown that orbits of the no - slip billiard system in an infinite strip domain are bounded . by extending and refining this fact to wedge regions we obtain local stability for periodic orbits in no - slip polygonal billiards . this is theorem [ polygon stability ] . we also give in section [ higher order polygon ] an exhaustive description of periodic orbits in wedge billiards . finally , in section [ curved section ] we consider linear stability of period-@xmath1 orbits in the presence of curvature . our results here extend those of @xcite for no - slip sinai billiards . wojtkowski makes in @xcite the following striking observation : for a special period-@xmath1 orbit in a sinai billiard ( corresponding in our study to angle @xmath10 ) there is a parameter defined in terms of the curvature of the circular scatterer that sets a threshold between elliptic and hyperbolic behavior . this is based on an analysis of the differential of the billiard map at the periodic trajectory . here we derive similar results for period-@xmath1 orbits in general . although the analysis is purely linear , we observe the occurrence of elliptic islands and stable behavior in systems that are the no - slip counterpart of fully chaotic standard billiards . the fast transition between stability and chaos near the threshold set by the curvature parameter obtained from the linear analysis is also very apparent numerically . as already observed by wojtkowski in @xcite , proving local stability in the presence of curvature would require a difficult kam analysis ( for reversible , but not hamiltonian systems ; cf .

_ this leads to the study of _ no - slip billiards_planar billiard systems in which the moving particle is a disc ( with rotationally symmetric mass distribution ) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain . in this paper we greatly extend previous results on boundedness of orbits ( broomhead and gutkin ) and linear stability of periodic orbits for a sinai - type billiard ( wojtkowski ) for no - slip billiards .

abstract rigid bodies collision maps in dimension two , under a natural set of physical requirements , can be classified into two types : the standard specular reflection map and a second which we call , after broomhead and gutkin , _ no - slip . _ this leads to the study of _ no - slip billiards_planar billiard systems in which the moving particle is a disc ( with rotationally symmetric mass distribution ) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain . in this paper we greatly extend previous results on boundedness of orbits ( broomhead and gutkin ) and linear stability of periodic orbits for a sinai - type billiard ( wojtkowski ) for no - slip billiards . we show among other facts that : ( i ) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than , no - slip billiard dynamics will always contain elliptic period- orbits ; ( ii ) polygonal no - slip billiards always admit small invariant open sets and thus can not be ergodic with respect to the canonical invariant billiard measure ; ( iii ) the no - slip version of a sinai billiard must contain linearly stable periodic orbits of period and , more generally , we provide a curvature threshold at which a commonly occurring period- orbit shifts from being hyperbolic to being elliptic ; ( iv ) finally , we make a number of observations concerning periodic orbits in a class of polygonal billiards .

0910.4172

i

a _ convex body _ is a compact convex set in @xmath3 with nonempty interior . let @xmath1 be a family of convex bodies . the _ packing number _ @xmath7 is the maximum cardinality of a set of pairwise - disjoint convex bodies in @xmath1 , and the _ transversal number _ @xmath0 is the minimum cardinality of a set of points that intersects every convex body in @xmath1 . let @xmath8 be the _ intersection graph _ of @xmath1 with one vertex for each convex body in @xmath1 and with an edge between two vertices if and only if the two corresponding convex bodies intersect . the _ independence number _ @xmath9 is the maximum cardinality of an independent set in @xmath8 . the _ clique partition number _ @xmath10 is the minimum number of classes in a partition of the vertices of @xmath8 into cliques . since a set of pairwise - disjoint convex bodies in @xmath1 corresponds to an independent set in @xmath8 , we have @xmath11 . also , since any subset of convex bodies in @xmath1 that share a common point corresponds to a clique in @xmath8 , we have @xmath12 . for the special case that @xmath1 is a family of axis - parallel boxes in @xmath3 , we indeed have @xmath13 since any subset of pairwise - intersecting boxes share a common point . in general , we clearly have the inequality @xmath14 , thus also @xmath15 . but what else can be said about the relation between @xmath0 and @xmath7 ? of axis - parallel unit squares . left : all squares that intersect the highest ( shaded ) square contain one of its two lower vertices . right : five squares form a 5-cycle . ] for example , let @xmath1 be any family of axis - parallel unit squares in the plane , and refer to figure [ fig : s ] . one can obtain a subset of pairwise - disjoint squares by repeatedly selecting the highest square that does not intersect the previously selected squares . then @xmath1 is pierced by the set of points consisting of the two lower vertices of each square in the subset . this implies that @xmath16 . the factor of @xmath17 can not be improved below @xmath18 since @xmath19 and @xmath20 for a family @xmath1 of five squares arranged into a 5-cycle @xcite . for a convex body @xmath2 in @xmath3 , @xmath21 , define @xmath22 where @xmath23 ranges over all families of translates of @xmath2 , and @xmath24 ranges over all families of ( positive ) homothets of @xmath2 . our previous discussion ( figure [ fig : s ] ) yields the bounds @xmath25 for any square @xmath2 . define @xmath26 ( resp . @xmath27 ) as the smallest number @xmath28 such that for any family @xmath1 of _ pairwise - intersecting _ translates ( resp . homothets ) of a convex body @xmath2 , there exists a set of @xmath28 points that intersects every member of @xmath1 . note that @xmath29 and @xmath30 generalize @xmath31 and @xmath32 . for any convex body @xmath2 , the four numbers @xmath5 , @xmath6 , @xmath26 , and @xmath27 are invariant under any non - singular affine transformation of @xmath2 , and we have the four inequalities @xmath33 , @xmath34 , @xmath35 , and @xmath36 . grnbaum @xcite showed that , for any convex body @xmath2 in @xmath3 , both @xmath26 and @xmath27 are bounded by functions of @xmath4 . deriving bounds on @xmath26 and @xmath27 for various types of convex bodies @xmath2 in @xmath3 is typical of classic gallai - type problems @xcite , and has been extensively studied . for example , a result by karasev @xcite states that @xmath37 for any convex body @xmath2 in the plane , i.e. , for any family of pairwise - intersecting translates of a convex body in the plane , there always exists a set of three points that intersects every member of the family . it is folklore that @xmath38 for any parallelogram @xmath2 ( see @xcite and the references therein ) . also , @xmath39 for any affinely regular hexagon @xmath2 ( * ? ? ? * example 2 ) , @xmath40 for any triangle @xmath2 @xcite for any triangle @xmath2 in appendix [ sec : t1 ] . ] , @xmath41 for any ( circular ) disk @xmath2 @xcite , and @xmath42 for any centrally symmetric convex body @xmath2 in the plane @xcite . perhaps the most celebrated result on point transversals of convex sets is alon and kleitman s solution to the hadwiger - debrunner @xmath43-problem @xcite . we refer to the two surveys @xcite and @xcite for more related results . the two numbers @xmath26 and @xmath27 bound the values of @xmath0 for special families @xmath1 of translates and homothets , respectively , of a convex body @xmath2 with @xmath44 . it is thus natural to study the general case @xmath45 , and to obtain estimates on @xmath5 and @xmath6 . despite the many previous bounds on @xmath26 and @xmath27 @xcite , first estimates on @xmath5 and @xmath6 have been only obtained recently @xcite . note that the related problem for families of @xmath4-intervals ( which are nonconvex ) has been extensively studied @xcite . kim et al . @xcite showed that @xmath5 is bounded by a function of @xmath4 for any convex body @xmath2 in @xmath3 , and gave the first bounds on @xmath5 for convex bodies @xmath2 in @xmath3 and on @xmath6 for convex bodies @xmath2 in the plane . in this paper , we show that @xmath6 is also bounded by a function of @xmath4 for any convex body @xmath2 in @xmath3 , and present new or improved bounds on both @xmath5 and @xmath6 for various convex bodies @xmath2 in @xmath3 for all dimensions @xmath4 . note that in the definitions of @xmath29 and @xmath30 , both the convexity of @xmath2 and the homothety of @xmath23 and @xmath24 are necessary for the values @xmath5 and @xmath6 to be bounded . to see the necessity of convexity , let @xmath2 be the union of a vertical line segment with endpoints @xmath46 and @xmath47 and a horizontal line segment with endpoints @xmath46 and @xmath48 , where the shared endpoint @xmath46 is the _ corner _ , and let @xmath1 be a family of @xmath49 translates of @xmath2 with corners at @xmath50 , @xmath51 @xcite . then at least @xmath52 points are required to intersect every member of @xmath1 . to see the necessity of homothety , let @xmath1 be a family of @xmath49 pairwise intersecting line segments ( or very thin rectangles ) in the plane such that no three have a common point . then again at least @xmath52 points are required to intersect every member of @xmath1 . [ [ definitions . ] ] definitions . + + + + + + + + + + + + for a convex body @xmath2 in @xmath3 , denote by @xmath53 the lebesgue measure of @xmath2 , i.e. , the area in the plane , or the volume in @xmath4-space for @xmath54 . for a family @xmath1 of convex bodies in @xmath3 , denote by @xmath55 the lebesgue measure of the union of the convex bodies in @xmath1 , i.e. , @xmath56 . for two convex bodies @xmath57 and @xmath58 in @xmath3 , denote by @xmath59 the minkowski sum of @xmath57 and @xmath58 . for a convex body @xmath2 in @xmath3 , denote by @xmath60 the _ scaled copy _ of @xmath2 by a factor of @xmath61 , denote by @xmath62 the _ reflexion _ of @xmath2 about the origin , and denote by @xmath63 the _ translate _ of @xmath2 by the vector from the origin to @xmath64 . write @xmath65 for @xmath66 . for two parallelepipeds @xmath67 and @xmath68 in @xmath3 that are parallel to each other ( but are not necessarily axis - parallel or orthogonal ) , denote by @xmath69 , @xmath70 , the length ratios of the edges of @xmath68 to the corresponding parallel edges of @xmath67 . then , for a convex body @xmath2 in @xmath3 , define @xmath71 where @xmath67 and @xmath68 range over all pairs of parallelepipeds in @xmath3 that are parallel to each other , such that @xmath72 . note that in this case @xmath73 for @xmath70 . we review some standard definitions of packing and covering densities in the following ; see ( * ? ? ? * chapter 1 ) . a family @xmath1 of convex bodies is a _ packing _ in a domain @xmath74 if @xmath75 and the convex bodies in @xmath1 are pairwise - disjoint ; @xmath1 is a _ covering _ of @xmath76 if @xmath77 . density _ of a family @xmath1 relative to a bounded domain @xmath76 is @xmath78 . if @xmath79 is the whole space , then the _ upper density _ and the _ lower density _ of @xmath1 are , respectively , @xmath80 where @xmath81 denote a ball of radius @xmath82 centered at the origin ( since we are taking the limit as @xmath83 , a hypercube of side length @xmath82 can be used instead of a ball of radius @xmath82 ) . for a convex body @xmath2 in @xmath3 , define the _ packing density _ of @xmath2 as @xmath84 where @xmath1 ranges over all packings in @xmath3 with congruent copies of @xmath2 , and define the _ covering density _ of @xmath2 as @xmath85 where @xmath1 ranges over all coverings of @xmath3 with congruent copies of @xmath2 . if the members of @xmath1 are restricted to translates of @xmath2 , then we have the _ translative _ packing and covering densities @xmath86 and @xmath87 . if the members of @xmath1 are further restricted to translates of @xmath2 by vectors of a lattice , then we have the _ lattice _ packing and covering densities @xmath88 and @xmath89 . note that the four densities @xmath87 , @xmath89 , @xmath86 , and @xmath88 are invariant under any non - singular affine transformation of @xmath2 . for any convex body @xmath2 in @xmath3 , we have the inequalities @xmath90 . for two convex bodies @xmath57 and @xmath58 in @xmath3 , denote by @xmath91 the smallest number @xmath92 such that @xmath57 can be covered by @xmath92 translates of @xmath58 . [ [ main - results . ] ] main results . + + + + + + + + + + + + + kim et al . @xcite recently proved that , for any family @xmath1 of translates of a convex body in @xmath3 , @xmath93 , in particular @xmath94 when @xmath95 , and moreover @xmath96 when @xmath97 . we improve these bounds for all dimensions @xmath4 in the following theorem : [ thm : conv ] for any family @xmath1 of translates of a convex body @xmath2 in @xmath3 , @xmath98 in particular , @xmath99 when @xmath95 , and @xmath100 when @xmath97 . for any parallelepiped @xmath2 in @xmath3 , we can choose two parallelepipeds @xmath67 and @xmath68 such that @xmath101 hence @xmath72 . then @xmath102 for @xmath70 , and @xmath103 . this implies the following corollary : [ cor : parallelepipedt ] for any family @xmath1 of translates of a parallelepiped in @xmath3 , @xmath104 . in contrast , for a family @xmath1 of ( not necessarily congruent or similar ) axis - parallel parallelepipeds in @xmath3 , the current best upper bound @xcite ( see also @xcite ) is @xmath105 kim et al . @xcite also proved that , for any family @xmath1 of translates of a centrally symmetric convex body in the plane , @xmath106 . the following theorem gives a general bound for any centrally symmetric convex body in @xmath3 and an improved bound ( if @xmath107 ) for any centrally symmetric convex body in the plane : [ thm : symm ] for any family @xmath1 of translates of a centrally symmetric convex body @xmath108 in @xmath3 , @xmath109 moreover , @xmath110 when @xmath95 , and @xmath111 when @xmath97 . for special types of convex bodies in the plane , the following theorem gives sharper bounds than the bounds implied by theorem [ thm : conv ] and theorem [ thm : symm ] . also , as we will show later , inequality below may give a better asymptotic bound than and for high dimensions . [ thm : greedyt ] let @xmath1 be a family of translates of a convex body @xmath2 in @xmath3 . then @xmath112 where @xmath113 ranges over all closed half spaces bounded by hyperplanes through the center of @xmath65 . moreover , @xmath114 if @xmath2 is a centrally symmetric convex body in the plane . also , 1 . if @xmath2 is a square , then @xmath115 , 2 . if @xmath2 is a triangle , then @xmath116 , 3 . if @xmath2 is a disk , then @xmath114 . having presented our bounds for families of translates , we now turn to families of homothets . kim et al . @xcite proved that , for any family @xmath1 of homothets of a convex body @xmath2 in the plane , @xmath117 and , if @xmath2 is centrally symmetric , @xmath118 . the following theorem gives a general bound for any convex body in @xmath3 , an improved bound for any centrally symmetric convex body in the plane , and additional bounds for special types of convex bodies in the plane : [ thm : greedyh ] let @xmath1 be a family of homothets of a convex body @xmath2 in @xmath3 . then @xmath119 in particular , @xmath120 if @xmath2 is a centrally symmetric convex body in the plane . moreover , 1 . if @xmath2 is a square , then @xmath121 , 2 . if @xmath2 is a triangle , then @xmath122 , 3 . if @xmath2 is a disk , then @xmath123 for any parallelepiped @xmath2 in @xmath3 , @xmath65 is a translate of @xmath124 and can be covered by @xmath125 translates of @xmath2 , thus @xmath126 . this implies the following corollary : [ cor : parallelepipedh ] for any family @xmath1 of homothets of a parallelepiped in @xmath3 , @xmath127 . both theorem [ thm : greedyt ] and theorem [ thm : greedyh ] are obtained by a simple greedy method , used also previously by kim et al . @xcite . although we have improved their bounds using new techniques in theorem [ thm : conv ] and theorem [ thm : symm ] , we show that a refined analysis of the simple greedy method yields even better asymptotic bounds for high dimensions in theorem [ thm : greedyt ] and theorem [ thm : greedyh ] . we will use the following lemma by chakerian and stein @xcite in our analysis : [ lem : cs ] for every convex body @xmath2 in @xmath3 there exist two parallelepipeds @xmath67 and @xmath68 such that @xmath72 , where @xmath67 and @xmath68 are homothetic with ratio at most @xmath4 . for any convex body @xmath2 in @xmath3 , let @xmath67 and @xmath68 be the two parallelepipeds in lemma [ lem : cs ] . since @xmath128 and @xmath129 , it follows that @xmath130 ; see also ( * ? ? ? * lemma 4 ) . the classic survey by danzer , grnbaum , and klee @xcite lists several other upper bounds due to rogers and danzer : ( i ) @xmath131 for any convex body @xmath2 in @xmath3 , ( ii ) @xmath132 and @xmath133 for any centrally symmetric convex body @xmath2 in @xmath3 . note that @xmath134 for any convex body @xmath2 in @xmath3 , according to a result of rogers @xcite . the following lemma summarizes the upper bounds on @xmath135 : [ lem : kappa ] for any convex body @xmath2 in @xmath3 , @xmath136 . moreover , if @xmath2 is centrally symmetric , then @xmath137 . from lemma [ lem : kappa ] and theorem [ thm : greedyh ] , it follows that @xmath6 is bounded by a function of @xmath4 , namely by @xmath138 , for any convex body @xmath2 in @xmath3 . since @xmath139 , lemma [ lem : kappa ] also provides upper bounds on @xmath140 in theorem [ thm : greedyt ] . as a result , implies an upper bound @xmath141 for any family @xmath1 of translates of a convex body in @xmath3 , which is better than the upper bound @xmath142 in when @xmath4 is sufficiently large . also , implies an upper bound @xmath143 for any family @xmath1 of translates of a centrally symmetric convex body @xmath108 in @xmath3 . schmidt @xcite showed that , for any centrally symmetric convex body @xmath108 , @xmath144 ; see also @xcite . hence implies the bound @xmath145 . note that @xmath146 . so may be also better than for high dimensions . table [ tab1 ] summarizes the current best upper bounds on @xmath5 and @xmath6 ( obtained by us and by others ) for various types of convex bodies @xmath2 in @xmath3 . .upper bounds on @xmath5 and @xmath6 for a convex body @xmath2 in @xmath3 . @xmath147by theorem [ thm : greedyh ] and lemma [ lem : kappa ] : for @xmath95 , @xmath148 and @xmath149 . [ cols= " < , > , < , > " , ]

according to a classical result of grnbaum , the transversal number of any family of pairwise - intersecting translates or homothets of a convex body in is bounded by a function of . denote by ( resp . ) the supremum of the ratio of the transversal number to the packing number over all families of translates ( resp . homothets ) of a convex body in . kim et al . recently showed that is bounded by a function of for any convex body in , and gave the first bounds on for convex bodies in and on for convex bodies in the plane . here we show that is also bounded by a function of for any convex body in , and present new or improved bounds on both and for various convex bodies in for all dimensions . * keywords * : geometric transversals , gallai - type problems , packing and covering , approximation algorithms .

according to a classical result of grnbaum , the transversal number of any family of pairwise - intersecting translates or homothets of a convex body in is bounded by a function of . denote by ( resp . ) the supremum of the ratio of the transversal number to the packing number over all families of translates ( resp . homothets ) of a convex body in . kim et al . recently showed that is bounded by a function of for any convex body in , and gave the first bounds on for convex bodies in and on for convex bodies in the plane . here we show that is also bounded by a function of for any convex body in , and present new or improved bounds on both and for various convex bodies in for all dimensions . our techniques explore interesting inequalities linking the covering and packing densities of a convex body . our methods for obtaining upper bounds are constructive and lead to efficient constant - factor approximation algorithms for finding a minimum - cardinality point set that pierces a set of translates or homothets of a convex body . * keywords * : geometric transversals , gallai - type problems , packing and covering , approximation algorithms .

0910.3085

c

we have presented several methods to encode sets of finite vertices as single vertices . in the first part , we used depth - first spanning trees to encode edges by vertices . as an application we were able to extend courcelle s result on the collapse of @xmath1 to @xmath2 on sparse hypergraphs from countable hypergraphs to hypergraphs of arbitrary cardinality . in the second part we used network flows to encode arbitrary finite sets by vertices . let us mention some open questions . considering the first part it would be interesting to find out whether sparse classes are the only examples where @xmath1 collapses to @xmath2 . is there a class @xmath26 that is not @xmath0-sparse , for any @xmath0 , such that over @xmath26 every @xmath1-sentence is equivalent to an @xmath2-sentence.08em ? the results of the second part are much less complete . it is unlikely that they are the best possible . improve theorem [ thm : coding sets by vertices ] by allowing more general classes of graphs or hypergraphs.08em ; more general classes of partial functions @xmath583 . our results were inspired by work of colcombet and lding @xcite . the question arises of whether we can also generalise the remaining results of that article . can we prove corollary 4.4 of for other graphs than trees.08em ?

according to a theorem of courcelle monadic second - order logic and guarded second - order logic ( where one can also quantify over sets of edges ) have the same expressive power over the class of all countable-sparse hypergraphs . in the first part of the present paper we extend this result to hypergraphs of arbitrary cardinality . in the second part , we present a generalisation dealing with methods to encode sets of vertices by single vertices .

according to a theorem of courcelle monadic second - order logic and guarded second - order logic ( where one can also quantify over sets of edges ) have the same expressive power over the class of all countable-sparse hypergraphs . in the first part of the present paper we extend this result to hypergraphs of arbitrary cardinality . in the second part , we present a generalisation dealing with methods to encode sets of vertices by single vertices .

1508.05240

c

in this paper , we have presented new alma observations of the four brightest stars in the globular cluster 47 tucanae , examining the co @xmath97 transitions . no source was detected . having modelled the interaction between the isrf and the circumstellar enviroment , we expect that the co line fluxes from the giant stars in 47 tuc are around two orders of magnitude below our observation limit . for any given globular cluster giant , if the wind is optically thin , then the radius of co dissociation should exhibit an approximately linear relation with metallicity and an approximately inverse relation with isrf strength ( section [ codisssect ] ) . the radiation field should be highly stochastic , driven by the temporally variable population of hot white dwarfs . the population of young white dwarfs is not well determined in most clusters , therefore it is not immediately possible to identify which clusters will be the best candidates for observation . however , the isrf can be expected to vary by several orders of magnitude within a cluster , both temporally and spatially ( table [ isrftable ] ) . targets far from the cluster centre may experience radiation fields that are weaker than those in the solar neighbourhood ( though they will still contain harder radiation ) . giant stars at large distances from their host clusters centres may therefore provide the best metal - poor stars around which to detect stellar winds and measure their expansion velocities . table [ isrftable ] indicates that v2 and v3 may occasionally experience isrfs that are weaker than the solar neighbourhood . while this seems not to be the case at present , we can hope that a similar situation can be found in other milky way globular clusters . since the higher stellar density forces the co envelopes to be smaller than their galactic counterparts , the co(32 ) line ( or higher rotational states ) may be considerably brighter than co(21 ) , as these are better populated in the warmer regions close to the star ( table [ modeltable ] ) . we therefore suggest observations to detect co line strengths and widths from globular cluster stars focus on : 1 . stars at large radii from their host clusters and 2 . high co rotational states emitted closer to the star . we also advocate further observations of the circumstellar dust . unexpectedly large amounts of dust are seen around many globular cluster stars @xcite . their spectra have strong silicate features @xcite , but also an underlying continuum , possibly explained by metallic iron @xcite . this contrasts with the notable absence of interstellar dust within clusters @xcite , and suggests dust is being destroyed while still in the circumstellar environment . the highest - energy photons ( @xmath040 ev , @xmath0300 ) are likely to penetrate into the dust - forming regions . while they are of insufficient number to dissociate a large fraction of co in this region , they may still be important in the photo - chemical formation of dust . finally , we note that this situation is unlikely to be limited to globular clusters . the dissociation of co around agb stars by nearby post - agb stars and white dwarfs is largely a function of stellar density ( although stellar age , elemental composition and the density of the surrounding ism are also important ) . this source of co dissociation may also become important in old open clusters , nuclear star clusters , and gas - poor galaxies . the implications for the formation , chemistry , evolution and survival of dust in such environments are significant .

alma observations show a non - detection of carbon monoxide around the four most luminous asymptotic giant branch ( agb ) stars in the globular cluster 47 tucanae . we estimate that the co envelopes will be truncated at a few hundred stellar radii from their host stars and that the line intensities are about two orders of magnitude below our current detection limits .

alma observations show a non - detection of carbon monoxide around the four most luminous asymptotic giant branch ( agb ) stars in the globular cluster 47 tucanae . stellar evolution models and star counts show that the mass - loss rates from these stars should be.23.5 10 m yr . we would navely expect such stars to be detectable at this distance ( 4.5 kpc ) . by modelling the ultraviolet radiation field from post - agb stars and white dwarfs in 47 tuc , we conclude co should be dissociated abnormally close to the stars . we estimate that the co envelopes will be truncated at a few hundred stellar radii from their host stars and that the line intensities are about two orders of magnitude below our current detection limits . the truncation of co envelopes should be important for agb stars in dense clusters . observing the co ( 32 ) and higher transitions and targeting stars far from the centres of clusters should result in the detections needed to measure the outflow velocities from these stars . [ firstpage ] stars : mass - loss circumstellar matter infrared : stars stars : winds , outflows globular clusters : individual : ngc 104 stars : agb and post - agb

1302.0983

i

silicon is at the heart of all modern microelectronics . its properties have allowed the semiconductor industry to follow moore s law for nearly half a century , delivering nowadays billions of nanometre - scale transistors per chip . remarkably , silicon is also an ideal material to manipulate quantum information encoded in individual electron spins @xcite . this is a consequence of the weak spin - orbit coupling and the existence of an abundant spin - zero isotope , which can be further enriched to obtain a `` semiconductor vacuum '' in which an electron spin can preserve a coherent quantum superposition state for exceptionally long times @xcite . + in order to define a robust spin-@xmath0/@xmath1 qubit hilbert space , it is necessary that the energy scale of the two - level system is well separated from higher excitations . in this respect , a major challenge for the use of silicon is represented by the multi - valley nature of its conduction band . in a bulk silicon crystal the conduction band minima are six - fold degenerate , but in a two - dimensional electron gas ( 2deg ) , the degeneracy is broken @xcite into a two - fold degenerate ground state ( @xmath2 valleys ) and a four - fold degenerate excited state ( @xmath3 valleys ) , due to vertical confinement of electrons with different effective mass along the longitudinal and transverse directions , respectively . furthermore , the @xmath2 valley degeneracy is generally lifted by a sharp perpendicular potential @xcite and the relevant energy separation is termed the valley splitting ( vs ) . + the valley splitting depends on physics at the atomic scale @xcite ( e.g. roughness , alloy and interface disorder ) , and so it is not surprising that experiments have revealed a large variability of splittings among devices , ranging from hundreds of @xmath4ev @xcite up to tens of mev in exceptional cases @xcite . at present , the lack of a reliable experimental strategy to achieve control over the vs is driving an intense research effort for the development of devices that can assure robust electron spin qubits by minimising multi - valley detrimental effects @xcite , or even exploit the valley degree of freedom @xcite for new types of qubits . + another crucial parameter to assess the suitability of a physical system to encode spin - based qubits is the relaxation time of spin excited states ( @xmath5 ) . spin lifetimes have been measured for gate - defined si quantum dots ( qds ) @xcite , si / sige qds @xcite and donors in si @xcite , reporting values which span from a few milliseconds to a few seconds . furthermore , the dependence of the spin relaxation rate ( @xmath6 ) on an externally - applied magnetic field ( @xmath7 ) has been investigated . different mechanisms apply to donors and qds , accounting for observed @xmath8 and @xmath9 dependencies @xcite , respectively . in principle , @xmath10 depends on the valley configuration and the details of the excited states above the spin ground state . however , until now , no experimental observation of the effects of a variable vs on the relaxation rate has been reported . + here , we demonstrate for the first time that the valley splitting in a silicon qd can be finely tuned by direct control of an electrostatic gate potential . we find that the dependence of the vs on vertical electric field at the si / sio@xmath1 interface is strikingly linear , and show that its tunability is in excellent agreement with atomistic tight - binding predictions . we demonstrate accurate control of the vs over a range of about 500 @xmath4ev and use it to explore the physics of spin relaxation for different qd occupancies ( @xmath11=1 , 2 , 3 ) . we probe both the regime where the vs is much larger than the zeeman splitting at all magnetic fields and that where the valley and spin splittings are comparable . we observe a dramatic enhancement of the spin decay rate ( relaxation hot - spot ) when spin and valley splittings coincide . to our knowledge , such hot - spots have been predicted for relaxation involving orbital states @xcite ( not valley states ) , but these are yet to be observed . we develop an analytic theory that explains the @xmath7-field dependence of the relaxation rates and the details of the relaxation hot - spot in terms of admixing of spin - valley states . this mechanism is seen to be significantly more prominent than the conventional spin - orbit hybridization @xcite . ) is sensitive to the qd charge state due to the qd / set capacitive coupling ( @xmath12 ) . ( c ) device cross - sectional schematic . an electron layer is formed underneath the positively biased gates : r1 and r2 define the qd reservoir ; p controls the qd population ; and st the sensor s island . the sio@xmath1 layer ( in purple ) thickness and plunger gate width are indicated . ( d ) energy diagram showing qualitatively the conduction band profile in the device . electrons accumulate wherever the gate bias lowers the conduction band below the fermi level , @xmath13 . ( e ) charging energy as a function of electron number . spikes corresponding to complete 2d shell filling are observed . ( f ) schematic of electron filling for two - valley 2d fock - darwin states . each state can hold two electrons of antiparallel spin and is identified by a pair of quantum numbers ( @xmath14 ) and its valley occupancy ( @xmath15 ) . ]

here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metal - oxide - semiconductor quantum dot , providing splittings spanning 0.3 - 0.8 mev . the splitting varies linearly with applied electric field , with a ratio in agreement with atomistic tight - binding predictions . we demonstrate single - shot spin readout and measure the spin relaxation for different valley configurations and dot occupancies , finding one - electron lifetimes exceeding 2 seconds . an analytical theory describes the magnetic field dependence of the relaxation rate , including the presence of a dramatic rate enhancement ( or hot - spot ) when zeeman and valley splittings coincide .

although silicon is a promising material for quantum computation , the degeneracy of the conduction band minima ( valleys ) must be lifted with a splitting sufficient to ensure formation of well - defined and long - lived spin qubits . here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metal - oxide - semiconductor quantum dot , providing splittings spanning 0.3 - 0.8 mev . the splitting varies linearly with applied electric field , with a ratio in agreement with atomistic tight - binding predictions . we demonstrate single - shot spin readout and measure the spin relaxation for different valley configurations and dot occupancies , finding one - electron lifetimes exceeding 2 seconds . spin relaxation occurs via phonon emission due to spin - orbit coupling between the valley states , a process not previously anticipated for silicon quantum dots . an analytical theory describes the magnetic field dependence of the relaxation rate , including the presence of a dramatic rate enhancement ( or hot - spot ) when zeeman and valley splittings coincide .

1610.04741

i

in a _ geometric drawing of a graph @xmath0 _ , the vertices of @xmath0 are represented by distinct points in the plane and each edge @xmath10 of @xmath0 is represented by the line segment between the pair of points that represent the vertices of @xmath10 . as usual , we identify the vertices and their images , as well as the edges and the line segments representing them . let @xmath11 be a finite set of points in the plane in _ general position _ , that is , there are no three collinear points in @xmath11 . the _ complete geometric graph _ @xmath12 is the geometric drawing of the complete graph @xmath13 with vertices represented by the points of @xmath11 . an _ obstacle _ is a polygon in the plane . an _ obstacle representation _ of a graph @xmath0 is a geometric drawing @xmath14 of @xmath0 together with a set @xmath15 of obstacles such that two vertices of @xmath0 are connected by an edge @xmath10 if and only if the line segment representing @xmath10 in @xmath14 is disjoint from all obstacles in @xmath15 . the _ obstacle number _ @xmath1 of @xmath0 is the minimum number of obstacles in an obstacle representation of @xmath0 . the _ convex obstacle number _ @xmath16 of a graph @xmath0 is the minimum number of obstacles in an obstacle representation of @xmath0 in which all the obstacles are required to be convex . clearly , we have @xmath17 for every graph @xmath0 . for a positive integer @xmath2 , let @xmath18 be the maximum obstacle number of a graph on @xmath2 vertices . in this paper , we provide the first nontrivial upper bound on @xmath18 ( theorem [ thm - generalupperbound ] ) . we also show a lower bound for the number of graphs with small obstacle number ( theorem [ thm - count - lb ] ) and a matching lower bound for the complexity of a collection of faces in an arrangement of line segments that share endpoints ( theorem [ thm - arrangements ] ) . all proofs of our results are based on so - called _ @xmath19-dilated bipartite drawings of @xmath20 _ , which we introduce in section [ sec : bipartitedrawings ] . in the following , we make no serious effort to optimize the constants . all logarithms in this paper are base 2 . the obstacle number of a graph was introduced by alpert , koch , and laison @xcite who showed , among several other results , that for every positive integer @xmath6 there is a graph @xmath0 with @xmath21 . using extremal graph theoretic tools , pach and sariz @xcite proved that the number of labeled @xmath2-vertex graphs with obstacle number at most @xmath6 is at most @xmath22 for every fixed integer @xmath6 . this implies that there are bipartite graphs with arbitrarily large obstacle number . mukkamala , pach , and sariz @xcite established more precise bounds by showing that the number of labeled @xmath2-vertex graphs with obstacle number at most @xmath6 is at most @xmath23 for every fixed positive integer @xmath6 . it follows that @xmath24 . later , mukkamala , pach , and plvlgyi @xcite improved the lower bound to @xmath25 . currently , the strongest lower bound on the obstacle number is due to dujmovi and morin @xcite who showed @xmath26 . surprisingly , not much has been done for the general upper bound on the obstacle number . we are only aware of the trivial bound @xmath27 for every graph @xmath0 on @xmath2 vertices . this follows easily , as we can consider the complete geometric graph @xmath12 for some point set @xmath11 of size @xmath2 and place a small obstacle @xmath28 on every _ non - edge _ @xmath10 of @xmath0 such that @xmath28 intersects only @xmath10 in @xmath12 . a non - edge of a graph @xmath29 is an element of @xmath30 . concerning special graph classes , fulek , saeedi , and sariz @xcite showed that the convex obstacle number is at most five for every outerplanar graph , and at most four for every bipartite permutation graph . chaplick et al . @xcite showed that @xmath31 whenever @xmath32 and that @xmath33 . they also found an @xmath34-vertex graph @xmath0 with @xmath35 such that in every its representation with a single obstacle , the obstacle lies in one of the bounded cells of the drawing of @xmath0 . berman et al . @xcite constructed a @xmath36-vertex planar graph @xmath0 with @xmath37 . alpert , koch , and laison @xcite asked whether the obstacle number of every graph on @xmath2 vertices can be bounded from above by a linear function of @xmath2 . we show that this is true for bipartite graphs and split graphs , even for the convex obstacle number . split graph _ is a graph for which the vertex set can be partitioned into an independent subset and a clique . [ thm - bipartiteupperbound ] if @xmath29 is a bipartite graph or a split graph , then we have @xmath38 where @xmath39 denotes the complement of @xmath0 . on the other hand , a modification of the proof of the lower bound by mukkamala , pach , and plvlgyi @xcite implies that there are bipartite graphs @xmath0 on @xmath2 vertices with @xmath40 for every positive integer @xmath2 . in contrast to the above question of alpert , koch , and laison on the existence of a linear upper bound , mukkamala , pach , and plvlgyi @xcite conjectured that the maximum obstacle number of @xmath2-vertex graphs is around @xmath41 . we refute this conjecture by showing the first non - trivial general upper bound on the obstacle number of graphs . in fact , we prove a stronger result that provides a general upper bound for the convex obstacle number . [ thm - generalupperbound ] for every positive integer @xmath2 and every graph @xmath0 on @xmath2 vertices , the convex obstacle number of @xmath0 satisfies @xmath42 the question whether the upper bound on @xmath18 can be improved to @xmath4 remains open . we can , however , prove @xmath43 provided that the chromatic number ( or even the _ subchromatic number _ ) of @xmath0 is bounded from above by a constant . let @xmath29 be a graph and let @xmath44 be a function that assigns a _ color _ @xmath45 to every vertex @xmath46 of @xmath0 . we call @xmath47 a _ subcoloring of @xmath0 _ if for every @xmath48 the color class @xmath49 induces a disjoint union of cliques in @xmath0 subchromatic number of @xmath0 _ , denoted by @xmath50 , is the least number of colors needed in any subcoloring of @xmath0 . we prove the following result that asymptotically implies theorems [ thm - bipartiteupperbound ] and [ thm - generalupperbound ] . [ thm - chromaticupperbound ] for every positive integer @xmath2 and every graph @xmath0 on @xmath2 vertices , the convex obstacle number of @xmath0 satisfies @xmath51 note that we have @xmath52 , thus theorem [ thm - chromaticupperbound ] gives the bound @xmath43 if the chromatic number of @xmath0 or the chromatic number of its complement @xmath53 is bounded from above by a constant . theorem [ thm - chromaticupperbound ] immediately implies an analogous statement for _ cochromatic number of @xmath0 _ , since the cochromatic number of a graph @xmath0 , first defined by lesniak and straight @xcite , is at most @xmath50 for every graph @xmath0 . for positive integers @xmath6 and @xmath2 , let @xmath54 be the number of labeled @xmath2-vertex graphs with obstacle number at most @xmath6 . the lower bounds on the obstacle number by mukkamala , pach , and plvlgyi @xcite and by dujmovi and morin @xcite are both based on the upper bound @xmath55 . in fact , any improvement on the upper bound for @xmath54 will translate into an improved lower bound on the obstacle number @xcite . dujmovi and morin @xcite conjectured @xmath56 where @xmath57 . we show the following lower bound on @xmath54 . [ thm - count - lb ] for every pair of integers @xmath2 and @xmath6 satisfying @xmath58 , we have @xmath59 an _ arrangement @xmath60 of line segments _ is a finite collection of line segments in the plane . the line segments of @xmath60 partition the plane into _ vertices _ , _ edges _ , and _ cells_. a vertex is a common point of two or more line segments . removing the vertices from the line segments creates a collection of subsegments which are called edges . the cells are the connected components of the complement of the line segments . a _ face _ of @xmath60 is a closure of a cell . note that every geometric drawing of a graph is an arrangement of line segments and vice versa . the edges of the graph correspond to the line segments of the arrangement and the vertices of the graph correspond to the endpoints of the line segments . a line segment @xmath61 of @xmath60 is _ incident _ to a face @xmath62 of @xmath60 if @xmath61 and @xmath62 share an edge of @xmath60 . the _ complexity of a face _ @xmath62 is the number of the line segments of @xmath60 that are incident to @xmath62 . if @xmath63 is a set of faces of @xmath60 , then the _ complexity of @xmath63 _ is the sum of the complexities of @xmath62 taken over all @xmath64 . an _ arrangement of lines _ is a finite collection of lines in the plane with faces and their complexity defined analogously . edelsbrunner and welzl @xcite constructed an arrangement of @xmath65 lines having a set of @xmath66 faces with complexity @xmath67 for every @xmath65 and @xmath68 . wiernik and sharir @xcite constructed an arrangement of @xmath65 line segments with a single face of complexity @xmath69 . these two constructions can be combined to provide the lower bound @xmath70 for the complexity of @xmath66 faces in an arrangement of @xmath65 line segments , where @xmath68 . the best upper bound for the complexity of @xmath66 faces in an arrangement of @xmath65 line segments is @xmath71 by aronov et al . @xcite . arkin et al . @xcite studied arrangements whose line segments share endpoints . that is , they considered the maximum complexity of a face when we bound the number of endpoints of the line segments instead of the number of the line segments . they showed that the complexity of a single face in an arrangement of line segments with @xmath2 endpoints is at most @xmath72 . an @xmath73 lower bound was then proved by matouek and valtr @xcite . arkin et al . @xcite asked what is the maximum complexity of a set of @xmath66 faces in an arrangement of line segments with @xmath2 endpoints . since every arrangement of line segments with @xmath2 endpoints contains at most @xmath74 line segments , the upper bound @xmath75 can be deduced from the bound by aronov et al . @xcite . together with the upper bound @xmath76 , which follows from the bound by arkin et al . @xcite on the complexity of a single face , we obtain an upper bound @xmath77 on the complexity of @xmath66 faces in an arrangement of line segments with @xmath2 endpoints . we give the following lower bound . [ thm - arrangements ] for every sufficiently large integer @xmath2 , there is an arrangement @xmath60 of line segments with @xmath2 endpoints such that for every @xmath66 satisfying @xmath78 there is a set of at most @xmath66 faces of @xmath60 with complexity @xmath79 . whenever @xmath80 , this lower bound matches the upper bound up to a multiplicative constant . for @xmath81 , the lower bound differs from the upper bound by at most an @xmath82 multiplicative factor . see also table [ tab : arrgbounds ] . .summary of the best known upper and lower bounds on the complexity of @xmath66 faces in an arrangement of segments with @xmath2 endpoints . [ cols="<,<,<,<",options="header " , ]

an _ obstacle representation _ of a graph is a set of points in the plane representing the vertices of , together with a set of polygonal obstacles such that two vertices of are connected by an edge in if and only if the line segment between the corresponding points avoids all the obstacles . the _ obstacle number of _ is the minimum number of obstacles in an obstacle representation of . we provide the first non - trivial general upper bound on the obstacle number of graphs by showing that every-vertex graph satisfies . this refutes a conjecture of mukkamala , pach , and plvlgyi . for-vertex graphs with bounded chromatic number , we improve this bound to . both bounds apply even when the obstacles are required to be convex . we also prove a lower bound on the number of-vertex graphs with obstacle number at most for and an asymptotically matching lower bound for the complexity of a collection of faces in an arrangement of line segments with endpoints .

an _ obstacle representation _ of a graph is a set of points in the plane representing the vertices of , together with a set of polygonal obstacles such that two vertices of are connected by an edge in if and only if the line segment between the corresponding points avoids all the obstacles . the _ obstacle number of _ is the minimum number of obstacles in an obstacle representation of . we provide the first non - trivial general upper bound on the obstacle number of graphs by showing that every-vertex graph satisfies . this refutes a conjecture of mukkamala , pach , and plvlgyi . for-vertex graphs with bounded chromatic number , we improve this bound to . both bounds apply even when the obstacles are required to be convex . we also prove a lower bound on the number of-vertex graphs with obstacle number at most for and an asymptotically matching lower bound for the complexity of a collection of faces in an arrangement of line segments with endpoints .

1609.00094

i

a fundamental challenge in studying complex networked systems is to reveal the interplay between network topology and functionality@xcite . here we tackle this challenge by investigating a classical notion in graph theory , i.e. articulation points . a node in a network is an articulation point ( ap ) if its removal disconnects the network or increases the number of connected components in the network@xcite . those ap nodes play important roles in ensuring the connectivity of many real - world networks . for example , in infrastructure networks such as air traffic networks or power grids , aps , if disrupted or attacked , pose serious risks to the infrastructure @xcite . in wireless sensor networks , failures of aps will block data transmission from one network component to others @xcite . in the yeast protein - protein interaction network , lethal mutations are enriched in the group of highly connected proteins that are aps @xcite . identification of aps also helps us better solve other challenging problems , e.g. the calculation of determinants of large matrices @xcite ; and the vertex cover problem on large graphs , which is a notorious np - complete problem @xcite . representing natural vulnerabilities of a network , aps are potential targets of attack if one aims for immediate damage to a network . for example , in the terrorist communication network of the 9/11 attacks on u.s . ( fig.1a ) , cutting off those ap members ( shown in red ) could have caused a great deal of disruption in the terrorists ability to communicate @xcite . note that the removal of an ap in a network may lead to the emergence of new aps in the remainder of the network . a brute - force strategy of network destruction is to _ iteratively _ remove the most destructive ap that will cause the most nodes disconnected from the giant connected component ( gcc ) of the current network . given a limited `` budget '' ( i.e. the number of nodes to be removed ) , this ap - targeted attack ( apta ) strategy is very efficient in reducing the gcc , compared with other existing strategies @xcite . indeed , we find that for a small fraction of removed nodes apta leads to the quickest reduction of gcc for a wide range of real - world networks from technological to infrastructure , biological , communication , and social networks ( see appendix [ sec.si-i-a ] ) . depending on the initial network structure , apta would either completely decompose the network or result in a residual gcc that occupies a finite fraction of the network . this residual gcc is a biconnected component ( or bicomponent ) , in which any two nodes are connected by at least two independent paths and hence no ap exists @xcite . for accuracy , we will call it residual giant bicomponent ( rgb ) hereafter . this rgb naturally represents a _ core _ that maintains the _ structural integrity _ of the network . identification and removal of aps also provide us a novel perspective on the organizational principles of complex networks . for example , in a terrorist communication network , each ap member can be considered as a _ messenger _ of a particular subnetwork , because any information exchange between that subnetwork and the rest of the network passes through the ap . all the aps and their associated subnetworks in the original network constitute the _ first layer _ of the terrorist network . after removing all the aps in the original network , the first layer is peeled off , new aps emerge and the _ second layer _ of the network is exposed . we can repeat this greedy aps removal ( gapr ) process until there is no ap left in the network . figure 1 illustrates this novel network decomposition process in the 9/11 terrorist communication network , which has 62 terrorists . we find that this network consists of 3 layers and an rgb of 26 nodes . ( note that the rgb associated with the gapr process is similar to but not necessarily the same as that of the apta process , see appendix [ sec.si-i-b ] ) interestingly , among those 26 rgb nodes , 16 of them are hijackers in the 9/11 terrorist attack , which in total has 19 hijackers . in a sense , this rgb serves as a _ core _ maintaining the _ functionality _ of this covert network , which has a particular goal hijacking . note that some of the hijackers in the rgb are not hubs ( i.e. highly connected nodes ) , but only have two or three neighbors in the network . hence they can not be easily identified through traditional network decompositions , e.g. the classical @xmath0-core decomposition @xcite . despite the importance of aps in ensuring the connectivity of many real - world networks , we still lack a deep understanding on the roles and properties of aps in many complex networks . we do nt have a theory to quantify if a given network has overrepresented or underrepresented aps comparing to its randomized counterparts . moreover , the implication of the rgb serving as a core that maintains the structural integrity and/or functionality of a network has never been explored before . in this article we offer an analytical framework to study those fundamental issues pertinent to aps in complex networks .

an articulation point in a network is a node whose removal disconnects the network . those nodes play key roles in ensuring connectivity of many real - world networks , from infrastructure networks to protein interaction networks and terrorist communication networks . despite their fundamental importance , a general framework of studying articulation points in complex networks is lacking . here we develop analytical tools to study key issues pertinent to articulation points , e.g. the expected number of them and the network vulnerability against their removal , in an arbitrary complex network . we find that a greedy articulation point removal process provides us a novel perspective on the organizational principles of complex networks .

an articulation point in a network is a node whose removal disconnects the network . those nodes play key roles in ensuring connectivity of many real - world networks , from infrastructure networks to protein interaction networks and terrorist communication networks . despite their fundamental importance , a general framework of studying articulation points in complex networks is lacking . here we develop analytical tools to study key issues pertinent to articulation points , e.g. the expected number of them and the network vulnerability against their removal , in an arbitrary complex network . we find that a greedy articulation point removal process provides us a novel perspective on the organizational principles of complex networks . moreover , this process is associated with two fundamentally different types of percolation transitions with a rich phase diagram . our results shed light on the design of more resilient infrastructure networks and the effective destruction of terrorist communication networks .

1412.1958

i

it has long been known that general relativity and electrodynamics exhibit certain formal similarities , see matte @xcite and bel @xcite , or for a more recent overview costa and natrio @xcite . on the other hand , petrov type d spacetimes can be thought of as spacetimes with coulomb - like sources , as already pointed out by szekeres @xcite ; see also stephani _ et al_. @xcite , p. 39 . in this sense , they are the closest scenario to electro / magneto - statics that general relativity has to offer . the plebaski demiaski solution @xcite is of type d , has seven free parameters , and can be used to model various axisymmetric electrovacuum spacetimes , in particular the kerr solution @xcite . overviews of original papers for various subclasses of the pd solution can be found in plebaski and demiaski @xcite , table 1 , or stephani _ et al_. @xcite , table 21.1 . for our work , we found particularly useful the reviews on kerr nut(a)desitter by mars and senovilla @xcite ; the ( rotating ) @xmath5-metric by hong and teo @xcite ; kerr - desitter by chambers @xcite ; and of course the kerr metric itself , see for example carter @xcite , defelice and bradley @xcite , and cherubini _ et al_. @xcite . a complete physical interpretation of the pd coordinates and its free parameters has been lacking for a long time , but it is possible to transform the pd coordinates to boyer lindquist - like coordinates , see griffiths and podolsk @xcite . the purpose of this paper is twofold : first of all , it is of general interest to calculate various geometric quantities of the pd solution in physically well - motivated coordinates . and secondly , the setting of the pd solution may endow the more general , formal similarities between gravitation and electrodynamics with an explicit physical understanding . we proceed as follows : in the first section , we fix our notation and demonstrate that the pd solution satisfies the einstein maxwell equations . in the second section , we briefly check the coordinate transformations introduced by griffiths and podolsk , and confirm that the pd solution , expressed in their boyer lindquist - like coordinates , still fulfills the einstein maxwell equations . in the third section , we decompose the curvature irreducibly and calculate its pieces for the pd solution in both coordinates . for the asymptotically flat case , the asymptotic structure of the weyl 2-form is analyzed and compared to classical electrodynamics . in the fourth section , we move on to the quadratic curvature invariants and recover the well - known analogies between general relativity and electrodynamics . drawing on these analogies , we provide a quick way to derive the recently introduced bel robinson 3-form in section 5 and confirm its correspondence to the bel robinson tensor . finally , we turn to cubic expressions in curvature and calculate the pieces of the irreducible decomposition of the kummer tensor , an important tensor related to the principal null directions of curvature , for a general type d spacetime . in the concluding section , we briefly summarize our findings .

robinson 3-form , from which the bel we also determine the kummer tensor , a tensor cubic in curvature , for a general type d solution for the first time , and calculate the pieces of its irreducible decomposition . the calculations are carried out in two coordinate systems : in the original polynomial pd coordinates , and in a modified boyer lindquist - like version introduced by griffiths and podolsk ( gp ) allowing for a more straightforward physical interpretation of the free parameters . _ file : 05_elm_inv_v9.tex , jul 2 , 2015 _

analogies between gravitation and electromagnetism have been known since the 1950s . here , we examine a fairly general type d solution the exact seven parameter solution of plebaski demiaski ( pd)to demonstrate these analogies for a physically meaningful spacetime : the two quadratic curvature invariants and are evaluated analytically . in the asymptotically flat case , the leading terms of and can be interpreted as gravitoelectric mass and gravitoelectric current of the pd solution , respectively , if there are no gravitomagnetic monopoles present . furthermore , the square of the bel robinson tensor reads for the pd solution , reminiscent of the square of the energy density in electrodynamics . by analogy to the energy - momentum 3-form of the electromagnetic field , we provide an alternative way to derive the recently introduced bel robinson 3-form , from which the bel robinson tensor can be calculated . we also determine the kummer tensor , a tensor cubic in curvature , for a general type d solution for the first time , and calculate the pieces of its irreducible decomposition . the calculations are carried out in two coordinate systems : in the original polynomial pd coordinates , and in a modified boyer lindquist - like version introduced by griffiths and podolsk ( gp ) allowing for a more straightforward physical interpretation of the free parameters . _ file : 05_elm_inv_v9.tex , jul 2 , 2015 _

1412.1958

c

in this paper , we have used the plebaski demiaski solution of general relativity to show that the formal analogy of invariants in general relativity and electrodynamics can be understood in a physical sense : the electric field @xmath202 ( charges ) corresponds to the mass parameter @xmath116 ( gravitational charge ) , and the magnetic field @xmath203 ( moving charges ) corresponds to the angular momentum @xmath204 ( moving gravitational charges ) for vanishing gravitomagnetic monopoles . secondly , we rederived the bel robinson 3-form in a novel way which uses the definition of electromagnetic energy - momentum . this derivation is physically straightforward finally , we calculated the irreducible pieces of the kummer tensor for any type d spacetime . these results are hoped to be helpful for determining the relation of the properties of the kummer tensor with the principal null directions of curvature for a generic type d spacetime .

analogies between gravitation and electromagnetism have been known since the 1950s . here , we examine a fairly general type d solution the exact seven parameter solution of plebaski demiaski ( pd)to demonstrate these analogies for a physically meaningful spacetime : the two quadratic curvature invariants and are evaluated analytically . in the asymptotically flat case , the leading terms of and can be interpreted as gravitoelectric mass and gravitoelectric current of the pd solution , respectively , if there are no gravitomagnetic monopoles present . robinson tensor reads for the pd solution , reminiscent of the square of the energy density in electrodynamics . by analogy to the energy - momentum 3-form of the electromagnetic field , we provide an alternative way to derive the recently introduced bel

analogies between gravitation and electromagnetism have been known since the 1950s . here , we examine a fairly general type d solution the exact seven parameter solution of plebaski demiaski ( pd)to demonstrate these analogies for a physically meaningful spacetime : the two quadratic curvature invariants and are evaluated analytically . in the asymptotically flat case , the leading terms of and can be interpreted as gravitoelectric mass and gravitoelectric current of the pd solution , respectively , if there are no gravitomagnetic monopoles present . furthermore , the square of the bel robinson tensor reads for the pd solution , reminiscent of the square of the energy density in electrodynamics . by analogy to the energy - momentum 3-form of the electromagnetic field , we provide an alternative way to derive the recently introduced bel robinson 3-form , from which the bel robinson tensor can be calculated . we also determine the kummer tensor , a tensor cubic in curvature , for a general type d solution for the first time , and calculate the pieces of its irreducible decomposition . the calculations are carried out in two coordinate systems : in the original polynomial pd coordinates , and in a modified boyer lindquist - like version introduced by griffiths and podolsk ( gp ) allowing for a more straightforward physical interpretation of the free parameters . _ file : 05_elm_inv_v9.tex , jul 2 , 2015 _

1611.08349

i

we continue a study ( @xcite ; article i ) on the 26 december 2001 event ( sol2001 - 12 - 26 ) . this solar eruptive - flare event produced a strong flux of solar energetic particles ( seps , mainly protons ) near earth and a 63rd ground - level enhancement of cosmic - ray intensity ( gle63 ) . gles represent the highest - energy extremity of seps ( see , _ e.g. _ , @xcite ; and references therein ) . gles are rare events ; seventy - two gles only have been recorded since 1942 . the rare occurrence of gles hampers understanding their origins and finding consistent patterns that might govern their appearance and properties . unlike electrons , whose signatures are manifold throughout the whole electromagnetic range , accelerated protons and heavier ions can only be detected on the sun from nuclear @xmath1-ray emission lines appearing in their interactions with dense material ( see , _ e.g. _ , @xcite ) . solar @xmath1-ray observations have been very limited in the past . no @xmath1-ray images were available before 2002 . due to the observational limitations , considerations of the origins of seps and , especially , gles mainly refer to the hypotheses proposed several years ago ( see , _ e.g. _ , @xcite for a review ) . on the other hand , studies based on recent observations @xcite indicate that some of these hypotheses might need refinement . gles are typically produced by major eruptive events associated with big flares ( mostly of the goes x - class ) , fast coronal mass ejections ( cmes ) , and strong microwave bursts . the common association with different solar energetic phenomena @xcite hampers the identification of the origins of seps and gles . all of these circumstances show how important the analysis of a solar source of each gle is . with a general correspondence between the magnitudes of seps , cme speeds , and flare manifestations , there are few outliers from the correlations @xcite . strong fluxes of high - energy protons were observed near earth after these events associated with moderate microwave bursts . the correlation between the proton fluences , on the one hand , and microwave and soft x - ray ( sxr ) fluences , on the other hand , is considerably higher . this is difficult to understand , if seps are exceptionally shock - accelerated . a flare - related contribution could also be significant in the events , whose proton productivity was enhanced for some unclear reasons . one of these events was the sol2001 - 12 - 26 eruptive - flare event responsible for gle63 . this solar event has not been comprehensively studied because of its limited observations . we are not aware of either low - coronal observations of an eruption or hard x - ray ( hxr ) data . on the other hand , the event was observed in microwaves by the _ siberian solar radio telescope _ ( ssrt : @xcite ) at 5.7ghz ; the _ nobeyama radioheliograph _ ( norh : @xcite ) at 17 and 34ghz , and in ultraviolet ( uv ) by the _ transition region and coronal explorer _ ( trace : @xcite ) . the ssrt observations have been studied in article i. its conclusions are : 1 . most likely , gle63 was caused by the m7.1 solar event in active region ( ar ) 9742 ( n08w54 ) . contribution from a concurrent far - side event is unlikely . the flare was much longer than other gle - related flares and consisted of two parts . the first , possibly eruptive , flare and a moderate microwave burst started at 04:30 and reached an m1.6 level . the main flare , up to m7.1 , with a much stronger burst started at 05:04 , when a cme was launched . 3 . the main flare involved strong magnetic fields presumably associated with a sunspot in the west part of ar 9742 . two non - thermal sources observed at 5.7ghz initially approached each other nearly along the magnetic neutral line , and then moved away from it like expanding ribbons , as if they were associated with the legs of the flare arcade . it was difficult to confirm and expand these indications in article i based solely on the ssrt data because of their insufficient spatial resolution and coalignment accuracy . to verify and elaborate these results , observations of the flare arcade or ribbons in a different spectral range , where they are clearly visible , should be compared with microwave data of a higher spatial resolution . of special interest is a conjectured localization of the non - thermal microwave sources in the legs of the flare arcade . this possibility does not contradict a commonly accepted view on the flare process ; however , hxr and microwaves almost always show a few simple non - thermal sources . ribbon - like hxr structures have been observed in exceptional events @xcite . the simplicity and confinement of non - thermal sources in impulsive flares suggested involvement of one to two loops @xcite . later observational studies extended this view up to some long - duration flares @xcite . a concept of a single microwave - emitting loop became dominant . one of the major purposes of our companion articles is to find the possible causes of the contrast between the rich proton outcome of this solar event and its moderate manifestations in microwaves . we consider _ a priori _ the contributions from the acceleration processes both in the flare and by the shock wave to be possible @xcite . accordingly , we examine the 26 december 2001 flare in this article ( article ii ) and eruptive event in article iii ( grechnev _ et al . _ , 2017 , in preparation ) . here we endeavor to shed further light on the listed issues related to the flare itself and its microwave emission , analyzing the observations carried out by trace in 1600 and by norh at 17 and 34ghz .

analysis of the observations of the sol2001 - 12 - 26 event related to ground - level - event gle63 , including microwave spectra and images from the _ nobeyama radioheliograph _ at 17 and 34ghz , _ siberian solar radio telescope _ at 5.7ghz , and _ transition region and coronal explorer _ in 1600 has led to the following results . a flare ribbon overlapped with the sunspot umbra , which is typical of large particle events .

analysis of the observations of the sol2001 - 12 - 26 event related to ground - level - event gle63 , including microwave spectra and images from the _ nobeyama radioheliograph _ at 17 and 34ghz , _ siberian solar radio telescope _ at 5.7ghz , and _ transition region and coronal explorer _ in 1600 has led to the following results . a flare ribbon overlapped with the sunspot umbra , which is typical of large particle events . atypical were : i ) long duration of the flare of more than one hour ; ii ) moderate intensity of a microwave burst , aboutsfu ; iii ) low peak frequency of the gyrosynchrotron spectrum , around 6ghz ; and its insensitivity to the flux increase by more than one order of magnitude . this was accompanied by a nearly constant ratio of the flux emitted by the volume in the high - frequency part of the spectrum to its elevated low - frequency part determined by the area of the source . with the self - similarity of the spectrum , a similarity was observed between the moving microwave sources and the brightest parts of the flare ribbons in 1600 . comparison of the 17ghz and 1600 images has confirmed that the microwave sources were associated with multiple flare loops , whose footpoints appeared in ultraviolet as intermittent bright kernels . to understand the properties of the event , we simulated its microwave emission using a system of several homogeneous gyrosynchrotron sources above the ribbons . the scatter between the spectra and sizes of the individual sources is determined by the inhomogeneity of the magnetic field within the ribbons . the microwave flux is mainly governed by the magnetic flux passing through the ribbons and the sources . an apparent simplicity of microwave structures is caused by a poorer spatial resolution and dynamic range of the microwave imaging . the results indicate that microwave manifestations of accelerated electrons correspond to the structures observed in thermal emissions , as well - known models predict .

1611.08349

c

analysis of the microwave and uv images has shown that each of the two microwave sources observed at 5.7 , 17 , and 34ghz was associated with one of the two ribbons located in opposite - polarity magnetic fields . the appearance of each microwave source corresponds to the whole related ribbon , as if it had been viewed by norh . both the ribbons and microwave sources exhibited nearly identical systematic motions . until the peak of the microwave burst , the two flare centers approached each other nearly along the neutral line , so that the distance and shear between them considerably decreased . after that , the motion occurred away from the neutral line , corresponding to the usual expansion of the ribbons . the expansion was measured from the ssrt data until 06:30 in article i. the correspondence of each microwave source to the flare ribbon , their nearly identical motions , and surprisingly persistent shape of the microwave spectrum , in spite of 18-fold flux variations , indicate that microwaves were emitted by the conjugate legs of multiple loops constituting the flare arcade . the relation of the magnetic - field strengths under the ribbons points to a strongly asymmetric configuration . magnetic - flux conservation in flare loops requires larger areas of the sources above the weaker - field east ribbon relative to their conjugate counterparts above the sunspot - associated west ribbon . to verify these considerations , we will attempt to reproduce the observed spectra by means of a simple model , using the magnetic fields actually measured . the microwave flux density [ @xmath38 at optically thin frequencies [ @xmath39 is controlled by the instantaneous total number of emitting high - energy electrons [ @xmath40 . self - similarity of the spectra in a wide range of microwave fluxes in figure [ f - norp_lear]e shows that the @xmath41 relation in this event also applied at lower frequencies [ @xmath42 . accelerated electrons are produced in a reconnection process ; therefore , @xmath43 is associated with a power of flare energy release . let us find which observable parameters of a flare indicate this association . the energy flux density entering the flare current sheet from one of its sides is governed by the poynting vector , @xmath44/{4\pi}= -[[\textbf{\textit{v}}\times\textbf{\textit{b}}]\times\textbf{\textit{b}}]/{4\pi}= [ \textbf{\textit{v}}b^{2}-\textbf{\textit{b}}(\textbf{\textit{b}}\cdot \textbf{\textit{v}})]/{4\pi}= { \textbf{\textit{v}}b^{2}}/{4\pi}$ ] [ erg@xmath45s@xmath6 ] . here @xmath47/c$ ] ; @xmath48 is a vertical magnetic field , _ i.e. _ @xmath49 ; @xmath50 is a horizontal velocity of the plasma inflow into the vertical current sheet , _ i.e. _ , @xmath51 . the total power released in the current sheet dimensioned @xmath52 by @xmath53 is @xmath54[ergs@xmath6 ] . here @xmath55 , and @xmath56 is the input rate of the magnetic flux . let @xmath57 be the lifetime of a point - like uv kernel in the footpoint of a thin magnetic tube during reconnection in the current sheet and afterward . a multitude of kernels constitutes an instant uv - emitting stripe corresponding to a narrow moving flare ribbon in the standard model . then , @xmath58 , which is magnetic flux within the ribbon stripe at time [ @xmath59 . presumably , @xmath57 is proportional to the lifetime of accelerated electrons in the magnetic tube , and @xmath60 is proportional to the magnetic flux across a gs source at time [ @xmath59 . a particular value of @xmath57 is not important , if it is much shorter than the burst . if the flare process operated self - similarly throughout the burst , then the ratio of energy released in the current sheet during @xmath57 , @xmath61 , to the total energy of electrons produced at the same time , @xmath62 , was constant , _ i.e. _ @xmath63 . this relation is correct , as long as the vertical size [ @xmath53 ] of the current sheet is constant , and magnetic field [ @xmath25 in its vicinity is uniform . the latter assumption is justified by rapid disappearance with an increasing height of small magnetic features , which reflect strong inhomogeneity of the magnetic field on the photosphere . with a power - law energy distribution of gs - emitting electrons [ @xmath65 ( @xmath66 ) ] their energy density is @xmath67 \epsilon_{0 } n \equiv \overline{\epsilon}n$][erg@xmath68 , @xmath69 . here @xmath70 is the number of electrons per @xmath71 in a unit interval of energy [ @xmath72 , @xmath73 is the total number of gs - emitting electrons in 1@xmath71 , and @xmath74 is an average energy of a single electron . since @xmath75 , from the condition @xmath76 we get a final equation for the total number of gs - emitting electrons , @xmath77 . note that @xmath78 is related here to the vicinity of the current sheet . in turn , @xmath60 is an instant magnetic flux within one of the ribbon stripes , @xmath79 . the relation between the emission flux and total number of emitting electrons , @xmath80 , is transformed to the form @xmath81 . this relation , similar evolutions of microwave sources observed by two radio heliographs and flare ribbons observed in uv , and similarity between microwave images and convolved uv images motivated our usage of a model source system referring to the ribbons to simulate gs emission in this event . the spectrum of the non - thermal microwave emission in the 26 december 2001 event has two conspicuous features : persistent shape with weak changes in the peak frequency under large flux variations and an enhanced low - frequency part . @xcite found the peak - frequency variations to be small in about one third of the events . the authors related this behavior to gs self - absorption around the peak of the burst and to the tsytovich razin suppression in its early rise and late decay . @xcite reported the latter effect indeed ; however , its importance at the rise of a long - duration flare is difficult to reconcile with chromospheric evaporation . it is quantified by the neupert effect @xcite , _ i.e. _ similarity between the soft x - ray flux ( directly dependent on the plasma density ) and antiderivative of the microwave burst . the plasma density is initially low , reducing the tsytovich razin effect at this stage . at the decay of our burst , _ e.g. _ at 05:29 , the net total area at 34ghz was @xmath82@xmath83 ( figure [ f - norh_timeprof]c ) , emission measure estimated from goes data @xmath84@xmath85 ( article i ) , and plasma density @xmath86@xmath85 . with a magnetic field strength of @xmath87 g estimated in article i at 05:20 , the razin frequency was @xmath88ghz ( figure [ f - delta]c ) . thus , the ideas of @xcite are unlikely to help us , because the magnetic fields of @xmath89 g they considered are too weak for our sunspot - associated flare . to account for the low - frequency increase in a gs spectrum , inhomogeneity of the source and superposition of multiple sources have been proposed ( _ e.g. _ @xcite ) . the major inhomogeneity in these models is related to the magnetic field in a flare loop of a varying cross - section . this undoubted inhomogeneity affects the shape of the spectrum , especially its optically thick part @xcite . it is difficult to understand why the spectrum from a single inhomogeneous loop had a constant shape , while indications of multiple sources are certain . we are not aware of inhomogeneous multi - loop models . to verify our interpretations , we are forced to use a tentative simplified modeling of gs emission from a set of several homogeneous sources . we are interested in general features of this system and need a very simple analytic description of gs emission , which the @xcite approximations present . their reduced accuracy at the lowest and highest harmonics of the gyrofrequency is not crucial for the task of the model to understand properties of our event . our model contains a considerable number of homogeneous gs sources , each with a different magnetic - field strength and volume . their number depends on the width and length of the brightest parts of the uv ribbons . the model should also demonstrate the direct dependence of the total flux and spectrum of the microwave emission on the total magnetic flux and its distribution over each of the ribbons . the model does not consider the influence of the ambient plasma on generation and propagation of the gs emission , _ i.e. _ the tsytovich razin effect and free free absorption , whose importance in our event is unlikely . the loop system constitutes an arcade rooted in the ribbons . each ribbon in our event is extended and inhomogeneous in brightness and width . we relate a set of microwave sources to a brightest , broadest stripe of each ribbon . its width [ @xmath90 corresponds to a typical transversal size of a loop , whose end is rooted in this ribbon . the width of a loop varies according to the magnetic field strength [ @xmath25 along it , being equal to @xmath91 , with superscript `` 0 '' related to the first ribbon . for a narrow ribbon stripe , the number of emitting loops [ @xmath92 should be about its length to width ratio . if the loops do not overlap , then their total flux [ @xmath38 is the sum of the fluxes emitted by all of the loops . each @xmath93th loop is represented by two homogeneous cubic sources in its legs above both ribbons , corresponding to the observations at 5.7 and 17ghz . magnetic fluxes in conjugate cubes are equal to each other , @xmath94 . the ratio of their sizes [ @xmath95 is determined by the ratio of the magnetic - field strengths in the east source [ @xmath96 and the west one [ @xmath97 , so that @xmath98 . it is convenient to use a set of @xmath99 loops , each of which encloses equal magnetic flux @xmath100 . this assumption ensures the balance of magnetic fluxes in conjugate legs of any loop , irrespective of its location , and facilitates partition into @xmath99 cubic sources . two methods of partition are possible . in the first method , the total magnetic flux @xmath101 is divided into @xmath99 equal parts on the magnetogram within each ribbon in a fixed direction . the widths of the pieces can be different , while their magnetic fluxes are equal to each other . each @xmath93th pair corresponds to a loop . the loops do not overlap , and the procedure to find @xmath102 and @xmath103 seems to be physically justified . a rather formal second method considers the histograms @xmath104 of the magnetic - field distribution within each ribbon , where @xmath105 is the number of pixels where the magnetic - field strength is equal to @xmath78 . the area under the histogram is divided into @xmath99 equal parts corresponding to equal magnetic fluxes @xmath106 , which is easy to calculate . the cubic sources obtained in this way are different , and their paired link is lost . on the other hand , the scatter of the size and magnetic - field strength is maintained , as in the first method of partition . we use the second method , which is simpler to implement . the spectral flux density @xmath107 from each @xmath93th unpolarized source is @xmath108 ( \nu^2/c^2)\ , a_i / r^2 $ ] , where @xmath109 is boltzmann s constant , @xmath110 the source area , @xmath111 au , and @xmath112 the optical thickness . the effective temperature [ @xmath113 and absorption coefficient [ @xmath114 are calculated following @xcite . in their figure 3 , the log log plots of @xmath115 and @xmath116 deviate at low @xmath117 from the quasi - linear parts into opposite directions that reduces the errors @xcite . the deviations are less for @xmath118 in our case . the total flux spectrum is a sum of @xmath119 spectra from all sources . the number density of microwave - emitting electrons [ @xmath120 and their power - law index @xmath121 are identical for all sources . the viewing angles of the sources in the legs of the loops above the east and west ribbons @xmath122 and @xmath123 are different , while their half - sum is the longitude of the flare site . the optimal number of the paired sources [ @xmath92 was adjusted iteratively to meet three conditions : i ) the sum of @xmath119 sources provides a gradual spectrum with a single peak , ii ) the value of @xmath99 is about the ribbon length to width ratio , and iii ) the model acceptably fits the observed spectrum . with an optimal number [ @xmath92 , the field strengths [ @xmath124 estimated from the photospheric magnetogram should be corrected to the coronal values . it is possible to use a constant scaling factor @xmath125 , so that magnetic field strengths in coronal sources are @xmath126 . to estimate @xmath125 , we refer to @xcite , who found an average magnetic field of @xmath127 g using a homogeneous gs source model and a scaling law between @xmath128 and the total area of a source . an intuitive option to calculate @xmath125 as a ratio of 400 g to an average field strength measured from the magnetogram within the ribbons leads to a biased estimate because of a nonlinear dependence of the microwave flux on magnetic field . with any number of the sources , the flux at optically thin frequencies [ @xmath129 is controlled by the total number of emitting electrons and their emissivity . thus , @xmath130 with @xmath131 ; @xmath132 is the viewing angle @xcite . the constancy of the optically thin total flux emitted by the same electron population with any number of the sources , up to a single large one ( subscript `` s '' ) , results in an equality @xmath133 . using the constancy of the total number of emitting electrons , @xmath134 , we obtain an average magnetic - field strength in an equivalent single source @xmath135 . finally , we have estimated @xmath136 g for the sources above each ribbon separately and obtained @xmath137 . in this approach , the magnetic flux [ @xmath138 retains , and the change from @xmath102 to @xmath139 results in a corresponding change in the size of each @xmath93th source from @xmath103 to @xmath140 . figure [ f - model_spectrum]a presents the results of the modeling by the gray line for the first flare ( episode 1 ) and the black line for the rise of the main flare ( episode 2 ) . these two episodes were soon after fresh injections , when emission of trapped electrons , which we do not consider , can be neglected , and an assumption of a constant @xmath141 is justified . the symbols denote the observed fluxes . an enhancement at 2ghz was due to a contribution from plasma emission . the average magnetic flux over the ribbon stripe [ @xmath142 used in the modeling is specified for each episode . the average field strength above the west ribbon stripe [ @xmath143 additionally affects the microwave flux at optically thin frequencies . the number density of electrons with energies @xmath144kev was @xmath145@xmath85 in both episodes . the total number of electrons was @xmath146 in episode 1 and @xmath147 in episode 2 . in section [ s - reconnection_power ] we obtained @xmath148 assuming the magnetic field to be uniform at both sides of the neutral line that is not realistic . nevertheless , the model takes account of a real inhomogeneity of the photospheric magnetic field and acceptably matches the real spectrum . the relation between the magnetic flux and microwaves can be generalized to a variable magnetic field , considering the intrinsic dependence of the microwave flux on the magnetic field strength , @xmath149 . the ratio of the optically thin microwave fluxes in the two episodes in figure [ f - model_spectrum]a is 8.9 , while the ratio of the magnetic fluxes within the ribbon stripes is 4.4 . with @xmath150 , the expected microwave flux ratio is @xmath151 , which agrees with the model result of 8.9 within @xmath152 . this seems to be acceptable with our simplified approach . figure [ f - model_spectrum]b demonstrates the influence on the spectrum of the number of loops for episode 1 . the increase of the number of loops elevates the low - frequency branch of the spectrum . the effect is similar to that of the source inhomogeneity . the ratio of the 16-loop spectrum to the single - loop one at 2ghz is 5.1 in this case . the lowest - frequency slope , 2.9 , corresponds to the classical gs spectrum . figure [ f - model_spectrum]c illustrates the role of asymmetry in the magnetic configuration with the same magnetic flux reconnection rate . the solid line corresponds to the real situation in episode 2 . the dashed line represents the spectrum for a hypothetical situation , when both microwave - emitting regions were located above identical ribbons corresponding to the actual west ribbon . the dotted line represents a similar experiment with two east ribbons . the strongest effect of asymmetry at optically thin frequencies is illustrated by two extremities . the high - frequency flux from a highly asymmetric configuration is determined by a single source , and the flux is doubled in a symmetric configuration ( two identical sources ) . the effect of asymmetry varies between a factor of one and two . the same occurs for the opposite asymmetry . thus , with the same magnetic - flux reconnection rate , the high - frequency flux can vary within a factor of four . the actual ratios of the optically thin fluxes in figure [ f - model_spectrum]c are 1.26 between the dashed and solid lines and 1.40 between the solid and dotted lines . the asymmetry in this event progressively increased . the west ribbon expanded into the sunspot umbra , the east ribbon developed into weaker - field regions . to balance magnetic flux , an increasing high - frequency emission from the stronger - field west regions must be accompanied by an increasing area of the weaker - field east region that elevated the low - frequency part of the spectrum . the spectrum shape remained nearly constant , in spite of large changes in the microwave flux . the relevance of a homogeneous source and simplified expressions by @xcite were discussed by @xcite . as they showed , the usage of a scaling law between the average magnetic field and total source area makes the homogeneous model sufficient to estimate statistical characteristics of microwave bursts such as the peak flux and frequency and spectral index . each elementary loop in our model is replaced by two homogeneous sources of different size and magnetic field . a set of 32 homogeneous sources reflects inhomogeneity of the ribbons . the scaling factor to shift from their magnetic fields to those in microwave sources is based on the results of @xcite . our model acceptably reproduces the spectra also around the peak and at lower frequencies . modeling of the circular polarization of the gs emission is complicated by the near - the - limb location of the flare site . since the west microwave source is visible through quasi - transversal magnetic fields associated with the arcade , polarization reversal is expected in a wide frequency range . thus , the polarization of each elementary source is a result of interplay between the optical thickness and polarization reversal issues , both frequency - dependent . this makes the analysis of the polarization of the west source and the total emission too complex . the polarized emission of the east source can only be extracted in norh 17ghz and ssrt 5.7ghz images . for the degree of polarization of each optically thin elementary source we used a corrected formula from @xcite , otherwise we assigned an opposite polarization of 15% . the results for episode 1 are ( all negative ; first observed , second model ) : @xmath153\%$ ] at 5.7ghz , @xmath154\%$ ] at 17ghz ; for episode 2 : @xmath155\%$ ] at 5.7ghz , @xmath156\%$ ] at 17ghz . the model reproduces the actual degree of polarization within a factor of two with a correct sign . the outcome of our simple modeling can be summarized as follows . 1 . the observed properties of the gs spectrum are consistent with the emission from a distributed multi - loop system , _ i.e. _ the flare arcade . the asymmetry of the magnetic configuration is important . the magnetic - flux balance requires larger areas at the weaker - field side that elevated the low - frequency part of the spectrum , shifting the peak frequency left . inhomogeneities of the individual sources could strengthen this effect . 3 . to reproduce the observed spectrum with realistic magnetic fields , a large increase of the magnetic flux is required in the main flare ( the average field strength changed insignificantly ) . this result is consistent with a temporal correlation between the magnetic - flux change rate and hxr emission found by @xcite , who measured magnetic flux within expanding parts of the ribbons . replacement of a distributed multi - loop system by a single loop is generally not equivalent and can result in a different shape of the spectrum , peak frequency , and their behaviors during the burst . this leads to the next item . modeling of a microwave - emitting loop initiated by @xcite has been developed into a powerful tool @xcite . the models use real data on magnetic field and take account of its inhomogeneity , anisotropy and spatial distribution of electrons . a next - step challenge is a realistic multi - loop model , at least , for simplified conditions . elements of the scheme presented here might be helpful in its development . as known from observations in the h@xmath2 line , flare kernels and ribbons initially approach each other along the neutral line , and then they move away from it . the expansion of the ribbons was explained by the two - dimensional ( 2d ) standard flare model . the motions along the neutral line have not been clearly visualized . various motions observed later in hxr were summarized by @xcite . the authors interpreted them in terms of the 2d model ; their cartoons implied a questionable rotation of the current sheet around the vertical axis . the motions along the neutral line have reasonably been considered as an intrinsically 3d effect . its scheme was presented by @xcite . the authors supposed contraction of reconnected loops to be a necessary element of the unshearing process , whereas the two phenomena seem to be different results of the flare reconnection not directly related with each other . @xcite reported microwave observations of the motions along the neutral line . the authors noticed that an m2.6 flare developed along the arcade visible in the h@xmath2 and extreme - uv images and that , at least , several loops were involved in the process . however , they considered a single loop for each instance . in the discussion of the unshearing motions they mostly follow @xcite . the motions of the microwave sources in our event corresponded to those of the brightest parts of the ribbons , which we call the ribbon stripes for brevity . figure [ f - scheme ] explains the relative displacement of the stripes . the intrinsic 3d geometry implies the presence in an extended current sheet of a zone of most efficient energy release , where the reconnection process is most similar to the 2d model . the formation of major stripes is associated with this zone . figures [ f - scheme]a and [ f - scheme]b present evolution of three magnetic loops , which reconnected in this zone . initially , these loops belonged to a certain layer of magnetic arcades above the pre - eruptive filament . the dotted line connects the bases of the central loop to show the shear . figure [ f - scheme]c shows the central loops of different layers . the higher a loop , the smaller is the shear of its bases . the loops evolve similarly to the central loop in figures [ f - scheme]a and [ f - scheme]b . the major stripes corresponding to these loops displace as the arrows show , reflecting the decrease of the shear with the increasing height of the pre - eruptive arcade . the major motions initially occurred nearly along the neutral line and away from it during the main flare phase and afterwards . this behavior ( not shown in figure [ f - scheme ] ) is associated with a non - uniform decrease of the shear away from the neutral line . a similar behavior is exhibited by helical lines of a cylindric nonlinear force - free magnetic field ( @xmath157 ) when @xmath158 with @xmath159 . non - thermal emissions are generated by energetic electrons and carry the most direct information about acceleration processes in flares . as noted in section [ s - introduction ] , non - thermal sources observed in hxr and microwaves are usually simple and confined , which favors their identification with one or two flaring loops . this view drawn from microwave observations of almost all impulsive flares has been generalized to some major long - duration flares @xcite . still stronger confinement is typical of hxr sources ; complex structures like extended ribbons were observed in exceptional events @xcite . by contrast , observations of flares in thermal emissions ( h@xmath2 line , uv , extreme uv , sxr ) typically show complex multi - component structures of a larger extent . this dissimilarity has led to different views on flaring structures drawn from different observations . on the other hand , synergy between the structures observed in thermal and non - thermal emissions has been conjectured previously , including studies in which some of us participated @xcite . the causes of the differences between non - thermal sources and configurations visible in thermal emissions were unclear . a key idea by @xcite explaining this difference has not been commonly perceived . the authors pointed out the limitations on the sensitivity and dynamic range ( typically @xmath160 ) in the hxr imaging . indeed , images in hxr and @xmath1-rays have been provided by the imagers of the fourier - synthesis type with an intrinsically limited coverage of the @xmath161-plane . these are the _ hard x - ray telescope _ ( hxt : @xcite ) onboard _ yohkoh _ and the _ reuven ramaty high - energy solar spectroscopic imager _ ( rhessi : @xcite ) . sources weaker than 10% of the brightest one are not detectable in their images @xcite . @xcite also found that hxr sources accompanied the h@xmath2 kernels in strongest magnetic fields only ; `` the hxr sources indicate where large energy release has occurred , while the h@xmath2 kernels show the precipitation sites of nonthermal electrons with higher spatial resolution '' . flares have been imaged in microwaves also almost exclusively with fourier - synthesis interferometers ( mainly norh ) . they provide a better coverage of the @xmath161-plane than hxr imagers . on the other hand , optically thin gs emission strongly depends on the magnetic field @xmath162 with @xmath163 . with a dynamic range of norh of @xmath164 @xcite , its opportunities to observe weaker non - thermal structures seem to be comparable to hxr imagers . also , microwave telescopes have a poorer spatial resolution than hxr imagers . for these reasons , the strongest non - thermal sources are only expected in hxr and microwave images without weaker structures due to instrumental limitations . configurations , in which accelerated electrons are manifested , and those visible in thermal emissions , must actually be closely associated with each other . both thermal and non - thermal emissions in our event originated in basically the same configuration . dissimilarities between the structures visible in microwaves and uv were mainly due to different spatial resolution and dynamic range of the instruments and different dependencies of the emissions on the magnetic field strength . @xcite also concluded that , at least , some of the single - loop configurations in norh images corresponded to multi - loop arcades observed with telescopes of a higher spatial resolution . an appropriate proxy for a configuration responsible for non - thermal emissions may be a structure observed in extreme ultraviolet or in soft x - rays . this expectation corresponds to widely accepted model concepts of eruptive flares ( processes in confined flares are unlikely different in nature see , _ e.g. _ @xcite ) . this should be thoroughly verified . if this is correct , then considerations of simple configurations are justified , when they appear so in thermal emissions . inevitable simplifications should be recognized to avoid inadequate conclusions .

atypical were : i ) long duration of the flare of more than one hour ; ii ) moderate intensity of a microwave burst , aboutsfu ; iii ) low peak frequency of the gyrosynchrotron spectrum , around 6ghz ; and its insensitivity to the flux increase by more than one order of magnitude . comparison of the 17ghz and 1600 images has confirmed that the microwave sources were associated with multiple flare loops , whose footpoints appeared in ultraviolet as intermittent bright kernels . the microwave flux is mainly governed by the magnetic flux passing through the ribbons and the sources .

analysis of the observations of the sol2001 - 12 - 26 event related to ground - level - event gle63 , including microwave spectra and images from the _ nobeyama radioheliograph _ at 17 and 34ghz , _ siberian solar radio telescope _ at 5.7ghz , and _ transition region and coronal explorer _ in 1600 has led to the following results . a flare ribbon overlapped with the sunspot umbra , which is typical of large particle events . atypical were : i ) long duration of the flare of more than one hour ; ii ) moderate intensity of a microwave burst , aboutsfu ; iii ) low peak frequency of the gyrosynchrotron spectrum , around 6ghz ; and its insensitivity to the flux increase by more than one order of magnitude . this was accompanied by a nearly constant ratio of the flux emitted by the volume in the high - frequency part of the spectrum to its elevated low - frequency part determined by the area of the source . with the self - similarity of the spectrum , a similarity was observed between the moving microwave sources and the brightest parts of the flare ribbons in 1600 . comparison of the 17ghz and 1600 images has confirmed that the microwave sources were associated with multiple flare loops , whose footpoints appeared in ultraviolet as intermittent bright kernels . to understand the properties of the event , we simulated its microwave emission using a system of several homogeneous gyrosynchrotron sources above the ribbons . the scatter between the spectra and sizes of the individual sources is determined by the inhomogeneity of the magnetic field within the ribbons . the microwave flux is mainly governed by the magnetic flux passing through the ribbons and the sources . an apparent simplicity of microwave structures is caused by a poorer spatial resolution and dynamic range of the microwave imaging . the results indicate that microwave manifestations of accelerated electrons correspond to the structures observed in thermal emissions , as well - known models predict .

1008.0554

i

there are more and more astrophysical observations justifying the existence of extremely compact massive objects with size close to their schwarzschild radius @xcite . it is widely accepted that these observations also provide indirect justifications of the existence of black holes ( bhs ) . nevertheless , there are also alternative ideas about claiming that exotic states of matter may exist which could stabilize extremely compact stars suiting to the aforementioned astrophysical observations ( see , e.g. , @xcite for a recent review ) . one of the most popular among these types of bh mimicking objects is the gravitational vacuum star ( gravastar ) model which has received considerable attention not least because its relation to the concept of dark energy . in this model of mazur and mottola @xcite an interior de sitter spacetime region is connected via three intermediate layers to an outer schwarzschild solution such that the radius of the outermost layer is supposed to be slightly larger than the schwarzschild radius of the system . it is worth mentioning that in advance to the gravastar model there were several constructions in which the matching of a de sitter region to the schwarzschild spacetime was applied . for instance , to get rid of the @xmath0 singularity of the schwarzschild spacetime , frolov , markov and mukhanov in @xcite proposed a matching of a de sitter interior to it at a small radius of the planck scale ensuring thereby that the curvature remains bounded everywhere in the yielded spacetime . dynamical investigation of this model was already carried out in @xcite . note that up to certain extent the model used in @xcite could be considered as the precursor of the gravastar model although in the latter the matching was made in a more elaborated way and the outermost matching surface , at the boundary of the schwarzschild region , was supposed to be arranged such that its radius is slightly larger than the pertinent schwarzschild radius . once such a model is set up the following questions manifest themselves : * what type of physical process may produce such a gravastar ? * is a gravastar stable ? * if it is , does it provide a viable alternative to bhs ? while the first question has not been tackled yet even the second question turned out to be too complex within the original model of mazur and mottola although in @xcite an argument claiming for its thermodynamical stability was given as it is composed by making use of three different types of regions with unspecified matter . to reduce the related ambiguities visser and wiltshire @xcite introduced a simplified three - layer gravastar model where the interior de sitter region is matched to the exterior schwarzschild spacetime via a single matter shell . this model is simple enough to carry out various analytic investigations by making use of the thin - shell formalism of israel @xcite . visser and wiltshire besides deriving the basic relations determining the evolution of gravastars also carried out the first investigation of their radial stability . since then the stability of gravastars has been studied by several authors within this simplified model or within its continuum correspondence @xcite . results relevant for radial stability can be found in @xcite , and in case of electrically charged gravastars in @xcite . the stability has also been investigated with respect to axial perturbations @xcite . in all of these investigations attention was restricted to the space of gravastars in equilibrium , i.e. the radial stability was investigated by determining the response of a gravastar to a slight formal change of the underlying effective potential . for instance , in @xcite the excursions of gravastars was investigated in such a way that their evolution started with carefully prepared initial conditions . in all of the pertinent investigations it was demonstrated that by suitably adjusting the equation of state ( eos ) of the matter forming the surface of the gravastar the subspace of ` stable ' gravastars may always be ensured to be of non - zero measure . nevertheless , it was also found that whenever the measure of the subspaces of the configuration space representing stable and unstable gravastars is compared the former is always found to be negligible with respect to the latter . this observation was commonly interpreted that gravastars may not offer a viable alternative to bhs . the main purpose of the present paper is to determine the response of a gravastar in equilibrium to the arrival of a dust shell onto its surface . this is done not merely by considering some formal change of the effective potential determining the state of a gravastar but also by making use of the full dynamical setup . for the sake of definiteness , we assume that the surface of the gravastar and the dust shell collide in a totally inelastic manner . in addition , concrete eoss are chosen and it is assumed that the dominant energy condition ( dec ) holds and the speed of sound in the surface of the gravastar does not exceed that of the light . then the relevant non - linear problems the basic equations of which are based on the dynamics of spherical shells are solved by using analytical and numerical approaches . in this way we can not only study the excursion of particular gravastar models but also determine the maximal mass of the dust shell colliding with the surface of the gravastar without converting the latter into a bh . this paper is organized as follows . in section [ visser - model ] some of the basics of the visser and wiltshire three - layer dynamics gravastar model are recalled using dimensionless variables . as a preparation for the aforementioned dynamical investigations a short survey of the configuration space of stable gravastars is also provided . ( although there are no completely new results in this section , we believe that this review provides a good reference frame for the results of the succeeding dynamical investigations . ) in section [ collision - section ] the dynamics of the system composed by the spherically symmetric dust shell falling onto a stable gravastar , along with their collision , is described . section [ results ] reports about our analytical and numerical results concerning the dynamics of maximally loaded gravastars , while section [ conclusions ] contains our concluding remarks . finally , in the appendix , a generic description of the totally inelastic collision of spherical shells in spherically symmetric spacetimes is provided . throughout this paper the geometrized units , with @xmath1 , are applied .

as a preparation for the dynamical investigations , this paper begins with a short review of the three - layer gravastar model with distinguished attention to the structure of the pertinent parameter space of gravastars in equilibrium . then the radial stability of these types of gravastars is studied by determining their response for the totally inelastic collision of their surface layer with a dust shell . it is assumed that the dominant energy condition holds and the speed of sound does not exceed that of the light in the matter of the surface layer . for those configurations which remain stable the excursion of their radius is assigned . it is also shown in addition , a generic description of the totally inelastic collision of spherical shells in spherically symmetric spacetimes is also provided in the appendix .

as a preparation for the dynamical investigations , this paper begins with a short review of the three - layer gravastar model with distinguished attention to the structure of the pertinent parameter space of gravastars in equilibrium . then the radial stability of these types of gravastars is studied by determining their response for the totally inelastic collision of their surface layer with a dust shell . it is assumed that the dominant energy condition holds and the speed of sound does not exceed that of the light in the matter of the surface layer . while in the analytic setup the equation of state is kept to be generic , in the numerical investigations three functionally distinct classes of equations of states are applied . in the corresponding particular cases the maximal mass of the dust shell that may fall onto a gravastar without converting it into a black hole is determined . for those configurations which remain stable the excursion of their radius is assigned . it is found that even the most compact gravastars can not get beyond the lower limit of the size of conventional stars , provided that the dominant energy condition holds in both cases . it is also shown independent of any assumption concerning the matter interbridging the internal de sitter and the external schwarzschild regions that the better is a gravastar in mimicking a black hole the easier is to get the system formed by a dust shell and the gravastar beyond the event horizon of the composite system . in addition , a generic description of the totally inelastic collision of spherical shells in spherically symmetric spacetimes is also provided in the appendix .

1008.0554

r

by inspection of the new effective potential the dynamics of the gravastar can be determined . if the new potential possesses the form of graph 2 of figure [ figure1 ] the motion of the gravastar surface remains bounded and we can determine the minimum and maximum value of the radius . in particular , the maximum mass which may be dropped onto the original gravastar without converting it into a bh can also be determined . first note that after collision the initial velocity points inward ( unless @xmath186 ) ; therefore , the investigation has to start by deriving whether the new effective potential has any root in the interval @xmath208 $ ] . if not then an event horizon will form and the gravastar collapses to a bh . the examination of the potential is made by a numerical maximum finding algorithm . if the maximum is found to be less than zero , the collapse to a bh will occur . using an interval bisection method , changing the initial gravitational mass of the dust shell to be dropped , the numerical value of the maximal gravitational mass , @xmath209 , which may fall onto the gravastar without converting it into a bh , can be determined . whenever @xmath191 and there is a root in the interval @xmath208 $ ] , the motion is bounded and it happens so that @xmath210 , where @xmath65 is the largest root that is smaller than @xmath53 and @xmath211 is the smallest root that is larger than @xmath53 . if @xmath186 then @xmath53 coincides with either the upper or lower bound of the motion . it is an inherent property of the gravastar model that we can not have information about the de sitter energy density inside the gravastar , @xmath17 , nor about the eos of the shell before and after the collision . to have some quantitative results suitable hypotheses have to be applied . in order to avoid the use of a flood of parameters each of which has to possess definite values in numerical investigations , as indicated in subsection [ eossubsec ] , in this paper we analyze three types of eos : homogeneous linear , broken linear and polytrop . if we fix an eos with all of its parameters , then , as a function of @xmath17 , the stable equilibrium radius , @xmath53 , is to be determined . note , that this radius can be determined by the intersection of the specific eos curve and the corresponding @xmath17-level curve in the figure [ figure3 ] . next we have to make an assumption about the eos after the collision . since we have no observational clues , as it was indicated above , we shall assume that the eos remains intact , which means that the dust shell melts into the matter of the gravastar shell yielding an increase of the pressure according to its mass contribution . note that in the case of an initially homogeneous linear eos the possibility of keeping the surface tension to be intact will also be investigated . this latter case might be a good approximation whenever the contribution of the dust shell to the new shell s composition is negligible . as was mentioned in subsection [ eossubsec ] , the simplest possible eos is the homogeneous linear one . if the value of @xmath17 and @xmath212 are given that of @xmath53 is determined by the implicit relation @xmath213 . it is clearly visible in figure [ figure3 ] that for a given value of @xmath214)$ ] the @xmath215 line intersects any @xmath17 constant curve at most at two points . one of these intersections corresponds to a stable configuration , with the smaller @xmath51 value , while the other is unstable . accordingly , once the choice for @xmath17 and @xmath212 is made first we determine the value of @xmath53 by numerically solving the implicit relation @xmath213 . then , for any choice of the pair @xmath216 , we determine the stable gravastar configuration , and as it was outlined above , the maximal mass of the shell that can be dropped onto it without converting it into a bh can also be determined . the results of the corresponding investigations are depicted by figure [ w0surf ] . it is worthy to mentioned that some of the special cases depicted on these panels play distinguished role in various arguments . for instance , the case @xmath217 corresponds to ultrarelativistic gas , which based on certain dimension reduction arguments that might not be straightforward to be applied in the present case is represented in @xcite by a system with @xmath218 . is shown as a function of @xmath17 and @xmath212 for the homogeneous linear eos , where @xmath53 stands for the maximal dimensionless radius allowed by the dec . on the top two panels @xmath185 , while on the bottom two ones @xmath186 . the panels to the left depict the intact eos cases , while in the right ones the surface tension was kept to be intact . , width=377 ] note that to have a stable configuration the value of @xmath219 has to be larger than @xmath220 given by ( [ omegamin ] ) . in particular , @xmath219 may take all the values from the interval @xmath221 $ ] only for @xmath168 , i.e. whenever the inner region is vacuum . also note that @xmath222 has to be smaller than @xmath223 . if @xmath224 occurred , the system composed of the gravastar and the dust shell would get beyond its own schwarzschild radius before the collision occurred . this makes the use of the rescalled quantity @xmath225 to be advantageous . before interpreting figures [ w0surf ] and [ v2p2 ] let us recall that a homogeneous linear eos , @xmath226 , has @xmath212 as the only parameter . therefore , it is also straightforward to investigate the case when both the surface tension and the functional form of the eos remain intact during the collision . it is done by determining the value of the new @xmath227 via the relation @xmath228 . if this happens the particles of the dust shell do not adopt the properties of the particles comprising the surface of the gravastar as only the rest mass of the surface will be increased by the collision . in figures [ w0surf ] and [ v2p2 ] four different subcases are considered . these subcases are yielded , on the one hand , by keeping the eos or the surface tension to be intact and , on the other hand , by choosing the velocity of the particles of the dust shell at the moment of the collision to be such that either @xmath185 or @xmath186 , as discussed in section [ collision - section ] . sections of the four panels of figure [ w0surf ] are shown providing a better comparison of @xmath222 for the considered configuration . the order of the curves from top to bottom coincides with that of the legend at the upper - right corner . , width=377 ] figure [ v2p2 ] depict only the @xmath100referred to as stiff matter in @xcite sections of the four cases indicated in figure [ w0surf ] which offers a better comparison of the corresponding four different dynamical subcases . figure [ bananas ] does also refer to these four basic configurations with @xmath100 . the horizontal sections , corresponding to an @xmath167 value , of the grey regions indicate the dynamical range of the radius of gravastars . for the particular value @xmath229 the pertinent new potentials are also indicated . considered in figure [ v2p2 ] . the @xmath167 sections of the gray regions represent the dynamical range of the radius relevant for the considered four subcases . the @xmath17 dependence of @xmath53 of the gravastars in advance to the collision is indicated by the dashed curves . whenever the initial velocity of the dust shell is nonzero ( e.g. , @xmath185 ) the dashed line is in the interior of the grey region , whereas whenever @xmath186see , e.g. , the right panels the gravastar stars its motion at the edge ( minimal or maximal @xmath29 value ) of the dynamical range . for the particular choice @xmath229 the corresponding potentials , are also plotted , where the zeros of the potentials are indicated by the dotted horizontal lines and the applied scales are the same on each of these panels . the dotted vertical line with the larger @xmath230 value correspond to the buchdahl limit , while the other one with @xmath231 indicates the corresponding lower bound to the size of most compact conventional stars satisfying the dec . , width=377 ] there are two interesting points to be mentioned that are indicated in figure [ bananas ] . first , there are two dotted vertical lines on each of the panels . the one with the larger @xmath29 value corresponds to the buchdahl limit with @xmath230 @xcite , while the other with a smaller @xmath29 value indicates the corresponding lower bound , with @xmath231 , to the size of most compact conventional stars . on these panels of figure [ bananas ] it is also visible that dynamical gravastars can not get below the buchdahl limit so they definitely remains less compact than the most compact conventional stars satisfying the dec . second , on the right two panels with @xmath186 the grey regions on one side are bounded by the dashed lines representing the @xmath17 dependence of @xmath53 of the initial gravastar . this is , on one hand , in accordance with intuition as the particles of the dust shell are simply simultaneously placed on the surface of the gravastar . what might be unexpected , on the other hand , is the following . whenever the eos remains intact ( see the right - top panel of figure [ bananas ] ) there is a subregion where the dashed line bounds from above and another where it bounds from below . from zero to @xmath222 first the radius increases . however , for configurations with @xmath232 there is a turning point in @xmath29 and before reaching @xmath222 the radius is smaller than the initial @xmath53 . ] these subregions are separated by a point @xmath233 where the initial gravastar remains apparently at rest for @xmath234 , while whenever the surface tension remains intact as it is indicated by the bottom - right panel of figure [ bananas] the dynamical range for @xmath234 is always bounded from above by the initial gravastar configurations . since the gravastar radius has always to decrease at the moment of the collision the surface density must increase at this moment . thus , the functional form of the new effective potential depends only on that of the eos relevant for mass density greater than @xmath51 . motivated by this observation we have used the following particular broken linear eos . @xmath194 was chosen to be the breaking point , i.e. the choice @xmath235 was made , and only the slope was varied for @xmath236 . since we have found that the higher the value of @xmath212 the more stable is a gravastar , in the broken linear eos we have chosen @xmath237 and @xmath63 to be varied form zero to 1 , in steps @xmath238 , as it is indicated on the right panel of figure [ w1dec ] . is chosen to be maximal such that the dec is still guaranteed to hold . on the graphs of the left panels the value of @xmath63 decreases form 1 to zero , in steps @xmath238 , while moving downward . , width=377 ] in figure [ w1dec ] the @xmath17 dependence of @xmath222 , i.e the maximal value of the gravitational mass of the shell that may be dropped onto a stable gravastar without converting it into a bh is shown for configurations with @xmath185 and @xmath186 . on both of the left panels the value of @xmath63 decreases from 1 to zero while moving from the top to the bottom . it is also visible that for each particular value of @xmath63 there is a maximal value of @xmath17 that is indicated by the vertical ticks on the horizontal axis of the the top - left panel . these figures do also justify for the pertinent linear approximation of the eos that for given @xmath17 and @xmath53 the larger the value of @xmath239 the more stable the gravastar becomes in the sense that the allowed value of @xmath222 increases together with @xmath240 . a polytrop eos of the form of ( [ polytrop ] ) has two parameters , @xmath74 and @xmath75 . given a point @xmath241with @xmath242it is not obvious to choose at all the parameters @xmath74 and @xmath75 to get an eos fitting to this point except for the choice @xmath243 . to overcome this slight technical difficulty and also because for a given value of @xmath74 the choice @xmath243 always yields the most stable configuration , we have chosen @xmath243 and varied only the value of @xmath74 . it is also informative to compare the behavior of a gravastar with a polytrop eos and with a linear one having the same slope at @xmath51 as it is indicated by the nested panel in figure [ kappa2 ] relevant for the particular choice @xmath243 . for any choice of @xmath17 the value of @xmath53 is chosen to be determined by the intersection of the @xmath167 line and the dashed - dotted curve in figure [ figure2 ] . note that if @xmath53 was chosen to be the maximal value for a given @xmath17 such that the dec holds , the polytrop eos would degenerate into the homogeneous linear one with @xmath100 . and the linear one , having the same slope at @xmath51 are illustrated . left : for any choice of @xmath17 the corresponding value of @xmath53 is determined by the intersection of the @xmath167 line and the dashed - dotted curve in figure [ figure2 ] . the @xmath17 dependence of @xmath225 is shown for two types of eos and for the two types of initial velocities , @xmath185 and @xmath186 . the order of the curves from top to bottom coincides with that of the legend at the upper right corner . , width=264 ] the @xmath17 dependence of the maximum of the normalized gravitational mass of the dust shell , @xmath225 , where @xmath222 is the maximal mass that may be dropped onto a gravastar without converting it into a bh is shown in figure [ kappa2 ] for two types of eos and for the two types of choices concerning the initial velocities , @xmath185 and @xmath186 . as it might have been anticipated , figure [ kappa2 ] justifies that a gravastar characterized by a polytrop eos is more stable than the corresponding one with the linear eos in consequence of the fact that the slope of the polytrop eos is larger than that of the linear one for @xmath236 .

while in the analytic setup the equation of state is kept to be generic , in the numerical investigations three functionally distinct classes of equations of states are applied . in the corresponding particular cases the maximal mass of the dust shell that may fall onto a gravastar without converting it into a black hole is determined . it is found that even the most compact gravastars can not get beyond the lower limit of the size of conventional stars , provided that the dominant energy condition holds in both cases .

as a preparation for the dynamical investigations , this paper begins with a short review of the three - layer gravastar model with distinguished attention to the structure of the pertinent parameter space of gravastars in equilibrium . then the radial stability of these types of gravastars is studied by determining their response for the totally inelastic collision of their surface layer with a dust shell . it is assumed that the dominant energy condition holds and the speed of sound does not exceed that of the light in the matter of the surface layer . while in the analytic setup the equation of state is kept to be generic , in the numerical investigations three functionally distinct classes of equations of states are applied . in the corresponding particular cases the maximal mass of the dust shell that may fall onto a gravastar without converting it into a black hole is determined . for those configurations which remain stable the excursion of their radius is assigned . it is found that even the most compact gravastars can not get beyond the lower limit of the size of conventional stars , provided that the dominant energy condition holds in both cases . it is also shown independent of any assumption concerning the matter interbridging the internal de sitter and the external schwarzschild regions that the better is a gravastar in mimicking a black hole the easier is to get the system formed by a dust shell and the gravastar beyond the event horizon of the composite system . in addition , a generic description of the totally inelastic collision of spherical shells in spherically symmetric spacetimes is also provided in the appendix .

1103.4003

i

we start the introduction by considering the following views of great men : @xmath0``although nobody is in doubt today of the validity of the remarkable interpretation of thermodynamics with which statistical mechanics , following the efforts of boltzmann and gibbs , has recently provided us , it still remains extremely difficult to give a completely accurate justification for it '' -louis de broglie@xcite . @xmath0``the problem [ defining the ensemble distribution depending upon the external conditions imposed on a system ] was solved by gibbs , although a rigorous justification of the distributions obtained is a complicated problem that is still not completely solved at the present time . it is not even clear to what extent this rigorous justification is possible '' -d . n. zubarev@xcite . @xmath0``there is probably no other well - established field of theoretical physical science that is as much plagued by paradox and criticism of its fundamental logic as is statistical mechanics '' -joseph edward mayer and maria goeppert mayer@xcite @xmath0 `` if we are describing only a state of knowledge about a single system , then clearly there can be nothing physically real about frequencies in the ensemble ; and it makes no sense to ask , ' ' which ensemble is the correct one ? `` ... gibbs understood this clearly ; and that , i suggest , is the reason why he does not say a word about ergodic theorems , ... '' e. t. jaynes@xcite @xmath0 there is lot of confusion regarding ergodic hypothesis in the literature ( regarding whether it is necessary for statistical mechanics or not ) and in the usual discussions of people . also the reasons how and why statistical mechanics works are not properly understood . the problem appears complicated as one has to take into consideration various ingredients of vary different nature ( from the character of experimental measurements , large number of degrees of freedom , and the microscopic dynamical properties ) involved in the statistical approach for dynamical systems . in the present review we try to dis - entangle various ingredients and present the resolution of the ergodic problem in a simpler and clear way . a detailed study of literature shows that there are mainly three camps at the foundations of statistical mechanics ( 1 ) ergodic school , ( 2 ) non - ergodic school , and ( 3 ) quantum foundations of statistical mechanics . we critically analyse all the three approaches to reach on a broader understanding of the foundations of statistical mechanics or we attempt to see the `` complete picture '' . aim of the present review is to show that ergodicity is not relevant for equilibrium statistical mechanics of macroscopic systems ( with tolman and landau ) . we will see that the solution of the problem is in the very peculiar nature of the macroscopic observables ( they are sum functions ) and with the very large number of the degrees of freedom involved as first pointed out by khinchin . similar arguments are used by landau based upon the approximate property of `` statistical independence '' . we critically review these ideas . we also review the role of chaos ( classical and quantum ) in the foundations of statistical mechanics , i.e. , where it is important and where it is not important . for students and beginners the purpose of the article is best served if they first understand the first chapter of landau - lifshitz s book on statistical physics and read the little book of khinchin on the mathematical foundations of statistical mechanics . _ again our motivation is to have a broader view of the subject and a pedagogical presentation . _ in the present section we present a brief summary of the essential topics in the form of questions and answers . in section ii we review the ergodic approach to the foundations of statistical mechanics by dividing it into further subsections . first defining the ergodic problem and then giving a historical account of how the ergodic problem came up with the works of maxwell and boltzmann . in subsection ii(c ) the reformulation of the ergodic problem by birkhoff is given and the resolution of the ergodic problem by khinchin is given in subsection d. the role of chaos , integrability , and non - integrability is discussed in subsection e. in section iii we present the non - ergodic approach . this section consists of two subsections one on landau - lifshitz s approach and another on e. t. jaynes approach with a critique . we also advance the plausibility arguments for equal - a - priori probability hypothesis . the last section is devoted to the quantum foundations of statistical mechanics . in this we review the eigenstate thermalization hypothesis , the quantum ergodic theory of von neumann , and other recent quantum - statistical typicality approaches . we will see that von neumann s quantum ergodic theorem is a general statement applicable to systems with many degrees of freedom and the eigenstate thermalization hypothesis is a consequence of quantum chaos , other approaches falling under typicality properties are also analyzed . the issue of compatibility of microscopic time reversibility and macroscopic time irreversibility is not considered here . this has been clarified in detail in the beautiful papers of jeol lebowitz@xcite(for irreversibility in quasi - isolated systems see@xcite ) . in their famous book@xcite lev landau wrote : `` in the discussion of the foundations of statistical physics , we consider from the start the distribution of small subsystems ...... this allows the complete avoidance of ergodic or similar hypotheses , which are in fact not essential as regard to these aims [ i.e. , the foundations of physical statistics ] '' @xmath0 as we know from the most used prescription of gibbs we calculate the phase space averages of dynamical quantities using appropriate ensembles and we find that these phase averages agree very well with experiments . clearly actual experiments are not done on a hypothetical ensemble they are done on the actual system in the laboratory and these experiments take a finite amount of time . thus it is usually argued that actual measurements are time averages and they are equal to phase averages due to ergodicity , @xmath1 now we will give the following reasons against ergodicity . \(1 ) if one assumes that during the measurement time the system samples all the microstates and @xmath2 limit of equation ( 1 ) is reached during the measurement time , then ones assumption is wrong . it is a well know fact that time taken by the system to sample all the available phase is fantastically large ( more than the age of the universe ) . for example@xcite , if we consider a nanoparticle with @xmath3 nuclei each with spin half and consider only the nuclear spin system . the total number of quantum state are @xmath4 . consider that these spins continuously flip from up spin to down spin and vice versa and this spin flipping is caused by phonons from the system and bath in which the nanoparicle is present . at room temperature this frequency is about @xmath5 cycles per second . let us assume that each spin makes a transition during this period . thus the total number of transitions per second is @xmath6 . the time taken by the system to go over all the states is @xmath7 , and the age of universe is @xmath8 . _ thus clearly during the laboratory measurement time the full ensemble is not realized , and only a very tiny fraction of the total number of the members of the ensemble is realized . _ note that the time scale diverges nearly exponentially with the number of atoms in the sample . \(2 ) if one accepts ergodic hypothesis ( and do not accept explanation ( 1 ) ) , that the laboratory measurements are time averages ( measurement time scale being much large as compared to microscopic dynamical time scale ( time taken to flip a spin @xmath9 ) , then one accepts equation ( 1 ) and accepts @xmath2 ) , then time average value of an observable ( or a result of measurement ) in a series of experiments ( repeating the same experiment again and again ) on the same system will agree with ensemble average . this is in fundamental contradiction with what we observe i.e. thermodynamical measurements are not strictly reproducible@xcite . fluctuations in thermodynamic measurements are the unavoidable consequence of our lack of control of microscopic dynamics and thus our lack of control of the initial micro - condition at the start of the measurement . the above discussion against ergodicity opens the question `` but how does statistical mechanics work ? '' and the theory has been so successful . the answer is : in statistical mechanics we concern with macroscopic systems and with macroscopic observables which are very special ones . special in the sense that they are `` sum functions '' i.e. , their value for the whole system is equal to the sum of equivalent functions for the small parts of the body ( for example , energy of the whole system is equal to the sum of energies of the small parts of the body ) . due to the property called `` statistical independence '' of small parts of the body , the sum functions take almost constant value on the energy hypersurface ( details are given in section ii(d ) ) . @xmath0the microscopic time taken by the measurement produces that `` same '' value . thus equality of time average and phase space average is imposed due to the fact of `` temporal constancy of special macroscopic observables in equilibrium '' . the magnitude of fluctuations goes as @xmath10 . thus these are very small for macroscopic systems . put differently , for an extremely large number micro - states ( called typical micro - states ) the macro - state of the system is the same . only very exceptional micro - states ( `` bad states '' with fantastically small probability ) give non - typical behaviour . to explain these lines let us consider a macroscopic system and measure an observable . repeat the experiment many many times , each time the starting micro - state is different but important point is that each time we get approximately the same value for the observable ( since huge number of micro - states are equilibrium micro - states ) . it is extremely rare that we get a value during a measurement which is very different from the value in other measurements . this `` typicality '' is at the heart of why statistical mechanics explain thermodynamic behaviour . here the probability theory inters into statistical description . the typicality explains its high predictability for macroscopic systems ( even though statistical mechanics has probabilistic basis ) . as the macro - observables are extremely insensitive to micro - condition and microscopic dynamics , the role of microscopic dynamics is very insignificant in equilibrium statistical mechanics . see section ii . d. for a detailed explanation , and the role of fermi - pasta - ulam problem in this connection ( section ii . also it is good idea to abandon 19th century philosophy of explaining everything we see around with microscopic dynamics called `` reductionism '' . boltzmann initially tried to explain 2nd law of thermodynamics with microscopic molecular dynamics but later on realized the need of of statistical laws . at each level of complexity new laws emerge not explainable `` purely '' by shroedinger equation@xcite .

as the title says we want to answer the question ; how and why does statistical mechanics work ? as we know from the most used prescription of gibbs we calculate the phase space averages of dynamical quantities and we find that these phase averages agree very well with experiments . clearly actual experiments are not done on a hypothetical ensemble they are done on the actual system in the laboratory and these experiments take a finite amount of time . thus it is usually argued that actual measurements are time averages and they are equal to phase averages due to ergodicity . aim of the present review is to show that ergodicity is not relevant for equilibrium statistical mechanics ( with tolman and landau ) . we will see that the solution of the problem is in the very peculiar nature of the macroscopic observables and with the very large number of the degrees of freedom involved in macroscopic systems as first pointed out by khinchin . similar arguments are used by landau based upon the approximate property of `` statistical independence '' . we review the role of chaos ( classical and quantum ) where it is important and where it is not important . we critically review various quantum approaches ( quantum - statistical typicality approaches ) to the foundations of statistical mechanics . analogies are seen in the khinchin s classical approach and in the modern quantum - statistical typicality approaches . `` unless the conceptual problems of a field have been clearly resolved , you can not say which mathematical problems are the relevant ones worth working on , and your efforts are more than likely to be wasted '' - e. t. jaynes .

as the title says we want to answer the question ; how and why does statistical mechanics work ? as we know from the most used prescription of gibbs we calculate the phase space averages of dynamical quantities and we find that these phase averages agree very well with experiments . clearly actual experiments are not done on a hypothetical ensemble they are done on the actual system in the laboratory and these experiments take a finite amount of time . thus it is usually argued that actual measurements are time averages and they are equal to phase averages due to ergodicity . aim of the present review is to show that ergodicity is not relevant for equilibrium statistical mechanics ( with tolman and landau ) . we will see that the solution of the problem is in the very peculiar nature of the macroscopic observables and with the very large number of the degrees of freedom involved in macroscopic systems as first pointed out by khinchin . similar arguments are used by landau based upon the approximate property of `` statistical independence '' . we review these ideas in detail and in some cases present a critique . we review the role of chaos ( classical and quantum ) where it is important and where it is not important . we criticise the ideas of e. t. jaynes who says that the ergodic problem is conceptual one and is related to the very concept of ensemble itself which is a by - product of frequency theory of probability , and the ergodic problem becomes irrelevant when the probabilities of various micro - states are interpreted with laplace - bernoulli theory of probability ( bayesian viewpoint ) . in the end we critically review various quantum approaches ( quantum - statistical typicality approaches ) to the foundations of statistical mechanics . the literature on quantum - statistical typicality is organized under four notions ( 1 ) kinematical canonical typicality , ( 2 ) dynamical canonical typicality , ( 3 ) kinematical normal typicality , and ( 4 ) dynamical normal typicality . analogies are seen in the khinchin s classical approach and in the modern quantum - statistical typicality approaches . `` unless the conceptual problems of a field have been clearly resolved , you can not say which mathematical problems are the relevant ones worth working on , and your efforts are more than likely to be wasted '' - e. t. jaynes .

cond-mat0612361

i

most of the current experiments involving trapped atomic fermi gases focus on the bec - bcs crossover . by changing the magnetic field around a feshbach resonance , the scattering length @xmath0 of the atoms can be varied from small positive values through very large values near the resonance to small negative values . for @xmath1 , @xmath2 ( where @xmath3 denotes the fermi momentum ) the system can be considered as a bose - einstein condensate ( bec ) of diatomic molecules . the crossover region , @xmath4 , is not yet very well understood from a theoretical point of view . finally , on the other side of the resonance , when @xmath5 , @xmath6 , the system should be in the bcs phase if the temperature is sufficiently low . however , the bcs critical temperature @xmath7 is extremely low , and very soon the magnetic field reaches the point where @xmath7 becomes smaller than the actual temperature @xmath8 , and the system undergoes the phase transition to the normal ( non - superfluid ) phase . one possibility to study the crossover experimentally is to measure the properties of certain collective oscillations . for example , the radial and axial breathing modes of a cigar - shaped trapped fermi gas have been measured over the whole crossover region @xcite . in these experiments one can observe how the frequencies and damping rates of the modes change from what one expects for a bec to what one expects for a collisionless normal fermi gas . assuming that , except in the collisionless normal phase , hydrodynamics is valid , the measured frequencies can give some information on the equation of state in the crossover region . however , this schematic picture is not completely accurate . since the system is in a trap , there is no sharp transition from the superfluid to the normal phase . this can be seen as follows : the bcs critical temperature @xmath7 depends on the atom density @xmath9 , and the density depends on the position @xmath10 . in the center of the trap , the density @xmath11 and hence the local critical temperature @xmath12 are higher than in the outer part of the trap . as a consequence , for a given temperature , the outer part gets already normal at a magnetic field where the inner part is still superfluid . to be more precise , a system in the bcs phase at finite temperature behaves effectively like a mixture of superfluid and normal components with densities @xmath13 and @xmath14 , respectively , which become @xmath15 , @xmath16 in the limit @xmath17 and @xmath18 , @xmath19 in the limit @xmath20 . as a consequence , if @xmath21 , the superfluid inner part of the trap behaves like a mixture of normal and superfluid components , while only the outer part with @xmath22 is completely normal @xcite . if the collision rate was high enough , also the normal component of the gas would behave hydrodynamically . such a system could be described by landau s two - fluid hydrodynamics which has been applied to collective modes in trapped superfluid gases at finite temperature @xcite . however , although in the recent experiments the transition to the normal phase seemed to occur at a value of @xmath23 @xcite ( i.e. , the bcs phase has not really been reached ) , the system behaved already like a collisionless normal fermi gas . hence it seems to be clear that the normal component can not be treated in terms of hydrodynamics , but a description in terms of a vlasov equation is required . we note that there are other approaches to the description of the collective modes at finite temperature . in particular , let us mention the quasiparticle random phase approximation ( qrpa ) @xcite , which can be seen as the linearized form of the time - dependent bogoliubov - de gennes ( bdg ) equations . however , for practical reasons this method is limited to systems with spherical symmetry and numbers of particles up to a few times @xmath24 . another disadvantage of this method is that it does not allow to include a collision term . for the case of clean superconductors , a semiclassical transport theory taking the coupling between normal and superconducting components into account has been developed by betbeder - matibet and nozires @xcite . transport theories of this type have also been used for describing the dynamics of superfluid @xmath25he @xcite . in a preceding paper @xcite , we derived the semiclassical transport equations for the case of trapped atomic fermi gases and applied them to the quadrupole mode of a gas in a spherical trap . we found that the presence of the normal component leads to a strong damping of the hydrodynamic collective mode . the same mechanism might explain the strong damping observed experimentally near the transition to the collisionless behavior @xcite . however , in ref . @xcite we had to replace the gap @xmath26 by a constant in order to find an analytical solution of the transport equations . due to this simplification , which can not really be justified , the damping of the hydrodynamic mode at a given temperature was much weaker than that obtained in qrpa calculations @xcite . in the present paper we will work out a numerical method which allows us to treat the realistic @xmath10-dependence of the gap . in addition , the method is very versatile and allows to treat much more general cases than can be solved analytically in the constant - gap approximation . the basic idea is to replace the continuous phase - space distribution function of the quasiparticles by a sum of a finite number of delta functions in phase space , called `` test particles . '' in the normal phase , the test - particle method is routinely used for solving the vlasov equation , e.g. for simulating heavy - ion collisions in nuclear physics @xcite . it has also been applied to the simulation of the dynamics of normal trapped atomic fermi gases with collision term @xcite and of bose - fermi mixtures @xcite . however , to our knowledge , the test - particle method has not yet been used in the context of superfluid systems , and in fact the numerical difficulties are quite different from those encountered in the usual applications . the article is organized as follows . in sec . [ transportbcs ] , we give a brief summary of the transport equations for the bcs phase and their linearization in the case of small deviations from equilibrium . we also give arguments why some terms which appear in the equations can be neglected . [ testparticlemethod ] we introduce the test - particle method for the case of small oscillations around equilibrium . we describe in detail a number of tricky points we encountered during the implementation of the method , in particular the calculation of the test - particle trajectories , the generation of the test - particle distribution in phase space , and the initialization after a delta - like perturbation . in sec . [ firstresults ] we present the first results obtained with the help of this method , again for the quadrupole mode in a spherical system . finally , in sec . [ conclusions ] , we summarize and draw our conclusions .

at finite temperature , the hydrodynamic collective modes of superfluid trapped fermi gases are coupled to the motion of the normal component , which in the bcs limit behaves like a collisionless normal fermi gas . the coupling between the superfluid and the normal components is treated in the framework of a semiclassical transport theory for the quasiparticle - distribution function , combined with a hydrodynamic equation for the collective motion of the superfluid component . we develop a numerical test - particle method for solving these equations in the linear response regime . as a first application we study the temperature dependence of the collective quadrupole mode of a fermi gas in a spherical trap . the coupling between the superfluid collective motion and the quasiparticles leads to a rather strong damping of the hydrodynamic mode already at very low temperatures . at higher temperatures the spectrum has a two - peak structure , the second peak corresponding to the quadrupole mode in the normal phase

at finite temperature , the hydrodynamic collective modes of superfluid trapped fermi gases are coupled to the motion of the normal component , which in the bcs limit behaves like a collisionless normal fermi gas . the coupling between the superfluid and the normal components is treated in the framework of a semiclassical transport theory for the quasiparticle - distribution function , combined with a hydrodynamic equation for the collective motion of the superfluid component . we develop a numerical test - particle method for solving these equations in the linear response regime . as a first application we study the temperature dependence of the collective quadrupole mode of a fermi gas in a spherical trap . the coupling between the superfluid collective motion and the quasiparticles leads to a rather strong damping of the hydrodynamic mode already at very low temperatures . at higher temperatures the spectrum has a two - peak structure , the second peak corresponding to the quadrupole mode in the normal phase

1312.6925

i

the scissors mode , which is a rotational mode of isovector character in deformed nuclei@xcite , is usually visualized by an out of phase orbital motion of protons against neutrons . this intuitive picture lead to the prediction of the scissors mode both in terms of nucleonic@xcite and bosonic@xcite degrees of freedom , and has formed the basis of many investigations @xcite . in some respect , the scissors mode can be thought of as an extension of the familiar isoscalar collective rotation@xcite to the isovector rotational motion . indeed , in a recent work @xcite we have used a simple field theory model for the collective orbital motion to show how an effective hamiltonian of the 2-rotor type emerges naturally from the cooperation of the isoscalar ( goldstone ) and isovector ( scissors ) degrees of freedom . in usual nuclear hamiltonians , the orbital angular momentum alone is not conserved . a familiar example is the spin - orbit potential , which explicitly breaks the separate spin and orbital symmetries on the level of the mean field hamiltonian . the explicit breaking of the spin symmetry on the mean field level has important consequences for the properties of nuclear rotational states , and we mention two of them here in more detail : first , if we use the mechanism of spontaneous symmetry breaking to generate a deformed mean field , it is the _ total _ angular momentum symmetry which will be restored by the collective rotation of the whole system . that is , the isoscalar goldstone modes are generated essentially by the total angular momentum @xmath0 , and their degrees of freedom will lead to an effective rotational hamiltonian of the schematic form @xmath1 , where @xmath2 is the total moment of inertia . in analogy to the situation found for the pure orbital case@xcite , we can then anticipate the existence of a mode at low excitation energy , which is generated by the isovector combination of @xmath3 and @xmath4 , and which cooperates with the goldstone mode so as to give a total effective rotational hamiltonian of the schematic 2-rotor form @xmath5 . the structure of the goldstone and scissors modes can therefore be expected to come out similar to the case of the pure orbital motion , but with the orbital angular momenta replaced by the total angular momenta . second , the explicit breaking of the spin symmetry on the mean field level leads to new types of correlation functions , because the spin operators ( @xmath6 and @xmath7 ) can generate particle - hole states of finite excitation energy . these new types of non - interacting correlators , which are visualized by the bubble graphs shown by fig . 2 of the next section with one of the external operators being a spin operator , will obviously modify the sum rules , because the latter ones can be represented in terms of correlation functions@xcite . in particular , we will see that the presence of these correlators implies that the spin part of the m1 operator can couple to the low energy scissors modes . this observation is of particular interest , because it is known that there exists a discrepancy between the theoretical and experimental values of the inverse energy weighted m1 sum rule , for which the low energy scissors mode is expected to give the dominant contribution@xcite . the purpose of the present paper is to consider a `` minimal model '' to get analytic insight into the role of the spin for the low energy rotational states in deformed nuclei . by `` minimal '' we mean that the spin dependence of the nuclear hamiltonian is taken into account only via the one - body spin - orbit potential , and not via the two - body interactions . besides simplicity , the reason to restrict ourselves to this `` minimal model '' is to keep the discussions distinct from the collective spin modes at higher energies , which are known to be strongly modified by interactions of the spin - spin type@xcite . in order to concentrate on the important role of the spin - orbit potential , we also will not include the effects of pairing correlations in this paper . the role of spin dependent interactions and pairing correlations for the m1 sum rules will be discussed in a separate work@xcite . in sect . ii we will briefly explain the model and the approximations ( mean field approximation and random phase approximation ( rpa ) ) . in sect . iii we will explain the constraints which follow from angular momentum conservation , and in sect . iv we will discuss the m1 sum rules , with a special emphasis on the role of the spin in the inverse energy weighted ( iew ) sum rule . v gives a summary of our results . more formal discussions and derivations are presented in four appendices .

background : : : the scissors mode is a rotational mode of isovector character in deformed nuclei . together with the isoscalar rotation , it is the prominent collective mode at low excitation energies . special emphasis is put on the coupling of the spin part of the m1 operator to the scissors mode . methods : : : we apply the mean field approximation and random phase approximation ( rpa ) to a model hamiltonian based on a simple quadrupole - quadrupole interaction .

background : : : the scissors mode is a rotational mode of isovector character in deformed nuclei . together with the isoscalar rotation , it is the prominent collective mode at low excitation energies . purpose : : : we use a simple field theory model to investigate the role of the nucleon spin for the magnetic sum rules associated with the low - lying scissors mode . special emphasis is put on the coupling of the spin part of the m1 operator to the scissors mode . methods : : : we apply the mean field approximation and random phase approximation ( rpa ) to a model hamiltonian based on a simple quadrupole - quadrupole interaction . the effects of the spin - orbit interaction are included in the mean field hamiltonian . ward - takahashi relations are used to derive the m1 sum rules . results : : : the presence of the spin - orbit interaction leads to interference terms in the inverse energy weighted sum rule . it is shown that the low - lying scissors mode , which is generated by the isovector combination of proton and neutron total angular momenta , gives the main contribution to the spectral sum for this case . conclusions : : : the basic concept of the scissors mode as an isovector vibrational rotation remains valid in the presence of the nucleon spin . the inverse energy weighted sum rule , however , receives non - trivial modifications because of interference terms .

hep-ph9604416

i

even though qcd yields a remarkably good description of the strong interaction , the low energy hadron physics has to be modelled phenomenologically . this is due to the fact that the usual perturbative approach in the coupling constant can not be applied to qcd below energies of the order of 1 gev . most of the phenomenological results were based on pcac and current algebra . however , in 1979 , weinberg @xcite showed how to reobtain many of these predictions by means of an effective lagrangian . the fields in that lagrangian are the light mesons , ( pions , kaons and etas ) which are understood as the goldstone bosons ( gb ) arising from the spontaneous breaking of chiral symmetry . the lagrangian is built as an expansion in derivatives , that respects the symmetry breaking pattern of qcd . indeed , the first term in the expansion is fixed by the symmetry requirements and accounts for the current algebra results . the next terms in the expansion produce further corrections , which depend on several phenomenological parameters but are always consistent with the qcd symmetry constraints . these techniques were later developed to one loop in a set of papers by gasser and leutwyler @xcite . they showed how to obtain amplitudes involving light mesons , as functions of their momenta , their masses and those few phenomenological parameters . by fitting these parameters from a few low energy experiments it is then possible to obtain successful predictions for other processes the whole approach is known as chiral perturbation theory ( chpt ) . very recently some partial higher order calculations @xcite have appeared in the literature as well as a complete two loop calculation of @xmath4 scattering @xcite , which will be needed in order to analyze new data to come from da@xmath8ne and brookhaven . for a general review of the available experimental data on pion physics and future prospects , we refer the reader to @xcite . nevertheless , there are some intrinsic limitations when applying chpt , namely , the fact that the amplitudes calculated within the chiral approach are only unitary in the perturbative sense , that is , up to the next order in the external momenta . such a breakdown of unitarity is most severe at high energies , where the external momenta is no longer a good expansion parameter , although it can also occur at moderate energies @xcite . as a result , it is not possible to reproduce resonant states , which are one of the most characteristic features of the strongly interacting regime . many different methods have been proposed in order to improve this behavior and thus to extend the applicability of chpt to higher energies ; among them : the use of pad approximants @xcite , the explicit introduction of resonances @xcite , the k - matrix @xcite , the large @xmath9 limit @xcite ( n being the number of gb ) or the inverse amplitude method ( iam ) @xcite . this work is devoted precisely to the last method , which can be justified within a dispersive approach and can easily reproduce the two lightest resonances : the @xmath2 in @xmath4 scattering @xcite and the @xmath10 in @xmath11 scattering @xcite . but not only that , the iam also improves considerably the fit to data even in non - resonant channels , almost up to the first two particle inelastic threshold ( the many particle inelastic thresholds can be neglected since they are suppressed by phase space factors ) . this fit provides a remarkably good parametrization that can be used for other processes . indeed , in a previous work @xcite , the authors showed how it can be used together with a simple unitarization prescription to obtain successful results on @xmath12 up to 700 mev . of course , it is also possible to obtain very good parametrizations @xcite of @xmath4 or @xmath11 elastic scattering by including all resonant states explicitly . however , our aim choosing the iam is to reproduce these phenomena just with the few phenomenological parameters present in the chpt lagrangian . in this way , even though their masses and widths will not be obtained with great accuracy , resonances can be regarded as real predictions . that is one of the relevant features of the iam since other very popular unitarization methods are not able to reproduce resonances unless they are explicitly introduced in the calculation . that is , for instance , the case with the k - matrix . the purpose of this work is , first , to study how high in energies the iam yields good results and what are its limitations . we would also like to know whether it is possible to reproduce further resonance states . it is clear that the best candidates are the lightest resonances whose dominant decay modes are @xmath4 or @xmath11 . we have listed them in table 1 . in case these resonances were not accommodated after our unitarization , it would be interesting to understand why . second , once we have a good fit to these resonances , we want to make a complete numerical analysis of several low - energy quantities of interest , like the chiral parameters or the scattering lengths , including estimations for their errors . as we will see below , we expect that the iam somehow will include effects that can not be obtained from the pure @xmath13 expansion . [ cols="^,^,^,^,^",options="header " , ] lightest resonances with @xmath4 or @xmath11 dominant decay modes . data taken from @xcite . 0.cm finally , we would like to comment on another motivation of the present work , which at first may not seem very related to the main topic . the philosophy of the chiral approach has also reached the description of the strongly interacting symmetry breaking sector ( sisbs ) of the standard model @xcite . the scalar sector of such a model displays the same symmetry breaking pattern as two flavor massless qcd . hence it is possible to build an effective lagrangian , much as it is done for chpt @xcite . although the electroweak gb are not physical , using this lagrangian it is possible to obtain predictions for the scattering of longitudinal gauge bosons @xcite at future colliders , like the lhc . indeed , there are already experimental proposals to measure the electroweak chiral parameters at cms @xcite . most of the works on the sisbs make use of the equivalence theorem @xcite , which allows us to read the observable amplitudes , in terms of longitudinal gauge bosons , directly from those with gb . this theorem has been recently proved in the chiral lagrangian formalism @xcite and seems to be severely constrained by the lack of unitarity . at this point is when the unitarization procedures come into play and it is crucial to know whether they are reliable , since what we are now looking for are real predictions and not elaborated fits to still unavailable data . in section 2 we review some basic aspects of exact and perturbative unitarity and we define the partial waves in elastic scattering . section 3 introduces the iam , first with a derivation from dispersion theory and then by studying the constraints to its applicability . section 4 and section 5 are organized in the same way , although they refer to @xmath0 and @xmath1 chpt , respectively : first we apply the iam to chpt with the chiral parameters obtained from low energy experiments in order to study the iam predictive power . next , we present an iam fit to the data . for the best @xmath1 fit we present the unitarized results for the scattering lengths and some other phenomenological parameters . then , in section 6 , we study the analytic structure on the complex plane of the iam amplitudes . in section 7 we present the conclusions . there is also an appendix where we give the elastic scattering formulae used in this work , as well as a discussion on perturbative unitarity .

numerically , we find that this unitarization technique yields the correct complex analytic structure in terms of cuts and poles . indeed , using the chiral parameter estimates obtained from low energy experiments we obtain the poles associated with the and resonances . we have obtained the chiral parameters and scattering lengths , which can be relevant for future experiments . + universidad complutense . pelez + theoretical physics group . lawrence berkeley laboratory + university of california . usa .

based on a dispersive approach , we apply the inverse amplitude method to unitarize one - loop and chiral perturbation theory . numerically , we find that this unitarization technique yields the correct complex analytic structure in terms of cuts and poles . indeed , using the chiral parameter estimates obtained from low energy experiments we obtain the poles associated with the and resonances . just by fixing their actual masses we obtain a parametrization of the and phase shifts in eight different channels . with this fit we have then calculated several low - energy phenomenological parameters estimating their errors . among others , we have obtained the chiral parameters and scattering lengths , which can be relevant for future experiments . pacs : 14.40.aq , 14.40.cs,11.80.et,13.75.lb 6.5 in 8.5 in epsf lbl-38645 ucm - ft 3/96 april 1996 revised nov96 to appear in : prd 1-sept-97 * the inverse amplitude method in chiral + perturbation theory * .5 cm a. dobado + departamento de fsica terica . + universidad complutense . 28040 madrid , spain + and .3 cm j.r . pelez + theoretical physics group . lawrence berkeley laboratory + university of california . berkeley , california 94720 . usa . .5 cm

hep-ph9604416

m

let us briefly review the standard derivation @xcite of the inverse amplitude method , since we will use it later in order to understand the applicability of the method . any partial wave obtained from a relativistic quantum field theory should present a characteristic analytic structure in the complex @xmath42 plane . indeed , the reaction threshold becomes a cut in the from @xmath43 to @xmath44 . due to crossing symmetry , there should be another left cut along the negative axis . if we now apply cauchy s theorem to our complex amplitudes we obtain integral equations known as dispersion relations . for instance , a three times subtracted dispersion relation is @xmath45 where we have not written explicitly the left cut ( lc ) contribution . the number of subtractions needed depends on how the amplitude behaves at infinity in order to ensure the vanishing of the contributions coming from closing the integral contour . we have chosen three subtractions since we are going to use @xmath39 chpt amplitudes which at high @xmath42 behave as @xmath46 . but our arguments remain valid for @xmath41 amplitudes when using four times subtracted dispersion relations , etc ... the chpt partial waves present both cuts and we can calculate both the subtraction constants @xmath47 and the integrand inside eq.[disp ] _ perturbatively _ @xmath48 where we have expanded the subtraction constants in terms of @xmath49 . the iam is based on the fact that the function @xmath50 displays the very same analytic structure of @xmath51 , apart from some possible pole contributions . for later convenience , we will make use of @xmath52 . notice that we have multiplied @xmath53 by a real function without singularities ; thus we keep the same analytic structure and we can write a very similar dispersion relation : @xmath54 where @xmath55 stands for possible pole contributions . the advantage of using @xmath56 is that , using eqs.[uni ] and [ punit ] , we can calculate exactly the integral over the right cut ( but not on the left , since those equations only hold on the elastic cut ) , as follows : @xmath57 note that we are denote by @xmath51 the exact amplitude , which is unknown , although we know its analytic properties . in contrast , the expressions for @xmath36 and @xmath38 , etc ... have been calculated explicitly . as we did before , we can also expand the @xmath58 subtraction coefficients in powers of @xmath59 , and then rewrite the dispersion relation for @xmath56 , which now reads @xmath60 where we have approximated @xmath61 on the left cut and we have neglected the pole contribution . in other words , @xmath62 this is the iam result that we are going to use in the present work . incidentally , eq.[iam ] can be understood as the formal @xmath63 $ ] pad approximant of the chpt amplitude . it is important to remark that if we expand again eq.[iam ] at low energies , we find @xmath64 that is , we recover the chpt expansion . hence , up to @xmath65 our method and chpt yield the same low energy results if the same chiral lagrangian parameters are used . let us review all the approximations made in the previous section , in order to comment how they will constraint the iam applicability : in eq.[preiam ] we have replaced the @xmath56 left cut integral by that of @xmath66 . as we have remarked in the preceeding discussion , eqs.[uni ] and [ punit ] are only exact on the right cut . on the left cut we can not write the chain of equalities that lead to eq.[img ] . nevertheless , if we use the chpt result as an approximation : @xmath67 we get @xmath68 notice that , in order to obtain the @xmath34 phase shifts , we are going to calculate @xmath69 for real @xmath70 . that means that the denominator @xmath71 inside the integrals is never going to be very small , which somehow will wash out the error on the left cut . but note also that treating differently the right and left cuts violates crossing symmetry . indeed , in @xcite it has already been pointed out that the pad approximants do not reproduce correctly the subleading logarithms that would appear at _ next order _ in the chiral expansion ( @xmath65 in this case ) . of course they would be obtained if we applied the iam to the chiral amplitudes at @xmath65 , but still the method would not yield the correct logarithms at @xmath72 and so on . at high energies chiral logarithms are not so relevant , but at low energies they are a very important feature of chpt and indeed they can give the dominant contribution in some channels . nevertheless , from eq.[recover ] we see that at low energies the iam yields the very same @xmath73 chpt expansion , _ including _ the _ dominant _ chiral logarithms . the contribution from the left cut and subleading logarithms is @xmath65 . as a consequence , if we try to make a low energy fit to the data , the parameters that we would obtain with the iam would not lie very far from those of chpt , but they will not be the same . that is the reason why , in the following sections , we will denote with a hat the parameters obtained from any iam fit . in eq.[preiam ] , we have neglected the contributions coming from zeros in the amplitude , that will appear as poles of the inverse function . there is no way to know _ a priori _ whether or not a partial wave will vanish for a given value of @xmath42 , although it is known that chiral amplitudes have zeros below threshold , which are known as adler zeros . their position is not known except for the @xmath74 channel , where the pole is located at threshold . in our derivation it is compensated by the same zero in the @xmath75 channel . that is not the case of the @xmath76 amplitudes and therefore we are neglecting the contribution of their residue . consequently , our amplitudes are not valid to obtain adler zeros and that will affect our results at low energy ( but no more than @xmath65 ) . that is another reason to differentiate the parameters obtained from our fit from those of the pure chiral expansion . this is apparently a harmless assumption in the above reasoning , although it dramatically affects the results of the iam . in fact , it can happen that @xmath77 . in the @xmath78 channels of @xmath4 scattering or in the @xmath79 in @xmath11 , this only occurs for isolated values of @xmath42 , at or below threshold . in particular , that means that the iam amplitudes will have the same zeros as the lowest order chiral amplitudes . however , every other partial wave vanishes at @xmath37 , for any @xmath42 . as a consequence , the formula in eq.[iam ] is no longer valid . nevertheless , we can generalize our previous derivation , in order to include those channels whose leading order is @xmath39 . we only have to go through the very same steps , although now we would write a dispersion relation for @xmath80 . but let us remember that the main improvement of the approach is that we are calculating exactly the integral of @xmath81 over the right cut . however , for that purpose we need an imaginary part , and by looking at eq.[punit ] we can see that @xmath77 implies that @xmath82 . therefore , unless we have a calculation up to @xmath83 , the corresponding imaginary part will vanish . hence when following the derivation of the iam if @xmath77 the best we can get is plain chpt again . at present , only @xmath84 calculations are available and we can only expect to obtain a real improvement with our approach in the six channels listed above . thus , we will not be able to reproduce the @xmath85 resonance . in order to obtain @xmath86 on the right cut , eq.[img ] , we have just made use of the elastic unitarity condition of eq.[uni ] . however , the right cut is composed of many superimposed cuts , each one corresponding to a different inelastic intermediate channel . actually , eq.[uni ] is only true below the first inelastic threshold , and the real unitarity condition reads @xmath87 the sum is over all the physically accessible intermediate states @xmath88 , whose phase space is @xmath89 . as far as we are neglecting electromagnetic interactions , the first inelastic channel in @xmath4 is the four pion intermediate state , at 550 mev . similarly , for @xmath11 is @xmath90 , whose threshold is @xmath91 910 mev . strictly speaking , only for lower energies the elastic approximation is exact . nevertheless , the contribution of these intermediate states is strongly suppressed by the four particle phase space and we expect the iam to provide a good approximation . unfortunately within the range of energies we are interested in , there are intermediate channels which are not suppressed by phase space . indeed , at approximately @xmath92 mev the inelastic @xmath93 threshold opens up . its phase space factor is the @xmath89 in eq.[sigma ] , with @xmath94 . therefore , above the two kaon threshold we have to reconsider the derivation of the iam . let us illustrate with @xmath4 scattering how inelastic effects modify our result . as the starting point , for @xmath95 , we have a new unitarity relation : @xmath96 where we have denoted by @xmath97 the generic @xmath51 pion elastic scattering amplitude and by @xmath98 the @xmath34 partial wave of the process @xmath99 . thus we now have , for @xmath100 , that @xmath101 which differs from eq.[img ] in the term coming from two kaon intermediate production . if we follow the very same steps of our previous derivation , we arrive at @xmath102 notice that , using chpt , @xmath103 . but at these high energies that is not negligible . besides , we are interested in the above integral for physical values of @xmath42 and therefore the denominator will be almost divergent for some @xmath104 . for these reasons we can not neglect this integral and then we should not trust the iam since it could miss some relevant physical features . that is indeed the case in pion scattering since , as it can be seen in table 1 , there is one resonance , the @xmath105 , whose nature is closely related to the @xmath93 threshold . nowadays , the interpretation of that resonance is still controversial : different authors propose different poles ( not always just one ) in the vicinity of the @xmath93 inelastic cut @xcite . as we will see later , our approach is not able to reproduce any of these poles , consistent with the fact that the iam makes use just of elastic unitarity . at this point we want to remark the importance of understanding why and when the method does no longer yield the right results . let us remember that we are also thinking in possible applications of this unitarization procedures to the electroweak chiral effective lagrangian , whose reference model is the standard model with a heavy higgs . in such case , one would expect to see a broad resonance in the scalar channel and we want to have a unitarization procedure whose predictions we can trust . throughout the derivation of the iam we have been using the chiral amplitudes up to @xmath39 . nevertheless , it is possible to extend the argument to include higher order terms , as for instance the @xmath41 contributions . in that case we would have started from a four times subtracted dispersion relation for the two - loop calculation . once more , the integral over the right cut would be related to the one for @xmath52 . working out the expansion of the subtraction constants , we would then arrive to @xmath106 again that is the formal [ 1,2 ] pad approximant , and it satisfies the elastic unitarity condition . as we have already mentioned two recent papers have appeared with @xmath41 calculations of @xmath4 scattering within @xmath0 chpt @xcite . we have not used these results , since , as we have just seen , they will not help us to overcome any of the preceeding objections to the iam . however it is quite likely that , have we used them , the parameters of the fits that we will present in the next sections would had been slightly modified .

based on a dispersive approach , we apply the inverse amplitude method to unitarize one - loop and chiral perturbation theory . pacs : 14.40.aq , 14.40.cs,11.80.et,13.75.lb 6.5 in 8.5 in epsf lbl-38645 ucm - ft 3/96 april 1996 revised nov96 to appear in : prd 1-sept-97 * the inverse amplitude method in chiral + perturbation theory * .5 cm a. dobado + departamento de fsica terica .

based on a dispersive approach , we apply the inverse amplitude method to unitarize one - loop and chiral perturbation theory . numerically , we find that this unitarization technique yields the correct complex analytic structure in terms of cuts and poles . indeed , using the chiral parameter estimates obtained from low energy experiments we obtain the poles associated with the and resonances . just by fixing their actual masses we obtain a parametrization of the and phase shifts in eight different channels . with this fit we have then calculated several low - energy phenomenological parameters estimating their errors . among others , we have obtained the chiral parameters and scattering lengths , which can be relevant for future experiments . pacs : 14.40.aq , 14.40.cs,11.80.et,13.75.lb 6.5 in 8.5 in epsf lbl-38645 ucm - ft 3/96 april 1996 revised nov96 to appear in : prd 1-sept-97 * the inverse amplitude method in chiral + perturbation theory * .5 cm a. dobado + departamento de fsica terica . + universidad complutense . 28040 madrid , spain + and .3 cm j.r . pelez + theoretical physics group . lawrence berkeley laboratory + university of california . berkeley , california 94720 . usa . .5 cm

cond-mat0211615

i

the pyrochlores with the chemical formula @xmath0mo@xmath1o@xmath2 ( @xmath3 being the divalent element ) exhibit a number of interesting and not completely understood phenomena,@xcite which present the main motivation for the analysis of electronic and magnetic structure of these compounds , which we undertake in this work . @xmath11 the pyrochlores @xmath0mo@xmath1o@xmath2 have rather puzzling magnetic phase diagram as a function of averaged ionic radius of @xmath3-sites , @xmath12 ( fig . [ fig.tcexp]).@xcite large @xmath12 ( @xmath13@xmath14@xmath8@xmath15 ) stabilizes the ferromagnetic ( fm ) ground state . typical examples of the fm pyrochlores are nd@xmath1mo@xmath1o@xmath2 and gd@xmath1mo@xmath1o@xmath2 . the curie temperature ( @xmath16 ) is of the order of 80 k and slowly increases with @xmath12 . smaller @xmath12 ( @xmath17@xmath14 ) give rise to the spin - glass ( sg ) behavior . the characteristic transition temperature into the sg state is of the order of 20 k. the canonical example of the sg compounds is y@xmath1mo@xmath1o@xmath2.@xcite now it is commonly accepted that the exchange coupling between nearest - neighbor ( _ nn _ ) mo spins changes the sign around @xmath14 and becomes antiferromagnetic ( afm ) in the sg region . the pyrochlores with afm interactions in the lattice present a typical example of geometrically frustrated systems with infinitely degenerate magnetic ground state.@xcite according to the recent experimental data,@xcite the sg behavior itself is caused by the combination of the geometrical frustrations and the disorder of the local lattice distortions . the latter is required in order to produce an inhomogeneity in the distribution of the _ nn _ interactions and freeze the random spin configuration . the origin of this behavior itself is very complicated , and we will not be able to address it properly in our work . however , our main concern will be in some sense more general : we will try to understand which parameter of the crystal structure controls the sign of the _ nn _ magnetic interactions , and which part of the electronic structure is responsible for the fm and afm interactions in these compounds ? contrary to the widespread point of view that the magnetic ground state is controlled by the mo - o - mo angle , we will argue that the key parameter is in fact the mo - mo distance , which is directly related with the unit cell volume and controls the strength of direct interactions between extended mo(@xmath6 ) orbitals . according to the experimental data , all mo pyrochlores which show the sg behavior are small - gap insulators . however , an opposite statement is generally incorrect and in different compounds the ferromagnetism is known to coexist with the metallic as well as insulating behavior.@xcite we will show that this trend can be naturally understood in terms of the mo - mo distances and the strength of the on - site coulomb interaction @xmath7 amongst the mo(@xmath6 ) electrons . @xmath11 the spin - orbit interaction ( soi ) in the pyrochlore lattice should generally lead to a non - collinear magnetic ordering associated with the modulation of single - ion anisotropy axes.@xcite the magnitude and the main source of this non - collinearity ( whether it is primarily associated with the mo or @xmath3 sublattices , and whether it reflects the ordering of spin or orbital magnetic moments ) is largely unresolved problem . in the present work we will try to rationalize this question by focusing on the magnetic structure of the mo sublattice . we will show that as long as the system is metallic ( that by itself depends on the strength of the coulomb interaction @xmath7 ) , the non - collinearity is very small . however , the transition into the insulating state is accompanied by the abrupt change of the spin and orbital magnetic structures , which become essentially non - collinear . @xmath11 some ferromagnetic ( or nearly ferromagnetic ) pyrochlores exhibit rather unusual anomalous hall effect ( ahe ) . a non - vanishing hall conductivity ( @xmath18 ) emerges near @xmath16 , then monotonously increases with the decrease of the temperature @xmath19 , and reaches the maximal value ( @xmath18@xmath8@xmath20 @xmath21 for nd@xmath1mo@xmath1o@xmath2 ) near @xmath19@xmath4@xmath22 . both magnitude and the temperature dependence of the ahe is in a drastic contrast with other oxide materials , such as colossal - magnetoresistive manganites , for example.@xcite the theory of the ahe ( originally called as the extraordinary hall effect ) has very long and rather controversial history.@xcite many controversies were related with the lack of the general transport theory and partly with the lack of good computer facilities in 1950s and 1960s , so that many discussions went around very simplified and not always justified approximations , such as the free - electron and weak scattering limit , the single - orbital tight - binding approach , the classical boltzmann transport equation , etc.@xcite the modern way to address the problem is to use the kubo formalism and relate @xmath18 with the real part of the antisymmetric off - diagonal element of the conductivity tensor.@xcite from this point of view there are no rigid constraints which would forbid the existence of @xmath18 , even for periodic systems , once the time - reversal symmetry is violated macroscopically.@xcite the main question is which mechanism leads to the finite value of @xmath18 . the first one was proposed by karplus and luttinger,@xcite who stressed the importance of soi in the problem . although the concrete scenario considered by karplus and luttinger was not correct,@xcite they were certainly right by arguing that the soi is essential in order to couple the spin polarization of the conduction electrons to the lattice and produce the right - left asymmetry of the stationary bloch states , which may lead to the hall current , perpendicular to both the electric field and the magnetization . however , an exceptionally large @xmath18 observed in the pyrochlore compounds urged several authors to search for a new and unconventional mechanism for the ahe . such mechanism was proposed to be due to the spin chirality ( basically , the solid angle subtended by three non - coplanar spins ) in essentially non - collinear magnetic structure.@xcite the chirality acts as an effective field , which in the combination with the peculiar lattice geometry is believed to yield a finite @xmath18 . although the mechanism itself is rather exotic and there is a great deal of cancellations between different elements of the non - collinear spin structure,@xcite these authors were certainly right by stressing the crucial role of the inter - band transitions in the problem of ahe . in this work we will go back to a more conventional picture and argue that the large value of @xmath18 in some mo pyrochlores can be explained by rather unique combination of three factors : ( i ) large soi associated with the mo(@xmath6 ) states in the nearly collinear fm structure ; ( ii ) strong enhancement of the soi by the on - site coulomb interactions in the @xmath5 electron systems,@xcite and ( iii ) half - metallic electronic structure of the fm pyrochlores.@xcite regarding the coulomb interaction itself , which in our work will be treated as a parameter , we will be able to present a very consistent picture by arguing that @xmath7@xmath8@xmath9-@xmath10 ev in mo pyrochlores accounts simultaneously for ( i ) metal - insulator transition ; ( ii ) fm - sg transition ; ( iii ) required enhancement for the ahe in the fm pyrochlore compounds . the rest of the paper is organized as follows . in sec . [ sec : main_details ] we remind the basic details of the crystal and electronic structure of @xmath0mo@xmath1o@xmath2 . we consider three characteristic compounds corresponding to @xmath3@xmath4 y , gd and nd . in sec . [ sec : model ] we describe the model hartree - fock ( hf ) approach , which combines the ideas of _ ab - initio _ electronic structure calculations with the physics of degenerate hubbard model on the pyrochlore lattice . [ sec : dos ] and [ sec : exchange ] summarize results of calculations without soi ( correspondingly , the basic changes of the electronic structure and the orbital ordering , and the behavior of inter - atomic magnetic interactions ) . secs . [ sec : nc ] and [ sec : hall ] are devoted to the analysis of soi related properties ( the non - collinear magnetic structure , the hall conductivity , and the magneto - optical effect ) . finally , in sec . [ sec : summary ] we discuss some subtle points related with the interpretation of experimental data and give the summary of the entire work .

the coulomb repulsion plays a very important role in the problem , and the semi - empirical value- ev accounts simultaneously for the metal - insulator ( m - i ) transition , the ferromagnetic ( fm ) - spin - glass ( sg ) transition , and for the observed enhancement of the anomalous hall effect ( ahe ) . the m - i transition is mainly controlled by . the sign of exchange interactions in the insulating state is controlled by the mo - mo distances . smaller distances favor the antiferromagnetic coupling , which preludes the sg behavior in the frustrated pyrochlore lattice .

being motivated by recent experimental studies , we investigate magnetic structures of the mo pyrochloresmoo ( y , nd , and gd ) and their impact on the electronic properties . the latter are closely related with the behavior of twelve mo( ) bands , located near the fermi level and well separated from the rest of the spectrum . we use a mean - field hartree - fock approach , which combines fine details of the electronic structure for these bands , extracted from the conventional calculations in the local - density approximation , the spin - orbit interaction , and the on - site coulomb interactions amongst the mo( ) electrons , treated in the most general rotationally invariant form . the coulomb repulsion plays a very important role in the problem , and the semi - empirical value- ev accounts simultaneously for the metal - insulator ( m - i ) transition , the ferromagnetic ( fm ) - spin - glass ( sg ) transition , and for the observed enhancement of the anomalous hall effect ( ahe ) . the m - i transition is mainly controlled by . the magnetic structure at the metallic side is nearly collinear fm , due to the double exchange mechanism . the transition into the insulating state is accompanied by the large canting of spin and orbital magnetic moments . the sign of exchange interactions in the insulating state is controlled by the mo - mo distances . smaller distances favor the antiferromagnetic coupling , which preludes the sg behavior in the frustrated pyrochlore lattice . large ahe is expected in the nearly collinear fm state , near the point of m - i transition , and is related with the unquenched orbital magnetization at the mo sites . we also predict large magneto - optical effect in the same fm compounds .

cond-mat0211615

i

we have presented results of model hf calculations for pyrochlores @xmath0mo@xmath1o@xmath2 ( @xmath3@xmath4 y , gd , and nd ) . the model combines details of lda electronic structure for the mo(@xmath5 ) bands located near the fermi level , the spin - orbit interaction , and the on - site coulomb interactions amongst the mo(@xmath6 ) electrons . the main results can be summarized as follows . @xmath11 the magnetic ground state of the @xmath0mo@xmath1o@xmath2 compounds depends on the mo - mo distance , which controls the strength of afm se interactions between the @xmath52 orbitals . the latter competes with the fm interactions between the @xmath53 orbitals , and dominates when the mo - mo distance is small . in the frustrated pyrochlore lattice , this corresponds to the transition into the sg state , though we can not specify precisely which mechanism freeze the random configuration of the mo spins . it could be the disorder of the mo - mo bondlengths proposed in refs . and . another possibility is the anisotropy of nearest - neighbor magnetic interactions caused by the orbital ordering.@xcite it is also important to note that the coulomb repulsion @xmath7 should be sufficiently strong in order to open the band gap and suppress the fm double exchange interactions between the @xmath53 orbitals . @xmath11 the metal - insulator transition in these systems does not necessary coincide with the fm - sg transitions . all metallic compounds are expected to be nearly collinear ferromagnets , due to the double exchange mechanism . however , the inter - atomic exchange coupling in the insulating state can be both ferromagnetic and antiferromagnetic , depending on the mo - mo bondlength . the finding is qualitatively consistent with the experimental data.@xcite the magnetic structure in the insulating state is expected to be strongly non - collinear , due to the competition between the isotropic exchange interactions and the single - ion anisotropies . @xmath11 the large hall conductivity can be explained by the combination of three factors in the nearly collinear fm state : large soi at the mo sites , which unquenches the orbital magnetization ; strong enhancement of this effect by the on - site coulomb interaction ; and the half - metallic electronic structure of the fm pyrochlores . we also expect large magneto - optical effect in the same compounds . a possible way to enhance these properties is to use heavy @xmath50 elements , which have large spin - orbit coupling . unfortunately , all known @xmath50 pyrochlores are non - magnetic , though some of them are presumably close to the point of magnetic instability.@xcite nevertheless , one can try to exploit an old idea known from the physics of intermetallic alloys,@xcite and replace part of the mo sites in @xmath0moo@xmath2 by the @xmath50 elements . then , the main question is whether the remaining mo sites will be magnetic in the new environment or not . if they will , they will also magnetize the @xmath50 states at the neighboring sites , and in this way enhance the non - reciprocal characteristics of the considered pyrochlore compounds . as a purely model example , we have considered the ordered gd@xmath1mowo@xmath2 alloy ( using the lattice parameters of gd@xmath1mo@xmath1o@xmath2 ) . according to the lsda calculations , the system is expected to be ferromagnetic . the values of ( spin only ) magnetic moments at the mo and w sites are 2.1 and 1.5 @xmath161 , respectively . the situation is highly unusual , because the examples when @xmath6 and @xmath50 elements form a magnetic solution are rather rare . although gd@xmath1mowo@xmath2 itself may be rather unrealistic example from the technological point of view , the idea to use mo atoms in order to magnetize the heavy @xmath50 elements in the pyrochlore lattice seems to be promising . in conclusion , the simple hf calculations allow to rationalize many unusual properties of the pyrochlore compounds @xmath0mo@xmath1o@xmath2 . it is also important that all these properties can be explained by the universal value of @xmath7@xmath8@xmath9-@xmath10 ev , which is of the same order of magnitude as the characteristic bandwidth for the mo(@xmath5 ) electrons . this will certainly place the considered compounds to the direct proximity of the metal - insulator transition , meaning that the quantum fluctuations beyond the mean - field hf approach will play an important role too . this is certainly a very important direction for the extension of the present analysis . finally , we would like to make several comments on the comparison between our calculations and the experimental magnetization@xcite and neutron diffraction@xcite data . \(1 ) the experimental value of the mo moment ( @xmath8@xmath247-@xmath248 @xmath161 ) seems to be smaller than @xmath8@xmath160 @xmath161 obtained in our mean - field calculations for the nearly collinear fm state , even after taking into account the orbital part ( @xmath8@xmath194 @xmath161 ) , which according to the third hund s rule is antiparallel to the spin one and partly compensate the latter . the small difference is presumably related with the quantum fluctuations at the proximity of the metal - insulator transition.@xcite \(2 ) the experimental data for nd@xmath1mo@xmath1o@xmath2 and sm@xmath1mo@xmath1o@xmath2 typically indicate at the existence of two antiferromagnetically coupled sublattices . these sublattices are usually regarded as mo and @xmath3 ones . however , according to the electronic structure point of view , the _ spin _ coupling between these two sublattices should be ferromagnetic,@xcite being actually consistent with the experimental data for gd@xmath1mo@xmath1o@xmath2.@xcite the controversies can be reconciled with each other if the local magnetic moment at the nd / sm sites has a large orbital counterpart , which is larger than the spin one . the afm orientation of the spin and orbital moments at the nd / sm sites is again the subject of the third hund s rule . then , the magnetization and neutron diffraction experiments presumably probe the afm coupling between _ spin _ magnetic moments at the mo sites and the _ orbital _ magnetic moments at the nd / sm sites . gd@xmath249 has no orbital moment , and therefore the experimental data for this compound indicate at the conventional fm coupling between two spin sublattices . then , the sharp decrease of the magnetization observed in nd@xmath1mo@xmath1o@xmath2 at low @xmath19,@xcite which is frequently tempted to connect with the anomaly of the hall conductivity is due to the orbital magnetic structure , not the spin one . the change of the _ orbital _ magnetic structure seems to be the right subject to concentrate at when trying to explain , for example , the temperature dependence of the ahe . in this respect , another interesting problem seems to be the behavior of _ spin and orbital _ magnetic moments at the mo sites , because already in the ground state these two sublattices from mutually non - collinear magnetic structures . therefore , it is reasonable to expect very unconventional spin and orbital dynamic at the elevated temperatures .

we use a mean - field hartree - fock approach , which combines fine details of the electronic structure for these bands , extracted from the conventional calculations in the local - density approximation , the spin - orbit interaction , and the on - site coulomb interactions amongst the mo( ) electrons , treated in the most general rotationally invariant form . the magnetic structure at the metallic side is nearly collinear fm , due to the double exchange mechanism . we also predict large magneto - optical effect in the same fm compounds .

being motivated by recent experimental studies , we investigate magnetic structures of the mo pyrochloresmoo ( y , nd , and gd ) and their impact on the electronic properties . the latter are closely related with the behavior of twelve mo( ) bands , located near the fermi level and well separated from the rest of the spectrum . we use a mean - field hartree - fock approach , which combines fine details of the electronic structure for these bands , extracted from the conventional calculations in the local - density approximation , the spin - orbit interaction , and the on - site coulomb interactions amongst the mo( ) electrons , treated in the most general rotationally invariant form . the coulomb repulsion plays a very important role in the problem , and the semi - empirical value- ev accounts simultaneously for the metal - insulator ( m - i ) transition , the ferromagnetic ( fm ) - spin - glass ( sg ) transition , and for the observed enhancement of the anomalous hall effect ( ahe ) . the m - i transition is mainly controlled by . the magnetic structure at the metallic side is nearly collinear fm , due to the double exchange mechanism . the transition into the insulating state is accompanied by the large canting of spin and orbital magnetic moments . the sign of exchange interactions in the insulating state is controlled by the mo - mo distances . smaller distances favor the antiferromagnetic coupling , which preludes the sg behavior in the frustrated pyrochlore lattice . large ahe is expected in the nearly collinear fm state , near the point of m - i transition , and is related with the unquenched orbital magnetization at the mo sites . we also predict large magneto - optical effect in the same fm compounds .

1005.1312

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the principal results of this paper are as follows . assuming only a _ large scale _ anisotropic distribution of crs ( generated , for example by a nearby accelerator , such as a snr ) and a @xcite ( gs ) cascade of alfvenic turbulence originating from some scale @xmath1 , which is the longest scale relevant for the wave - particle interactions , we calculated the propagation of the crs down their gradient along the interstellar magnetic field . it is found that the cr distribution develops a characteristic angular shape consisting of a large scale anisotropic part ( first eigenfunction of the pitch - angle scattering operator ) superposed by a beam , tightly focused in the momentum space in the local field direction . the large scale anisotropy carries the _ momentum dependence _ of the source , _ _ while both the beam angular width and its fractional excess ( with respect to the large scale anisotropic component ) grow with momentum ( as @xmath338 and @xmath318 , respectively ) . apart from the width and the fractional excess of the beam , the theory predicts its maximum momentum on the ground that beyond this momentum the beam destroys itself . all the three quantities are completely determined by the turbulence scale @xmath1 . even if @xmath1 is considered unknown , it can be inferred from any of the three independent milagro measurements . these are the width , the fractional excess and the maximum energy of the beam , and all the three consistently imply the same scale @xmath3391 pc . the calculated beam maximum momentum encouragingly agrees with that measured by milagro ( ~10 tev / c ) . the theoretical value for the angular width of the beam is found to be @xmath340 , where @xmath341 . the beam fractional excess related to the large scale anisotropic part of the cr distribution is @xmath342 . both quantities also match the milagro results for @xmath343 tev . so the beam has a momentum scaling that is one power shallower than the cr carrier , it is drawn from . this finding will receive a due discussion . obviously , the determination of the absolute value of the beam excess would require the source intensity . for the lack of such information , an indirect inference was made in sec.[sub : observational - consequences ] about the galactic cr ( gcr ) large scale anisotropy being of the same order as the large scale anisotropy responsible for the beam . below the rationale for this admission is given which we open with the following notations : * @xmath344 and @xmath345 are the large scale anisotropic and isotropic parts of the galactic ( not assumed to be related to the source of the beam ) distribution , respectively * @xmath346 and @xmath347 ( i.e. @xmath112 in sec.[sec : particle - propagation ] ) are the similar quantities related to the source of the beam at the distance @xmath301 * @xmath237 is the beam distribution on the top of @xmath346 unless the source of the beam is also responsible for the gcr , the quantities @xmath346 and @xmath344 are independent of each other and can not be related since the source intensity , the distance to it , @xmath301 and the losses are unknown . even if the beam and its carrier @xmath346 propagate without significant losses ( or suffer similar losses ) , the current theory determines only a fractional excess @xmath348 ( independent of @xmath301 ) . for the same reasons we do not know what is the source contribution @xmath347 into the total isotropic cr background @xmath349 . however , since the beam a is not observed at the minimum of the ( measured ) total large scale @xmath350 as it would , were @xmath351 the case ( see fig.[fig : eigenfunction ] ) , we infer @xmath352 . furthermore , since @xmath353 is calculated and @xmath350 is measured along with @xmath237 , the both quantities @xmath346 and @xmath344 can also be determined . we found that @xmath354 for @xmath2pc which was , in turn , deduced from two other independent measurements ( beam width @xmath355 and its maximum energy @xmath356 ) . since milagro measurements indicate that @xmath357 , we conclude that @xmath358 . a more specific relation between the two would not be meaningful since the measurements of @xmath359 are rather limited . if particle losses from the flux tube are negligible , it follows that @xmath360 for @xmath361 ( @xmath362- being the source position ) . these findings allow us to speculate about the possible source of the beam . first , if the source is an active accelerator that emits strongly anisotropic particle flux , the last relation implies that @xmath363 . since by observations @xmath364 , such source can not contribute significantly to the knee region at @xmath365 pev . in this case our inferences of @xmath1 from three independent measurements all strikingly pointing at the @xmath366 pev accelerator cut - off energy ( with @xmath367) must be either a coincidence or a different mechanism couples the galactic knee particles with the scale of the turbulence that generates the beam . if it is not a coincidence and the source contributes significantly to the observed cr background , the escaping particle flux should be quasi - isotropic , @xmath368 ( to allow for @xmath369 ) . in combination with the assumption that particles escape in the wide range 1tev -3 pev ( to both form the beam and to inject mhd energy at the scale @xmath1 ) , the source is unlikely to be an active accelerator , but rather a region of an enhanced cr density , with a steep cut - off at @xmath365 pev . the near isotropy at the source is not inconsistent with a currently working accelerator , but escape in such a broad energy range probably is . indeed , at least the available ( known to us ) mechanisms , that offer a broad energy escape from a snr along with the spectrum steepening ( spectral break , starting 1 - 2 orders of magnitude below the cut - off , @xcite ) seem to fall short to cover three orders of magnitude in energy . moreover for the source to be a recent accelerator ( such as a recent snr , suggested by @xcite with the spectrum @xmath370 ) the mechanism should be found that makes the spectrum of the escaping particles at least @xmath371 steeper ( and still steeper if the acceleration was strongly nonlinear ) . combined with the nonlinear acceleration ( which is required in @xcite model ) this would make an acceptable spectrum but again , it is not clear how these particles can initiate the mhd cascade at such a long scale , to ensure the required value of @xmath1 . our argument against the beam and the bulk cr @xmath344 coming from the same source is that the observed beam is not located at the minimum of the angular distribution of the first eigenfunction , so we need to allow for a second component . this is largely a technical limitation , stemming from 1-d transport model , in which @xmath237 and @xmath346 are coupled , as wells as from the single eigenfunction approximation . by removing this latter simplification alone ( which is probably even necessary for an accurate description of @xmath346 at tev- energies , sec.[sec : distance - to - the ] ) the above constraint can be relaxed . another possibility is a lateral diffusion and drifts of @xmath346-component from the flux tube . an interesting obvious conjecture from the common origin of the beam and @xmath344 would be that the proton knee at @xmath365 pev is also of the same origin as the beam . however , the beam spectrum is calculated to be one power flatter than its carrier . according to milagro the beam index is about @xmath372 , so that the carrier should have an index @xmath373 which is closer to the gcr than to a hypothetical recent snr. in particular , this would not support the single source hypothesis of the gcr knee @xcite . equally problematic would be an active accelerator scenario , unless the steepening mechanisms of the run - away cr mentioned above can be adopted after due modifications . all told , the beam is likely to be at least partly drawn from the gcr ( due to the relation between the indices @xmath374 ) but the gs- turbulence that creates the beam must be driven by considerably more energetic , @xmath375 pev particles ( due to the constraint , @xmath376 ) , unless the spiral - arm 1pc value @xcite for @xmath1 is , indeed , acceptable . to explore the possibility of the gcr origin of the beam , an extension of the above model is necessary . at a minimum , the model should include the transport of energetic particles across the flux tube . on the other hand , the particle beaming processes should remain similar to that described in sec.[sec : pitch - angle - scattering - of ] . however , such consideration is out of the scope of this paper , particularly because the transport across the flux tube requires a separate study . yet another possibility is that the required gs cascade starts in the local interstellar cloud ( lic ) . it has a suitable size of @xmath377 pc @xcite and there would be no problem with the spectrum slope since the beam would be drawn from the gcr with the right spectral index @xmath378 . whether the turbulence energy can be injected at the required scale remains to be studied . if it can , the above transport and beam focusing mechanism would be applicable since a parsec wave length and the gs cascade are the only requirements to draw the beam out of the background cr distribution . to conclude , the model presented in this paper offers an explanation of the most pronounced milagro beam a , while there are two more . one of them is the beam b , @xmath379 away from beam a and the second one is in the cygnus loop area @xmath380 away . any attempt to incorporate those two beams into our current model would be speculative . we merely note that the local ism environment is complicated indeed thus offering many possibilities in explaining various cr anomalies ( @xcite ) . approaches to the explanation of all three beams based on such a complexity , could hardly pass the occam s razor test . in contrast , the model suggested in this paper is devoid of free parameters , if the knee energy at @xmath381pev can be associated with the maximum cr energy of the source of the beam a. even though such an association is not proven , our propagation model predicts the following three beam characteristics : its width , fractional excess and maximum energy to be the functions of a single quantity , the longest wave - particle interaction scale @xmath1 . they all give the correct milagro values for @xmath382 pc , which is unlikely to be coincidental . however , the exact origin of this particular value remains unclear . ld and mm acknowledge the hospitality of kitp in santa barbara during the program particle acceleration in astrophysical plasmas , july 26-october 3 , 2009 . the work of pd and mm is supported by nasa under the grants nnx 07ag83 g and nnx09at94 g as and by the department of energy , grant no . de - fg02 - 04er54738 .

both the acceleration of cosmic rays ( cr ) in supernova remnant shocks and their subsequent propagation through the random magnetic field of the galaxy deem to result in an almost isotropic cr spectrum . the key assumption is that crs are scattered by a strongly anisotropic alfven wave spectrum formed by the turbulent cascade across the local field direction . the width , the fractional excess and the maximum momentum of the beam are calculated from a systematic transport theory depending on a single scale which can be associated with the longest alfven wave , efficiently scattering the beam . possible approaches to determination of the scale from the characteristics of the source are discussed .

both the acceleration of cosmic rays ( cr ) in supernova remnant shocks and their subsequent propagation through the random magnetic field of the galaxy deem to result in an almost isotropic cr spectrum . yet the milagro tev observatory discovered a sharp ( arrival anisotropy of cr nuclei . we suggest a mechanism for producing a weak and narrow cr beam which operates en route to the observer . the key assumption is that crs are scattered by a strongly anisotropic alfven wave spectrum formed by the turbulent cascade across the local field direction . the strongest pitch - angle scattering occurs for particles moving almost precisely along the field line . partly because this direction is also the direction of minimum of the large scale cr angular distribution , the enhanced scattering results in a weak but narrow particle excess . the width , the fractional excess and the maximum momentum of the beam are calculated from a systematic transport theory depending on a single scale which can be associated with the longest alfven wave , efficiently scattering the beam . the best match to all the three characteristics of the beam is achieved atpc . the distance to a possible source of the beam is estimated to be within a few 100pc . possible approaches to determination of the scale from the characteristics of the source are discussed . alternative scenarios of drawing the beam from the galactic cr background are considered . the beam related large scale anisotropic cr component is found to be energy independent which is also consistent with the observations .

gr-qc0002092

i

this article presents the calculation , using continuum and lattice methods , of boundary terms in 3-dimensional gravity . the gravity theory is presented in first order palatini form , this being a particular example of the general class of bf models @xcite as this is the most convenient presentation for deriving the discretization . we find a variety of boundary conditions , and discuss the significance of these for different types of boundaries in space - time . the bulk theory of three - dimensional gravity is well known to be a topological field theory , however it is also well known that three - dimensional topological field theories can give rise to non- topological boundary degrees of freedom , the classic example being the cs theory giving rise to a wzw model on the boundary @xcite . in the case of three dimensional gravity with cosmological constant , one can utilize a trick that relates the action to the difference of two cs actions , and then use the standard cs - wzw relationship , however the actual boundary conditions are a little more subtle . in three dimensions this is relevant to the @xmath0 space , or more generally to btz black hole solutions . in this paper we wish to understand in the context of discretization of quantum gravity the boundary degrees of freedom that correspond to black hole entropy . this paper is directed towards a longer study of boundary terms in gravity theories , ultimately in @xmath1 dimensions , with the hope of understanding directly in a theory of quantum gravity , the possibile origin of holographic phenomena , and of the microscopic details of black hole entropy , in particular well out of the supersymmetric and extremal limits which have been very well studied in the framework of string theory . the actual type of discretization that we consider here is maybe at first sight a bit unusual . the approach is originally due to ponzano and regge @xcite where they considered a simplicial decomposition of a three - manifold and the path - integral is then defined as a summation over the possible sets of lengths of the edges of the dual lattice . the alternative of course is to fix the size of the simplices and to form the path integral by summation over possible simplicial decompositions . for the major part of this paper , we will be discussing three dimensional models that have a topological invariance in the bulk and thus the fixed decomposition is somewhat innocuous but again the use of this simplicial decomposition also for the boundary where in general we believe there are physical degrees of freedom needs to be considered more cautiously . in addition we eventually need to extend our results to the realistic case of four - dimensional gravity where we do not even have topological invariance in the bulk making things more intricate though hopefully still manageable . we begin however , in the context of euclidean three - dimensional gravity where already we find some interesting results concerning the boundary theories . we will start off with a discussion of a discretization of the bf theory that corresponds to three - dimensional euclidean gravity in the framework of the ponzano - regge discretization , that is a discretization into tetrahedra with edges labelled by so(3 ) spins , and each tetrahedron then weighted in the path integral ( sum ) by the corresponding @xmath2 symbol . from this discretization we can then derive a boundary action and will compare this to what we may expect from the corresponding bf theory . we in fact find that there are two simple types of boundary conditions , one leads to a topological boundary theory and the other to a dynamical boundary theory . in addition we discuss mixed boundary conditions which are relevant for the boundary at infinity in @xmath0 for example . we discuss modifications to these boundary actions that arise when one replaces the group @xmath3 with @xmath4 which would correspond to gravity with lorentzian signature . we also discuss the regularization via quantum groups and find some interesting relationships to work on string theory and @xmath0/cft duality . finally we make some suggestions for understanding black hole entropy in this context and we discuss briefly the extension of these methods to four - dimensional quantum gravity .

boundary actions for three - dimensional quantum gravity in the discretized formalism of ponzano - regge are studied with a view towards understanding the boundary degrees of freedom . these degrees of freedom postulated in the holography hypothesis are supposed to be characteristic of quantum gravity theories . in particular it is expected that some of these degrees of freedom reside on black hole horizons . this paper is a study of these ideas in the context of a theory of quantum gravity that requires no additional structure such as supersymmetry or special gravitational backgrounds .

boundary actions for three - dimensional quantum gravity in the discretized formalism of ponzano - regge are studied with a view towards understanding the boundary degrees of freedom . these degrees of freedom postulated in the holography hypothesis are supposed to be characteristic of quantum gravity theories . in particular it is expected that some of these degrees of freedom reside on black hole horizons . this paper is a study of these ideas in the context of a theory of quantum gravity that requires no additional structure such as supersymmetry or special gravitational backgrounds . lorentzian as well as euclidean regimes are examined . some surprising relationships to liouville theory and string theory in are found . gr - qc/0002092

1211.5799

i

during last decades , the interest in diffusive processes has grown tremendously because of their universality in diverse physical areas ; ranging from condensed matter to elementary particle physics and gravitation @xcite . in particular , it has emerged an intense activity in the study of brownian motion in curved manifolds motivated by problems coming from biophysics @xcite . for instance , the lateral diffusion of proteins and lipids occurring inside cell membranes are interesting and complex since they determine the flux of nutrients between the cell and its exterior affecting , in consequence , the cell functionality @xcite . from the theoretical point of view , it is difficult to describe this phenomenon because the interactions with the remaining components of the membrane and the protein finite - size effects @xcite . besides , there are also curvature contributions @xcite and thermal fluctuations that produce shape undulations @xcite coupled to the lateral motion @xcite . and on top of that , protein diffusion is also affected by changing membrane thickness @xcite . the simplest approach to study this problem is to consider the brownian motion of a punctual particle on a frozen two - dimensional regular surface that represents the membrane @xcite . in this approximation , both thermal shape fluctuations and finite - size effects have not been taken into account explicitly but as an effective result reflected in the parameters of the model . as discussed below and although the results presented here are quite general for the brownian motion on a manifold , this work is primarily motivated by the aforementioned transport phenomena . although the understanding of brownian motion was established a century ago , it is noteworthy to mention that the study of brownian motion on manifolds started three decades ago . since the seminal work of n. g. van kampen @xcite , the fundamental equations of brownian motion on manifolds were established and the manifolds introduced , like in classical mechanics , as a result of the appropriate canonical transformations involved in the system with certain holonomic constraints . manifolds also appeared naturally in the dynamics of polymers in solution @xcite , when the polymer is modeled by means of the theory of brownian motion with constraints ( see , e.g. , @xcite , and more recently @xcite for a review ) . in this case , the number of constraints that take into account the bonds between monomers establishes the dimension of the manifold . however , in a real situation , the rigid constraints represent idealizations of stiff potentials that limit the motion in a certain spatial domain @xcite , whereas by including either thermal or statistical fluctuations the rigid constraints , in general , will no longer represent idealizations of elastic potentials @xcite . nonetheless , albeit the fluctuations are present , the rigid constraints may emulate real molecular bonds as illustrative toy models and , in some cases , realistic models can be also represented through a coarse - grain or large - scale description , like in the rigid - rod and wormlike chain models @xcite . in addition , brownian dynamics on curved manifolds becomes a natural framework to study diffusion on crystals with topological disorder , where the torsion of the manifold is crucial to quantify the degree of disorder @xcite . furthermore , smerlak has found that the eckart s heat flux in general relativity and the generalization of the tolman - ehrenfest relation to non - equilibrium stationary states , as well as gravitational corrections , can be best understood through the mean - square displacements of hypothetic particles in static isotropic curved space - times @xcite . although the diffusion equation is suited to study the brownian motion of free particles on curved surfaces , a more complete description is provided by the langevin equation . the latter is based on the newton s equation of motion but including a rapidly fluctuating force , gaussian distributed , representing the interaction among the particle and the solvent . it is well - known that in euclidean open spaces the mean - square displacement ( msd ) calculated from the langevin equation reproduces the standard einstein kinematical relation . in this kind of spaces , both langevin and diffusion equations describe the same dynamical behavior at the diffusive time regime , i.e. , @xmath0 , where @xmath1 is the friction coefficient of the solvent , @xmath2 is the particle mass and @xmath3 the momentum relaxation time @xcite . in a curved space , one might ask whether this property is preserved and , in general , what is the dependence of the dynamics on the geometry of the space . these points have been recently addressed by m. polettini @xcite whom posed a langevin equation , derived by a gauge principle and proved that its overdamped limit corresponds to the diffusion equation in curved manifolds . we here discuss the aforementioned points , but taking the damped ( @xmath4 ) and overdamped ( @xmath5 ) limits in the msd and look at its behaviour as a function on the geometrical properties of the space . in this work , we write down the langevin equation for manifolds following the same method introduced by e. j. hinch @xcite . the starting point is the newton s equation for free particles in a @xmath6-dimensional hypersurface @xmath7 . free means here that particles do not interact between each other and non external force is acting on them . nevertheless , they are restricted to move on @xmath7 . for the local momenta @xmath8 and local coordinates @xmath9 , the resulting langevin equations are , @xmath10 where @xmath11 is the riemannian metric tensor and @xmath12 the christoffel symbols . it turns out that these equations are the same found by kleinert and shabanov @xcite , who discussed its generalization to connections with torsion , as well as those derived in the work of m. polettini using a gauge principle ( invariance under local rotations ) @xcite . besides , the global version of these equations were obtained by e. j. hinch @xcite for the particular case of two monomers , one of them excessively massive , with one constraint . we also find that the msd , up to first order in curvature , calculated from ( [ locall-2 ] ) is given by @xmath13\nonumber\\ & -&\frac{2r_{g}}{3}\left(\tau_{b}d_{0}\right)^{2}\mathcal{j}\left(t/\tau_{b}\right)+\cdots , \label{result1}\end{aligned}\ ] ] where the terms in the square parenthesis are found to be the standard msd for the particle dynamics in the euclidean @xmath14 space and @xmath15 is a non - dimensional function that characterizes the particle dynamics coupled to the curvature ( see below at appendix b for its definition ) . @xmath16 is the ricci scalar curvature and @xmath17 is the geodesic distance of the general riemannian manifold . taking this equation , we are able to investigate the particle dynamics at different time scales : @xmath18 and for @xmath19 . @xmath20 is a characteristic time for the positions and momenta relaxation of the solvent molecules , at which the langevin description is not longer valid , and @xmath21 is the time scale when curvature effects become evident . it is shown that equation ( [ msd0 - 1 ] ) reproduces the same leading curvature effects in the diffusion regime as in ref . @xcite , which is based on the diffusion equation on curved manifolds . the geometrical properties become evident at times @xmath22 much more longer than @xmath21 . when this happens , the system reaches the thermodynamical equilibrium . in this regime , we reproduce the free - particle dynamics based on the diffusion equation on curved manifolds . the latter is explicitly compared with our computer simulations . using the well - known result that for compact supports ( @xmath23 ) the spectra of the laplace - beltrami operator , @xmath24 , is discrete . thus , it is easy to find that the expectation value for any observable @xmath25 in the overdamped limit is , @xmath26 where @xmath27 is the first non - zero eigenvalue of @xmath24 and @xmath28 is the volume element of the riemannian geometry @xcite . it is remarkable that the leading term obtained in this way allows us to determine the steady spacial density @xmath29 where @xmath30 and @xmath31 the volume of @xmath7 . this density ( see appendix d ) is consistent with the original calculation performed by kramers @xcite and recently discussed in @xcite . we explicitly analyze the dynamics of particles confined along a circle , as well as on a sphere . we test equations ( [ msd0 - 1 ] ) and ( [ obev1 ] ) by means of brownian dynamics computer simulations based on an heuristic adaptation of the ermak - mccammon algorithm @xcite to the langevin equation along curves , as well as on the standard algorithm . in the first case , which is here only applied to the circle , the particles are allowed to move in any direction with equal probability , but the geodesic distances they travel are gaussian randomly distributed . in the second case , the particles are subjected under the action of a spring - like force field in the @xmath32 ( @xmath33)-dimensional euclidean space , where the corresponding potential , with a mexican hat " shape , has its minima at the same points of the circle ( sphere ) . in the limit case of very stiff springs , we get the same results from both numerical routes , and analytical one , as we will see further below . we should mention that the inclusion of a spring - like potential to reproduce the holonomic constrain is a controversial issue because the agreement between theory and simulations establishes a clear example where the particles dynamics with lagrange constraints is equivalent with that using the stiff elastic potential even in systems with fluctuations . moreover , we should point out that this is not in contradiction with the work done by e. j. hinch @xcite and kampen and lodder @xcite . in particular , it is shown that the single canonical partition function using the stiff potential posed is the same for the single canonical partition function on the circle ( sphere ) in the limit of very stiff springs field , as far as the spring - like constant @xmath34 scales with the square of temperature . after the introduction , the manuscript is organized as follows . in section ii we present the langevin equation for curved manifolds , written in both global and local coordinates . in addition , we study the curvature effects on the msd at the following time regimes : @xmath18 and @xmath19 . in section iii we study the particle dynamics on the geometrical regime ( @xmath35 ) by means of the diffusion equation on curved manifolds . in section iv we explicitly compare the predictions for particles restricted to move along a circle and on a sphere with brownian dynamics computer simulations . finally , in section v we summarize some concluding remarks and perspectives of our work .

brownian motion of free particles on curved surfaces is studied by means of the langevin equation written in riemann normal coordinates . in the diffusive regime we find the same physical behavior as the one described by the diffusion equation on curved manifolds [ j. stat . our findings are corroborated by means of brownian dynamics computer simulations based on a heuristic adaptation of the ermak - mccammon algorithm to the langevin equation along the curves , as well as on the standard algorithm , but for particles subjected to an external harmonic potential , deep and narrow , that possesses a

brownian motion of free particles on curved surfaces is studied by means of the langevin equation written in riemann normal coordinates . in the diffusive regime we find the same physical behavior as the one described by the diffusion equation on curved manifolds [ j. stat . mech . ( 2010 ) p08006 ] . therefore , we use the latter in order to analytically investigate the whole diffusive dynamics in compact geometries , namely , the circle and the sphere . our findings are corroborated by means of brownian dynamics computer simulations based on a heuristic adaptation of the ermak - mccammon algorithm to the langevin equation along the curves , as well as on the standard algorithm , but for particles subjected to an external harmonic potential , deep and narrow , that possesses a mexican hat " shape , whose minima define the desired surface . the short - time diffusive dynamics is found to occur on the tangential plane . besides , at long times and compact geometries , the mean - square displacement moves towards a saturation value given only by the geometrical properties of the surface .

1403.7007

i

the demand for high - definition video streaming services such as youtube and netflix is driving the rapid growth of internet traffic . in order to mitigate the effect of this increased load on the underlying communication infrastructure , content delivery networks deploy storage memories or caches throughout the network . these caches can be populated with some of the content during off - peak traffic hours . this cached content can then be used to reduce the network load during peak traffic hours when users make the most requests . content caching has a rich history , see for example @xcite and references therein . more recently , it has been studied in the context of video - on - demand systems for which efficient content placement schemes have been proposed in @xcite among others . the impact of different content popularities on the performance of caching schemes has been investigated for example in @xcite . a common feature among the caching schemes studied in the literature is that those parts of a requested file that are available at nearby caches are served locally , whereas the remaining files parts are served via orthogonal transmissions from an origin server hosting all the files . recently , @xcite proposed a new caching approach , called _ coded caching _ , that exploits cache memories not only to deliver part of the content locally , but also to create coded multicasting opportunities among users with different demands . it is shown there that the reduction in rate due to these coded multicasting opportunities is significant and can be on the order of the number of users in the network . the setting considered in @xcite consists of a single layer of caches between the origin server and the end users . the server communicates directly with all the caches via a shared link , and the objective is to minimize the required transmission rate by the server . for this basic network scenario , coded caching is shown there to be optimal within a constant factor . these results have been extended to nonuniform demands in @xcite and nonuniform cache - access in @xcite , and to online caching systems in @xcite . in practice , many caching systems consist of not only one but multiple layers of caches , usually arranged in a tree - like hierarchy with the origin server at the root node and the users connected to the leaf caches @xcite . each parent cache communicates with its children caches in the next layer , and the objective is to minimize the transmission rates in the various layers . there are several key questions when analyzing such hierarchical caching systems . a first question is to characterize the optimal tradeoff between the cache memory sizes and the rates of the links connecting the layers . one particular point of interest is if there is any tension between the rates in the different layers in the network . in other words , if we reduce the rate in one layer , does it necessarily increase the rate in other layers ? if there is no such tension , then both layers can simultaneously operate at minimum rate . a second question is how to extend the coded caching approach to this setting . can we can simply apply the single - layer scheme from @xcite in each layer separately or do we need to apply coding across several layers in order to minimize transmission rates ? files is connected to @xmath0 mirrors each able to store @xmath1 of the files . each of the mirrors , in turn , is connected to @xmath2 caches each able to store @xmath3 of the files . a single user is attached to each of these caches . once the mirrors and caches are filled , each user requests one of the @xmath4 files . the aim is to minimize the rate @xmath5 from the server to the mirrors and the rate @xmath6 from the mirrors to the caches . in the figure , @xmath7 , @xmath8 , @xmath9 , and @xmath10 . ] in this work , we focus on a hierarchical caching system with two layers of caches as depicted in fig . [ fig : setup ] . for simplicity , we will refer to the first layer of caches as mirrors . we propose a new caching scheme exploiting two types of coded caching opportunities : the first type involves only a single layer at a time , i.e. , it operates between a node and its direct children . these single - layer coding opportunities are available over the link connecting the origin server to the mirrors and also in the link connecting each mirror to the user caches . the second type involves two layers at a time . these two - layer opportunities are available between the origin server and the user caches . we show that , by striking the right balance between these two types of coded caching opportunities , the proposed caching scheme attains the approximately optimal memory - rate tradeoff to within a constant additive and multiplicative gap . due to the possible interaction between the two cache layers , the network admits many different prefetching and delivery approaches . it is thus perhaps surprising that a combination of these two basic schemes is sufficient to achieve the approximately optimal memory - rate tradeoff . furthermore , investigating the achievable rates also reveals that there is no tension between the rates over the first and second layers up to the same aforementioned gap . thus , both layers can simultaneously operate at approximately minimum rate . the remainder of the paper is organized as follows . we describe the problem setting in section [ sec : problemsetup ] and provide some preliminaries in section [ sec : singlelayer ] . section [ sec : results ] presents our main results and discusses their engineering implications . section [ sec : achievableschemes ] introduces the proposed caching scheme and characterizes its performance . the proofs of our main results are discussed in section [ sec : proofs ] and their details are provided in the appendices . appendix [ sec : lowerbounds ] proves information - theoretic bounds on the performance of any caching scheme . the proof of the constant multiplicative and additive gap between the performance of the proposed scheme and the optimal caching scheme is provided in appendices [ sec : gapr1 ] and [ sec : gapr2 ] .

caching of popular content during off - peak hours is a strategy to reduce network loads during peak hours . recent work has shown significant benefits of designing such caching strategies not only to deliver part of the content locally , but also to provide coded multicasting opportunities even among users with different demands . exploiting both of these gains we propose a new caching scheme that combines two basic approaches . the first approach provides coded multicasting opportunities within each layer ; the second approach provides coded multicasting opportunities across multiple layers . by striking the right balance between these two approaches , we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap . we further show that there is no tension between the rates in each of the two layers up to the aforementioned gap . thus , both layers can simultaneously operate at approximately the minimum rate .

caching of popular content during off - peak hours is a strategy to reduce network loads during peak hours . recent work has shown significant benefits of designing such caching strategies not only to deliver part of the content locally , but also to provide coded multicasting opportunities even among users with different demands . exploiting both of these gains was shown to be approximately optimal for caching systems with a single layer of caches . motivated by practical scenarios , we consider in this work a hierarchical content delivery network with two layers of caches . we propose a new caching scheme that combines two basic approaches . the first approach provides coded multicasting opportunities within each layer ; the second approach provides coded multicasting opportunities across multiple layers . by striking the right balance between these two approaches , we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap . we further show that there is no tension between the rates in each of the two layers up to the aforementioned gap . thus , both layers can simultaneously operate at approximately the minimum rate .

1305.4685

i

this account of physics starts by considering _ configuration space _ @xmath1 , i.e. the space of all possible configurations @xmath2 that a physical system can take . in ordinary mechanics , the configurations are particle positions @xcite . in field theories they are the values taken by the field on a fixed spatial slice . in gr they are the values taken by the 3-metrics on a spatial slice with fixed spatial topology , @xmath3 @xcite . one then builds composite objects from the configurations , one s first goal being to write down an action for one s theory . ( other composite objects include notions of distance , of information and of correlation @xcite . ) _ temporal relationalism _ @xcite is then the classical precursor of the well - known frozen formalism problem facet of the problem of time . it concerns the leibnizian ` no time for the universe as a whole ' idea @xcite . this is mathematically implemented by use of geometrical actions that happen to be parametrization - irrelevant , is the metric on configuration space . for mechanics , @xmath4 , where @xmath5 is the potential energy and @xmath6 is the total energy . the @xmath7 are the momenta conjugateto the @xmath8 . for gr ( minisuperspace models for now ) , @xmath9 ric and @xmath10 is ( a truncation of ) the inverse undensitized gr supermetric , @xmath11 . = = s s : = |||| _ = . [ srel ] the first form involves the physical line element @xmath12 . on the other hand , the second expression contains the conformally - related configuration space geometrical line element @xmath13 . for mechanics , ( [ srel ] ) is jacobi s formulation @xcite , and for minisuperspace it is misner s formulation @xcite . parametrization - irrelevant actions must lead , by dirac s argument @xcite , to primary constraints . these include the well - known hamiltonian constraint of gr and the energy constraint of relational particle mechanics ( rpm ) models . both of these constraints are purely quadratic in the corresponding momenta due to the square - root form of @xmath13 . the above timelessness is then to be resolved by mach s ` time is to be abstracted from change ' . three alternatives for this involve ` any change ' ( rovelli @xcite ) , ` all change ' ( barbour @xcite ) , or my sufficient totality of locally significant change ( stlrc ) @xcite . in the last case , a generalized local ephemeris time ( glet ) @xcite emerges . to fulfil the true content of the stlrc implementation of mach s time principle , all change is given opportunity to contribute to the timestandard . however only changes that do so in practise to within the desired accuracy are actually kept . moreover , this approximation requires a curious indirect procedure . i.e. one can not simply compare the sizes of the various contributions to the energy equation . one must rather @xcite assess terms at the level of the resulting force terms that arise upon variation . the emergent jacobi barbour bertotti ( jbb ) time ( in a form suitable for mechanics or minisuperspace ) is given by /t^ ( ) = /s ^ ( ) = s/. . [ plain - tem ] here , @xmath14 , thus incorporating a ` choice of ` calendar year zero ' . a constant scaling ` constant tick length ' can also be included @xcite . using @xmath15 simplifies both the momentum change relations @xmath16 and the jacobi counterpart of the euler lagrange equations of motion , @xmath17 moreover , @xmath15 leads to a relational recovery of what is , in various suitable contexts , newtonian time , proper time and cosmic time . finally , @xmath15 is , on first sight , built from an ` all change ' expression , but , upon practical consideration @xcite , it is a stlrc . thus this timestandard itself is a local generalization of the astronomers ' ephemeris time @xcite . explicit forms for this have been worked out for 1- and 2-@xmath18 rpm s @xcite and for minisuperspace @xcite . this resolution of the frozen formalism problem facet of the problem of time then meets two complications . \1 ) the emergent jbb time fails to unfreeze the quantum wave equation . the riposte to 1 ) is to consider a machian semiclassical approach which gives rise to a semiclassical machian timestandard @xmath19 . moreover this timestandard is indeed _ expected _ to be be different from it on machian grounds . this is because there there are now quantum , rather than classical , light ( @xmath20)-degrees of freedom to be given the opportunity to contribute . @xcite and the present article are the first to comment on the extent to which the semiclassical approach is a ) machian and b ) has a well - known ( and also machian ) classical precursor . \2 ) a second facet interferes . in the case of classical gr , this most usually termed the thin sandwich problem @xcite . moreover , it generalizes to a wider range of theories as barbour s best matching problem . it furthermore generalizes as regards at which level it is tackled . the thin sandwich is specifically at the ` lagrangian ' level , or , in the fully relational formulation , at the jacobi level : in terms of @xmath2 , d@xmath2 variables . on the other hand , _ configurational relationalism _ can be at other levels , such as the classical hamiltonian level or at some quantum level . the interference of this second facet is clear from the action now containing auxiliary @xmath21-variables for @xmath21 a group of physically irrelevant transformations . moreover , one now takes one s emergent time object to be a @xmath21-extremization of one s action , = diff(@xmath3 ) and the @xmath22 are presented in the form @xmath23 ( frame variables ) . ] s _ = _ ||_g|| _ [ tjbb ] then ^ ( ) = ( ||_g||_/ ) . [ g - tem ] n.b . ( [ tjbb ] ) s local character : gr time is a function of local position . moreover , for 1- and 2-@xmath18 rpm s @xcite _ and for the below halliwell hawking arena that they model _ @xcite , this expression is explicitly evaluable via best matching / thin sandwich being resolved . the above also ensures a set of classical beables , thus also resolving a third problem of time facet @xcite : the problem of beables . observables , or beables following john bell @xcite : a more cosmologically - appropriate notion than observables are hard to come by in classical and quantum gr . resolved best matching readily implies possession of a full set of classical beables , i.e. quantities that poisson - brackets - commute with the classical linear constraints . for more on the problem of time , see @xcite . this is an example of nontrivial configurational relationalism @xcite , and is the main concrete example used in this paper . the action for scaled rpm is are mass - weighted jacobi inter - particle cluster relative coordinates @xcite with conjugate momenta @xmath24 . these are the most convenient relative coordinates due to their diagonalizing the kinetic term . @xmath25 is a rotational auxiliary variable . the lower - case latin letters are relative particle ( cluster ) labels running from 1 to @xmath26 = @xmath27 1 for @xmath27 the number of particles . @xmath28 , for potential @xmath5 and fixed total energy of the model universe @xmath29 . the @xmath30 are general @xmath26-sphere coordinates and @xmath31 , @xmath32 are spherical coordinates on triangleland ( the 3-particle rpm in dimension 2 or higher ) . i.e. a function of the ratio between the base and the median and the angle between the base and the median respectively . as explained in @xcite , much of the notation and concepts for rpm s come from theoretical molecular physics kinematics and from kendall - type shape theory @xcite . b s : = || - || . [ uuno ] the quadratic energy constraint : = ||||^2/2 + v ( ) = e _ [ encon ] then follows as a primary constraint , i.e. purely due to the form of the action with no variation performed @xcite . @xmath33 and gr s hamiltonian constraint , @xmath34 , are denoted collectively by quad , which emphasizes their quadraticity in the momenta . furthermore , variation with respect to the auxiliary @xmath35 variables produces a zero total angular momentum constraint that is linear in the momenta , : = _ _ ^n ^i _ i = 0 . [ zam ] the specific examples of rpm s used in this paper are all scaled : they are @xmath27-stop metroland ( @xmath27 particles on a line ) , in particular 3-stop metroland , and triangleland .- body problems . the latter carry implications of being a small subsystem within a larger universe whereas ours are whole - universe models . this leads to mathematical differences between the two at the quantum level @xcite . ] reduction can be performed for these ( and in fact for all @xmath27-a - gonlands and for all the pure - shape i.e. shape alone and not scale versions of all of these also ) . equivalently by @xcite , one can set up a mechanics on the configuration space geometry . i refer to the common outcome of these two procedures as the _ r - formulation_. in the case of @xmath27-stop metroland , the action is s_n- = s , s : = , corresponding to the configuration space geometry being @xmath36 . on the other hand , for triangleland , = s , s : = , corresponding to the configuration space geometry being @xmath37 with a non - flat ( but conformally flat ) metric . here , @xmath38 is the total moment of inertia of the model universe . the advantages of considering triangleland are that it incorporates nontrivial configurational relationalism . rpm s generalize previously - studied absolute particle models of the semiclassical approach @xcite by inclusion of auxiliary terms and subsequently of linear constraints . see @xcite for a minisuperspace model arena treatment of the present paper s approach . i provide below an overview of the standard semiclassical approach . suppose that @xcite there are slow , heavy ` @xmath39 ' variables that provide an approximate timestandard with respect to which the other fast , light ` @xmath20 ' degrees of freedom evolve . the semiclassical approach is not only an emergent time strategy toward resolving the problem of time . it is also used along the lines of e.g. halliwell and hawking @xcite ) in acquiring more solid foundations for other aspects of quantum cosmology . this halliwell hawking model is for the quantum - cosmological origin of cosmological fluctuations observed today ( microwave background hot - spots and galaxies ) . these are treated as small inhomogeneous perturbations ( @xmath20 ) about the spatially homogeneous and isotropic @xmath40 universe ( @xmath39 ) . the present paper studies rpm s as a simpler model arena for understanding the halliwell hawking model . in particular , i give a qualitative study of a more general perturbation scheme that is to be a robustness test for the halliwell hawking model ( where many terms were droppod from the equations with little comment ) . rpm s then already succeed in illustrating the qualitative differences upon keeping these various terms , whilst having equations that are around 5 times shorter and more tractable than the corresponding generalization of the halliwell hawking model . [ this factor of 5 comes from the halliwell hawking model having scalar , vector and tensor gravitational modes and inhomogeneous scalar field modes , which form a total of five singlet or even odd doublet modes . ] i concentrate in this article on the case of particular cosmological significance : scale = @xmath39 , shape = @xmath20 splits . for gr , @xmath39 is the scalefactor ( and homogeneous matter modes ) and @xmath20 are inhomogeneities ( treated perturbatively in the halliwell hawking scheme @xcite ) . whereas for rpm s @xmath39 is ( the square root of ) the moment of inertia for the whole universe and @xmath20 are pure - shape degrees of freedom . the semiclassical approach then involves making \i ) the born oppenheimer and wkb anstze are , respectively , ( h , l ) = ( h)|(h , l ) , [ bo ] ( h ) = ( is(h)/ ) . [ wkb ] ( in each case there are a number of associated approximations covered in secs 2 , 3 and appendix a. ) \ii ) one forms the @xmath39-equation , @xmath41 then , under a number of simplifications ( including , for later reference , neglect of the light subsystem s kinetic term , @xmath42 ) this yields a hamilton jacobi equation @xmath43 where @xmath44 is the @xmath39-part of the potential . one way of solving this is for an approximate emergent semiclassical time , @xmath45 . \iii ) next , one forms the @xmath20-equation , @xmath46 this fluctuation equation can be recast ( modulo further approximations ) into an emergent - wkb - tdse ( time - dependent schrdinger equation ) for the l - degrees of freedom . the mechanics / rpm form of this is i|/t^ ( ) = _ l | . [ tdse2 ] the emergent - time - dependent left - hand side arises from the cross - term @xmath47 . @xmath48 is the remaining piece of the quantum energy constraint @xmath49 , acting as a hamiltonian for the @xmath20-subsystem . we shall see there are similar forms to ( [ plain - tem ] , [ g - tem ] ) for @xmath50 . sec [ + temjbb ] s machian classical @xmath39@xmath20 split ( level 1 of the current program ) is furtherly motivated as a simplification of semiclassical scheme associated with well - known physics . it is already machian , and is level 1 of the current program . sec [ semicl ] gives more detail of the semiclassical approach , including of how it too can be cast in machian form . this is level 2 of the current program . n.b . that the working leading to a time - dependent schrdinger equation ceases to work in the absence of making the wkb ansatz and approximation @xcite . thus for quantum cosmology , this is not known to be a particularly strongly supported ansatz and approximation to make . propping this up requires considering one or two further problem of time strategies from the classical level upwards ( see the conclusion ) . @xmath50 aligns with @xmath51 at least to first approximation . as outlined in the conclusion ( see @xcite for more details ) , this justifies the wkb ansatz leads one to a machian semiclassical histories timeless records combined scheme : level 3 of the current program . the present article covers levels 1 and 2 . the first approximation for the emergent time coincides in the classical and semiclassical workings . however , this is rather un - machian in the sense that it abstracts its change just from the scale . ( sometimes this is alongside homogeneous isotropic matter modes , or , more widely , from the usually - small subset of @xmath39 degrees of freedom . ) however , in the second approximation , the @xmath20 degrees of freedom _ are _ given the opportunity to contribute to the corrected emergent time . moreover , they do so differently in the classical and semiclassical cases . in this paper , this is an rpm pure - shape change , though in subsequent papers @xcite it is a minisuperspace anisotropy and a perturbative gr inhomogeneity . finally , the number of approximations concurrently made in the semiclassical approach is large ( ` multiple approximations problem ' @xcite ) . there are non - adiabaticities , other ( including higher ) emergent time derivatives and averaged terms @xcite . including the last of these parallels the use of hartree fock self - consistent iterative schemes . however , in the present context the system is now more complex via involving a chroniferous ( ` time providing ' ) quantum - average - corrected hamilton jacobi equation . these have the effect of obscuring tests of the validity of the wkb approximation the truth involves vast numbers of different possible regimes . thus tests of validity are likely contingent on a whole list of approximations made . these approximations are covered in secs 2 and 3 as they arise . i note that combined classical and quantum perturbation schemes are unusual , as are perturbation schemes for fixing the timestandard rather than built on a presupposed timestandard .

i provide perturbative schemes for these in which the timefunction is to be determined rather than assumed . this paper is useful modelling as regards the halliwell hawking arena for the quantum origin of the inhomogeneous cosmological fluctuations . * edward anderson * pacs : 04.60kz , 04.20.cv . ea212@cam.ac.uk

classical and semiclassical schemes are presented that are timeless at the primary level and recover time from mach s ` time is to be abstracted from change ' principle at the emergent secondary level . the semiclassical scheme is a machian variant of the semiclassical approach to the problem of time in quantum gravity . the classical scheme is barbour s , cast here explicitly as the classical precursor of the semiclassical approach . thus the two schemes have been married up , as equally - machian and necessarily distinct , since the latter s timestandard is abstracted in part from quantum change . i provide perturbative schemes for these in which the timefunction is to be determined rather than assumed . this paper is useful modelling as regards the halliwell hawking arena for the quantum origin of the inhomogeneous cosmological fluctuations . * edward anderson * pacs : 04.60kz , 04.20.cv . ea212@cam.ac.uk

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classical and semiclassical schemes are presented that are timeless at the primary level and recover time from mach s ` time is to be abstracted form change ' principle at the emergent secondary level . this paper considers this for relational particle mechanics ( rpm ) model arenas . see @xcite for the minisuperspace counterparts ; perturbative midisuperspace counterparts of this are forthcoming . the classical scheme is barbour s , cast here explicitly as the classical precursor of the semiclassical approach by use of the @xmath39@xmath20 split in the quantum cosmological analogue case . ( the square root of moment of inertia is @xmath39 and pure shape is @xmath20 . ) the semiclassical scheme is a machian variant of the semiclassical approach to the problem of time ( problem of time ) in quantum gravity . @xmath167 to zeroth ( non - machian ) order . they differ to first order . are necessarily distinct , since _ quantum _ change is part of from where the latter s timestandard is abstracted . moreover , @xmath50 is rectified as a second application of equation simplifying . see @xcite for a minisuperspace counterpart of the present paper . the present paper gives a complete machian resolution of the classical and semiclassical problem of time for 1- and 2-@xmath18 rpm s , modulo two caveats . \a ) this analysis has not covered the possible need to construct dirac beables . \b ) this analysis has not justified the crucial wkb ansatz in the first place @xcite . it is not natural compared to a superposition of such wavefunctions @xcite . justification of wkb in ordinary qm follows from the pre - existence of a surrounding classical large system @xcite . but this no longer applies for the whole universe . nor does quantum cosmology possess pure incoming wave laboratory set - up " . moreover , not being able to justify the wkb ansatz in the semiclassical approach to the problem of time is a particular problem @xcite . this is since its its trick by which the chroniferous cross - term becomes the time - derivative part of a tdse is exclusive to wkb ansatz wavefunctions . \a ) and b ) are then resolved by a combined ( semi)classical histories timeless records scheme , as per @xcite ( built upon the non - machianly interpreted @xcite ) . this begins with b ) being addressed by _ decoherence_. ( some support for but also reservations about this have been expressed in e.g. @xcite . ) this is quantum - cosmological decoherence , which exhibits some differences from the qm concept @xcite . histories theory is the most likely source of such decoherence in quantum cosmology . the question of what decoheres what then leads to consideration of timeless records as well . then e.g. halliwell s way of phrasing timeless propositions leads to quantities commuting with quad . if built out of kucha beables , also commute with lin@xmath168 and hence constitute dirac beables . i have also given an improved presentation of the semiclassical approach , with qualitative physical analysis of neglected terms and associated regimes of study . some papers @xcite investigate quantum cosmology by expanding in 1 parameter . there are however multiple parameters , as pointed out by padmanabhan @xcite and investigated explicitly in the present article . moreover , the current paper s examples are comparable against outcome of exact quantization ( a useful and relatively unusual feature ) . we included proposing a scheme for a quantum - cosmological generalized local ephemeris time . one might think of this in terms of bare and dressed quantities , though the type of dress is somewhat unusual . e.g. it is classical , though fluid mechanics has an effective mass concept too . most of all it is a machian dress . i provided a nontrivial machian perturbation theory to first order for classical and semiclassical schemes . * acknowledgements*. i thank those close to me for support . jeremy butterfield , sean gryb , philip hoehn , pui ip , chris isham , sophie kneller , marc lachieze - rey , matteo lostaglio , flavio mercati , brian pitts and the anonymous referees for discussions . i was funded by a grant from the foundational questions institute ( fqxi ) fund , a donor - advised fund of the silicon valley community foundation on the basis of proposal fqxi - rfp3 - 1101 to the fqxi . i thank also theiss research and the cnrs for administering this grant , held at apc universit paris 7 .

classical and semiclassical schemes are presented that are timeless at the primary level and recover time from mach s ` time is to be abstracted from change ' principle at the emergent secondary level . the semiclassical scheme is a machian variant of the semiclassical approach to the problem of time in quantum gravity . the classical scheme is barbour s , cast here explicitly as the classical precursor of the semiclassical approach . thus the two schemes have been married up , as equally - machian and necessarily distinct , since the latter s timestandard is abstracted in part from quantum change .

classical and semiclassical schemes are presented that are timeless at the primary level and recover time from mach s ` time is to be abstracted from change ' principle at the emergent secondary level . the semiclassical scheme is a machian variant of the semiclassical approach to the problem of time in quantum gravity . the classical scheme is barbour s , cast here explicitly as the classical precursor of the semiclassical approach . thus the two schemes have been married up , as equally - machian and necessarily distinct , since the latter s timestandard is abstracted in part from quantum change . i provide perturbative schemes for these in which the timefunction is to be determined rather than assumed . this paper is useful modelling as regards the halliwell hawking arena for the quantum origin of the inhomogeneous cosmological fluctuations . * edward anderson * pacs : 04.60kz , 04.20.cv . ea212@cam.ac.uk

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* the rise of cloud apps : * + the popularity of consumer cloud storage providers ( csps ) over the previous decade has been on a roll . dropbox , google drive , and one drive have each amassed hundreds of millions of users . in order to further appeal to their users , the csps have been transitioning from being pure _ service providers _ to becoming _ app ecosystems_. hence , they now offer apis for developers to import and process users files stored in the cloud . consider , for example , a web app called https://pandadoc.com[pandadoc ] , which allows creating , editing , and signing documents online . when a user uses pandadoc from her laptop browser , she can import files stored in her google drive instead of her hard drive . such a pattern is increasingly more prevalent with the growing number of 3rd party cloud apps ( or 3pc apps ) that are tightly integrated with cloud storage services . dropbox alone claims that hundreds of thousands of apps have been integrated with its platform . even in the enterprise setting , 3rd party cloud apps are on the rise . this is first because companies are officially adopting the likes of _ dropbox business _ , _ onedrive for business _ , and _ google drive for work_. second , it is due to employees utilizing their personal cloud accounts to share company s files ( a.k.a shadow it ) . various reports from cloud application security providers state that organizations use from 10 to 20 times more cloud apps than their it department thinks @xcite . * risks in 3rd party cloud apps : * + however , in our previous work , we have shown that 76% of the 3rd party google drive apps featured on google chrome store request full access to users google drive data @xcite . around 64% of these apps are _ over - privileged _ : they require more permissions than are needed for them to function . accordingly , users are now faced with a new kind of privacy adversary : the 3rd party app vendors . with every app authorization decision that users make , they are trusting a new vendor with their data and increasing the potential attack surface . elastica , the cloud application security provider , estimates that the average financial impact on a company as a result of a cloud - storage data breach is $ 13.85 m , including remediation costs @xcite . in 2015 , the data breach at anthem , a us insurance company , has reportedly cost more than $ 100 m , with 80 m unencrypted health records leaked . this was a result of an exfiltration exploit leveraging a popular public cloud storage application @xcite . even on the personal level , the risk extends from breaches exposing financial information and health records to unnoticeable , continuous profiling based on stored files . * exposure through collaboration : * + an additional intricacy is that when users grant access to a 3rd party cloud app , they are not only sharing their personal data but also others data . this is because cloud storage providers are inherently collaborative platforms where users share and cooperate on common files . hence , protecting these files is not solely in the hands of the user . skyhigh networks , another provider of cloud security software , reports that 37.2% of documents ( across 23 million users ) are shared with at least one other user . in organizations , documents are shared , on average , with accounts from 849 external domains @xcite . moreover , around 23% of cloud documents were found by elastica to be `` broadly shared '' , which means that they are shared ( a ) among all employees , ( b ) with external partners and clients , or ( c ) with the public @xcite . interestingly , 12% of those documents contained compliance - related or confidential data . this further highlights what has been termed as the _ interdependent privacy problem _ @xcite , where the decisions of friends can affect the user s privacy and vice - versa . this concept was initially proposed in the context of third - party social networking apps , such as facebook . however , while 1.92% of facebook apps request friends personal information , this is much more pronounced in 3rd party cloud apps , where all apps accessing one s files get access to the part which is shared too . moreover , unlike facebook apps , due to the collaborative nature of cloud apps , the csps do not provide an option for users to control whether their collaborators apps can get access to data they own . * research questions : * + so far , the main approach to reducing the risk of 3pc apps has been focused on discovering over - privileged apps and deterring users from installing them @xcite . even then , a lot of users would still install such apps as they prioritize short - term utility over long - term risk aversion or due to the absence of alternatives . furthermore , that approach relies on manually inspecting each app by experts and on applying a plethora of machine learning algorithms to visualize the various risks for users . these issues could present a hurdle towards a wide - scale deployment by csps . in this work , we address the wider problem of minimizing the risk of all 3pc apps , regardless of whether they are over - privileged or least - privileged . we are further driven by the rationale that users will inevitably continue to install apps to achieve various services . hence , instead of stopping them , we aim to lead them to select apps from vendors in a way that minimizes their privacy risk . we achieve this by leading users to take what we term as _ history - based decisions_. such decisions account for the vendors who previously obtained access to the user s data , whether directly ( with her consent ) or via her collaborators . our strategy consists of introducing privacy indicators to the current permissions interfaces that help users minimize the number of vendors with access to their data . our `` usable privacy '' approach is guided via a data - driven study and is evaluated via a data - driven simulation . in essence , we tackle the following research questions : * from a practical perspective , are the collaborators decisions significant enough to be accounted for in users app adoption decisions ? * do users already account for entities with access to their data ? if not , to what extent can the usage of privacy indicators lead to users taking history - based decisions ? * how significant is the effect of adopting these privacy indicators in the case of large networks of users and teams ? * contributions : * towards addressing these questions , we make the following contributions : * in section [ sec : dataset ] , we analyze a real - world dataset of google drive users , and we show that the median privacy loss that collaborators cause by installing apps can be much higher than that inflicted by the user s own app adoption decisions ( 39% higher with 5% of shared files and 523% higher with 60% of shared files ) . to our knowledge , this is the first usage of a real - world dataset to give a concrete evaluation of interdependent privacy in any ecosystem . * driven by the significant impact of collaborators , we design new privacy indicators for helping users mitigate the privacy risk via history - based decisions ( cf . section [ sec : study ] ) . we assess these indicators via a web experiment with 141 users . we show that they significantly increase the likelihood that users choose the option with minimal privacy loss , even if not all of these users are motivated by privacy per se . to the best of our knowledge , this is also the first work to investigate a usable privacy approach to mitigating the problem of interdependent privacy . the few studies on this problem have mainly approached it from a theoretical perspective , such as developing game - theoretic or economic models @xcite or from a behavioral perspective , such as studying the factors affecting real users monetary valuation of others privacy @xcite . * we explore the potential of history - based decisions by performing a simulation on two large user networks . we show that the network - effects of our approach result in curtailing the growth privacy loss by 70% in a synthetic google drive - based collaboration network and by 40% in a real author collaboration network . we also simulate the effect of such decisions in a teams network . we demonstrate that teams can reduce the privacy loss by up to 45% by solely accounting for team members decisions ( cf . section [ sec : simulation ] ) .

such apps often request full access to users files , including files shared with collaborators . hence , whenever a user grants access to a new vendor , she is inflicting a privacy loss on herself and on her collaborators too . based on analyzing a real dataset of 183 google drive users and 131 third party apps , we discover that collaborators inflict a privacy loss which is at least 39% higher than what users themselves cause . we take a step toward minimizing this loss by introducing the concept of _ history - based decisions_. simply put , users are informed at decision time about the vendors which have been previously granted access to their data . thus , they can reduce their privacy loss by not installing apps from new vendors whenever possible . next , we realize this concept by introducing a new privacy indicator , which can be integrated within the cloud apps authorization interface . via a web experiment with 141 participants recruited from crowdflower , we show that our privacy indicator can significantly increase the user s likelihood of choosing the app that minimizes her privacy loss . finally , we explore the network effect of history - based decisions via a simulation on top of large collaboration networks . we demonstrate that adopting such a decision - making process is capable of reducing the growth of users privacy loss by 70% in a google drive - based network and by 40% in an author collaboration network . this is despite the fact that we neither assume that users cooperate nor that they exhibit altruistic behavior . to our knowledge , our work is the first to provide quantifiable evidence of the privacy risk that collaborators pose in cloud apps . we are also the first to mitigate this problem via a usable privacy approach . < ccs2012 > < concept > < concept_id>10002978.10003029.10011703</concept_id > < concept_desc > security and privacy usability in security and privacy</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002978.10003018.10003021</concept_id > < concept_desc > security and privacy information accountability and usage control</concept_desc > < concept_significance>300</concept_significance > < /concept > < concept > < concept_id>10002978.10003029.10011150</concept_id > < concept_desc > security and privacy privacy protections</concept_desc > < concept_significance>300</concept_significance > < /concept > < concept > < concept_id>10003120.10003121.10003122.10010854</concept_id > < concept_desc > human - centered computing usability testing</concept_desc > < concept_significance>300</concept_significance > < /concept > < /ccs2012 >

cloud storage services , like dropbox and google drive , have growing ecosystems of 3rd party apps that are designed to work with users cloud files . such apps often request full access to users files , including files shared with collaborators . hence , whenever a user grants access to a new vendor , she is inflicting a privacy loss on herself and on her collaborators too . based on analyzing a real dataset of 183 google drive users and 131 third party apps , we discover that collaborators inflict a privacy loss which is at least 39% higher than what users themselves cause . we take a step toward minimizing this loss by introducing the concept of _ history - based decisions_. simply put , users are informed at decision time about the vendors which have been previously granted access to their data . thus , they can reduce their privacy loss by not installing apps from new vendors whenever possible . next , we realize this concept by introducing a new privacy indicator , which can be integrated within the cloud apps authorization interface . via a web experiment with 141 participants recruited from crowdflower , we show that our privacy indicator can significantly increase the user s likelihood of choosing the app that minimizes her privacy loss . finally , we explore the network effect of history - based decisions via a simulation on top of large collaboration networks . we demonstrate that adopting such a decision - making process is capable of reducing the growth of users privacy loss by 70% in a google drive - based network and by 40% in an author collaboration network . this is despite the fact that we neither assume that users cooperate nor that they exhibit altruistic behavior . to our knowledge , our work is the first to provide quantifiable evidence of the privacy risk that collaborators pose in cloud apps . we are also the first to mitigate this problem via a usable privacy approach . < ccs2012 > < concept > < concept_id>10002978.10003029.10011703</concept_id > < concept_desc > security and privacy usability in security and privacy</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002978.10003018.10003021</concept_id > < concept_desc > security and privacy information accountability and usage control</concept_desc > < concept_significance>300</concept_significance > < /concept > < concept > < concept_id>10002978.10003029.10011150</concept_id > < concept_desc > security and privacy privacy protections</concept_desc > < concept_significance>300</concept_significance > < /concept > < concept > < concept_id>10003120.10003121.10003122.10010854</concept_id > < concept_desc > human - centered computing usability testing</concept_desc > < concept_significance>300</concept_significance > < /concept > < /ccs2012 >

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the most popular model for the central engines of long gamma - ray - burst ( grb ) jets is based on the `` failed supernova '' scenario of stellar collapse , or `` collapsar '' , where the iron core of progenitor star forms a bh @xcite . if the specific angular momentum in the equatorial part of progenitor exceeds that of the marginally bound orbit of the bh then the collapse becomes highly anisotropic . while in the polar region it may proceed more or less uninhibited , at least for a while , the equatorial layers form dense and massive accretion disk . the gravitational energy released in this disk can be very large , more then sufficient to overturn the collapse of outer layers and drive grb outflows @xcite . a similar configuration can be produced via inspiraling of a bh or a neutron star into the companion star during the common envelope phase of a close binary @xcite . in any case , the observed duration of long grbs , between 2 and 1000 seconds , imposes a strong constraint on the size of progenitors - their light crossing time should be significantly shorter then the burst duration . this implies that the progenitors must be compact stars stripped of their hydrogen envelopes . this conclusion agrees with the absence of hydrogen lines in the spectra of supernovae identified with grbs ( moreover , since some of these supernovae are of type ic their progenitors may have lost their helium envelopes as well . ) . massive solitary stars can lose their envelopes via strong stellar winds . however , this may lead to unacceptably low rotation rates by the time of stellar collapse @xcite . in the case of binary system the loss of envelope can be caused by the gravitational interaction with companion . the currently most popular mechanism of powering grb jets is the heating via annihilation of neutrinos and anti - neutrinos produced in the disk @xcite . the energy deposited in this way is a strong function of the mass accretion rate which must be above @xmath0 in order to agree with the observational constraints on the energetics of grbs @xcite . such high accretion rates can be provided in the collapsar scenario and the numerical simulations by @xcite and @xcite have demonstrated that sufficiently large energy deposition in the polar region above the disk may indeed result in fast collimated jets . however , we have to wait for simulations with proper implementation of neutrino transport before making final conclusion on the suitability of this model the long and complicated history of numerical studies of neutrino - driven supernova explosions teaches us to be cautious . the first attempt to include neutrino transport in collapsar simulations was made by @xcite and they did not see neutrino - driven polar jets - the heating due to neutrino annihilation could not overcome the cooling due to neutrino emission.s , whereas the jet simulation of @xcite started at @xmath1s when the density in the polar regions above the black hole has dropped down to @xmath2 leading to much lower cooling rates . ] in any case , it is unlikely that this model can explain the bursts that are longer than @xmath3s as by this time the mass accretion rate is expected to drop significantly . for example the accretion rate in the simulations by @xcite has already drifted below the critical @xmath4 by @xmath5s . the two alternatives to the neutrino mechanism which are also frequently mentioned in connection with grbs are the magnetic braking of the accretion disk @xcite and the magnetic braking of the central black hole @xcite . a number of groups have studied the potential of magnetic mechanism in the collapsar scenario using newtonian mhd codes and implementing the paczynski - witta potential in order to approximate the gravitational field of central bh @xcite . in this approach it is impossible to capture the blandford - znajek effect and only the magnetic braking of accretion disk can be investigated . the general conclusion of these studies is that the accretion disk can launch magnetically - driven jets provided the magnetic field in the progenitor core is sufficiently strong . this field is further amplified in the disk , partly due to simple winding of poloidal component and partly due to the magneto - rotational instability ( mri ) , until the magnetic pressure becomes very large and pushes out the surface layers of the disk . in particular , @xcite studied the collapse of a star with a similar structure to that considered @xcite and with purely monopole initial magnetic field of strength @xmath6 g at @xmath7 . the corresponding total magnetic flux , @xmath8 , is in fact within the observational constraints on the field of magnetic stars , including white dwarfs @xcite , ap - stars @xcite and massive o - stars @xcite . they used realistic equation of state and included the neutrino cooling but not the heating . the simulations lasted up to @xmath9s in physical time during which a poynting - dominated polar jet has developed in the solution . the jet power seemed to show strong systematic decline , from @xmath10 down to @xmath11 or even less , which might signal transient phenomenon . @xcite studied the collapse of rotating stellar cores and formation of bhs and their disks in the collapsar scenario using the full grmhd approximation . their simulations show powerful explosions soon after the accretion disk is formed and the free falling plasma of collapsing star begins to collide with this disk . however , they did not account for the neutrino cooling and the energy losses due to photo - dissociation of atomic nuclei so these explosions could be similar in nature to the `` successful '' prompt explosions of early supernova simulations @xcite . @xcite carried out grmhd simulations in the time - independent space - time of a central bh . their computational domain did not capture the bh ergosphere and thus they could not study the role of the blandford - znajek effect @xcite . in addition , the energy gains / losses due to the neutrino heating / cooling were not included and the equation of state ( eos ) was a simple polytrope . the simulations run for a rather short time , @xmath12 where @xmath13 , and jets were formed almost immediately , presumably due to the extremely strong initial magnetic field . the original theory of the blandford - znajek mechanism was developed within the framework of magnetodynamics ( the singular limit of rmhd approximation where the inertia of plasma particles is fully ignored ) . the pioneering numerical studies of this mechanism have shown that it can also operate within the full rmhd approximation , at least in the regime where the energy density of matter is much smaller compared to that of the electromagnetic field , and that it can be captured by modern numerical techniques @xcite . however , no systematic numerical study has been carried out in order to establish the exact conditions for activation of the bz mechanism . the first simulations demonstrating the potential of the bz mechanism in the collapsar scenario were carried out by @xcite who also used simple polytropic eos and considered purely adiabatic case , but captured the effects of black hole ergosphere . the initial configuration included a torus with poloidal magnetic and a free - falling stellar envelope . the results show ultrarelativistic jets with the lorentz factor up to 10 emerging from the black hole indicating that the blandford - znajek effect may play a key role in the production of grb jets . @xcite added a realistic eos and included the energy losses due to neutrino emission ( assuming optically thin regime ) and photo - dissociation of nuclei . they also reported the development of relativistic outflows powered by the central black hole via the blandford - znajek mechanism ( the magnetic braking of accretion disk did not seem to contribute much to the jet power though ) . the black hole - driven magnetic stellar explosions and relativistic jets observed in these exploratory computer simulations invite a deeper analysis of this model . obviously , such an outcome can not be a generic feature of core - collapse supernovae . indeed , only a very small fraction of sne ib / ic seem to produce grbs @xcite . so what are the relevant conditions and can they be reproduced as the results of stellar evolution ? these are the main issues we address in this paper .

we explore whether this process can explain the stellar explosions and relativistic jets associated with long gamma - ray - bursts . in however , this energy deficit can be recovered via additional power provided by the disk . black hole physics supernovae : general gamma - rays : bursts methods : numerical mhd relativity

collapse of massive stars may result in formation of accreting black holes in their interior . the accreting stellar matter may advect substantial magnetic flux onto the black hole and promote release of its rotational energy via magnetic stresses ( the blandford - znajek mechanism ) . in this paper we explore whether this process can explain the stellar explosions and relativistic jets associated with long gamma - ray - bursts . in particularly , we show that the blandford - znajek mechanism is activated when the rest mass - energy density of matter drops below the energy density of magnetic field in the very vicinity of the black hole ( within its ergosphere ) . we also discuss whether such a strong magnetic field is in conflict with the rapid rotation of stellar core required in the collapsar model and suggest that the conflict can be avoided if the progenitor star is a component of close binary . in this case the stellar rotation can be sustained via spin - orbital interaction . in an alternative scenario the magnetic field is generated in the accretion disk but in this case the magnetic flux through the black hole ergosphere is not expected to be sufficiently high to explain the energetics of hypernovae by the bz mechanism alone . however , this energy deficit can be recovered via additional power provided by the disk . black hole physics supernovae : general gamma - rays : bursts methods : numerical mhd relativity

0902.2881

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in this paper we continued our study into the potential of the blandford - znajek mechanism as main driver in the central engines of gamma ray bursts . in particularly , we analysed the conditions for activation of the mechanism in the collapsar model where accumulation of magnetic flux by the central black hole is likely to be a byproduct of stellar collapse . it appears that the rotating black hole begins to pump energy into the surrounding and overpowers accretion only when the rest mass energy density of matter drops below the energy density of the electromagnetic field near the ergosphere ( see eq.[cond-1 ] ) . only under this condition the mhd waves generated in the ergosphere can transport energy and angular momentum away from the black hole . in the case of spherical accretion the above criterion can be written as a condition on the total magnetic flux accumulated by the black hole and the total mass accretion rate ( see eq.[cond-2 ] ) . however , in the case of nonspherical accretion , characteristic for the collapsar model , this integral condition becomes less applicable due to the spacial separation of magnetic field and accretion flow inside the accretion shock . in fact , our numerical simulations show explosions for significantly lower magnetic fluxes compared to those expected from the integral criterion . we have not included the neutrino heating in our numerical models . this is another factor that can make activation of the bz mechanism in the collapsar setup easier . both the general energetic considerations and the condition for activation of the blandford - znajek mechanism require the central black hole to accumulate magnetic flux comparable to the highest observed values for magnetic stars , @xmath190 . current evolutionary models of massive solitary stars indicate that fossil field of such strength should slow down the rotation of helium cores below the rate required for formation of accretion disks around black holes of failed supernovae . the conflict between strong fossil magnetic field and rapid stellar rotation required by the bz mechanism may be resolved in the models where the grb progenitor is a component of close binary . in this case the high spin of the progenitor can be sustained at the expense of the orbital rotation . moreover , the strong gravitational interaction between components of close binary helps to explain the compactness of grb progenitors deduced from the observed durations of bursts . the magnetic field can also be generated in the accretion disk and its corona but in this case the magnetic flux through the black hole ergosphere is not sufficiently high to explain the power of hypernovae by the bz mechanism alone . on the other hand , in this case the disk is expected to drive mhd wind more powerful than the bz jet . although such a wind is unlikely to reach the high lorentz factors deduced from the observations of grb jets ( due to high mass loading ) it could be behind the observed energetics of hypernovae . , willmarth d.w.,2000 , in eds . et al . , _ stellar astrophysics _ , kluwer , p.175 aloy m.a . , muller e. , abner j.m . , marti j.m . , macfadyen a.i . , 2000 , apj , 531 , l119 ardeljan n.v . , bisnovatyi - kogan g.s . , moiseenko s.g . , 2005 , mnras , 359 , 333 barkov m.v . , komissarov s.s , 2008a , mnras , 385 , l28 barkov m.v . , komissarov s.s , 2008b , in _ high energy gamma - ray astronomy _ , aip conference proceedings , v.1085 , p.608 barkov m.v , komissarov s.s , in prerparation . begelman m.c . , blandford r.d . , rees m.j . , 1984 , rev.modern phys . , 56 , 255 bethe h.a . , 1990 , rev.mod.phys.,62,801 bezchastnov v.g . , haensel p. , kaminker a.d . , yakovlev d.g . , 1997 , a&ap , 328,409 bisnovatyi - kogan g.s . , ruzmaikin a.a . , 1974 , ap&ss , 28 , 45 bisnovatyi - kogan g.s . , ruzmaikin a.a . , 1976 , ap&ss , 42 , 401 blandford r.d . and znajek r.l . , 1977 , mnras , 179 , 433 blandford r.d . , payne d.g . , 1982 , mnras , 199 , 883 blondin j.m , mezzacappa a. , demarino c. , 2003 , apj , 584,971 braithwaite j. , spruit h.c . , 2004 , nature , 431 , 819 buras r. , janka h .- th . , rampp m. , kifonidis k. , 2006 , a&a 457 , 281 burrows a. , dessart l. , livne e. , ott c.d . , murphy j. , 2007 , apj,664,416 camenzind m. , in _ accretion disks and magnetic fields in astrophysics _ , g.belvedere , kluwer , dordrecht , p.129 , 1989 donati j .- f . , babel j. , harries t.j . , howarth i.d . , petit p. and semel m. , 2002 , mnras , 333 , 55 ferrario l. , wickramasinghe d.t . , 2005 , mnras , 356 , 615 frayer c.l , woosley s.e . , 1998 , apj lett . , 502 , l9 fujimoto s. , kotake k. , yamada s. , hashimoto m. , sato k. , 2006 , apj , 644 , 1040 ghosh p. , abramowicz m.a . , 1997 , mnras , 292 , 887 heger a. , langer n. , woosley s.e . , 2000 , apj , 528 , 368 heger a. , woosley s.e . , spruit h.c . , 2005 , apj , 626 , 350 mszros p. , rees m.j . , 1997 , apj lett . , 482 , l29 hirschi r. , meynet g. , maeder a , 2004 , a&a , 425 , 649 hirschi r. , meynet g. , maeder a , 2005 , a&a , 443 , 581 heyvaerts j. , priest e.r . , bardou a. , 1996 , apj , 473 , 403 howarth i.d . , siebert k.w . , hussain g.a . , prinja r.k , 1997 , mnras , 284 , 265 igumenshchev i.v . , 2008 , apj , 677 , 317 ivanova l.n . , imshennik v.s . , nadezhin d.k . , 1969 , sci.inf.astr.council.acad.sci , 13,3 koide s.,2004,apj lett . , 606,l45 komissarov s.s . , 1999 , mnras , 303 , 343 komissarov s.s . , 2001,mnras,326,l41 komissarov s.s . , 2004a , mnras,350,427 komissarov s.s . , 2004b , mnras , 350 , 1431 komissarov s.s . , 2006 , mnras , 368 , 993 komissarov s.s . , 2008 , arxiv0804.1912 lee h.k . , brown g.e . , wijers r.a.m.j , 2000 , apj , 536 , 416 livio m. , ogilvie g.i . , pringle j.e . , 1999 , apj , 512 , 100 lubow s.h . , papaloizou j.c.b . , pringle j.e . , 1994 , mnras , 267 , 235 macdonald d.a . and thorne k.s . , 1982 , mnras , * 198 * , 345 . macfadyen a.i . & woosley s.e , 1999 , apj , 524 , 262 maeder a. , meynet g. , 2005 , a&a , 440 , 1041 mckinney j.c . , gammie c.f . , 2004 , apj , 611 , 977 mckinney j.c . , 2006 , mnras,368,1561 mignone a. , bodo g. , 2006 , mnras , 368 , 1040 misner c.w . , thorne k.s . , wheeler j.a . , _ gravitation _ , san francisco : w.h . freeman and co. , 1973 mizuno y. , yamada s. , koide s. , shibata k. , 2004a , apj , 606 , 395 mizuno y. , yamada s. , koide s. , shibata k. , 2004b , apj , 615 , 389 moiseenko s.g . , bisnovatyi - kogan g.s . , ardeljan n.v . , 2006 , mnras , 370 , 501 moss d.,1987 , mnras , 226 , 297 nagakura h. , yamada s. , 2009 , arxiv:0901.4053 nagataki s. , takahashi r. , mizuta a. , takiwaki t. , 2007 , apj , 659 , 512 nagataki s. , 2009 , arxiv:0902.1908 narayan r. , paczyski b. , piran t. , 1992 , apj lett . , 395 , l8 obergaulinger m. , aloy m.a . , dimmelmeier h. , muller e. , 2006,a&a,457,209 penny l.r . , 1996 , apj . , 463 , 737 piran t. , 2005 , rev.mod.phys . , 76 , 1143 popham r. , woosley s.e . and fryer c.l . , 1999 , apj , 518 , 356 proga d. , macfadyen a.i . , armitage p.j . , begelman m.c . , 2003 , apj , 629 , 397 rdler k .- h . , 1968 , z.naturforsch . , a , 23 , 1851 robstein d.m . , lovelace r.v.e . , 2008 , apj , 677 , 1221 schinder p.j . , schramm d.n . , wiita p.j . , margolis s.h . , tubbs d.l . , 1987 , apj , 313 , 531 . sekiguchi y. , shibata m.,2007,prog.theor.phys,117,1029 scheck l. , janka h .- th . , foglizzio t. , kifonidis , k. , 2008 , a&a , 477 , 931 schmidt et al . , 2003 , apj , 595 , 1101 spruit h.c . , 2002 , a&a , 381 , 923 spruit h.c . , uzdensky d.a . , 2002 , apj , 629 , 960 takahashi m. , niita s. , tatematsu y. , tomimatsu a. , 1990 , apj , 363 , 206 . tao l. , proctor m.r.e . , weiss n.o . , 1998 , mnras , 300 , 907 taam r.e . , sandquist e.l . , 2000 , ann.rev.a&a , 38 , 113 tayler r. , 1973 , mnras , 161 , 365 timmes f.x . , swesty f.d . , 2000 , apjss , 126 , 501 tout c.a . , pringle j.e . , 1996 , mnras , 281 , 219 tutukov a. , iungelson l. , in _ mass loss and evolution of o - type stars _ , redel , dordrecht , p.401 , 1979 uzdensky d.a . & macfadyen a.i . , 2006 , apj , 647 , 1192 woosley s.e . , 1993 , apj , 405 , 273 woosley s.e . , baron e. , 1992 , apj , 391 , 228 woosley s.e . , bloom j.s . , 2006 , ann.rew.a&a , 44 , 507 woosley s.e . , heger a. , 2006 , apj , 637 , 914 yoon s .- c . , langer n. , norman c. , 2006 , a&a , 460 , 199 . yoon s .- c . , cantiello m. , langer n.,2008 , arxiv:0801.4373 zeldovich ya.b . , 1957 , sov.phys.jetp , 4 , 460 zhang w. , fryer c.l . , 2001 , apj , 550 , 357

particularly , we show that the blandford - znajek mechanism is activated when the rest mass - energy density of matter drops below the energy density of magnetic field in the very vicinity of the black hole ( within its ergosphere ) . we also discuss whether such a strong magnetic field is in conflict with the rapid rotation of stellar core required in the collapsar model and suggest that the conflict can be avoided if the progenitor star is a component of close binary . in this case the magnetic field is generated in the accretion disk but in this case the magnetic flux through the black hole ergosphere is not expected to be sufficiently high to explain the energetics of hypernovae by the bz mechanism alone .

collapse of massive stars may result in formation of accreting black holes in their interior . the accreting stellar matter may advect substantial magnetic flux onto the black hole and promote release of its rotational energy via magnetic stresses ( the blandford - znajek mechanism ) . in this paper we explore whether this process can explain the stellar explosions and relativistic jets associated with long gamma - ray - bursts . in particularly , we show that the blandford - znajek mechanism is activated when the rest mass - energy density of matter drops below the energy density of magnetic field in the very vicinity of the black hole ( within its ergosphere ) . we also discuss whether such a strong magnetic field is in conflict with the rapid rotation of stellar core required in the collapsar model and suggest that the conflict can be avoided if the progenitor star is a component of close binary . in this case the stellar rotation can be sustained via spin - orbital interaction . in an alternative scenario the magnetic field is generated in the accretion disk but in this case the magnetic flux through the black hole ergosphere is not expected to be sufficiently high to explain the energetics of hypernovae by the bz mechanism alone . however , this energy deficit can be recovered via additional power provided by the disk . black hole physics supernovae : general gamma - rays : bursts methods : numerical mhd relativity

1604.06244

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the casimir effect ( for reviews see @xcite ) is among the most interesting boundary - induced effects in quantum field theory . nearly seventy years after its discovery , the effect continues to be area of active theoretical and experimental research today . this is due to of fundamental and practical importance of the casimir effect in various fields of modern physics and technology . the imposition of boundary conditions on a quantum field shifts the expectation values of physical observables . the particular features of the effect depend on the spin of the field , on the geometries of boundaries and background spacetime , and on the specific boundary conditions . an interesting topic in the investigations of the casimir effect is the dependence of the physical characteristics , such as the vacuum energy and forces acting on the boundaries , on the dimension of the background spacetime . in particular , this is motivated by the role of the casimir effect in higher - dimensional models . the dependence of the casimir energy on the size of the extra dimensions provides a mechanism for their stabilization . the latter is an important issue in both the kaluza - klein - type and braneworld models . in addition , the casimir energy related to extra dimensions can serve as a model of dark energy driving the accelerating expansion of the universe at recent epoch . in the present paper we investigate combined effects of boundaries and background geometry on the local characteristics of the electromagnetic vacuum in an arbitrary number of spatial dimensions . as a background spacetime we consider @xmath0-dimensional de sitter ( ds ) spacetime . this choice is motivated by several reasons . the ds spacetime has maximal number of symmetries and because of this a large number of physical problems can be solved exactly on its background . a better understanding of the influence of classical gravitational field on quantum matter in ds spacetime serves as a way to deal with less symmetric geometries . the quantum effects in ds bulk play an important role in inflationary scenarios ( see @xcite and mart13 for recent reviews ) . quantum fluctuations in the inflationary universe with ds geometry produce observable effects in the form of small density perturbations in the post - inflationary stage . these perturbations have nearly scale - invariant spectrum and form the base of the currently most popular mechanism for the generation of the seeds for galaxy formation . the corresponding predictions are in good agreement with the recent observational data on the temperature anisotropies of the cosmic microwave background radiation . the importance of ds spacetime as a gravitational background further increased after the discovery of the accelerating expansion of the universe at recent epoch @xcite . within the framework of general relativity , among the most popular cosmological models is the one with a positive cosmological constant as a driving source behind the accelerating expansion . in this model the ds spacetime is the future attractor for the geometry of our universe . an important ingredient in the problems of the casimir effect is the geometry of boundaries . here we consider a cylindrical shell in background of ds spacetime , described by inflationary coordinates . on the shell the electromagnetic field obeys the generalized perfect conductor boundary condition . along with the boundaries of planar and spherical symmetry , the cylindrical boundaries are among the most popular geometries in the investigations of the casimir effect ( see @xcite and references therein ) . the patch of ds spacetime covered by inflationary coordinates is conformally flat and the corresponding results for the casimir effect in the case of conformally invariant fields are obtained from the expressions for the minkowski bulk by using the standard conformal relation between two problems ( see , for instance , @xcite ) . in particular , this is the case for the electromagnetic field in 4-dimensional spacetime . the case of conformally coupled massless scalar field obeying dirichlet boundary condition on a spherical shell in 4-dimensional ds spacetime has been considered in @xcite . for conformally non - invariant fields the investigation of the casimir effect in ds spacetime requires a separate consideration . in this case , at distances from the boundaries larger than the curvature radius of the background spacetime , the influence of the gravitational field may lead to a behavior of the local characteristics of the vacuum essentially different from that in the minkowski bulk . the scalar casimir effect for a massive field with general curvature coupling and with robin boundary conditions on planar boundaries in the background of @xmath0-dimensional ds spacetime has been investigated in @xcite . the corresponding problem with a spherical boundary was discussed in @xcite . an important feature for conformally non - invariant fields is that the vacuum energy - momentum tensor , in addition to the diagonal components , acquires an off - diagonal component corresponding to the energy flux along the direction normal to the boundary . the casimir densities induced by a spherical ds bubble in the minkowski bulk has been studied in @xcite . for a scalar field , the geometry with a cylindrical boundary with robin boundary condition was discussed in saha15 . depending on the mass of the field , at distances from the boundaries larger than the ds curvature radius , two different regimes are realized with monotonic or oscillatory decay of the vacuum expectation values . the electromagnetic casimir effect for planar boundaries with generalized perfect conductor boundary conditions in ds bulk , having an arbitrary number of spatial dimensions , has recently been studied in saha14 ( for the propagators of vector fields on ds background in the absence of boundaries see @xcite ) . the background geometry of friedmann - robertson - walker cosmologies with power - law scale factors was discussed in @xcite . the organization of the paper is the following . in the next section we consider the complete set of cylindrical modes for the electromagnetic field in ds spacetime , obeying the generalized perfect conductor boundary condition on a cylindrical shell . by using these modes , in section sec : e2 , the vacuum expectation value ( vev ) of the electric field squared inside the shell is evaluated . various asymptotic regions of the parameters are discussed . the vev of the energy - momentum tensor is studied in section [ sec : emt ] . the vevs of the electric field squared and of the energy - momentum tensor in the exterior region are investigated in section [ sec : ext ] . section [ sec : conc ] summarizes the main results of the paper . in appendix [ sec : app1 ] the cylindrical modes in @xmath0-dimensional minkowski spacetime are discussed . the integrals , appearing in the expressions for the vevs for the special case @xmath1 , are evaluated in appendix [ sec : app2 ] . in the numerical evaluations of the vevs we have considered this special case .

complete set of cylindrical modes is constructed for the electromagnetic field inside and outside a cylindrical shell in the background of-dimensional ds spacetime . on the shell , the field obeys the generalized perfect conductor boundary condition . for the bunch - davies vacuum state the vacuum energy - momentum tensor , in addition to the diagonal components , has a nonzero off - diagonal component corresponding to the energy flux along the direction normal to the shell . pacs numbers : 04.62.+v , 03.70.+k , 11.10.kk

complete set of cylindrical modes is constructed for the electromagnetic field inside and outside a cylindrical shell in the background of-dimensional ds spacetime . on the shell , the field obeys the generalized perfect conductor boundary condition . for the bunch - davies vacuum state , we evaluate the expectation values ( vevs ) of the electric field squared and of the energy - momentum tensor . the shell - induced contributions are explicitly extracted . in this way , for points away from the shell , the renormalization is reduced to the one for the vevs in the boundary - free ds bulk . as a special case , the vevs are obtained for a cylindrical shell in the-dimensional minkowski bulk . we show that the shell - induced contribution in the electric field squared is positive for both the interior and exterior regions . the corresponding casimir - polder forces are directed toward the shell . the vacuum energy - momentum tensor , in addition to the diagonal components , has a nonzero off - diagonal component corresponding to the energy flux along the direction normal to the shell . this flux is directed from the shell in both the exterior and interior regions . for points near the shell , the leading terms in the asymptotic expansions for the electric field squared and diagonal components of the energy - momentum tensor are obtained from the corresponding expressions in the minkowski bulk replacing the distance from the shell by the proper distance in the ds bulk . the influence of the gravitational field on the local characteristics of the vacuum is essential at distances from the shell larger than the ds curvature radius . the results are extended for confining boundary conditions of flux tube models in qcd . pacs numbers : 04.62.+v , 03.70.+k , 11.10.kk

1604.06244

c

in the investigations of the casimir effect the cylindrically symmetric boundaries are among the most popular geometries . in the present paper we have investigated the local casimir densities for the electromagnetic field inside and outside a cylindrical shell in background of @xmath0-dimensional ds spacetime . on the shell , the field tensor obeys the boundary condition ( [ bc1 ] ) . in the special case @xmath21 this corresponds to the perfect conductor boundary condition . the procedure , we employed for the evaluation of the vevs bilinear in the field , is based on the mode - sum formula ( vev ) . in this procedure the complete set of cylindrical mode functions for the electromagnetic field , obeying the boundary condition , is required . in the problem under consideration one has a single mode of the te type and @xmath291 modes of the tm type . for the bunch - davies vacuum state the corresponding vector potentials are given by the expressions ( [ a1 ] ) and ( [ a2 ] ) with the radial functions ( [ radin ] ) and ( [ radout ] ) for the exterior and interior regions , respectively . we have investigated the combined effects of a cylindrical boundary and background gravitational field on the vevs of the electric field squared and of the energy - momentum tensor . in the interior region the eigenvalues of the quantum number @xmath50 are expressed in terms of the zeros of the bessel function @xmath54 for the tm modes and in terms of the zeros of the derivative @xmath292 in the case of the te mode . for the summation of the series over these zeros we have used the generalized abel - plana summation formula ( [ sumform ] ) . this allowed us to extract from the vevs the parts corresponding to the boundary - free ds spacetime and to present the shell - induced contributions in terms of strongly convergent integrals , for points away from the boundary . with this separation , the renormalization of the vevs is reduced to the one for the boundary - free geometry . as a result , inside the shell , the vevs are decomposed as ( e2dec ) and ( [ tdec ] ) with the shell - induced parts given by ( [ e2b ] ) for the electric field squared and by ( [ tiib ] ) for the diagonal components of the energy - momentum tensor . a similar decomposition is provided for the exterior region . the expressions for the shell - induced parts in this region differ from the ones inside the shell by the replacements @xmath241 of the modified bessel functions ( see ( [ e2bout2 ] ) and ( [ tiibout ] ) ) . for both the interior and exterior regions the shell - induced contributions to the vev of the electric field squared are positive . in addition to the diagonal components , the vev of the energy - momentum tensor has nonzero off - diagonal component @xmath293 . it corresponds to the energy flux along the radial direction and is given by the expressions ( [ t01b ] ) and ( [ t10bout ] ) for the exterior and interior regions . the off - diagonal component is negative inside the shell and positive in the exterior region . this means that the energy flux is directed from the shell in both the regions . on the axis of the shell the flux vanishes . we have considered various special cases of general formulas . in the limit @xmath294 , the vevs inside and outside a cylindrical shell in the background of @xmath0-dimensional minkowski spacetime are obtained . the corresponding expressions generalize the results previously known for @xmath137 to an arbitrary number of spatial dimensions . note that for @xmath21 the electromagnetic field is conformally invariant and the shell - induced vevs in the ds bulk are obtained from those in minkowski spacetime by the standard conformal transformation . for points near the cylindrical boundary the contribution of small wavelengths dominates in the shell - induced vevs . the leading terms in the corresponding asymptotic expansions for the field squared and diagonal components of the energy - momentum tensor coincide with those for a cylindrical shell in the minkowski bulk with the distance from the shell replaced by the proper distance in ds bulk . the leading term in the energy flux is given by the relation ( [ t10near ] ) . the effects of the background gravitational field on the shell - induced vevs are essential at distances from the boundary larger than the curvature radius of the ds spacetime . in particular , for the numerical example considered by us in the case @xmath1 , at large distances the shell - induced contribution to the vacuum energy density is negative for the minkowski bulk and positive for ds background . near the shell , the energy density is negative in both these cases . as a consequence , for the ds bulk it vanishes for some intermediate value of the radial coordinate . another boundary condition , used for the confinement of gauge fields in bag models of hadrons and in flux tube models of qcd , is the one given by ( bc2 ) . the corresponding expressions for the vevs of the field squared and energy - momentum tensor are obtained from those for generalized perfect conductor boundary condition by the replacement ( [ replqcd ] ) . in this case , the boundary - induced contribution on the vev of the squared electric field is neagtive in both the interior and exterior regions .

, we evaluate the expectation values ( vevs ) of the electric field squared and of the energy - momentum tensor . , the renormalization is reduced to the one for the vevs in the boundary - free ds bulk . as a special case , this flux is directed from the shell in both the exterior and interior regions . for points near the shell , the leading terms in the asymptotic expansions for the electric field squared and diagonal components of the energy - momentum tensor are obtained from the corresponding expressions in the minkowski bulk replacing the distance from the shell by the proper distance in the ds bulk . the influence of the gravitational field on the local characteristics of the vacuum is essential at distances from the shell larger than the ds curvature radius .

complete set of cylindrical modes is constructed for the electromagnetic field inside and outside a cylindrical shell in the background of-dimensional ds spacetime . on the shell , the field obeys the generalized perfect conductor boundary condition . for the bunch - davies vacuum state , we evaluate the expectation values ( vevs ) of the electric field squared and of the energy - momentum tensor . the shell - induced contributions are explicitly extracted . in this way , for points away from the shell , the renormalization is reduced to the one for the vevs in the boundary - free ds bulk . as a special case , the vevs are obtained for a cylindrical shell in the-dimensional minkowski bulk . we show that the shell - induced contribution in the electric field squared is positive for both the interior and exterior regions . the corresponding casimir - polder forces are directed toward the shell . the vacuum energy - momentum tensor , in addition to the diagonal components , has a nonzero off - diagonal component corresponding to the energy flux along the direction normal to the shell . this flux is directed from the shell in both the exterior and interior regions . for points near the shell , the leading terms in the asymptotic expansions for the electric field squared and diagonal components of the energy - momentum tensor are obtained from the corresponding expressions in the minkowski bulk replacing the distance from the shell by the proper distance in the ds bulk . the influence of the gravitational field on the local characteristics of the vacuum is essential at distances from the shell larger than the ds curvature radius . the results are extended for confining boundary conditions of flux tube models in qcd . pacs numbers : 04.62.+v , 03.70.+k , 11.10.kk

0911.1777

i

a bayesian data analysis specifies joint probability distributions to describe the relationship between the prior information , the model or hypotheses , and the data . using bayes theorem , the posterior distribution is uniquely determined from the conditional probability distribution of the unknowns given the observed data . the posterior probability is usually stated as follows : @xmath0 where @xmath1 is the marginal likelihood . the symbol @xmath2 denotes the assumption of a particular model and the parameter vector @xmath3 . for physical models , the sample space @xmath4 is most often a continuous space . in words , equation ( [ eq : bayes ] ) says : the probability of the model parameters given the data and the model is proportional to the prior probability of the model parameters and the probability of the data given the model parameters . the posterior may be used , for example , to infer the distribution of model parameters or to discriminate between competing hypotheses or models . the latter is particularly valuable given the wide variety of astronomical problems where diverse hypotheses describing heterogeneous physical systems is the norm ( see * ? ? ? * for a thorough discussion of bayesian data analysis ) . for parameter estimation , one often considers @xmath5 to be an uninteresting normalization constant . however , equation ( [ eq : evidence ] ) clearly admits a meaningful interpretation : it is the support or _ evidence _ for a model given the data . this see this , assume that the prior probability of some model , @xmath6 say , is @xmath7 . then by bayes theorem , the probability of the model given the data is @xmath8 . the posterior odds of model @xmath9 relative to model @xmath10 is then : @xmath11 if we have information about the ratio of prior odds , @xmath12 , we should use it , but more often than not our lack of knowledge forces a choice of @xmath13 . then , we estimate the relative probability of the models given @xmath14 over their prior odds by the bayes factor @xmath15 ( see * ? ? ? * for a discussion of additional concerns ) . when there is no ambiguity , we will omit the explicit dependence on @xmath16 of the prior distribution , likelihood function , and marginal likelihood for notational convenience . the bayes factor has a number of attractive advantages for model selection @xcite : ( 1 ) it is a consistent selector ; that is , the ratio will increasingly favor the true model in the limit of large data ; ( 2 ) bayes factors act as an occam s razor , preferring the simpler model if the `` fits '' are similar ; and ( 3 ) bayes factors do not require the models to be nested in any way ; that is , the models and their parameters need not be equivalent in any limit . there is a catch : direct computation of the marginal likelihood ( eq . [ eq : evidence ] ) is intractable for most problems of interest . however , recent advances in computing technology together with developments in markov chain monte carlo ( mcmc ) algorithms have the promise to compute the posterior distribution for problems that have been previously infeasible owing to dimensionality or complexity . the posterior distribution is central to bayesian inference : it summarizes all of our knowledge about the parameters of our model and is the basis for all subsequent inference and prediction for a given problem . for example , current astronomical datasets are very large , the proposed models may be high - dimensional , and therefore , the posterior sample is expensive to compute . however , once obtained , the posterior sample may be exploited for a wide variety of tasks . although dimension - switching algorithms , such as reversible - jump mcmc @xcite incorporate model selection automatically without need for bayes factors , these simulations appear slow to converge for some of our real - world applications . moreover , the marginal likelihood may be used for an endless variety of tests , ex post facto . @xcite presented a formula for estimating @xmath17 from a posterior distribution of parameters . they noted that a mcmc simulation of the posterior selects values of @xmath3 distributed as @xmath18 and , therefore , @xmath19 or @xmath20_{p({\boldsymbol{\theta}}|{{\bf d } } ) } , \label{eq : hmean}\ ] ] having suppressed the explicit dependence on @xmath16 for notational clarity . this latter equation says that the marginal likelihood is the harmonic mean of the likelihood with respect to the posterior distribution . it follows that the harmonic mean computed from a sampled posterior distribution is an estimator for the marginal likelihood , e.g. : @xmath21^{-1}. \label{eq : hsamp}\ ] ] unfortunately , this estimator is prone to domination by a few outlying terms with abnormally small values of @xmath22 ( e.g. see * ? ? ? * and references therein ) . @xcite describes convergence criteria for equation ( [ eq : hsamp ] ) and @xcite present augmented approaches with error estimates . alternative approaches to computing the marginal likelihood from the posterior distribution have been described at length by @xcite . of these , the laplace approximation , which approximates the posterior distribution by a multidimensional gaussian distribution and uses this approximation to compute equation ( [ eq : evidence ] ) directly , is the most widely used . this seems to be favored over equation ( [ eq : harmonic ] ) because of the problem with outliers and hence because of convergence and stability . in many cases , however , the laplace approximation is far from adequate in two ways . first , one must identify all the dominant modes , and second , modes may not be well - represented by a multidimensional gaussian distribution for problems of practical interest , although many promising improvements have been suggested ( e.g. * ? ? ? * ) . @xcite explored the use of the savage - dickey density ratio for cosmological model selection ( see also * ? ? ? * for a full review of the model selection problem for cosmology ) . finally , we may consider evaluation of equation ( [ eq : evidence ] ) directly . the mcmc simulation samples the posterior distribution by design , and therefore , can be used to construct volume elements in @xmath23-dimensional parameter space , @xmath24 , e.g. when @xmath25 . although the volume will be sparsely sampled in regions of relatively low likelihood , these same volumes will make little contribution to equation ( [ eq : evidence ] ) . the often - used approach from computational geometry , delaney triangulation , maximizes the minimum angle of the facets and thereby yields the `` roundest '' volumes . unfortunately , the standard procedure scales as @xmath26 for a sample of @xmath27 points . this can be reduced to @xmath28 using the flip algorithm with iterative construction @xcite but this scaling is prohibitive for large @xmath27 and @xmath23 typical of many problems . rather , in this paper , we consider the less optimal but tractable kd - tree for space partitioning . in part , the difficulty in computing the marginal likelihood from the sampled posterior has recently led ( * ? ? ? * nesting sampling ) to suggest an algorithm to simulate the marginal likelihood rather than the posterior distribution . this idea has been adopted and extended by cosmological modelers @xcite . the core idea of nesting sampling follows by rewriting equation ( [ eq : zdef0 ] ) as a double integral and swapping the order of integration , e.g. @xmath29 the nested sampler is a monte carlo sampler for the likelihood function @xmath30 with respect to the prior distrbution @xmath31 so that @xmath32 . the generalization of the construction in equation ( [ eq : nest ] ) for general distributions and multiple dimensions is the lebesgue integral ( see [ sec : intgr ] ) . clearly , this procedure has no problems with outliers with small values of @xmath33 . of course , any algorithm implementing nested sampling must still thoroughly sample the multidimensional posterior distribution and so retains all of the intendant difficulties that mcmc has been designed to solve . in many ways , the derivation of the nested sampler bears a strong resemblance to the derivation of the harmonic mean but without any obvious numerical difficulty . this led me to a careful study of equations ( [ eq : evidence ] ) and ( [ eq : zdef0 ] ) to see if the divergence for small value of likelihood could be addressed . indeed they can , and the following sections describe two algorithms based on each of these equations . these new algorithms retain the main advantage of the harmonic mean approximation ( hma ) : direct incorporation of the sampled posterior without any assumption of a functional form . in this sense , they are fully and automatically adaptive to any degree multimodality given a sufficiently large sample . we begin in [ sec : intgr ] with a background discussion of lebesgue integration applied to probability distributions and monte carlo ( mc ) estimates . we apply this in [ sec : evid ] to the marginal likelihood computation . this development both illuminates the arbitrariness in the hma from the numerical standpoint and leads to an improved approach outlined in [ sec : algo ] . in short , the proposed approach is motivated by methods of numerical quadrature rather than sample statistics . examples in [ sec : examples ] compare the application of the new algorithms to the hma and the laplace approximation . the overall results are discussed and summarized in [ sec : summary ] .

computation of the marginal likelihood from a simulated posterior distribution is central to bayesian model selection but is computationally difficult . the often - used harmonic mean approximation uses the posterior directly but is unstably sensitive to samples with anomalously small values of the likelihood . the laplace approximation is stable but makes strong , and often inappropriate , assumptions about the shape of the posterior distribution . it is useful , but not general . , we may convert the harmonic mean approximation from a sample statistic to a quadrature rule . as a quadrature , in addition , i demonstrate that the integral expression for the harmonic - mean approximation converges slowly at best for high - dimensional problems with uninformative prior distributions . these observations lead to two computationally - modest families of quadrature algorithms that use the full generality sample posterior but without the instability . this eliminates convergence issues . the first algorithm is analogous to standard quadrature but can only be applied for convergent problems . the second is a hybrid of cubature : it uses the posterior to discover and tessellate the subset of that sample space was explored and uses quantiles to compute a representative field value . neither algorithm makes strong assumptions about the shape of the posterior distribution and neither is sensitive to outliers . based on numerical tests , we recommend a combined application of both algorithms as consistency check to achieve a reliable estimate of the marginal likelihood from a simulated posterior distribution . * keywords : * bayesian computation , marginal likelihood , algorithm , bayes factors , model selection

computation of the marginal likelihood from a simulated posterior distribution is central to bayesian model selection but is computationally difficult . the often - used harmonic mean approximation uses the posterior directly but is unstably sensitive to samples with anomalously small values of the likelihood . the laplace approximation is stable but makes strong , and often inappropriate , assumptions about the shape of the posterior distribution . it is useful , but not general . we need algorithms that apply to general distributions , like the harmonic mean approximation , but do not suffer from convergence and instability issues . here , i argue that the marginal likelihood can be reliably computed from a posterior sample by careful attention to the numerics of the probability integral . posing the expression for the marginal likelihood as a lebesgue integral , we may convert the harmonic mean approximation from a sample statistic to a quadrature rule . as a quadrature , the harmonic mean approximation suffers from enormous truncation error as consequence . this error is a direct consequence of poor coverage of the sample space ; the posterior sample required for accurate computation of the marginal likelihood is much larger than that required to characterize the posterior distribution when using the harmonic mean approximation . in addition , i demonstrate that the integral expression for the harmonic - mean approximation converges slowly at best for high - dimensional problems with uninformative prior distributions . these observations lead to two computationally - modest families of quadrature algorithms that use the full generality sample posterior but without the instability . the first algorithm automatically eliminates the part of the sample that contributes large truncation error . the second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly . this eliminates convergence issues . the first algorithm is analogous to standard quadrature but can only be applied for convergent problems . the second is a hybrid of cubature : it uses the posterior to discover and tessellate the subset of that sample space was explored and uses quantiles to compute a representative field value . qualitatively , the first algorithm improves the harmonic mean approximation using numerical analysis , and the second algorithm is an adaptive version of the laplace approximation . neither algorithm makes strong assumptions about the shape of the posterior distribution and neither is sensitive to outliers . based on numerical tests , we recommend a combined application of both algorithms as consistency check to achieve a reliable estimate of the marginal likelihood from a simulated posterior distribution . * keywords : * bayesian computation , marginal likelihood , algorithm , bayes factors , model selection

cond-mat0004182

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theoretical research on electronic properties in mesoscopic condensed matter systems has focussed primarily on the charge degrees of freedom of the electron , while its spin degrees of freedom have not yet received the same attention . however , an increasing number of spin - related experiments @xcite show that the spin of the electron offers unique possibilities for finding novel mechanisms for information processing and information transmission most notably in quantum - confined nanostructures with unusually long spin dephasing times @xcite approaching microseconds , as well as long distances of up to @xmath0 @xcite over which spins can be transported phase - coherently . besides the intrinsic interest in spin - related phenomena , there are two main areas which hold promises for future applications : spin - based devices in conventional @xcite as well as in quantum computer hardware @xcite . in conventional computers , the electron spin can be expected to enhance the performance of quantum electronic devices , examples being spin - transistors ( based on spin - currents and spin injection ) , non - volatile memories , single spin as the ultimate limit of information storage etc . @xcite . on the one hand , none of these devices exist yet , and experimental progress as well as theoretical investigations are needed to provide guidance and support in the search for realizable implementations . on the other hand , the emerging field of quantum computing @xcite and quantum communication @xcite requires a radically new approach to the design of the necessary hardware . as first pointed out in ref . , the spin of the electron is a most natural candidate for the qubit the fundamental unit of quantum information . we have shown @xcite that these spin qubits , when located in quantum - confined structures such as semiconductor quantum dots or atoms or molecules , satisfy all requirements needed for a scalable quantum computer . moreover , such spin - qubits being attached to an electron with orbital degrees of freedom can be transported along conducting wires between different subunits in a quantum network @xcite . in particular , spin - entangled electrons can be created in coupled quantum dots and as mobile einstein - podolsky - rosen ( epr ) pairs @xcite provide then the necessary resources for quantum communication . for both spin - related areas conventional computers and quantum computers similar and sometimes identical physical concepts and tools are needed , the common short - term goal being to find ways to control the coherent dynamics of electron spins in quantum - confined nanostructures . it is this common goal that makes research on the electron spin in nanostructures spintronics a highly attractive area . while we advance our basic knowledge about spin physics in many - body systems , we gain insights that promise to be useful for future technologies . we have remarked earlier @xcite that there have been almost as many proposals for solid state implementations of quantum computers as all the other proposals put together . a clear reason for this is that solid state physics is a most versatile branch of physics , in that almost any phenomenon possible in physics can be embodied in an appropriately designed condensed matter system . a related reason is that solid state physics , being so closely allied with computer technology , has exhibited great versatility in the creation of artificial structures and devices . this has been exploited to produce ever more capable computational devices . it appears natural to expect that this versatility will extend to the creation of solid state quantum computers as well ; the plethora of proposals would indicate that this is indeed true , although only time can tell whether any of these proposals will actually provide a successful route to a quantum computer . in the following we will review the current status of our theoretical efforts towards the goal of implementing quantum computation and quantum communication with electron spins in quantum - confined nanostructures . most of the results presented here have been discussed at various places in the literature to which we refer the interested reader for more details . the long - term goal of our investigations is quantum information processing with electron spins . thus , a brief description of this emerging research field and its goals are in order . quantum computing has attracted much interest recently as it opens up the possibility of outperforming classical computation through new and more powerful quantum algorithms such as the ones discovered by shor @xcite and by grover @xcite . there is now a growing list of quantum tasks @xcite such as cryptography , error correcting schemes , quantum teleportation , etc . that have indicated even more the desirability of experimental implementations of quantum computing . in a quantum computer each quantum bit ( qubit ) is allowed to be in any state of a quantum two - level system . all quantum algorithms can be implemented by concatenating one- and two - qubit gates . there is a growing number of proposed physical implementations of qubits and quantum gates . a few examples are : trapped ions @xcite , cavity qed @xcite , nuclear spins @xcite , superconducting devices @xcite , and our qubit proposal @xcite based on the spin of the electron in quantum - confined nanostructures . coupled quantum dots provide a powerful source of deterministic entanglement between qubits of localized but also of delocalized electrons @xcite . e.g. , with such quantum gates it is possible to create a singlet state out of two electrons and subsequently separate ( by electronic transport ) the two electrons spatially with the spins of the two electrons still being entangled the prototype of an epr pair . this opens up the possibility to study a new class of quantum phenomena in electronic nanostructures @xcite such as the entanglement and non - locality of electronic epr pairs , tests of bell inequalities , quantum teleportation @xcite , and quantum cryptography @xcite which promises secure information transmission . in the present work , quantum dots play a central role and thus we shall make some general remarks about these systems here . semiconductor quantum dots are structures where charge carriers are confined in all three spatial dimensions , the dot size being of the order of the fermi wavelength in the host material , typically between @xmath1 and @xmath2 @xcite . the confinement is usually achieved by electrical gating of a two - dimensional electron gas ( 2deg ) , possibly combined with etching techniques , see fig . [ figarray ] . precise control of the number of electrons in the 2 conduction band of a quantum dot ( starting from zero ) has been achieved in gaas heterostructures @xcite . the electronic spectrum of typical quantum dots can vary strongly when an external magnetic field is applied @xcite , since the magnetic length corresponding to typical laboratory fields @xmath3 is comparable to typical dot sizes . in coupled quantum dots coulomb blockade effects @xcite , tunneling between neighboring dots @xcite , and magnetization @xcite have been observed as well as the formation of a delocalized single - particle state @xcite .

control over electron - spin states , such as coherent manipulation , filtering and measurement promises access to new technologies in conventional as well as in quantum computation and quantum communication . we review our proposal of using electron spins in quantum confined structures as qubits and discuss the requirements for implementing a quantum computer . we discuss recently proposed schemes for using a single quantum dot as spin - filter and spin - memory device . considering electronic epr pairs needed for quantum communication we show that their spin entanglement can be detected in mesoscopic transport measurements using metallic as well as superconducting leads attached to the dots . 2

control over electron - spin states , such as coherent manipulation , filtering and measurement promises access to new technologies in conventional as well as in quantum computation and quantum communication . we review our proposal of using electron spins in quantum confined structures as qubits and discuss the requirements for implementing a quantum computer . we describe several realizations of one- and two - qubit gates and of the read - in and read - out tasks . we discuss recently proposed schemes for using a single quantum dot as spin - filter and spin - memory device . considering electronic epr pairs needed for quantum communication we show that their spin entanglement can be detected in mesoscopic transport measurements using metallic as well as superconducting leads attached to the dots . 2

1405.5311

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in recent years there has been a huge explosion in the variety of sensors and the dimensionality of the data produced by these sensors and this has been in a large number of applications ranging from imaging to other scientific applications.the total amount of data produced by the sensors is much more than the available storage . so we often need to store a subset of the data . we want to reconstruct the entire data from it . the famous nyquist - shannon sampling theorem [ 5 ] tells us that if we can sample a signal at twice its highest frequency we can recover it exactly . in applications this often results in too many samples which must be compressed in order to store or transmit . an alternative is compressive sampling ( cs ) which provides a more general data acquisition protocol by reducing the signal directly into a compressed representation by taking linear combinations . in this paper we present a brief of the conventional approach of compressive sampling and propose a new approach that makes use of the em algorithm to reconstruct the entire signal from the compressed signals .

conventional approaches of sampling signals follow the celebrated theorem of nyquist and shannon . compressive sampling , introduced by donoho , romberg and tao , is a new paradigm that goes against the conventional methods in data acquisition and provides a way of recovering signals using fewer samples than the traditional methods use . here we suggest an alternative way of reconstructing the original signals in compressive sampling using em algorithm .

conventional approaches of sampling signals follow the celebrated theorem of nyquist and shannon . compressive sampling , introduced by donoho , romberg and tao , is a new paradigm that goes against the conventional methods in data acquisition and provides a way of recovering signals using fewer samples than the traditional methods use . here we suggest an alternative way of reconstructing the original signals in compressive sampling using em algorithm . we first propose a naive approach which has certain computational difficulties and subsequently modify it to a new approach which performs better than the conventional methods of compressive sampling . the comparison of the different approaches and the performance of the new approach has been studied using simulated data .

1011.3041

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weak gravitational lensing is a powerful technique for studying the distribution of matter in the universe due to its ability to model the matter distribution in foreground structures , independent of the nature of the matter present . as the light from background sources is bent around foreground lenses , the galaxy images get distorted by the tidal gravitational field . the first - order distortion is known as shear and is essentially an elongation of the image causing the source galaxy to appear stretched in one direction . this type of distortion measurement has been used in a wide variety of cosmological studies ranging from modeling the large - scale structure using cosmic shear ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * for reviews ) to determining galaxy halo shapes using galaxy - galaxy lensing @xcite . first described by @xcite , the second - order distortion is a relatively new addition which has since been named flexion @xcite . there are two types of flexion relevant to weak lensing studies : the first flexion induces a skewness of the brightness profile whilst the second flexion is a three - pronged distortion . in combination with shear , these distortions cause the well - known banana shape of lensed source images . as flexion is effectively the gradient of shear , it is sensitive on small scales . this makes it an important complement to shear which is sensitive on relatively large scales only . by virtue of this , and of the orthogonality of the three measurements , flexion is highly beneficial to investigations of the inner profiles of dark matter haloes , where baryons become important , and to the detection of substructure in cluster haloes . indeed , it was recently shown @xcite that mass reconstructions profit from the use of flexions in combination with shear , and flexion has already been used to constrain the halo mass distribution and to detect substructure in clusters of galaxies @xcite . to provide more information on substructure and mass profiles , there are currently new statistical flexion tools being developed ( eg . * ; * ? ? ? * ; * ? ? ? another application , as discussed in @xcite , is to use both flexions in combination with shear to significantly tighten the constraints on galaxy halo ellipticities compared to using shear alone . the shape measurement technique known as shapelets @xcite works by decomposing a galaxy image into a series of 2d hermite polynomials . these provide a simple framework for describing the main galaxy image distortion operators , such as shear and flexion , and the convolution with the point - spread function ( psf ) . due to the flexible treatment of the psf , the shapelets formalism has an advantage over the currently most widely used shape measurement method , ksb ( from * ? ? ? * ) , since ksb uses an idealised model for the psf whilst shapelets is more versatile . the ksb equivalent for flexion is known as holics @xcite . since the field of weak lensing is relatively new , lensing measurements are continuously being improved in accuracy and applicability . being a statistical technique , however , the accuracy of the weak lensing results depends heavily on the amount of data available . galaxy - galaxy flexion has been tentatively observed @xcite using the ground - based deep lens survey ( dls ) , but to further investigate galaxy - size haloes more and better data is needed . with large surveys such as the canada - france - hawaii telescope legacy survey ( cfhtls ) and the red sequence cluster surveys ( rcs , rcs2 ) available , and new surveys like the 1500 square degree kilo - degree survey ( kids ) imminent , the future looks bright . however , a space - based data set provides better resolution and such a data set is already accessible to us : the hst cosmos survey . using this data we will in this paper improve on the galaxy - galaxy flexion measurements of @xcite . this paper is organised as follows : in section [ sec : flexion ] we review the formalism for shear and flexion , whilst we review the shapelets method in section [ sec : shapelets ] with a description of our implementation ( dubbed the mv pipeline ) in section [ sec : shapelets_mv ] . in section [ sec : tests ] we test the mv pipeline on simulations and in section [ sec : cosmos ] the pipeline is applied to data from the cosmos survey . we conclude in section [ sec : discussion ] . throughout this paper we assume the following cosmology ( wmap7 ; * ? ? ? * ) : @xmath0

current theories of structure formation predict specific density profiles of galaxy dark matter haloes , and with weak gravitational lensing we can probe these profiles on several scales . on small scales , higher - order shape distortions known as flexion add significant detail to the weak lensing measurements . [ firstpage ] cosmology : observations gravitational lensing : weak galaxies : haloes dark matter

current theories of structure formation predict specific density profiles of galaxy dark matter haloes , and with weak gravitational lensing we can probe these profiles on several scales . on small scales , higher - order shape distortions known as flexion add significant detail to the weak lensing measurements . we present here the first detection of a galaxy - galaxy flexion signal in space - based data , obtained using a new shapelets pipeline introduced here . we combine this higher - order lensing signal with shear to constrain the average density profile of the galaxy lenses in the hubble space telescope cosmos survey . we also show that light from nearby bright objects can significantly affect flexion measurements . after correcting for the influence of lens light , we show that the inclusion of flexion provides tighter constraints on density profiles than does shear alone . finally we find an average density profile consistent with an isothermal sphere . [ firstpage ] cosmology : observations gravitational lensing : weak galaxies : haloes dark matter

1011.3041

c

we have shown a significant detection of galaxy - galaxy f flexion for the first time with shapelets using the space - based cosmos data set . we used this flexion signal in conjunction with the shear to constrain the average density profile of the galaxy haloes in our lens sample . we found a power - law profile consistent with an sis . furthermore , we showed that the inclusion of f flexion provides tighter constraints on both power - law and nfw profiles , an important proof of concept . the galaxy - galaxy f flexion signal measured in cosmos is slightly higher than expected from the shear signal , especially if we apply the multiplicative bias correction . there is however no indication from the cross term that there are systematics present . the discrepancy could be partly due to insufficient nearby object light removal , but this is unlikely to explain the full offset . substructure in galaxy haloes may cause excess f flexion compared to what the shear measures . however , a large fraction of the galaxy halo mass has to be in substructure in order for the effect to become significant . we note that @xcite also find shear and f flexion signals that are inconsistent with each other ; the velocity resulting from an sis profile fit to their f flexion signal is nearly twice that found using shear . this is qualitatively consistent with our findings , which leads us to believe that there is something more fundamental affecting the signal . in the near future we would like to further investigate the dependence of these discrepancies on lens properties . we measure a galaxy - galaxy g flexion signal that is consistent with the predicted profile , but due to the large measurement errors it is also consistent with zero . this measurement is a lot noisier than the other two , an effect most likely caused by the fact that there is less information available in the higher @xmath65-order shapelets for fainter sources . to measure a g flexion signal we thus require many well - resolved sources , an extravagance not yet awarded us . future large space - based surveys such as euclid will enable us to investigate g flexion further , but for now f flexion is a promising tool in its own right . the software introduced in this paper , the mv pipeline , is able to detect these higher order lensing distortions . we have shown that in practice , the shapelets f flexion measure is affected by light from nearby bright objects and detailed a way to correct for this effect . this bor does require further sophistication to account for large , well resolved galaxies , galaxies which are not well described by the single srsic light profile employed here . from the flashes simulations it is clear that there is more work required in order to improve the accuracy of the f flexion measurements for future surveys . noise related biases are particularly significant for this type of shape measure , and we have modeled these biases in cosmos . in the future we hope to measure flexion on a larger survey , enabling us to reduce the noise so that we can investigate the trend with e.g. redshift and lens mass . a larger number of sources would also enable us to further tighten the profile constraints in the inner regions of dark matter haloes where baryons become important . it is not yet clear how well we can measure flexion on ground - based data , but surveys like kids , cfhtls and rcs2 should provide an excellent test - bed .

we present here the first detection of a galaxy - galaxy flexion signal in space - based data , obtained using a new shapelets pipeline introduced here . we combine this higher - order lensing signal with shear to constrain the average density profile of the galaxy lenses in the hubble space telescope cosmos survey . we also show that light from nearby bright objects can significantly affect flexion measurements . after correcting for the influence of lens light , we show that the inclusion of flexion provides tighter constraints on density profiles than does shear alone . finally we find an average density profile consistent with an isothermal sphere .

current theories of structure formation predict specific density profiles of galaxy dark matter haloes , and with weak gravitational lensing we can probe these profiles on several scales . on small scales , higher - order shape distortions known as flexion add significant detail to the weak lensing measurements . we present here the first detection of a galaxy - galaxy flexion signal in space - based data , obtained using a new shapelets pipeline introduced here . we combine this higher - order lensing signal with shear to constrain the average density profile of the galaxy lenses in the hubble space telescope cosmos survey . we also show that light from nearby bright objects can significantly affect flexion measurements . after correcting for the influence of lens light , we show that the inclusion of flexion provides tighter constraints on density profiles than does shear alone . finally we find an average density profile consistent with an isothermal sphere . [ firstpage ] cosmology : observations gravitational lensing : weak galaxies : haloes dark matter

astro-ph0512261

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in the sections above , we have shown that curves derived as powers of ellipse functions , called p - ellipses , provide simple , yet surprisingly accurate approximations to orbits in a range of power - law potentials , including the well - known logarithmic potential , and softened power - law potentials . for a given power - law potential the p - ellipse function is nearly as simple as an ellipse , but the family of p - ellipses extends continuously across a physically interesting range of potentials . there are at least two reasons why p - ellipses are good orbital approximates across this range of potentials , and are likely to be the best simple , analytic functions to do so . first , p - ellipses match the tendency for orbits of a given energy and angular momentum to become more circular as the potential changes from keplerian to solid - body ( and @xmath10 decreases ) . in this sense the p - ellipses adjust well to the appropriate orbital shape in a given potential . secondly , the precession rate obtained by demanding that the p - ellipse satisfy the equation of motion in a given potential to first order in eccentricity is the identical to that given by newton s theorem . thus , the p - ellipses precess correctly . moreover , a second order approximation yields an eccentricity dependence of the precession rate that is in qualitative agreement with the @xcite semi - analytic results . a number of the results obtained above for p - ellipses , like the apsidal precession rates in power - law potentials , are not new , and good approximations to individual orbits can be obtained with epicycles or numerical integration . however , a family of simple curves like the p - ellipses allow us to readily see trends across a range of potentials , so they provide a simple , conceptual picture for orbital variations , as discussed in the introduction . moreover , as shown in the previous sections the p - ellipses provide a very powerful tool for studying characteristics like the occurrence of ( non - circular ) closed orbits as a function of eccentricity in different potentials . the key feature of the p - ellipses in this regard is a relatively simple approximate formulae for precession rates as a function of @xmath10 and @xmath54 . newton s theorem is valid in the limit of small eccentricity , and valluri et al.s extensions involve integrals that must be evaluated numerically . equations ( 19 ) and ( 20 ) , though approximate , offer convenience . ( also see eq . ( 25 ) . ) the example of the softened power - law potentials holds out the hope that the p - ellipses can provide useful orbit approximations in other non - power - law potentials . non - axisymmetric potentials have not been considered in this paper , but it seems reasonable to hope that p - ellipses could provide good approximations to loop orbits in such potentials . this issue deserves more study . in section 5.3 i described several examples of how the systematics of p - ellipses might shed light on important astrophysical problems . this conclusion will be even more general if p - ellipses prove to be good orbital approximations in more types of potential . orbit theory is more general than celestial mechanics , and the p - ellipse approximations should also be relevant to any field than involves orbits in general potentials . electron orbits in general , steady ( gradient ) electric and magnetic fields are obvious examples . in sum , there seems to be room for a great deal more development of the theory and application of these simple curves . their simplicity may allow us to address a number of complex issues that would be hard to study directly through the accumulation of numerical examples . this research was partially supported by nasa spitzer grant 1263961 . this research had made use of nasa s astrophysics data system abstract service . i am greatful for helpful input on an earlier version from bev smith , scott tremaine , and s. r. valluri . the referee , fred adams , provided many suggestions that significantly improved the paper .

curves in a family derived from powers of the polar coordinate formula for ellipses are found to provide good fits to bound orbits in a range of power - law potentials . this range includes the well - known ( keplerian ) and logarithmic potentials . the p - ellipses make very useful tools for studying trends among power - law potentials , and especially the occurence of closed orbits .

curves in a family derived from powers of the polar coordinate formula for ellipses are found to provide good fits to bound orbits in a range of power - law potentials . this range includes the well - known ( keplerian ) and logarithmic potentials . these approximate orbits , called p - ellipses , retain some of the basic geometric properties of ellipses . they satisfy and generalize newton s apsidal precession formula , which is one of the reasons for their surprising accuracy . because of their simplicity the p - ellipses make very useful tools for studying trends among power - law potentials , and especially the occurence of closed orbits . the occurence of closed or nearly closed orbits in different potentials highlights the possibility of period resonances between precessing , eccentric orbits and circular orbits , or between the precession period of multi - lobed closed orbits and satellite periods . these orbits and their resonances promise to help illuminate a number of problems in galaxy and accretion disk dynamics .

1508.05245

c

by exploiting adiabatic elimination techniques , semi - classical methods , and c - mop theory , we have provided a theoretical analysis of the dompo which works even at the critical point . we have focused on the region where the optical field is fully quantum , showing that such a quantum - correlated field with no coherent component can induce significant mechanical cooling through a cooling - by - heating mechanism . c - mop techniques have allowed us to check the validity of the optical adiabatic elimination as well as the semi - classical approximation , whose predictions have indeed been shown to break down at threshold , showing the potential of c - mop to treat dissipative quantum - optical problems in the vicinity of critical points . we thank yue chang , tao shi , alessandro farace , eugenio roldn , germn j. de valcrcel , florian marquardt , christoph marquardt , and j. ignacio cirac for useful discussions and comments . this work has been supported by the german research foundation ( dfg ) via the crc 631 and the grant ha 5593/3 - 1 . acknowledge funding from the alexander von humbolt foundation through their fellowship for postdoctoral researchers . acknowledges support by uk engineering and physical sciences research council ( epsrc ) under ep / n009428/1 .

we introduce a theoretical approach that is capable of describing this regime , even at the critical point itself . we find that the down - converted field can induce significant mechanical cooling and identify the process responsible of this as a cooling - by - heating mechanism . our approach opens the possibility for analyzing further hybrid dissipative quantum systems in the vicinity of critical points .

degenerate optomechanical parametric oscillators are optical resonators in which a mechanical degree of freedom is coupled to a cavity mode that is nonlinearly amplified via parametric down - conversion of an external pumping laser . below a critical pumping power the down - converted field is purely quantum - mechanical , making the theoretical description of such systems very challenging . here we introduce a theoretical approach that is capable of describing this regime , even at the critical point itself . we find that the down - converted field can induce significant mechanical cooling and identify the process responsible of this as a cooling - by - heating mechanism . moreover , we show that , contrary to naive expectations and semi - classical predictions , cooling is not optimal at the critical point , where the photon number is largest . our approach opens the possibility for analyzing further hybrid dissipative quantum systems in the vicinity of critical points .

1202.6037

i

diagnostic sonography allows visualization of body tissues , by radiating them with acoustic energy pulses , which are transmitted from an array of transducer elements . the image typically comprises multiple scanlines , each constructed by integrating data collected by the transducers , following the transmission of an energy pulse along a narrow beam . as the pulse propagates , echoes are scattered by density and propagation - velocity perturbations in the tissue @xcite , and detected by the transducer elements . averaging the detected signals , after their alignment with appropriate time - varying delays , allows localization of the scattering structures , while improving the signal to noise ratio ( snr ) @xcite . the latter process is referred to as beamforming . performed digitally , beamforming requires that the analog signals , detected by the transducers , first be sampled . confined to classic nyquist - shannon sampling theorem @xcite , the sampling rate must be at least twice the bandwidth , in order to avoid aliasing . as imaging techniques develop , the amount of elements involved in each imaging cycle typically increases . consequently , the rates of data which need to be transmitted from the system front - end , and then processed by the beamformer , grow significantly . the growth in transmission and processing rates inevitably effects both machinery size and power consumption . consequently , in recent years there has been growing interest in reducing the amounts of data as close as possible to the system front - end . in fact , such reduction is already possible within the classical sampling framework : state of the art devices digitally downsample the data at the front - end , by exploiting the fact that the signal is modulated onto a carrier , so that the spectrum essentially occupies only a portion of its entire base - band bandwidth . the preliminary sample rate remains unchanged , since the demodulation is performed in the digital domain . nevertheless , a key to significant data compression lies beyond the classical sampling framework . indeed , the emerging compressive sensing ( cs ) framework @xcite states , that sparse signals may be accurately reconstructed from a surprisingly small amount of coefficients . complementary ideas rise from the finite rate of innovation ( fri ) framework @xcite , in which the signal is assumed to have a finite number of degrees of freedom per unit time . many classes of fri signals can be recovered from samples taken at the rate of innovation @xcite . for a detailed review of previously proposed fri methods , the reader is referred to @xcite . combining the latter notions with classical sampling methods , the developing xampling framework @xcite involves methods for fully capturing the information carried by an analog signal , by sampling it far below the nyquist - rate . following the spirit of xampling , al . proposed in @xcite , that ultrasound signals be described within the fri framework . explicitly , they assume that these signals , formed by scattering of a transmitted pulse from multiple reflectors , may be modeled by a relatively small number of pulses , all replicas of some known pulse shape . denoting the number of reflected pulses by @xmath0 , and the signal s finite temporal support by @xmath1 , the detected signal is completely defined by @xmath2 degrees of freedom , corresponding to the replicas unknown time delays and amplitudes . based on @xcite , the authors formulate the relationship between the signal s fourier series coefficients , calculated with respect to @xmath1 , and its unknown parameters , in the form of a spectral analysis problem . the latter may be solved using existing techniques , given a subset of fourier series coefficients , with a minimal cardinality of @xmath2 . the sampling scheme is thus reduced to the problem of extracting a small subset of the detected signal s frequency samples . two robust schemes are derived in @xcite , extracting such a set of coefficients from samples of the signal , taken at sub - nyquist rates . the system presented in @xcite employs a single processing channel , in which the analog signal is filtered by an appropriate sampling kernel and then sampled with a standard low - rate analog to digital converter ( adc ) . the method of @xcite employs multiple processing channels , each comprising a modulator and an integrator . these approaches were shown to be more robust than previous fri techniques and also allow for arbitrary pulse shapes . the initial motivation for our work stems from the need to translate the ultrasound xampling scheme proposed in @xcite , into one which achieves the final goal of reconstructing a two - dimensional ultrasound image , by integrating data sampled at multiple transducer elements . in conventional ultrasound imaging , such integration is achieved by the beamforming process . the question is how may we implement beamforming , using samples of the detected signals taken at sub - nyquist rates . a straightforward approach is to replace the nyquist - rate sampling mechanism , utilized in each receiver element , by an fri xampling scheme . having estimated the parametric representation of the signal detected in each individual element , we could reconstruct it digitally . the reconstructed signals can then be further processed via beamforming . however , the nature of ultrasound signals reflected from real tissues , makes such an approach impractical . this is mainly due to the detected signals poor snr , which results in erroneous parameter extraction by the xampling scheme , applied to each element independently . our approach is to generalize the fri xampling scheme proposed in @xcite , such that it integrates beamforming into the low - rate sampling process . the result is equivalent to that obtained by xampling the beamformed signal , which exhibits significantly better snr . furthermore , beamforming practically implies that the array of receivers is dynamically focused along a single scanline . consequently , the resulting signal depicts reflections originating in the intersection of the radiated medium with a vary narrow beam . such a signal better suits the fri model proposed in @xcite , which assumes the reflections to be caused by isolated , point - like scatterers . we refer to our scheme by the term compressed beamforming , as it transforms the beamforming operator into the compressed domain @xcite . applied to real cardiac ultrasound data obtained from a ge breadboard ultrasonic scanner , our approach successfully images macroscopic perturbations in the tissue while achieving a nearly eight - fold reduction in sampling rate , compared to standard imaging techniques . the paper is organized as follows : in section [ sec:02 ] , we summarize the general principles of beamforming in ultrasound imaging . in section [ sec:03 ] we outline the fri model and its contribution to sample rate reduction in the ultrasound context . we motivate compressed beamforming in section [ sec:04 ] , considering the nature of ultrasound signals reflected from biological tissues . beamforming and fri xampling are combined in section [ sec:05 ] , where we propose that the signal obtained by beamforming may be treated within the fri framework . following this observation , we derive our first compressed beamforming scheme , which operates on low - rate samples taken at the individual receivers . this approach is then further simplified in section [ sec:06 ] . in section [ sec:07 ] we focus on image reconstruction from the parametric representation obtained by either xampling scheme . in this context , we generalize the signal model proposed in @xcite , allowing additional unknown phase shifts of the detected pulses . we then discuss an alternative recovery approach , based on cs . simulations comparing the performance of several recovery methods are provided in section [ sec:08 ] . finally , experimental results obtained for cardiac ultrasound data are presented in section [ sec:09 ] .

emerging sonography techniques often require increasing the number of transducer elements involved in the imaging process . consequently , larger amounts of data must be acquired and processed . the significant growth in the amounts of data affects both machinery size and power consumption . within the classical sampling framework , state of the art systems reduce processing rates by exploiting the bandpass bandwidth of the detected signals . it has been recently shown , that a much more significant sample - rate reduction may be obtained , by treating ultrasound signals within the finite rate of innovation framework . this often results in erroneous parameter extraction , bringing forward the need to enhance the snr of the low - rate samples . in our work , we achieve snr enhancement , by beamforming the sub - nyquist samples obtained from multiple elements . we refer to this process as compressed beamforming " . applying it to cardiac ultrasound data , we successfully image macroscopic perturbations , while achieving a nearly eight - fold reduction in sample - rate , compared to standard techniques . array processing , beamforming , compressed sensing ( cs ) , finite rate of innovation ( fri ) , ultrasound , xampling

emerging sonography techniques often require increasing the number of transducer elements involved in the imaging process . consequently , larger amounts of data must be acquired and processed . the significant growth in the amounts of data affects both machinery size and power consumption . within the classical sampling framework , state of the art systems reduce processing rates by exploiting the bandpass bandwidth of the detected signals . it has been recently shown , that a much more significant sample - rate reduction may be obtained , by treating ultrasound signals within the finite rate of innovation framework . these ideas follow the spirit of xampling , which combines classic methods from sampling theory with recent developments in compressed sensing . applying such low - rate sampling schemes to individual transducer elements , which detect energy reflected from biological tissues , is limited by the noisy nature of the signals . this often results in erroneous parameter extraction , bringing forward the need to enhance the snr of the low - rate samples . in our work , we achieve snr enhancement , by beamforming the sub - nyquist samples obtained from multiple elements . we refer to this process as compressed beamforming " . applying it to cardiac ultrasound data , we successfully image macroscopic perturbations , while achieving a nearly eight - fold reduction in sample - rate , compared to standard techniques . array processing , beamforming , compressed sensing ( cs ) , finite rate of innovation ( fri ) , ultrasound , xampling

gr-qc9701059

i

understanding the origin of the black - hole entropy is one of the most fascinating problems in black - hole physics @xcite . the concept of the black - hole entropy traces back to the work by bekenstein , who pointed out that the behavior of the basic physical quantities describing stationary black holes has an analogous structure to that of ordinary thermodynamical systems @xcite . this thermodynamical structure inherent in the black - hole theory is usually called the ` black - hole thermodynamics ' . in analogy with ordinary material systems , then , it is natural to expect that the black - hole entropy comes from microscopic degrees of freedom of a system including a black hole . this suggests that quantum theory of gravity should inevitably take part in the black - hole thermodynamics . in this sense , understanding the origin of the black - hole entropy shall provide us with important information on quantum gravity . this is one among the several reasons why the black - hole entropy needs to be understood at the fundamental level . let us recall basic properties of the black - hole thermodynamics by taking a simple example . we consider the one - parameter family of schwarzschild black holes parameterized by the mass @xmath0 . here and throughout this paper , we assume that the relation analogous to the first law of thermodynamics holds for a black - hole system . in the present example , there is only one parameter @xmath0 characterizing a black hole . therefore , this relation should be of the simplest form @xmath1 where @xmath2 , @xmath3 and @xmath4 are quantities that are identified with the energy , the entropy and the temperature of a black hole , respectively . the relation eq.([eqn:1st - law ] ) is called the first law of the black - hole thermodynamics @xcite . thus , if two of the quantities @xmath2 , @xmath3 and @xmath4 are given , @xmath5 determines the remaining quantity . for simplicity , let us call this procedure of defining energy , entropy and temperature which satisfy the first law ( eq.([eqn:1st - law ] ) ) the ` construction of thermodynamics ' . in the present example , @xmath0 is the only parameter characterizing the family of black holes . therefore the simplest combination which yields the dimension of energy is @xmath6 this is the energy of the black hole . there is also a natural choice for @xmath4 @xcite . hawking showed that a black hole with surface gravity @xmath7 emits thermal radiation of a matter field ( which plays the role of a thermometer ) at temperature @xmath8 . moreover one can show that if any matter field in a thermal - equilibrium state at any temperature is scattered by a black hole , it goes to another thermal - equilibrium state at a temperature closer to @xmath4 @xcite . thus it is natural to define the temperature of a schwarzschild black hole with mass @xmath0 by @xmath9 since @xmath10 @xcite . from eqs.([eqn:1st - law])-([eqn : t - bh ] ) , we can construct the thermodynamics for the schwarzschild black holes . thus we get an expression for @xmath3 given by @xmath11 where @xmath12 is the area of the event horizon and @xmath13 is some constant . since a value of @xmath13 is not essential in our discussions , we shall set hereafter @xmath14 it is well - known that classically the area of the event horizon does not decrease in time just as the ordinary thermodynamical entropy . the result eq.([eqn : s - bh ] ) looks reasonable in this sense . indeed this observation was the original motivation for the introduction of the black - hole thermodynamics @xcite . however , it is not clear to what extent @xmath3 is related with the information as the ordinary thermodynamical entropy is . at this stage we would like to point out that the third law of thermodynamics does not hold for a black hole irrespective of the choice of the value for @xmath13 as is seen from eqs.([eqn : t - bh ] ) and ( [ eqn : s - bh ] ) . we shall come back to this point in [ subsection : discrepancy ] . in any case , understanding the origin of @xmath3 is an important problem in black - hole physics . there have been many attempts to understand the origin of the black - hole entropy @xcite . among them we shall concentrate only on the so - called entanglement entropy @xcite . the aim of this paper is to judge whether entanglement entropy can be regarded as the origin of black - hole entropy . for this purpose it is effective to investigate the whole structure of the thermodynamics obtained from the entanglement entropy , rather than just to examine the apparent structure of the entropy alone . thus we shall construct the ` thermodynamics of entanglement ' and compare it with the black - hole thermodynamics . as is expected by the above example of the black - hole thermodynamics , we have to define either energy or temperature to construct the thermodynamics of entanglement . combining it with the entanglement entropy , which is already at hand @xcite , the other is automatically defined by means of eq.([eqn:1st - law ] ) . in this paper we shall choose the option to define the entanglement energy firstly , deriving the entanglement temperature afterwards . we shall consider two possible definitions of entanglement energy . therefore we can construct two different kinds of thermodynamics by combining each of these definitions of entanglement energy with the entanglement entropy . we show that neither of these thermodynamics is compatible with the black - hole thermodynamics if no gravitational effects are taken into account . after that , we see how inclusion of gravity alter the thermodynamics of the entanglement . we give a suggestive argument that they have a common behavior if gravitational effects are taken into account properly . thus the entanglement entropy passes a non - trivial check to be the origin of the black - hole entropy . the rest of the paper is organized as follows . in section [ sec : sent ] we review the concept of the entanglement entropy . in section [ sec : eent ] we propose two natural definitions of entanglement energy and present general formulas for calculating the energy . in section [ sec : example ] explicit expression for the entanglement energy are derived for some tractable models with the help of the formulas prepared in section [ sec : eent ] . in section [ sec : discussion ] we construct the thermodynamics of entanglement and compare it with the black hole thermodynamics from various angles . section [ sec : summary ] is devoted to the summary of our results .

entanglement entropy is often speculated as a strong candidate for the origin of the black - hole entropy . to judge whether this speculation is true or not , it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy , rather than just to examine the apparent structure of the entropy alone or to compare it with that of the black hole entropy . it is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system . from this point of view , we construct a ` thermodynamics of entanglement ' by introducing an entanglement energy and compare it with the black - hole thermodynamics . we consider two possible definitions of entanglement energy . then we construct two different kinds of thermodynamics by combining each of these different definitions of entanglement energy with the entanglement entropy . we find that both of these two kinds of thermodynamics show significant differences from the black - hole thermodynamics if no gravitational effects are taken into account . finally we see how inclusion of gravity alter the thermodynamics of the entanglement . thus the entanglement entropy passes a non - trivial check to be the origin of the black - hole entropy .

entanglement entropy is often speculated as a strong candidate for the origin of the black - hole entropy . to judge whether this speculation is true or not , it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy , rather than just to examine the apparent structure of the entropy alone or to compare it with that of the black hole entropy . it is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system . from this point of view , we construct a ` thermodynamics of entanglement ' by introducing an entanglement energy and compare it with the black - hole thermodynamics . we consider two possible definitions of entanglement energy . then we construct two different kinds of thermodynamics by combining each of these different definitions of entanglement energy with the entanglement entropy . we find that both of these two kinds of thermodynamics show significant differences from the black - hole thermodynamics if no gravitational effects are taken into account . these differences are in particular highlighted in the context of the third law of thermodynamics . finally we see how inclusion of gravity alter the thermodynamics of the entanglement . we give a suggestive argument that the thermodynamics of the entanglement behaves like the black - hole thermodynamics if the gravitational effects are included properly . thus the entanglement entropy passes a non - trivial check to be the origin of the black - hole entropy .

1311.3975

c

through a combination of analytical calculations and quantum monte carlo simulations , we have investigated the behavior of valence bond probability distributions and correlations in ground states of the antiferromagnetic heisenberg model on chain , square and cubic lattices . our analytic results provide a clear ground for the probability distribution @xmath3 for a valence bond to join two sites ( existing on different sublattices ) separated by @xmath81 to scale as @xmath309 where @xmath34 is the lattice dimension . this result , which was observed in previous numerical simulations and justified by a mean - field ansatz , is now understood within a more trustworthy analytical framework : bosonisation in 1d , and the non - linear @xmath2 model in @xmath91 . this will be useful for variational calculations , or for constructing guiding wave functions in monte carlo simulations , which aim at targeting an antiferromagnetic state on a bipartite lattice . moreover , our results provide a formal justification for the factorization ansatz eq . introduced by liang , douot and anderson , as long as a non - linear @xmath2 model description is valid . indeed , our results , corroborated by large - scale qmc calculations in the 2d case , clearly indicate the absence of correlations between valence bonds at long distances . for the quasi - long - range ordered ground state of the heisenberg chain , the situation is more subtle as correlations are present between valence bonds , as clearly observed in fig . [ fig : vbcpumps ] . this may be understood as a signature of the strong ( power - law - decaying ) dimer correlations which are known @xcite to be present in the heisenberg af chain . we would also like to point out that the existence of long - range antiferromagnetic order does not necessarily imply a distribution of valence bonds with a distribution law @xmath310 with no correlations . for instance , the wave functions composed of the equal - weight linear combinations of nearest - neighbor valence bonds on the simple cubic or diamond lattices have been shown to sustain af long - range order @xcite ( see also ref . ) . however , their description is most certainly beyond the non - linear @xmath2 model approach , which probably can not describe the dipolar dimer correlations observed in these wave functions @xcite . finally , we suggest several further investigations as possible extensions of our work . for one - dimensional systems , the bosonization analysis of sec . [ sec : bos ] carries over for the critical phase of the @xmath311 anisotropic spin chain . while the su(2 ) symmetry is lost in this more general case , analytical predictions for mixed expectation values of the type @xmath312 [ such as in eq . ] can be made and tested through , for instance , dmrg or qmc calculations . in general , the power - law exponent will depend on the anisotropy parameter through the compactification radius @xmath242 . another interesting problem would be to calculate analytically logarithmic corrections to the power - law decays of valence bond occupations and correlations in one dimension , which most certainly exist in the su(@xmath8 ) case . we note , however , that these are presumably small , hard to detect , effects as the qmc results are already very well described by the pure power - law decays . in dimensions larger than 1 , the non - linear @xmath2 model approach may also be used to describe the behavior of valence bond distributions and correlations at a quantum critical point between the antiferromagnet and a paramagnet , such as for a bilayer heisenberg model @xcite . this could be useful in explaining and improving variational approaches based on valence bonds that aim to describe this quantum phase transition ( such as , _ e.g. _ , ref . ) . more generally , the current formalism based on the overlap eq . could be used with any effective field theory which describes a particular quantum phase or a quantum critical point of quantum antiferromagnets on bipartite lattices . finally , it would be very interesting to see whether an approach similar to the one developed here could be applied to describe the `` spinon detection '' procedure recently advocated by tang and sandvik @xcite , which is also based on a valence bond description of a spin system with one or two unpaired spins ( _ i.e. _ , that do not belong to a valence bond ) . this is a more challenging case as one would first need to derive equations similar to eq . or for this situation .

, we study the bipartite valence bond distributions and their correlations within the ground state of the heisenberg antiferromagnet on bipartite lattices . in terms of field theory , a non - linear model analysis reveals that at long distances the probability distribution of valence bond lengths decays as and that valence bonds are uncorrelated . by a bosonization analysis the analytical results are confirmed by high - precision quantum monte carlo simulations in , and .

every singlet state of a quantum spin- system can be decomposed into a linear combination of valence bond basis states . the range of valence bonds within this linear combination as well as the correlations between them can reveal the nature of the singlet state , and are key ingredients in variational calculations . in this work , we study the bipartite valence bond distributions and their correlations within the ground state of the heisenberg antiferromagnet on bipartite lattices . in terms of field theory , this problem can be mapped to correlation functions near a boundary . in dimension , a non - linear model analysis reveals that at long distances the probability distribution of valence bond lengths decays as and that valence bonds are uncorrelated . by a bosonization analysis , we also obtain in despite the different mechanism . on the other hand , we find that correlations between valence bonds are important even at large distances in , in stark contrast to . the analytical results are confirmed by high - precision quantum monte carlo simulations in , and . we develop a single - projection loop variant of the valence bond projection algorithm , which is well designed to compute valence bond probabilities and for which we provide algorithmic details .

1310.1360

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the discovery of many extrasolar planets ( exoplanets ) by radial velocity , transit , and imaging techniques has stimulated observational and theoretical studies to characterize their atmospheric chemistry and physical properties and to investigate whether these exoplanets could sustain life forms ( e.g. , * ? ? ? * ; * ? ? ? * ) . as density decreases with height in exoplanet atmospheres , photolysis ( photodissociation of molecules and photoionization of atoms ) will eventually dominate over thermal equilibrium . this typically occurs where the atmospheric pressure is less than 1 mbar . recently photochemical models have been computed for terrestrial planets and super - earths ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , hot - neptunes @xcite , and hot - jupiters @xcite . far ultraviolet ( fuv ) radiation at wavelengths below 170 nm and , in particular , the very bright ly@xmath0 emission line ( 121.6 nm ) , control the photodissociation of such important molecules as h@xmath1o , ch@xmath2 , and co@xmath1 , which can increase the mixing ratio of oxygen @xcite . ozone ( o@xmath3 ) has been called a potential biosignature in super - earth atmospheres @xcite , but it is important to assess the extent to which photolysis of o@xmath1 and subsequent chemical reactions rather than biological processes can control its abundance . future photochemical models based on realistic host star uv emission including intrinsic ly@xmath0 fluxes are needed to address questions of the reliability of biosignatures and atmospheric chemical abundances . recent models , such as those cited above , show that the c / o ratio , quenching reactions , thermal structure , and diffusion also play important roles in determining mixing ratios for important molecules in exoplanet atmospheres , but the short wavelength radiation of the host star is critically important . atmospheric chemistry models require as input the fuv ( 117170 nm ) radiation from the host star . spectra obtained with the cosmic origins spectrograph ( cos ) and space telescope imaging spectrograph ( stis ) instruments on _ hubble space telescope ( hst ) _ are providing these data ( e.g. , * ? ? ? * ) including m dwarf stars @xcite , which many authors believe are the most favorable candidate host stars with nearby earth - like exoplanets @xcite . the _ galaxy evolution explorer ( galex ) _ instrument is also providing broadband fuv ( not including the ly@xmath0 line ) and nuv fluxes of exoplanet host stars @xcite . while the ly@xmath0 line is the most important fuv emission feature for solar - type stars and is as bright as the entire 120320 nm spectrum of m dwarfs @xcite , the entire core of this line is absorbed by interstellar hydrogen . the intrinsic flux in the ly@xmath0 line can be reconstructed from high - resolution spectra @xcite or predicted from correlations with other emission lines @xcite . at wavelengths of 1091.2 nm , extreme - ultraviolet ( euv ) radiation from the host star photoionizes hydogen creating an ionosphere @xcite and heats the outer layers of these atmospheres , thereby inflating the atmosphere and driving mass loss . @xcite computed models that describe how photoionization heating of hot - jupiter atmospheres by euv radiation drives transonic hydrodynamic outflows ( also called hydrodynamic blow - off ) . these outflows are analogous to the parker - type solar wind @xcite , except that the heating is from above rather than below . for a hot - jupiter exoplanet like hd 209458b located at 0.05 au from its solar - type host star , the outflowing plasma is heated almost entirely by the kinetic energy of protons ( and their subsequent collisions ) after hydrogen atoms are ionized by host star euv photons with a small contribution of x - ray photons at @xmath4 nm @xcite . hot - jupiters and neptune - like exoplanets with hydrogen - rich atmospheres and weak magnetic moments can also lose mass when photoionized and charge - exchanged atomic and molecular hydrogen are picked - up by the stellar wind and coronal mass ejection plasma outside of the magnetopause standoff distance where the stellar wind pressure exceeds the planet s magnetic pressure ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the expanding atmospheres of exoplanets in orbit around older solar - like stars are heated by the incident euv radiation and cooled by expansion ( pdv work ) . these winds are described as _ energy limited_. @xcite showed that the mass - loss rates of such winds are proportional to the incident euv flux , @xmath5 . when the host star has a far larger euv flux , for example a t tauri star , x - ray heating also becomes important and radiative recombination rather than expansion cools the denser wind . the mass - loss rate for such winds is proportional to @xmath6 and @xcite call such winds _ radiation / recombination - limited_. they argue that the proper way to describe mass loss from the hydrogen - rich atmospheres of exoplanets located close to their host stars is by transonic hydrodynamic winds heated by euv photons rather than by radiation pressure @xcite . @xcite , @xcite , @xcite , and others have also computed transonic hydrodynamic outflows for hot - jupiters , and @xcite have computed such models for super - earths with hydrogen - rich upper atmospheres . super - earths have much smaller mass loss rates than hot - jupiters . roche lobe overflow can enhance the mass loss rate @xcite , but the ram and magnetic pressure of a strong stellar wind on the day side of the exoplanet can suppress a transonic outflow , producing instead a subsonic outflow often called a jeans - type stellar breeze . the supersonic orbital speed of a hot - jupiter moving through a stellar wind can produce a nonspherical shock front ahead of the exoplanet s motion with properties that depend on the stellar wind , magnetic field , and exoplanet s mass loss rate ( e.g. , * ? ? ? for all of these cases , the host star s unobservable euv flux and the model - dependent fraction of this flux that is converted to heat ( e.g. , * ? ? ? * ) are essential input parameters for computing realistic models of exoplanet atmospheres . @xcite calculated the response of the earth s oxygen and nitrogen atmosphere to changes in the euv flux from its host star , the present day and the young sun . with a 1-d hydrodynamic model coupled to a code that describes euv photoionization and heating by secondary electrons , they find that illumination by the present day sun predicts that the upper thermosphere is in hydrostatic equilibrium , but an increase in the euv flux by only a factor of 4.6 is sufficient to produce a hydrodynamic outflow that becomes the dominant cooling mechanism . a factor of 10 increase in the euv flux predicted for the young sun produces a transonic outflow . estimating the euv emission of host stars is , therefore , an essential but difficult problem to solve because interstellar hydrogen absorbs essentially all of the spectrum between 40 and 91.2 nm , even for the nearest stars , and there are spectra in the 1040 nm range for only a few stars observed with the _ extreme ultraviolet explorer ( euve ) _ satellite @xcite . despite these problems , several authors have developed methods for estimating the euv flux for solar - type stars . @xcite estimated euv fluxes and photodissociation rates from the observed fuv and x - ray fluxes of solar - type stars and discussed the effect of enhanced euv emission from the young sun on the early martian atmosphere . following a similar approach , @xcite estimated euv fluxes for solar - type stars from the fuv and x - ray emission of the sun and six stars with spectral types g0 v to g5 v and a range of ages and thus activity . this work is appropriate for solar - type stars , but its applicability to other spectral type stars is not discussed in their papers . recently , @xcite developed a technique for estimating the euv to ir flux of the sun as a function of age ( 0.66.7 gyr ) by computing relative flux multipliers for different wavelength intervals using observations of the g - type stars @xmath7 cet and ek dra to test the multipliers at earlier ages when the sun was more active . however , @xcite do not extend their approach to estimating fluxes for stars much different in spectral type from the sun . @xcite computed synthetic euv spectra of many f m stars from emission measure distribution analyses of their x - ray spectra . the euv fluxes computed from their synthetic spectra may not accurately include emission in the hydrogen lyman continuum , important for the 7091.2 nm region , the he i and he ii continua , and may exclude some of the emission lines seen in solar spectra . we will compare our results with those of @xcite later in this paper . in this paper our objective is to develop a different kind of technique for estimating the euv emission of host stars with spectral types f m that is relatively insensitive to stellar activity and variability . our technique is similar to that used by @xcite in that we estimate the ratios of euv fluxes in different wavelength bands to the emission in a representative emission line , in this case ly@xmath0 . since both the euv and ly@xmath0 fluxes increase and decrease together ( but not necessarily at the same rate ) with the magnetic heating rate that depends on stellar rotation , age , and magnetic field properties , euv / ly@xmath0 flux ratios should change rather slowly with the ly@xmath0 flux . in support of this hypothesis , @xcite show that the number of ly@xmath0 photons equals the total number of photons emitted below 170 nm by the sun at all ages from the zero age main sequence to the present , while the ratio of ly@xmath0 flux to the total solar flux below 170 nm increases smoothly from 20 to 36.5% over this time interval . we also find that euv / ly@xmath0 flux ratios vary slowly with activity and that the flux ratios in solar data and recent solar irradiance models are representative of stars with a wide range of spectral type and activity . observations with the _ euve _ satellite in the 1040 nm wavelength range and the _ far ultraviolet spectroscopic explorer ( fuse ) _ satellite in the 91.2117 nm range provide empirical tests of our method . when the reconstructed ly@xmath0 flux is not available for a given star , it may be estimated using the techniques described by @xcite .

since stellar euv fluxes can not now be measured and interstellar neutral hydrogen completely obscures stellar radiation between 40 and 91.2 nm , even for the nearest stars , we must estimate the unobservable euv flux by indirect methods . we provide formulae for predicting euv flux ratios based on the _ euve _ and _ fuse _ stellar data and on the solar models , which are essential input for modelling the atmospheres of exoplanets .

extreme ultraviolet ( euv ) radiations ( 10117 nm ) from host stars play important roles in the ionization , heating , and mass loss from exoplanet atmospheres . together with the host star s ly and far - uv ( 117170 nm ) radiation , euv radiation photodissociates important molecules , thereby changing the chemistry in exoplanet atmospheres . since stellar euv fluxes can not now be measured and interstellar neutral hydrogen completely obscures stellar radiation between 40 and 91.2 nm , even for the nearest stars , we must estimate the unobservable euv flux by indirect methods . new non - lte semiempirical models of the solar chromosphere and corona and solar irradiance measurements show that the ratio of euv flux in a variety of wavelength bands to the ly flux varies slowly with the ly flux and thus with the magnetic heating rate . this suggests and we confirm that solar euv / ly flux ratios based on the models and observations are similar to the available 1040 nm flux ratios observed with the _ euve _ satellite and the 91.2117 nm flux observed with the _ fuse _ satellite for f5 v m5 v stars . we provide formulae for predicting euv flux ratios based on the _ euve _ and _ fuse _ stellar data and on the solar models , which are essential input for modelling the atmospheres of exoplanets .

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euv fluxes from host stars control the photochemistry , heating , and mass loss from the outer atmospheres of exoplanets , especially for those exoplanets located close to their host stars . the objective of this study is to develop a useful technique for predicting the euv fluxes of f5m5 dwarf stars , since there are only a few measurements of stellar euv fluxes and interstellar absorption prevents measurements between 40 and 91.2 nm for all nearby dwarfs stars except for the sun . our technique employs ratios of euv fluxes in wavelength bands to the ly@xmath0 flux , because models of the solar chromosphere , transition region , and corona show that the euv flux scales with the ly@xmath0 flux . moreover , models of solar regions with different amounts of magnetic heating show temperature structures with similar shapes but displaced deeper into the atmosphere ( and thus higher densities ) with increasing magnetic heating . these empirical and theoretical arguments gives us confidence that the ratios of the euv flux in various wavelength bands to the ly@xmath0 flux should vary smoothly with stellar activity at least for stars that do not differ too greatly from the sun in spectral type . models of stellar chromospheres and transition regions comparable in detail with the solar models of @xcite are needed to confirm the range of stars for which our technique is useful . until such models are available , observations of the few stars in the 91.2117.0 nm range by the _ fuse _ satellite and in the 1040 nm range by the _ euve _ satellite show that our ratio technique is useful for f5m5 dwarfs . table 5 summarizes our recommended formulae for predicting the log [ f(@xmath33)/f(ly@xmath0 ) ] ratios in nine wavelength bands . for the 1020 nm , 2030 nm , and 3040 nm wavelength bands , we recommend using the formulae based on the stellar fluxes observed by _ euve_. our formulae predict flux ratios similar to the observations and to the emission measure analysis predictions of sf2011 , as indicated by comparison with the _ euve _ data for @xmath16 eri . for the 4091.2 nm wavelength range , where there are no reliable stellar observations to compare with the solar fluxes or models , we suggest using the formulae based on the @xcite solar models to predict flux ratios for f7k7 dwarf stars . for m stars , we suggest adding 0.2 dex to the solar ratios , the mean displacement of the m stars from the warmer stars in the 2030 and 3040 nm bands . in the 91.2117 nm band , the agreement between the _ fuse _ data and our model predictions shown in figure 3 suggests that the ratios for m stars may be the same as for the warmer stars . fits to the flux ratios based on the _ fuse _ data and our formulae are in good agreement . on the other hand , the predictions of the emission measure analysis models in the x - exoplanets website are far below our models and the _ fuse _ observations of @xmath16 eri . figures 79 and table 5 show that for the f5k5 stars the mean deviations from the fit lines lie in the range 2048% , and for the nonflaring m stars the mean deviations lie in the range 1224% . thus our formulae should be useful in predicting the euv flux ratios for f5m5 dwarf stars . we note that the flux ratios based on the @xcite solar models closely match the solar data , as expected , but they also come remarkably close to matching the stellar flux ratios . this work is supported by nasa through grants nnx08ac146 , nas5 - 98043 , and hst - go-11687.01-a to the university of colorado at boulder . kf acknowledges support through the nasa nancy grace roman fellowship during a portion of this work . we thank jorge sanz - forcada for calling attention to an error in our reduction of the _ euve _ data , martin snow and tom woods for providing the solar irradiance data , and jurgen schmitt for providing x - ray luminosities for m dwarf stars . we appreciate the availability of _ hst _ data through the mast website hosted by the space telescope science institute , stellar data though the simbad database operated at cds , strasbourg , france , and the x - exoplanets archive at the cab @xcite . finally , we thank the referee for his insightful and very useful comments . _ facilities : _ , . woods , t. & rottman , g. 2002 , in atmospheres in the solar system : comparative aeronomy , geophysical monograph 130 , ed . m. mendilloi , a. nagy , & hunter waite . washington , d.c . : american geophysical union , pp . 221234 to 91.2 nm divided by the ly@xmath0 flux for the @xcite semiempirical models 13x0 to 13x8 ( asterix symbols from left to right ) . the solid line is a least - squares fit to the data points . the sirs flux ratio is represented by the solar symbol . ] flux at 1 au scaled by the ratio of stellar radii , @xmath20 . the solid - line - connected asteriskes ( red line ) are the total flux in this passband for the @xcite semi - empirical models 13x0 to 13x8 ( from left to right ) . flux ratios for five stars based on _ fuse _ spectra and reconstructed lyman series fluxes are shown as @xmath21% error bar symbols . the sun symbol is the ratio for the sirs quiet sun data set . the dash - dot ( blue ) line is the least - squares fit to the stellar and sirs data . the `` m '' and `` m '' symbols are the solar minimum and maximum data obtained with the _ see _ instrument on the _ timed _ spacecraft @xcite . ] nm@xmath10 ) at a distance of 1 au for the quiet sun sirs data set and for the semiempirical solar irradiance models of @xcite . x - symbols are fluxes for spectral regions with no apparent emission lines . box symbols are for the sirs data . solid red lines are least - squares fits to the data . ] the symbols are total fluxes in each wavelength band for @xcite models 13x0 to 13x8 ( left to right ) . the solid lines are least - squares fits to each data set . the sun symbols are for the sirs quiet sun fluxes in these wavelength bands , and the `` m '' and `` m '' symbols are the solar minimum and maximum data obtained with the _ see _ experiment on the _ timed _ spacecraft for the wavelength bands . the 8091.2 nm passband is mainly lyman continuum flux , whereas the 7080 nm passband contains strong transition - region lines of o iii , o iv , and n iv that are likely overestimated in the models . ] flux vs. the reconstructed ly@xmath0 flux at 1 au scaled by the ratio of stellar radii , @xmath20 . the solid line - connected diamonds are the total flux ratios in this passband for the @xcite semiempirical models 13x0 to 13x8 ( from left to right ) . flux ratios for one f star ( cyan ) , four g stars ( black ) , four k stars ( red ) , and five m stars ( plum ) based on _ euve _ spectra are shown as @xmath21 error bar symbols . the dash - dot ( black ) line is the least - squares fit to the solar and f , g , and k star ratios . the plum dash - dot line is the mean of the m star ratios excluding the ev lac flare and au mic flare data . flux ratios for ev lac and au mic during flares ( blue ) are plotted two ways . the upper left symbols are ratios of euv flare fluxes to quiescent ly@xmath0 fluxes . dashed lines extending to the lower right indicate the ratios for increasing ly@xmath0 flux . the symbols at the lower end of the dashed lines are ratios obtained using the most likely values of the ly@xmath0 fluxes during flares ( see text ) . the `` m '' and `` m '' symbols are the solar minimum and maximum data obtained with the see instrument on the _ timed _ spacecraft @xcite . the sun symbol is the ratio for the sirs quiet sun data set . ] ly@xmath0 & 5.95 & & 5.78 & & 11.5 & + 1020 nm & 0.451 & 0.0758 & 0.440 & 0.0761 & 1.35 & 0.118 + 2030 nm & 0.276 & 0.0465 & 0.422 & 0.0730 & 1.64 & 0.143 + 3040 nm & 0.548 & 0.0921 & 0.514 & 0.0889 & 1.33 & 0.115 + 4050 nm & 0.0788 & 0.0132 & 0.0718 & 0.0124 & 0.316 & 0.0144 + 5060 nm & 0.134 & 0.0225 & 0.0977 & 0.0169 & 0.166 & 0.0145 + 6070 nm & 0.112 & 0.0188 & 0.0890 & 0.0154 & 0.141 & 0.0123 + 7080 nm & 0.115 & 0.0193 & 0.0721 & 0.0125 & 0.0995 & 0.00865 + 8091.2 nm & 0.287 & 0.0483 & 0.204 & 0.0354 & 0.426 & 0.0370 + 91.2117 nm & 0.502 & 0.0844 & 0.527 & 0.0911 & 1.060 & 0.0922 + 117130nm - ly@xmath0 & & & 0.538 & 0.0930 & 0.779 & 0.0677 + 130140 nm & & & 0.543 & 0.0939 & 0.811 & 0.0705 + 140150 nm & & & 0.558 & 0.0965 & 0.689 & 0.0599 + 150160 nm & & & 1.367 & 0.237 & 1.634 & 0.142 + 160170 nm & & & 3.174 & 0.549 & 3.631 & 0.316 + 170180 nm & & & 9.831 & 1.701 & 11.23 & 0.977 + lcccccc spectral type & f5 iv - v & g2 v & g2 v & k0 v & k1 v & m0 v + d(pc ) & 3.50 & & 1.325 & 1.255 & 3.216 & 9.91 + age(gyr ) & 1.85 & 4.566 & @xmath34 & @xmath34 & 0.43 & @xmath35 + f(ly@xmath0 ) & 77.1 & 5.95 & 7.54 & 10.1 & 21.5 & 43.0 + f(_fuse _ data without lyman lines ) & 6.46 & & 0.374 & 0.168 & 0.650 & 2.61 + f(lyman series ) & 5.41 & 0.242 & 0.239 & 0.354 & 0.976 & 2.47 + f(91.2117.0 nm ) & 11.87 & 0.507 & 0.613 & 0.522 & 1.626 & 5.08 + f(91.2117.0 nm)/f(ly@xmath0 ) & 0.154 & 0.0852 & 0.0813 & 0.0517 & 0.0756 & 0.118 + procyon ( f5 iv - v ) & 2.03 & procyon__9403122334n & 77.1 & 18.06 & 1.100 & 1.032 & 1.223 & 1.058 & 1.376 & 1.040 + @xmath36 ori ( g0 v ) & 0.98 & chi1_ori__9301261159n & 41.6 & 17.93 & 0.780 & 0.736 & 0.713 & 0.577 & 0.890 & 0.631 + @xmath0 cen ( g2 v+k0 v ) & 1.50 & alpha_cen_9703100800n & 17.64 & 17.61 & 1.052 & 1.028 & 1.100 & 1.042 & 1.316 & 1.203 + @xmath7 cet ( g5 v ) & 0.99 & kappa_cet__9510061036n & 30.0 & 17.89 & 0.645 & 0.608 & 0.738 & 0.622 & 0.985 & 0.739 + @xmath37 boo ( g8 v+k4 v ) & 1.16 & xi_boo__9704200202n & 35.3 & 17.92 & 0.849 & 0.806 & 0.807 & 0.681 & 0.958 & 0.715 + 70 oph ( k0 v+k4 v ) & 1.13 & gj_702__9307021144n & 23.6 & 18.06 & 0.942 & 0.878 & 0.960 & 0.790 & 1.190 & 0.839 + @xmath16 eri ( k1 v ) & 0.78 & eps_eri__9509051851n & 21.5 & 17.88 & 1.104 & 1.063 & 1.024 & 0.909 & 1.270 & 1.046 + ep eri ( k2 v ) & 0.93 & gj_117__9412020500n & 27.6 & 18.05 & 0.976 & 0.916 & 0.853 & 0.678 & 1.051 & 0.713 + cc eri ( k7 v ) & 0.66 & cc_eri__9509130049n & ( 54 ) & ( 18.1 ) & 0.690 & 0.640 & 1.211 & 1.046 & 1.433 & 1.064 + au mic flare ( m0 v ) & 0.61 & au_mic__9207141227n & 43.0 & 18.36 & 0.034 & + 0.050 & 0.564 & 0.238 & 0.894 & 0.233 + au mic ( m0 v ) & 0.61 & mean & 43.0 & 18.36 & 0.663 & 0.571 & 1.068 & 0.691 & 1.362 & 0.708 + yy gem ( dm1e+dm1e ) & 0.88 & yy_gem__9502201531n_1 & ( 50.0 ) & ( 18.0 ) & 0.511 & 0.471 & 0.614 & 0.463 & 0.784 & 0.489 + ev lac flare ( m3.5 v ) & 0.35 & ev_lac__9309091718n & 3.07 & 17.97 & 0.103 & 0.061 & 0.272 & 0.134 & 0.556 & 0.269 + ad leo ( m3.5 v ) & 0.38 & mean & 9.33 & 18.47 & 0.602 & 0.481 & 0.949 & 0.465 & 1.408 & 0.543 + yz cmi ( m4.5 v ) & 0.30 & mean & 6.7 & ( 17.8 ) & 0.421 & 0.395 & 0.584 & 0.482 & 0.848 & 0.662 + prox cen ( m5.5 v ) & 0.15 & proxima_cen__9305211911n & 0.301 & 17.61 & 0.580 & 0.560 & 0.775 & 0.712 & 0.771 & 0.646 + lccccccccc ly@xmath11 & 102.57 & 0.0655 & 0.0422 & 0.0599 & 0.128 & 0.296 & 0.670 & 1.302 & 2.094 + ly@xmath38 & 97.25 & 0.0155 & 0.0191 & 0.0267 & 0.0584 & 0.135 & 0.301 & 0.610 & 1.20 + ly@xmath39 & 94.97 & 0.0081 & 0.0119 & 0.0165 & 0.0369 & 0.0863 & 0.206 & 0.420 & 0.844 + ly@xmath16 & 93.78 & 0.00487 & 0.00765 & 0.0109 & 0.0244 & 0.0563 & 0.137 & 0.279 & 0.546 + ly7 & 93.08 & 0.00323 & 0.00496 & 0.00692 & 0.0160 & 0.0376 & 0.0988 & 0.208 & 0.416 + ly8 & 92.62 & 0.00186 & 0.00302 & 0.00426 & 0.0102 & 0.0251 & 0.0709 & 0.157 & 0.327 + ly9&92.31 & 0.0010 & 0.0019 & 0.0025 & 0.0067 & 0.0175 & 0.0556 & 0.128 & 0.279 + ly10 & 92.10 & & 0.000738 & 0.000781&0.00320 & 0.0101 & 0.0404 & 0.0978 & 0.232 + ly11+rest & 91.94 & & 0.00555 & 0.00849 & 0.0238 & 0.0686 & 0.226 & 0.583 & 1.288 + sum & & 0.114 & 0.0969 & 0.137 & 0.308 & 0.732 & 1.80 & 3.78 & 7.22 + ly@xmath0 & & 5.95 & 3.96 & 5.35 & 8.30 & 15.17 & 33.11 & 59.52 & 94.68 + sum / ly@xmath0 & & 0.0192 & 0.0245 & 0.0256 & 0.0371 & 0.0483 & 0.0545 & 0.0636 & 0.0763 + ly@xmath11/ly@xmath0 & & 0.0110 & 0.0107 & 0.0112 & 0.0155 & 0.0195 & 0.0202 & 0.0219 & 0.0221 + ly@xmath0/ly@xmath11 & & 90.84 & 93.86 & 89.33 & 64.64 & 51.22 & 49.45 & 45.70 & 45.22 + f(91.2117.0 nm ) & & 0.507 & 0.433 & 0.598 & 0.973 & 1.85 & 3.55 & 7.43 & 14.14 + f(91.2117.0)/f(ly@xmath0 ) & & 0.0852 & 0.109 & 0.112 & 0.117 & 0.122 & 0.107 & 0.125 & 0.149 + lycont & & 0.307 & 0.178 & 0.296 & 0.858 & 2.56 & 9.42 & 23.9 & 68.2 + lycont / ly@xmath0 & & 0.0516 & 0.0449 & 0.553 & 0.103 & 0.169 & 0.285 & 0.402 & 0.720 + ly@xmath0/lycont & & 19.4 & 22.2 & 18.1 & 9.67 & 5.93 & 3.51 & 2.49 & 1.39 + t(color ) ( k ) & & 12,210 & 12,640 & 13,230 & 14,360 & 14,930 & 15,160 & 15,480 & 15,390 + 1020 nm ( stars ) & 1.357 + 0.344 log[f(ly@xmath0 ) ] & 0.491 & + 2030 nm ( stars ) & 1.300 + 0.309 log[f(ly@xmath0 ) ] & 0.548 & + 3040 nm ( stars ) & 0.882 & 0.602 & + 4050 nm ( models ) & & & -2.294 + 0.258 log[f(ly@xmath0 ) ] + 5060 nm ( models ) & & & -2.098 + 0.572 log[f(ly@xmath0 ) ] + 6070 nm ( models ) & & & -1.920 + 0.240 log[f(ly@xmath0 ) ] + 7080 nm ( models ) & & & -1.894 + 0.518 log[f(ly@xmath0 ) ] + 8091.2 nm ( models ) & & & -1.811 + 0.764 log[f(ly@xmath0 ) ] + 91.2117 nm ( models ) & & & -1.004 + 0.065 log[f(ly@xmath0 ) ] + lyman series ( models ) & & & -1.798 + 0.351 log[f(ly@xmath0 ) ] + 91.2117 nm ( stars ) & & & 1.025 + 1020 nm mean deviation & 20.5% & 12.6% & + 1020 nm rms deviation & 29.0% & 15.1% & + 2030 nm mean deviation & 47.6% & 24.3% & + 2030 nm rms deviation & 56.6% & 26.9% & + 3040 nm mean deviation & 37.1% & 18.9% & + 3040 nm rms deviation & 41.0% & 20.5% & + 91.2117 nm mean deviation & & & 29.5% + 91.2117 nm rms deviation & & & 35.0% + lccccc & & & & & + model & 1.046 & 1.025 & 1.065 & 0.583 & 0.560 + sanz - forcada & @xmath40 & 1.092 & 0.621 & 2.385 & 1.621 + euve & & 1.063 & & & + & & & & & + model & 1.127 & 1.126 & 1.127 & 0.909 & 0.945 + sanz - forcada & @xmath41 & 0.822 & 0.661 & 1.985 & 1.411 + euve & & 0.909 & & & + & & & & & + model & 1.153 & 1.186 & 1.124 & 1.003 & 1.029 + sanz - forcada & @xmath42 & 0.972 & 0.681 & 1.435 & 1.051 + euve & & 1.046 & & & + & & & & & + model & 1.374 & 1.374 & 1.374 & 1.074 & 1.074 + sanz - forcada & @xmath43 & 1.195 & 0.896 & 1.306 & 0.988 + & & & & & + model & 1.237 & 1.237 & 1.237 & 0.942 & 0.942 + sanz - forcada & @xmath44 & 1.290 & 0.921 & 1.165 & 0.861 + & & & & & + model & 0.475 & 0.522 & 0.474 & 0.166 & 0.167 + sanz - forcada & @xmath45 & 0.342 & 0.041 & 0.775 & 0.411 + & & & & & + model & 0.926 & 0.917 & 0.934 & 0.991 & 1.029 + x - exoplanets & 2.482 & 2.006 & 1.688 & 1.906 & 1.619 + fuse+ly series & & 1.122 & & & +

extreme ultraviolet ( euv ) radiations ( 10117 nm ) from host stars play important roles in the ionization , heating , and mass loss from exoplanet atmospheres . together with the host star s ly and far - uv ( 117170 nm ) radiation , euv radiation photodissociates important molecules , thereby changing the chemistry in exoplanet atmospheres . new non - lte semiempirical models of the solar chromosphere and corona and solar irradiance measurements show that the ratio of euv flux in a variety of wavelength bands to the ly flux varies slowly with the ly flux and thus with the magnetic heating rate . this suggests and we confirm that solar euv / ly flux ratios based on the models and observations are similar to the available 1040 nm flux ratios observed with the _ euve _ satellite and the 91.2117 nm flux observed with the _ fuse _ satellite for f5 v

extreme ultraviolet ( euv ) radiations ( 10117 nm ) from host stars play important roles in the ionization , heating , and mass loss from exoplanet atmospheres . together with the host star s ly and far - uv ( 117170 nm ) radiation , euv radiation photodissociates important molecules , thereby changing the chemistry in exoplanet atmospheres . since stellar euv fluxes can not now be measured and interstellar neutral hydrogen completely obscures stellar radiation between 40 and 91.2 nm , even for the nearest stars , we must estimate the unobservable euv flux by indirect methods . new non - lte semiempirical models of the solar chromosphere and corona and solar irradiance measurements show that the ratio of euv flux in a variety of wavelength bands to the ly flux varies slowly with the ly flux and thus with the magnetic heating rate . this suggests and we confirm that solar euv / ly flux ratios based on the models and observations are similar to the available 1040 nm flux ratios observed with the _ euve _ satellite and the 91.2117 nm flux observed with the _ fuse _ satellite for f5 v m5 v stars . we provide formulae for predicting euv flux ratios based on the _ euve _ and _ fuse _ stellar data and on the solar models , which are essential input for modelling the atmospheres of exoplanets .

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quasi - potential equations ( qpes ) , despite their success as a basis for hadronic phenomenology , can not be improved upon systematically . this is because whenever they are used beyond first - order they predict amplitudes with unphysical singularities . these singularities also appear whenever attempts are made to boost the interaction appearing in the qpe from one frame to another . they arise from these equations use of delta - function constraints on the relative four - momentum . because of this shortcoming of qpes , in section [ sec - section2 ] of this paper we sought , and found , a systematic procedure for deriving three - dimensional equations from four - dimensional ones which does not involve the use of delta functions . as shown in section [ sec - section3 ] , this procedure is akin to the work of klein . it allows the derivation of a three - dimensional kernel which includes , in a systematic expansion which may be pursued to any desired accuracy , the effects of the relative - energy integration present in the four - dimensional integral equation . if the procedure is applied to infinite order , a result equivalent to the original four - dimensional equation is obtained . at any order the kernel derived has a simple interpretation in terms of the diagrams of time - ordered perturbation theory , except that the presence of negative - energy states requires additional rules . the resultant three - dimensional equations therefore include more of the meson - retardation effects than quasi - potential equations , and incorporate the effects of a dynamical boost . since the bse with any kernel which is a finite sum of feynman graphs does not have the correct one - body limit , if the procedure of section [ sec - section2 ] is directly applied to any solvable bse a three - dimensional equation without the correct one - body limit is found . however , as demonstrated by wallace @xcite , and recapitulated here in section [ sec - section4 ] , a bethe - salpeter equation with the correct one - body limit may be reorganized so that the pieces of the kernel which contribute at leading order in the one - body limit take on an iterative form . hence a four - dimensional equation , eq . ( [ eq:3.19 ] ) , with the correct one - body limit and meson - production thresholds is derived . a further approximation leads to a simple four - dimensional equation to which the procedure of section [ sec - section2 ] may be applied , eq . ( [ eq:3.20 ] ) . in making that approximation some of the meson - production thresholds are displaced . nevertheless , the equation still has exactly the correct cut - structure for @xmath257 . in section [ sec - section5 ] the procedure of section [ sec - section2 ] was applied to the four - dimensional equation ( [ eq:3.20 ] ) . the result is a simple three - dimensional equation with the correct one - body limit which has straightforward boost properties . the main shortcoming of this equation is the fact that certain thresholds of the original bethe - salpeter wave function are modified in the three - dimensional wave function . however , this modification is due to the approximation in moving from ( [ eq:3.19 ] ) to ( [ eq:3.20 ] ) , rather than to any deficiencies of our procedure for deriving three - dimensional equations . it could be systematically corrected for , at the price of complicating the three - dimensional kernel . finally , in section [ sec - section6 ] we compared and contrasted the bound - state properties predicted by six different bound - state equations in a scalar field theory . a first group of equations contained the ladder bethe - salpeter equation and two three - dimensional equations based on it : the salpeter equation , and the equation obtained by applying our method to first order which we referred to as the klein equation , since the formula obtained for the three - dimensional interaction appears in klein s work . in the second group were eq . ( [ eq:3.20 ] ) , the so - called equal - time equation and the equation derived in section [ sec - section6 ] . it was found that in each group of three equations the equation derived by the klein - like delta - function - free reduction technique of section [ sec - section2 ] approximated its parent four - dimensional equation better than the corresponding instant equation did . however , the bound - state spectra from the instant et equation lies closest to the bound - state spectrum obtained from the sum of ladder and crossed - ladder graphs , as calculated by nieuwenhuis and tjon @xcite . this can be understood on physical grounds and , we argued , is not necessarily a recommendation for the use of the et equation in physical systems . indeed , when the wave functions of a two - body system in a scalar field theory are examined it is seen that the equation derived in section [ sec - section5 ] does a much better job of reproducing the integrated four - dimensional wave function than the et equation wave function does . the results of section [ sec - section6 ] are in harmony with recent results for phase shifts in scalar - scalar scattering obtained by lahiff and afnan @xcite . they found that the klein method reproduces the phase shifts obtained from the corresponding bethe - salpeter equation much better than a blankenbecler - sugar calculation with an instant interaction . these formal and numerical results indicate that the procedure of section [ sec - section2 ] is successful in its goal of providing a way to systematically obtain three - dimensional equations from four - dimensional ones without the use of delta functions . in particular , when applied to the four - dimensional equation ( [ eq:3.20 ] ) this procedure yields a three - dimensional equation which , unlike the corresponding quasi - potential equation ( [ eq:4.3 ] ) , has a well - defined boost . this would seem to make such an equation a good starting point for calculations in few - hadron systems .

firstly , a systematic procedure is derived for obtaining three - dimensional bound - state equations from four - dimensional ones . unlike `` quasi - potential approaches '' this procedure does not involve the use of delta - function constraints on the relative four - momentum . in the absence of negative - energy states , the kernels of the three - dimensional equations derived by this technique may be represented as sums of time - ordered perturbation theory diagrams . consequently , such equations have two major advantages over quasi - potential equations : they may easily be written down in any lorentz frame , and they include the meson - retardation effects present in the original four - dimensional equation . secondly , a simple four - dimensional equation with the correct one - body limit is obtained by a reorganization of the generalized ladder bethe - salpeter kernel . is then tested by calculating bound - state properties in a scalar field theory using six different bound - state equations . it is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four - dimensional equations significantly better than the corresponding quasi - potential equations do .

firstly , a systematic procedure is derived for obtaining three - dimensional bound - state equations from four - dimensional ones . unlike `` quasi - potential approaches '' this procedure does not involve the use of delta - function constraints on the relative four - momentum . in the absence of negative - energy states , the kernels of the three - dimensional equations derived by this technique may be represented as sums of time - ordered perturbation theory diagrams . consequently , such equations have two major advantages over quasi - potential equations : they may easily be written down in any lorentz frame , and they include the meson - retardation effects present in the original four - dimensional equation . secondly , a simple four - dimensional equation with the correct one - body limit is obtained by a reorganization of the generalized ladder bethe - salpeter kernel . thirdly , our approach to deriving three - dimensional equations is applied to this four - dimensional equation , thus yielding a retarded interaction for use in the three - dimensional bound - state equation of wallace and mandelzweig . the resulting three - dimensional equation has the correct one - body limit and may be systematically improved upon . the quality of the three - dimensional equation , and our general technique for deriving such equations , is then tested by calculating bound - state properties in a scalar field theory using six different bound - state equations . it is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four - dimensional equations significantly better than the corresponding quasi - potential equations do .

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in this article we have studied the novel scenario proposed by refs . @xcite where the vacuum expectation values of susy flat directions generate a large mass for gauge bosons and gauginos . due to the large gauge boson masses thermalization is delayed after the end of inflation . in the non - thermal universe gravitino production in refs . @xcite is suppressed because of the dilute nature of the plasma . the delay in thermalization also leads to a low reheat temperature and so gravitino production after thermalization is also suppressed . thus it is argued that the gravitino problem of susy models is avoided in this scenario . in this article we have carried out a more careful analysis of gravitino production in the non - thermal universe scenario of refs . we have identified the initial state particle distribution functions in the gravitino production process for the eras before kinetic equilibration and after kinetic equilibration in a non - thermal universe ( the appropriate particle distribution function for the pre - kinetic equilibration era was not used in refs . these have then been included in the calculation of the collision integral in the integrated boltzmann equation . we have also explicitly obtained the conditions for obtaining a non - thermal universe after inflaton decay . we have further investigated final state phase space suppression in the context of heavy final state gluons and gluinos in both the non - thermal and thermal eras . we have investigated two cases ( i ) where the flat direction starts oscillating after inflaton decay ( @xmath133 ) and ( ii ) where the flat direction starts oscillating before inflaton decay ( @xmath401 ) . in the first case ( @xmath133 ) we find that there is no gravitino production during the non - thermal phase because of phase space suppression due to the large gluon and gluino mass . gravitino production is suppressed even after thermalization till the flat direction condensate decays at @xmath42 . however the temperature @xmath285 is low and as such any gravitino abundance produced after @xmath42 is small and is within the standard cosmological bounds . in the second case ( @xmath134 ) there is gravitino production before thermalization . the gravitino abundance generated is consistent with cosmological constraints . after thermalization , as in the case @xmath229 , there is no gravitino production before the flat direction condensate decays . after the decay of the condensate the temperature @xmath285 is low and does not lead to conflict with cosmological bounds .

flat directions in generic supersymmetric theories can change the thermal history of the universe . a novel scenario was proposed earlier where the vacuum expectation value of the flat directions induces large masses for all the gauge bosons and gauginos . we find that the total gravitino abundance generated is consistent with cosmological constraints .

flat directions in generic supersymmetric theories can change the thermal history of the universe . a novel scenario was proposed earlier where the vacuum expectation value of the flat directions induces large masses for all the gauge bosons and gauginos . this delays the thermalization of the universe after inflation and solves the gravitino problem . in this article we perform a detailed calculation of the above scenario . we include the appropriate initial state particle distribution functions , consider the conditions for the feasibility of the non - thermal scenario , and investigate phase space suppression of gravitino production in the context of heavy gauge bosons and gauginos in the final state . we find that the total gravitino abundance generated is consistent with cosmological constraints . physics of the early universe , supersymmetry and cosmology

0905.2215

c

we expect not only the drinfeld double @xmath2 but also the pair @xmath5 , with @xmath1 being a @xmath2-module algebra , to play a fundamental role on the quantum group side of the logarithmic kazhdan lusztig duality . based on the general recipe in sec . [ sec : thm ] , the contents of sec . [ sec : sl2 ] must have a counterpart for the quantum group @xmath185 that is kazhdan lusztig - dual to the @xmath186 logarithmic conformal field models @xcite ; hopefully , a `` truncation '' of the appropriate drinfeld double would also allow its dual version for the corresponding heisenberg double , yielding the pair @xmath187 , where @xmath188 is a @xmath185-module algebra . this paper was finished in the very inspiring atmosphere of the lcft meeting at eth , rh.21 zurich ( may 2009 ) , and it is a pleasure to thank m. gaberdiel for the kind hospitality . i am grateful to j. fuchs , a. gainutdinov , v. gurarie , p. mathieu , j. rasmussen , p. ruelle , i. runkel , and c. schweigert for stimulating discussions . special thanks , also for stimulation , go to g. mutafyan . i thank the referee for the useful comments . this work was supported in part by the rfbr grant 07 - 01 - 00523 , the rfbr cnrs grant 09 - 01 - 93105 , and the grant lss-1615.2008.2 .

, we work out the case where is the taft hopf algebra related to the quantum group that is kazhdan lusztig - dual to logarithmic conformal models . the corresponding pair is `` truncated '' to , where is a module algebra that turns out to have the form{\otimes}{\mathbb{c}}[\lambda]/(\lambda^{2p}-1)$ ] , where $ ] is the-module algebra with the relations , , and .

for a hopf algebra , we endow the heisenberg double with the structure of a module algebra over the drinfeld double . based on this property , we propose that is to be the counterpart of the algebra of fields on the quantum - group side of the kazhdan lusztig duality between logarithmic conformal field theories and quantum groups . as an example , we work out the case where is the taft hopf algebra related to the quantum group that is kazhdan lusztig - dual to logarithmic conformal models . the corresponding pair is `` truncated '' to , where is a module algebra that turns out to have the form{\otimes}{\mathbb{c}}[\lambda]/(\lambda^{2p}-1)$ ] , where $ ] is the-module algebra with the relations , , and .

quant-ph0309021

i

in classical mechanics , ensembles , such as the microcanonical and canonical ensembles , are represented by probability distributions on the phase space . in quantum mechanics , ensembles are usually represented by density matrices . it is natural to regard these density matrices as arising from probability distributions on the ( normalized ) wave functions associated with the thermodynamical ensembles , so that members of the ensemble are represented by a random state vector . there are , however , as is well known , many probability distributions which give rise to the same density matrix , and thus to the same predictions for experimental outcomes @xcite .. the measure that gives equal weight to these two states corresponds to the same density matrix as the one giving equal weight to @xmath5 and @xmath6 . however the physical situation corresponding to the former measure , a mixture of two grotesque superpositions , seems dramatically different from the one corresponding to the latter , a routine mixture . it is thus not easy to regard these two measures as physically equivalent . ] moreover , as emphasized by landau and lifshitz @xcite , the energy levels for macroscopic systems are so closely spaced ( exponentially small in the number of particles in the system ) that `` the concept of stationary states [ energy eigenfunctions ] becomes in a certain sense unrealistic '' because of the difficulty of preparing a system with such a sharp energy and keeping it isolated . landau and lifshitz are therefore wary of , and warn against , regarding the density matrix for such a system as arising solely from our lack of knowledge about the wave function of the system . we shall argue , however , that despite these caveats such distributions can be both useful and physically meaningful . in particular we describe here a novel probability distribution , to be associated with any thermal ensemble such as the canonical ensemble . while probability distributions on wave functions are natural objects of study in many contexts , from quantum chaos @xcite to open quantum systems @xcite , our main motivation for considering them is to exploit the analogy between classical and quantum statistical mechanics @xcite . this analogy suggests that some relevant classical reasonings can be transferred to quantum mechanics by formally replacing the classical phase space by the unit sphere @xmath7 of the quantum system s hilbert space @xmath2 . in particular , with a natural measure @xmath8 on @xmath7 one can utilize the notion of typicality , i.e. , consider properties of a system common to `` almost all '' members of an ensemble . this is a notion frequently used in equilibrium statistical mechanics , as in , e.g. , boltzmann s recognition that typical phase points on the energy surface of a macroscopic system are such that the empirical distribution of velocities is approximately maxwellian . once one has such a measure for quantum systems , one could attempt an analysis of the second law of thermodynamics in quantum mechanics along the lines of boltzmann s analysis of the second law in classical mechanics , involving an argument to the effect that the behavior described in the second law ( such as entropy increase ) occurs for typical states of an isolated macroscopic system , i.e. for the overwhelming majority of points on @xmath7 with respect to @xmath9 . probability distributions on wave functions of a composite system , with hilbert space @xmath2 , have in fact been used to establish the typical properties of the reduced density matrix of a subsystem arising from the wave function of the composite . for example , page @xcite considers the uniform distribution on @xmath7 for a finite - dimensional hilbert space @xmath2 , in terms of which he shows that the von neumann entropy of the reduced density matrix is typically nearly maximal under appropriate conditions on the dimensions of the relevant hilbert spaces . given a probability distribution @xmath1 on the unit sphere @xmath7 of the hilbert space @xmath2 there is always an associated density matrix @xmath10 @xcite : it is the density matrix of the mixture , or the statistical ensemble of systems , defined by the distribution @xmath1 , given by @xmath11 for any projection operator @xmath12 , @xmath13 is the probability of obtaining in an experiment a result corresponding to @xmath12 for a system with a @xmath1-distributed wave function . it is evident from that @xmath10 is the second moment , or covariance matrix , of @xmath1 , provided @xmath1 has mean 0 ( which may , and will , be assumed without loss of generality since @xmath14 and @xmath15 are equivalent physically ) . while a probability measure @xmath1 on @xmath7 determines a unique density matrix @xmath0 on @xmath2 via , the converse is not true : the association @xmath16 given by is many - to - one .- dimensional hilbert space the uniform probability distribution @xmath17 over the unit sphere has density matrix @xmath18 with @xmath19 the identity operator on @xmath2 ; at the same time , for every orthonormal basis of @xmath2 the uniform distribution over the basis ( which is a measure concentrated on just @xmath20 points ) has the same density matrix , @xmath21 . an exceptional case is the density matrix corresponding to a pure state , @xmath22 , as the measure @xmath1 with this density matrix is almost unique : it must be concentrated on the ray through @xmath14 , and thus the only non - uniqueness corresponds to the distribution of the phase . ] there is furthermore no unique `` physically correct '' choice of @xmath1 for a given @xmath0 since for any @xmath1 corresponding to @xmath0 one could , in principle , prepare an ensemble of systems with wave functions distributed according to this @xmath1 . however , while @xmath0 itself need not determine a unique probability measure , additional facts about a system , such as that it has come to thermal equilibrium , might . it is thus not unreasonable to ask : which measure on @xmath7 corresponds to a given thermodynamic ensemble ? let us start with the _ microcanonical _ ensemble , corresponding to the energy interval @xmath23 $ ] , where @xmath24 is small on the macroscopic scale but large enough for the interval to contain many eigenvalues . to this there is associated the spectral subspace @xmath25 , the span of the eigenstates @xmath26 of the hamiltonian @xmath27 corresponding to eigenvalues @xmath28 between @xmath29 and @xmath30 . since @xmath25 is finite dimensional , one can form the _ microcanonical density matrix _ @xmath31 with @xmath32}(h)$ ] the projection to @xmath25 . this density matrix is diagonal in the energy representation and gives equal weight to all energy eigenstates in the interval @xmath23 $ ] . but what is the corresponding _ microcanonical measure _ ? the most plausible answer , given long ago by schrdinger @xcite and bloch @xcite , is the ( normalized ) uniform measure @xmath33 on the unit sphere in this subspace . @xmath34 is associated with @xmath35 via . note that a wave function @xmath36 chosen at random from this distribution is almost certainly a nontrivial superposition of the eigenstates @xmath37 with random coefficients @xmath38 that are identically distributed , but not independent . the measure @xmath35 is clearly stationary , i.e. , invariant under the unitary time evolution generated by @xmath27 , and it is as spread out as it could be over the set @xmath39 of allowed wave functions . this measure provides us with a notion of a `` typical wave function '' from @xmath25 which is very different from the one arising from the measure @xmath40 that , when @xmath27 is nondegenerate , gives equal probability @xmath41 to every eigenstate @xmath42 with eigenvalue @xmath43 $ ] . the measure @xmath40 , which is concentrated on these eigenstates , is , however , less robust to small perturbations in @xmath27 than is the smoother measure @xmath35 . our proposal for the canonical ensemble is in the spirit of the uniform microcanonical measure @xmath35 and reduces to it in the appropriate cases . it is based on a mathematically natural family of probability measures @xmath1 on @xmath7 . for every density matrix @xmath0 on @xmath2 , there is a unique member @xmath1 of this family , satisfying for @xmath44 , namely the _ gaussian adjusted projected measure _ @xmath3 , constructed roughly as follows : eq . ( i.e. , the fact that @xmath10 is the covariance of @xmath1 ) suggests that we start by considering the gaussian measure @xmath45 with covariance @xmath0 ( and mean 0 ) , which could , in finitely many dimensions , be expressed by @xmath46 ( where @xmath47 is the obvious lebesgue measure on @xmath2 ) . this is not adequate , however , since the measure that we seek must live on the sphere @xmath7 whereas @xmath45 is spread out over all of @xmath2 . we thus adjust and then project @xmath45 to @xmath7 , in the manner described in section [ sec : defmeasure ] , in order to obtain the measure @xmath3 , having the prescribed covariance @xmath0 as well as other desirable properties . it is our contention that _ a quantum system in thermal equilibrium at inverse temperature @xmath48 should be described by a random state vector whose distribution is given by the measure @xmath49 associated with the density matrix for the canonical ensemble , _ @xmath50 in order to convey the significance of @xmath3 as well as the plausibility of our proposal that @xmath49 describes thermal equilibrium , we recall that a system described by a canonical ensemble is usually regarded as a subsystem of a larger system . it is therefore important to consider the notion of the distribution of the wave function of a subsystem . consider a composite system in a pure state @xmath51 , and ask what might be meant by the wave function of the subsystem with hilbert space @xmath52 . for this we propose the following . let @xmath53 be a ( generalized ) orthonormal basis of @xmath54 ( playing the role , say , of the eigenbasis of the position representation ) . for each choice of @xmath55 , the ( partial ) scalar product @xmath56 , taken in @xmath54 , is a vector belonging to @xmath52 . regarding @xmath55 as random , we are led to consider the random vector @xmath57 given by @xmath58 where @xmath59 is the normalizing factor and @xmath60 is a random element of the basis @xmath53 , chosen with the quantum distribution @xmath61 we refer to @xmath62 as the _ conditional wave function _ @xcite of system 1 . note that @xmath62 becomes doubly random when we start with a random wave function in @xmath63 instead of a fixed one . the distribution of @xmath62 corresponding to ( [ psi1def ] ) and ( [ marg ] ) is given by the probability measure on @xmath64 , @xmath65 where @xmath66 denotes the `` delta '' measure concentrated at @xmath67 . while the density matrix @xmath68 associated with @xmath69 always equals the reduced density matrix @xmath70 of system 1 , given by @xmath71 the measure @xmath72 itself usually depends on the choice of the basis @xmath73 . it turns out , nevertheless , as we point out in section [ sec : typicality ] , that @xmath74 is a universal function of @xmath70 in the special case that system 2 is large and @xmath14 is typical ( with respect to the uniform distribution on all wave functions with the same reduced density matrix ) , namely @xmath75 . thus @xmath3 has a distinguished , universal status among all probability measures on @xmath7 with density matrix @xmath0 . to further support our claim that @xmath49 is the right measure for @xmath76 , we shall regard , as is usually done , the system described by @xmath76 as coupled to a ( very large ) heat bath . the interaction between the heat bath and the system is assumed to be ( in some suitable sense ) negligible . we will argue that if the wave function @xmath14 of the combined `` system plus bath '' has microcanonical distribution @xmath35 , then the distribution of the conditional wave function of the ( small ) system is approximately @xmath77 ; see section [ sec : hb1 ] . indeed , a stronger statement is true . as we argue in section [ sec : hb2 ] , even for a typical _ fixed _ microcanonical wave function @xmath14 of the composite , i.e. , one typical for @xmath35 , the conditional wave function of the system , defined in , is then approximately @xmath49-distributed , for a typical basis @xmath53 . this is related to the fact that for a typical microcanonical wave function @xmath14 of the composite the reduced density matrix for the system is approximately @xmath76 @xcite . note that the analogous statement in classical mechanics would be wrong : for a fixed phase point @xmath78 of the composite , be it typical or atypical , the phase point of the system could never be random , but rather would merely be the part of @xmath78 belonging to the system . the remainder of this paper is organized as follows . in section [ sec : defmeasure ] we define the measure @xmath3 and obtain several ways of writing it . in section [ sec : prop ] we describe some natural mathematical properties of these measures , and suggest that these properties uniquely characterize the measures . in section [ sec : hb1 ] we argue that @xmath49 represents the canonical ensemble . in section [ sec : typicality ] we outline the proof that @xmath3 is the distribution of the conditional wave function for _ most _ wave functions in @xmath63 with reduced density matrix @xmath0 if system 2 is large , and show that @xmath49 is the typical distribution of the conditional wave function arising from a fixed microcanonical wave function of a system in contact with a heat bath . in section [ sec : rem ] we discuss other measures that have been or might be considered as the thermal equilibrium distribution of the wave function . finally , in section [ sec : two ] we compute explicitly the distribution of the coefficients of a @xmath49-distributed state vector in the simplest possible example , the two - level system .

for a quantum system , a density matrix that is not pure can arise , via averaging , from a distribution of its wave function , a normalized vector belonging to its hilbert space . while itself does not determine a unique , additional facts , such as that the system has come to thermal equilibrium , might . it is thus not unreasonable to ask , which , if any , corresponds to a given thermodynamic ensemble ? to answer this question we establish some nice properties of and show that this measure arises naturally when considering macroscopic systems . in particular , we argue that it is the most appropriate choice for systems in thermal equilibrium , described by the canonical ensemble density matrix . may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems , where distributions on are often used . key words : canonical ensemble in quantum theory ; probability measures on hilbert space ; gaussian measures ; density matrices .

for a quantum system , a density matrix that is not pure can arise , via averaging , from a distribution of its wave function , a normalized vector belonging to its hilbert space . while itself does not determine a unique , additional facts , such as that the system has come to thermal equilibrium , might . it is thus not unreasonable to ask , which , if any , corresponds to a given thermodynamic ensemble ? to answer this question we construct , for any given density matrix , a natural measure on the unit sphere in , denoted . we do this using a suitable projection of the gaussian measure on with covariance . we establish some nice properties of and show that this measure arises naturally when considering macroscopic systems . in particular , we argue that it is the most appropriate choice for systems in thermal equilibrium , described by the canonical ensemble density matrix . may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems , where distributions on are often used . key words : canonical ensemble in quantum theory ; probability measures on hilbert space ; gaussian measures ; density matrices .

1007.3662

i

for an r.v . @xmath0 with density @xmath2 , mean @xmath3 and finite variance @xmath4 , goldstein and reinert @xcite showed the identity ( see also @xcite ) @xmath5,\ ] ] which holds for any absolutely continuous function @xmath6 with a.s . derivative @xmath7 such that the right - hand side is finite . in ( [ eq1.1 ] ) , @xmath8 is defined to be the r.v . with density @xmath9 , @xmath10 . identity ( [ eq1.1 ] ) extends the well - known stein identity for the standard normal @xcite ; a discrete version of ( [ eq1.1 ] ) can be found in , for example , @xcite , where the derivative has been replaced by the forward difference of @xmath11 . in particular , identities of the form ( [ eq1.1 ] ) have many applications to variance bounds and characterizations @xcite , and to approximation procedures @xcite . several extensions and applications can be found in @xcite . in @xcite , the ( continuous ) pearson family is parametrized by the fact that there exists a quadratic @xmath12 such that @xmath13 typically , the usual definition of a pearson r.v . is related to the differential equation @xmath14 , with @xmath15 being a polynomial of degree at most 2 . in fact , the set - up of ( [ eq1.2 ] ) ( including , e.g. , the standard uniform distribution with @xmath16 ) will be the framework of the present work and will hereafter be called `` the pearson family of continuous distributions '' . it is easily seen that under ( [ eq1.2 ] ) , the support of @xmath0 , @xmath17 , must be an interval , say @xmath18 with @xmath19 , and @xmath20 remains strictly positive for @xmath21 . clearly , under ( [ eq1.2 ] ) , the covariance identity ( [ eq1.1 ] ) can be rewritten as @xmath22={\mathbb{e}}[q(x)g'(x)].\ ] ] it is known that , under appropriate moment conditions , the functions @xmath23,\qquad x\in(r , s),\ k=0,1,\ldots , m\ ] ] ( where @xmath24 can be finite or infinite ) are orthogonal polynomials with respect to the density @xmath2 so that the quadratic @xmath20 in ( [ eq1.2 ] ) generates a sequence of orthogonal polynomials by the rodrigues - type formula ( [ eq1.4 ] ) . in fact , this approach is related to the sturm liouville theory , @xcite , section 5.2 ; see also @xcite . in the present paper , we provide an extended stein - type identity of order @xmath1 for the pearson family . this identity takes the form @xmath25= { \mathbb{e}}\bigl [ q^k(x)g^{(k)}(x)\bigr],\ ] ] where @xmath26 is the @xmath1th derivative of @xmath11 ( since @xmath27 , ( [ eq1.5 ] ) for @xmath28 reduces to ( [ eq1.3 ] ) ) . identity ( [ eq1.5 ] ) provides a convenient formula for the @xmath1th fourier coefficient of @xmath11 , corresponding to the orthogonal polynomial @xmath29 in ( [ eq1.4 ] ) . for its proof , we make use of a novel `` rodrigues inversion '' formula that may be of some interest in itself . an identity similar to ( [ eq1.5 ] ) holds for the discrete pearson ( ord ) family . application of ( [ eq1.5 ] ) and its discrete analog yields the corresponding lower variance bounds , obtained in section [ sec4 ] . the lower bound for the @xmath30 distribution , namely , @xmath31\ ] ] ( cf . @xcite ) and the corresponding one for the normal@xmath32 distribution @xcite , @xmath33,\ ] ] are particular examples ( examples [ ex4.1 ] and [ ex4.5 ] ) of theorems [ th4.1 ] and [ th4.2 ] , respectively . both ( [ eq1.6 ] ) and ( [ eq1.7 ] ) are particular cases of the finite form of bessel s inequality and , under completeness , they can be extended to the corresponding parseval identity . in section [ sec5 ] , we show that this can be done for a fairly large family of r.v.s , including , of course , the normal , the poisson and , in general , all the r.v.s of the pearson system which have finite moments of any order . for instance , when @xmath0 is @xmath34 , inequality ( [ eq1.7 ] ) ( and identity ( [ eq1.5 ] ) ) can be strengthened to the covariance identity @xmath35= \sum_{k=1}^{\infty } \frac{(\sigma^2)^{k}}{k ! } { \mathbb{e}}\bigl[g_1^{(k)}(x)\bigr ] { \mathbb{e}}\bigl[g_2^{(k)}(x)\bigr],\ ] ] provided that for @xmath36 , @xmath37 , @xmath38 , @xmath39 and that @xmath40 ^ 2<\infty$ ] . similar identities hold for poisson , negative binomial , beta and gamma distributions . these kinds of variance / covariance expressions may sometimes be useful in inference problems see , e.g. , the applications [ ap5.1 ] and [ ap5.2 ] at the end of the paper .

for an absolutely continuous ( integer - valued ) r.v . of the pearson ( ord ) family , we show that , under natural moment conditions , a stein - type covariance identity of order holds ( cf . [ goldstein and reinert , _ j. theoret . this identity is closely related to the corresponding sequence of orthogonal polynomials , obtained by a rodrigues - type formula , and provides convenient expressions for the fourier coefficients of an arbitrary function . application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v . , expressions that seem to be known only in particular cases ( for the normal , see [ houdr and kagan , _ j. theoret .

for an absolutely continuous ( integer - valued ) r.v . of the pearson ( ord ) family , we show that , under natural moment conditions , a stein - type covariance identity of order holds ( cf . [ goldstein and reinert , _ j. theoret . probab . _ * 18 * ( 2005 ) 237260 ] ) . this identity is closely related to the corresponding sequence of orthogonal polynomials , obtained by a rodrigues - type formula , and provides convenient expressions for the fourier coefficients of an arbitrary function . application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v . , expressions that seem to be known only in particular cases ( for the normal , see [ houdr and kagan , _ j. theoret . probab . _ * 8 * ( 1995 ) 2330 ] ; see also [ houdr and prez - abreu , _ ann . probab . _ * 23 * ( 1995 ) 400419 ] for corresponding results related to the wiener and poisson processes ) . some applications are also given .

1604.00019

c

in order to quantitatively interpret the radio light curves and spectra of isne , one needs to understand the evolution of the shocked gas . one popular model for shock evolution is the ( * ? ? ? * c82 ) one - dimensional self - similar interaction model , which assumes a continuous csm . here we explored moving beyond these models to the case of csm shells ( e.g. due to novae ) in the low - density case where the shocked csm does not cool and is optically thin . this removes the self - similar model assumption that the interaction has had time to reach self - similarity ( i.e. that the csm is close to the progenitor star or is very extended ) . we created a suite of models to explore the parameter space of shell properties , focusing on shell density , thickness , and time of impact . in our `` fiducial '' models , we enforce an initial csm - to - ejecta density ratio of @xmath178 at the contact discontinuity ( motivated by c82 ; equation [ eqn : rcss ] ) . if the shell were infinite , these models would evolve to the c82 self - similar solution ( [ sec : mods ] ) ; however , because of the finite extent of the shells , self - similarity is never reached in our models . we calculated synchrotron radio light curves for these models , since this emission traces the evolution of the shocked region only . we found a similar behavior among the radio light curves of fiducial models they peak at a predictable time ( equation [ eqn : plaw_peak ] ) , have the same shape ( equation [ eqn : asyml ] ) , and peak at a luminosity that can be calculated from shell characteristics ( equation [ eqn : empirl ] ) . therefore these light curves can be parameterized in a way that allows csm properties to be inferred readily from an observed light curve ( [ sec : shell_lcs ] ) . in fact we find that shell properties can be approximated even if there is a higher initial density ratio ( i.e. faster , lower density ejecta colliding with the shell ) than is assumed in the fiducial set ( [ sec : high_vel_ej ] ) . we then showed how one can use the fiducial models to better interpret radio non - detections of sne ia by applying our work to the radio observations of sn 2011fe and sn 2014j ( [ sec : obscomp ] ) . we presented a two - shell model to illustrate that our single shell light curves do not apply in that case , as a double shell light curve is much longer lived because of an effect that we call `` shock sloshing '' that occurs in the first shell ( [ sec : multishell ] ) . shock sloshing produces elbows in the light curve that may be an observable signature of this behavior ; exploring the diversity in multiple shell collisions is an enticing direction for future work . our fiducial model set is a new tool for studying the interaction of sne ia with a csm , with the goal of constraining the supernova progenitor system . with the fiducial set , one can easily make observational predictions and analyze radio observations . future work will focus on calculations of the radiation transport in the optical and x - ray , to show how data at those wavelengths can reinforce radio constraints on supernovae interaction .

however , while such models have been successfully applied to core - collapse sne , the assumption of continuity may not be accurate for sne ia , as shells of csm could be formed by pre - supernova eruptions ( novae ) . in this work , we model the interaction of sne with a spherical , low density , finite - extent csm and create a suite of synthetic radio synchrotron light curves . the relations obeyed by the fiducial models can be used to deduce csm properties from radio observations ; we demonstrate this by applying them to the non - detections of sn 2011fe and sn 2014j .

for decades , a wide variety of observations spanning the radio through optical and on to the x - ray have attempted to uncover signs of type ia supernovae ( sne ia ) interacting with a circumstellar medium ( csm ) . the goal of these studies is to constrain the nature of the hypothesized sn ia mass - donor companion . a continuous csm is typically assumed when interpreting observations of interaction . however , while such models have been successfully applied to core - collapse sne , the assumption of continuity may not be accurate for sne ia , as shells of csm could be formed by pre - supernova eruptions ( novae ) . in this work , we model the interaction of sne with a spherical , low density , finite - extent csm and create a suite of synthetic radio synchrotron light curves . we find that csm shells produce sharply peaked light curves , and identify a fiducial set of models that all obey a common evolution and can be used to generate radio light curves for interaction with an arbitrary shell . the relations obeyed by the fiducial models can be used to deduce csm properties from radio observations ; we demonstrate this by applying them to the non - detections of sn 2011fe and sn 2014j . finally , we explore a multiple shell csm configuration and describe its more complicated dynamics and resultant radio light curves .

1611.08503

i

many mathematical models of physical phenomena deal with systems of volterra integral equations with singular kernels ( e.g. @xcite ) . in this paper , motivated by some nonlinear volterra integral equations arising in quantum mechanics , we investigate the properties of convolution operators with kernels possibly more singular than the more known abel ones . namely , given a generic positive , locally integrable function @xmath0 , we study the action of the operator @xmath7 defined by @xmath8 on intervals @xmath9 $ ] , with @xmath10 ( this assumption being understood in the whole paper ) . precisely , we prove its regularizing effect in hlder and lebesgue spaces and its `` contractive '' effect in sobolev spaces ( where with `` contractive '' we mean that the sobolev norm of @xmath11 on @xmath9 $ ] can be estimated by the norm of @xmath12 times a constant that gets smaller as @xmath13 ) . it is also worth highlighting that the assumption of local integrability of @xmath0 is the minimum requirement so that definition make sense in general . in fact , the aim of the paper ( even though some results will require additional hypothesis ) is to work with the least set of assumptions that are necessary in order to detect remarkable effects from the application of the operator @xmath3 . a particular relevance in applications is acquired by the case @xmath14 ( figure [ fig : nu ] ) , where the operator @xmath3 reads @xmath15 we observe that , if we denote by @xmath16 the _ volterra functions _ defined by @xmath17 then @xmath18 coincides with @xmath19 , which is the so - called volterra function _ of order _ @xmath20 ( see @xcite , section 18.3 ) , that is discussed in section [ sec - volt ] . in addition , recalling that a kernel @xmath21 is said a _ sonine kernel _ if it is a divisor of the unit with respect to the convolution operation , that is , if there exists another kernel @xmath22 such that @xmath23,\ ] ] then , one can prove that @xmath24 is a sonine kernel , with @xmath25 , @xmath26 representing the euler - mascheroni constant ( see eq . ) . the class of sonine kernels is wide and there are many papers ( see e.g. @xcite and references therein ) , starting with the pioneering one by sonine ( @xcite ) , where embedding theorems for integral operators with kernels displaying singularities at the origin of the type @xmath27 are discussed . however , we stress that the results proved in the present paper are more general since they take into account also kernels that are more singular in a neighborhood of the origin , such as , indeed , the volterra function @xmath24 , whose asymptotyc expansion near @xmath28 is given by @xmath29 ( see ) . it is also worth mentioning that a first discussion on the operator @xmath30 is present in @xcite , whereas similar integral operators , but with more regular kernels , have been investigated more recently by @xcite . more in detail , in @xcite it is analyzed the case of a certain class of _ almost decreasing _ sonine kernels in terms of _ weighted generalized _ hlder spaces , while in @xcite an `` inverse '' operator is discussed within the framework of @xmath31 spaces . we also recall that in @xcite some relevant features of volterra functions are pointed out , such as asymptotic expansions and some striking relations with the ramanujan integrals . the interest of the operator @xmath30 is mainly due to its applications in quantum mechanics , and precisely in the study of the schrdinger equation with nonlinear point interactions in @xmath32 . we recall briefly that a schrdinger equation with a _ linear _ point interaction with strength @xmath33 , placed at @xmath34 , is @xmath35 where @xmath36 is a differential operator with domain @xmath37 \displaystyle \hspace{7 cm } \lim_{{\mathbf{x}}\to { \mathbf{y } } } \phi_\lambda({\mathbf{x } } ) = \left(\alpha+\tfrac{1}{2\pi } \log \tfrac{\sqrt{\lambda}}{2 } + \tfrac{\gamma}{2\pi}\right)q\bigg\ } \end{array}\ ] ] ( @xmath38 denoting the green s function of @xmath39 in @xmath32 ) and action @xmath40 for a complete discussion on the solution of this equation through the theory of self - adjoint extension , we refer the reader to @xcite . in addition , it is well known that , given an initial datum @xmath41 , the solution of the associated cauchy problem reads @xmath42 where @xmath43 is the the _ propagator _ of the free schrdinger equation in @xmath32 ( with integral kernel @xmath44 ) and @xmath45 ( with a little abuse of notation that is usual in the literature ) is a complex - valued function satisfying the so - called _ charge _ equation @xmath46 now , a _ nonlinear _ point interaction arises when one assumes that the strength of the interaction depends in fact on the function @xmath45 , and in particular when one sets @xmath47 ( @xmath48 , @xmath49 ) in , thus obtaining @xmath50 ( see @xcite ) . since in the nonlinear case no theory of self - adjoint extensions is available , the relevance of the operator @xmath30 is clear : the well - posedness of the associated cauchy problem is strictly related to the study of the existence and uniqueness of solutions of , which strongly depends on the properties of @xmath30 . even though the application presented above concerns complex - valued functions , this papers only manages real - valued functions . however , one can check that the results of section [ sob ! ] ( which are actually required in @xcite ) can be easily generalized to complex - valued functions . another topical example of integral kernels that are included in our general framework are the well known _ abel kernels _ , which correspond to the choice @xmath51 in . these ones are very important in the theory of _ fractional integration _ and _ generalized differentiation _ ( @xcite ) and , again , in quantum mechanics . in the study of nonlinear point interactions in @xmath52 and @xmath53 , indeed , the resulting integral equations present the kernel , with @xmath54 , in place of @xmath24 ( see @xcite ) . finally , we describe briefly the main results of the paper . they concern , as we told at the beginning , the properties of the operator @xmath3 in hlder spaces , @xmath31 spaces and sobolev spaces . preliminarily , since it is crucial in the following , we define the integral function of the kernel @xmath0 @xmath55 since @xmath0 is always supposed positive and locally integrable , it turns out that @xmath4 is a positive , increasing and absolutely continuous function with @xmath56 as @xmath57 . in the case of hlder spaces , it is well known ( @xcite , theorem 4.2.1 p. 70 ) that when the kernel is @xmath58 , @xmath59 , the operator @xmath3 transforms @xmath60 functions into @xmath61 ones , improving this way the regularity of the modulus of continuity . as a consequence of our main result of section [ sectionhold ] ( theorem [ casoholder ] ) , we will see that more generally the improvement is at least given by the integral function of the kernel : the phenomenon that the power @xmath62 gives as improvement the exponent @xmath63 is therefore true also for any locally integrable kernel which is assumed just equivalent to a decreasing function in a neighborhood of the origin and not blowing too much ( derivative bounded above , for instance ) in its domain . in the case of @xmath31 spaces , it is well known ( @xcite , theorem 4.1.4 p. 67 ) that when the kernel is @xmath58 , the operator @xmath3 transforms @xmath64 functions , @xmath65 , into @xmath66 functions . to a minor integrability of the kernel corresponds a minor gain of integrability for @xmath67 , and apparently the gain disappears when the kernel is just @xmath68 . as a consequence of our main result of section [ sectionellepi ] ( theorem [ casolp ] ) , we will show that _ any kernel locally integrable ( again , we assume that it is equivalent to a decreasing function in a neighborhood of the origin ) gives an improvement of integrability , measured in terms of orlicz spaces . the improvement is strictly linked to the orlicz integrability of the kernel , hence it always exists : a classical , remarkable theorem in orlicz spaces theory ( see e.g. @xcite , p. 60 ) tells that any function @xmath68 is always in some orlicz space strictly contained in @xmath68 . furthermore , in the case @xmath69 , we show ( proposition [ pro : infinity ] ) , under the unique assumption of local integrability , that @xmath3 transforms @xmath70 functions in continuous functions and that the @xmath70 norm of @xmath11 on @xmath9 $ ] is controlled by the norm of @xmath12 times @xmath71 . _ finally , in the case of sobolev spaces , it is well known ( @xcite , theorem 4.2.2 p. 73 ) that when the kernel is @xmath58 , the operator @xmath3 transforms @xmath72 functions , with @xmath73 , in @xmath74 functions . analogous results for @xmath75 functions are discussed in @xcite . in this case , the minor integrability of the kernel yields a minor gain in the sobolev index , which disappears when the kernel is just @xmath68 ( also the preservation of the index is not straightforward ) . as a consequence of our main result of section [ sob ! ] ( theorem [ contr_lemma ] ) , we will show that when @xmath76 , provided @xmath77 , the sobolev index is in fact preserved and the sobolev norm of @xmath11 is bounded , up to a multiplicative constant , by the norm of @xmath12 times @xmath71 . furthermore , we will prove that ( almost ) the same result holds for @xmath78 functions , but just in the case @xmath79 ( theorem [ contr_lemma_2 ] ) , and for @xmath80 functions ( theorem [ contr_lemma_3 ] ) .

for kernels which are positive and integrable we show that the operator on a finite time interval enjoys a regularizing effect when applied to hlder continuous and lebesgue functions and a `` contractive '' effect when applied to sobolev functions . for hlder continuous functions , we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel , namely by the factor . for functions in lebesgue spaces , keywords : _ volterra functions , singular kernels , volterra integral equations , sonine kernels , orlicz integrability . _ msc 2010 : _ 26a33 , 47g10 , 45e99 , 44a99 , 46e30 .

for kernels which are positive and integrable we show that the operator on a finite time interval enjoys a regularizing effect when applied to hlder continuous and lebesgue functions and a `` contractive '' effect when applied to sobolev functions . for hlder continuous functions , we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel , namely by the factor . for functions in lebesgue spaces , we prove that an improvement always exists , and it can be expressed in terms of orlicz integrability . finally , for functions in sobolev spaces , we show that the operator `` shrinks '' the norm of the argument by a factor that , as in the hlder case , depends on the function ( whereas no regularization result can be obtained ) . these results can be applied , for instance , to abel kernels and to the volterra function , the latter being relevant for instance in the analysis of the schrdinger equation with concentrated nonlinearities in . _ keywords : _ volterra functions , singular kernels , volterra integral equations , sonine kernels , orlicz integrability . _ msc 2010 : _ 26a33 , 47g10 , 45e99 , 44a99 , 46e30 .

1503.03936

r

in this section , we report numerical experiments to illustrate the regularizing effects of minres and mr - ii and make a number of comparisons . we justify our theory : ( i ) minres has only the partial regularization , independent of the degree of ill - posedness , and a hybrid minres is generally needed ; ( ii ) the @xmath0th minres iterate @xmath102 is always more accurate than the @xmath1th mr - ii iterate @xmath123 until the semi - convergence of minres ; ( iii ) mr - ii has only the partial regularization for mildly ill - posed problems , and a hybrid mr - ii is needed . in the meantime , experimentally , we demonstrate a remarkable and attractive feature , stronger than our theory predicts : mr - ii has the full regularization for severely and moderately ill - posed problems and its iterates at semi - convergence is as accurate as the best tsvd solutions for these two kinds of problems . we will use the function @xmath266 in @xcite to depict the l - curves . in order to simulate exact arithmetic , the lanczos process with reorthogonalization is used in minres and mr - ii . table [ table ] lists test problems and their degree of ill - posedness , all of which are symmetric and arise from the discretization of the first kind fredholm integral equations ; see hansen s regularization toolbox @xcite for details . for each problem except the 2d image deblurring problem ` blur ' , we use the corresponding code in @xcite to generate a @xmath267 matrix @xmath3 , the true solution @xmath28 and noise - free right - hand @xmath25 . in order to simulate the noisy data , we generate the white noise vector @xmath9 whose entries are normally distributed with mean zero , such that the relative noise level @xmath268 , respectively . to simulate exact arithmetic , the full reorthogonalization is used during the lanczos process . we remind that , as far as ill - posed problem is concerned , our primary goal consists in justifying the regularizing effects of iterative solvers , which are _ unaffected by sizes _ of ill - posed problems and only depends on the degree of ill - posedness . therefore , for this purpose , as extensively done in the literature ( see , e.g. , @xcite and the references therein ) , it is enough to test not very large problems . indeed , for @xmath89 large , say , 1,0000 , we have observed completely the same behavior as that for @xmath89 not large , e.g. , @xmath269 used in this paper except for the problem ` blur ' with @xmath270 . a reason for using @xmath89 not large is because such choice makes it practical to fully justify the regularization effects of lsqr by comparing it with the tsvd method , which suits only for small and/or medium sized problems for computational efficiency . all the computations are carried out using matlab 7.8 with the machine precision @xmath271 under the microsoft windows 7 64-bit system . 12.1cm@lll problem & description & ill - posedness + shaw & one - dimensional image restoration model & severe + foxgood & severely ill - posed test problem & severe + gravity & one - dimensional gravity surveying problem & severe + phillips & phillips `` famous '' test problem & moderate + deriv2 & computation of second derivative & mild + blur & 2d image deblurring test problem & mild / moderate + [ table ] we now compare minres and mr - ii and justify our theory : ( i ) the mr - ii iterate is always more accurate than the minres iterate at their respective semi - convergence , meaning that minres can not obtain best possible regularized solutions and has only the partial regularization , independent of the degree of ill - posedness ; ( ii ) the minres iterates @xmath102 are always more accurate that the mr - ii iterates @xmath123 until the semi - convergence of minres . in this subsection , we only report the results for the noise level @xmath272 . results for the other two @xmath273 are analogous and thus omitted . by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for shaw ( left ) and foxgood ( right).,width=264,height=188 ] ( a ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for shaw ( left ) and foxgood ( right).,width=264,height=188 ] ( b ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for shaw ( left ) and foxgood ( right).,width=264,height=188 ] ( c ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for shaw ( left ) and foxgood ( right).,width=264,height=188 ] ( d ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for gravity ( left ) and phillips ( right ) . , width=264,height=188 ] ( a ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for gravity ( left ) and phillips ( right ) . , width=264,height=188 ] ( b ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for gravity ( left ) and phillips ( right ) . , width=264,height=188 ] ( c ) by minres and mr - ii ; ( c)-(d ) : plots of the singular values ( circles for minres , stars for mr - ii ) of the projected matrices and the ones ( solid lines ) of @xmath3 for gravity ( left ) and phillips ( right ) . , width=264,height=188 ] ( d ) figures [ fig1 ] and [ fig2 ] display numerous curves for severely and moderately ill - posed problems . clearly , all the mr - ii ierates are always more accurate than the minres iterates at their respective semi - convergence . this indicates that minres has only the partial regularization . as elaborated previously , this is because that a small singular value of the projected matrix @xmath97 appears before a regularized solution becomes best , causing that its error does not reach the same error level as that obtained by mr - ii . for instance , we see from figure [ fig1 ] ( a ) and ( c ) that all the singular values of @xmath87 in mr - ii are excellent approximations to the @xmath0 large singular values of @xmath3 in natural order for @xmath274 . we see that the semi - convergence of mr - ii occurs at iteration @xmath275 . by the comments in the end of section [ sectionrev ] and the explanations after , this clearly justifies the full regularization of mr - ii , and the best possible regularized solution by mr - ii includes _ seven _ dominant spectral or svd components . on the other hand , it is clearly seen from figure [ fig1 ] ( c ) that the smallest singular value of @xmath276 in minres is smaller than @xmath277 , making the relative error starts to increase dramatically at iteration 5 and minres have only the partial regularization . similar phenomena are observed for foxgood , and mr - ii has the full regularization with @xmath278 , as indicated by figure [ fig1 ] ( b ) and ( d ) , where the smallest singular value of @xmath279 lies between @xmath280 and @xmath281 and the best iterate @xmath102 by minres at semi - convergence is considerably less accurate than the best iterate @xmath129 by mr - ii at semi - convergence , meaning that minres has only the partial regularization . we have analogous findings for gravity and phillips , as shown by figure [ fig2 ] ( b ) and ( d ) , which again demonstrate that mr - ii has the full regularization but minres has only the partial regularization . as for the mildly ill - posed problem deriv2 , we also see from figure [ fig3 ] ( a ) that the relative error obtained by mr - ii clearly reaches the lower minimum level than that by minres , indicating that mr - ii has better regularizing effects than minres . the above experiments have illustrated that mr - ii always obtains more accurate regularized solutions than minres does for the test severely , moderately and mildly problems . this justifies our theory that minres only has the partial regularization , independent of the degree of ill - posedness . therefore , one must use a hybrid minres with some regularization method applied to the projected problems in order to remove the effects of small singular values of @xmath97 and improve the accuracy of regularized solutions until a best regularized solution is found . it is clear from figures [ fig1][fig2 ] and figure [ fig3 ] ( a ) that , for each test problem , the minres iterates @xmath102 are more accurate than the corresponding mr - ii iterates @xmath123 until the semi - convergence of minres . afterwards , the regularized solutions @xmath102 are deteriorated more and more seriously . this confirms our theory in section [ sectioncom ] , i.e. , assertion ( ii ) in the beginning of this subsection . we first test mr - ii , minres and their hybrid variants for the mildly ill - posed problem deriv2 , and justify our theory that mr - ii has only the partial regularization and one must use its hybrid variant to compute a best possible regularized solution . by the pure minres and mr - ii as well as the hybrid minres and mr - ii ; ( b ) : the l - curves of minres and mr - ii for deriv2.,width=264,height=188 ] ( a ) by the pure minres and mr - ii as well as the hybrid minres and mr - ii ; ( b ) : the l - curves of minres and mr - ii for deriv2.,width=264,height=188 ] ( b ) for deriv2 , figure [ fig3 ] ( a ) shows that the relative errors of regularized solutions obtained by the hybrid minres and mr - ii with the tsvd regularization method applied to the projected problems reach a considerably smaller minimum level than those by minres and mr - ii themselves . for this problem , before minres or mr - ii captures all the dominant spectral components needed , a small singular value of @xmath97 or @xmath87 appears and starts to deteriorate the regularized solutions . in contrast , their hybrid variants expand krylov subspaces until enough dominant spectral components are captured and the tsvd regularization method effectively dampens the svd components corresponding to small singular values of the projected matrices @xmath97 by minres and @xmath87 by mr - ii . for example , we see from figure [ fig3 ] ( a ) that the semi - convergence of mr - ii occurs at iteration @xmath282 , which is also observed by the corner of the l - curve depicted by figure [ fig3 ] ( b ) . however , as shown by figure [ fig3 ] ( a ) , such regularization of mr - ii is not enough , and the hybrid mr - ii uses a larger six dimensional krylov subspace @xmath283 to improve the solutions and get a best possible regularized solution , whose residual norm is smaller than that obtained by the pure mr - ii . after @xmath284 , the regularized solutions almost stabilize with the minimum error as @xmath0 increases . we observe similar phenomena for minres and its hybrid variant , where we find that the relative error by the hybrid minres reaches the same minimum level as that by the hybrid mr - ii . next we test mr - ii , minres and their hybrid variants for severely and moderately ill - posed problems . we attempt to get more insight into the regularizing effects of mr - ii . as a matter of fact , we have already justified the full regularization of mr - ii for the four test problems in section [ seccom ] . in what follows , we will give more details and justifications on the full regularization of mr - ii . we show that ( i ) the relative error obtained by the hybrid minres reaches the same minimum level as that by the hybrid mr - ii ; ( ii ) mr - ii has the full regularization effects : at semi - convergence , the regularized solution by the pure mr - ii is as accurate as that by the hybrid mr - ii with the tsvd regularization used within projected problems ; ( iii ) mr - ii generates near best rank @xmath0 approximations @xmath96 to @xmath3 , i.e. , the relation @xmath285 holds with different noise levels . keep in mind and . this means that @xmath96 generated by mr - ii plays the same role as @xmath99 , the best rank @xmath0 approximation to @xmath3 , so that mr - ii has the full regularization . by mr - ii , and hybrid mr - ii and minres with additional tsvd regularization for shaw , foxgood , gravity , phillips ( from top left to bottom right ) . , width=264,height=188 ] ( a ) by mr - ii , and hybrid mr - ii and minres with additional tsvd regularization for shaw , foxgood , gravity , phillips ( from top left to bottom right ) . , width=264,height=188 ] ( b ) by mr - ii , and hybrid mr - ii and minres with additional tsvd regularization for shaw , foxgood , gravity , phillips ( from top left to bottom right ) . , width=264,height=188 ] ( c ) by mr - ii , and hybrid mr - ii and minres with additional tsvd regularization for shaw , foxgood , gravity , phillips ( from top left to bottom right ) . , width=264,height=188 ] ( d ) for mr - ii and the hybrid mr - ii , we observe from figure [ fig4 ] that mr - ii reaches the same error level as the hybrid mr - ii , and the tsvd regularization applied to projected problems simply makes the regularized solutions with the minimum error almost stabilize and does not improve the regularized solution by mr - ii at semi - convergence . this justifies the full regularization of mr - ii . compared with figures [ fig1][fig2 ] , we find from figure [ fig4 ] that the hybrid minres improves on minres substantially and the relative errors of iterates by the hybrid minres reach the same minimum level as mr - ii and the hybrid mr - ii . these phenomena again justify our assertion in section [ sectioncom ] that the hybrid minres is necessary , independent of the degree of ill - posedness , and the hybrid minres is as effective as the hybrid mr - ii . figure [ fig5 ] and figure [ fig7 ] display the curves of sequences @xmath235 with the noise levels @xmath286 , respectively , for the four severely and moderately problems . we see that @xmath285 , almost independent of noise level @xmath273 . we point out that , due to the round - offs in finite precision arithmetic , they level off at the level of @xmath287 when @xmath288 for shaw , @xmath289 for foxgood and @xmath290 for gravity . the results indicate that the @xmath96 are near best rank @xmath0 approximations to @xmath3 with the approximate accuracy @xmath245 so that @xmath87 does not become ill - conditioned before @xmath116 . as a result , the regularized solutions @xmath129 become increasingly better approximations to @xmath28 until iteration @xmath40 , and they are deteriorated after that iteration . at iteration @xmath40 , @xmath101 captures the @xmath40 dominant spectral or equivalent svd components of @xmath3 and is a best possible regularized solution , i.e. , mr - ii has the full regularization for the severely ill - posed problems tested . figure [ fig6 ] and figure [ fig8 ] plot the relative errors @xmath291 with different noise levels for these four severely and moderately ill - posed problems . for smaller noise levels , mr - ii gets more accurate best regularized solutions at cost of more iterations . this is expected since , from and @xmath292 , a bigger @xmath40 is needed for a smaller noise level @xmath224 . moreover , mr - ii needs more iterations to achieve semi - convergence for moderately ill - posed problems with the same noise level , since @xmath55 does not decay as fast as that for a severely ill - posed problem . and @xmath293 for shaw with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for foxgood with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( a ) and @xmath293 for shaw with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for foxgood with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( b ) and @xmath293 for shaw with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for foxgood with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( c ) and @xmath293 for shaw with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for foxgood with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( d ) with respect to @xmath286 for shaw ( left ) and foxgood ( right ) by mr - ii.,width=264,height=188 ] ( a ) with respect to @xmath286 for shaw ( left ) and foxgood ( right ) by mr - ii.,width=264,height=188 ] ( b ) and @xmath293 for gravity with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for phillips with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( a ) and @xmath293 for gravity with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for phillips with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( b ) and @xmath293 for gravity with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for phillips with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( c ) and @xmath293 for gravity with @xmath294 ( left ) and @xmath272 ( right ) by mr - ii ; ( c)-(d ) : plots of decaying behavior of the sequences @xmath235 and @xmath293 for phillips with @xmath272 ( left ) and @xmath295 ( right ) by mr - ii.,width=264,height=188 ] ( d ) with respect to @xmath286 for gravity ( left ) and phillips ( right ) by mr - ii.,width=264,height=188 ] ( a ) with respect to @xmath286 for gravity ( left ) and phillips ( right ) by mr - ii.,width=264,height=188 ] ( b ) figures [ figpl ] display the decreasing curves of quantities @xmath296 , @xmath252 and @xmath297 , @xmath298 . from figure [ figpl ] ( a ) , we see that , for the severely ill - posed problem shaw , @xmath296 and @xmath252 decrease as fast as @xmath299 and the three quantities level off at the level of @xmath287 for @xmath0 no more than 20 , and after that these quantities are purely round - offs and not reliable any more . similar phenomena are also observed for the other two severely ill - posed problems foxgood and gravity , as indicated by figure [ figpl ] ( b ) and ( c ) . figure [ figpl ] ( d ) illustrates that @xmath252 decreases as fast as @xmath299 but @xmath296 decays as fast as @xmath297 in the first iterations and then considerably faster than @xmath297 as @xmath0 increases in the later stage for moderately ill - posed problem phillips . , @xmath252 and @xmath299 for shaw , foxgood , gravity , phillips ( from top left to bottom right ) with @xmath272 by mr - ii.,width=264,height=188 ] ( a ) , @xmath252 and @xmath299 for shaw , foxgood , gravity , phillips ( from top left to bottom right ) with @xmath272 by mr - ii.,width=264,height=188 ] ( b ) , @xmath252 and @xmath299 for shaw , foxgood , gravity , phillips ( from top left to bottom right ) with @xmath272 by mr - ii.,width=264,height=188 ] ( c ) , @xmath252 and @xmath299 for shaw , foxgood , gravity , phillips ( from top left to bottom right ) with @xmath272 by mr - ii.,width=264,height=188 ] ( d ) finally , we report some comparison results on lsqr , the hybrid lsqr and mr - ii , the hybrid mr - ii . as already proved in @xcite , a hybrid lsqr should be used to compute best possible regularized solutions for mildly ill - posed problems . it has also been experimentally justified in @xcite that lsqr has the full regularization for severely and moderately ill - posed problems . we have tested lsqr and the hybrid lsqr , and compared their effectiveness and efficiency with mr - ii and the hybrid mr - ii . we have found that , for each of the severely and moderately ill - posed problems in table [ table ] and with the same noise level , both the pure mr - ii and lsqr obtain the best regularized solutions with almost the same accuracy using almost the same iterations . for deriv2 , the hybrid mr - ii and lsqr compute the best possible regularized solutions using almost the same iterations . these results tell us two things : ( i ) as an iterative regularization method , mr - ii is as effective as lsqr for an ill - posed problem ; ( ii ) mr - ii is twice as efficient as lsqr . the problem blur is a 2d image deblurring problem and more complex than the other five 1d problems in table [ table ] . it arises in connection with the degradation of digital images by atmospheric turbulence blur . we use the code @xmath300 in @xcite to generate an @xmath301 matrix @xmath3 , the true solution @xmath28 and noise - free right - hand @xmath25 . the vector @xmath28 is a columnwise stacked version of a simple test image , while @xmath302 holds for a columnwise stacked version of the blurred image . the blurring matrix @xmath3 is block toeplitz with toeplitz blocks , which has two parameters @xmath303 and @xmath304 ; the former specifies the half - bandwidth of the toeplitz blocks , and the latter controls the shape of the gaussian point spread function and thus the amount of smoothing . we generate a blurred and noisy image @xmath121 by adding a while noise vector @xmath9 . the goal is to restore the true image @xmath28 from @xmath4 . we take @xmath305 and the relative noise level @xmath306 , giving rise to @xmath3 with order @xmath307 . it is known that the larger the @xmath304 , the less ill - posed the problem . purely for an experimental purpose , we computed all the singular values of a few @xmath3 with @xmath308 using the matlab function svd . since the degree of ill - posedness is the same for different large @xmath309 , we have deduced from the computed singular values for these matrices @xmath3 that @xmath310 ( the default setting ) generates mildly ill - posed problems , while @xmath311 gives rise to moderately ill - posed problems . we next test minres , mr - ii and their hybrid variants for these two problems , and verify the regularizing effects similar to the previous mildly and moderately ill - posed problems : deriv2 and phillips . by minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath310.,width=226,height=170 ] ( a ) by minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath310.,width=264,height=188 ] ( b ) by minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath310.,width=264,height=188 ] ( c ) by minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath310.,width=264,height=188 ] ( d ) with respect to minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath311.,width=226,height=170 ] ( a ) with respect to minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath311.,width=264,height=188 ] ( b ) with respect to minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath311.,width=264,height=188 ] ( c ) with respect to minres , hybrid minres , mr - ii , and hybrid mr - ii ; ( b ) : the original image ; ( c ) : the blurred and noisy image ; ( d ) : the restored image with @xmath311.,width=264,height=188 ] ( d ) figure [ fig11 ] ( a ) shows that minres and mr - ii have the partial regularization for the mildly ill - posed problem blur . the semi - convergence of the two methods occurs at the very first iteration , then regularized solutions are progressively deteriorated , while the hybrid minres finds the best possible regularized solution at iteration @xmath312 and the hybrid mr - ii does so at @xmath313 . moreover , we see that the hybrid minres and mr - ii reaches the same minimum error level . figure [ fig11 ] exhibits the restoration performance , where the restored image is chosen by the regularized solution at the iteration where the hybrid mr - ii first reaches the minimum error level . we observe from figure [ fig11 ] ( d ) that the outline of original image is restored quite well by the restored image . from figure [ fig12 ] ( a ) , we see that the semi - convergence of mr - ii occurs at the first iteration and the regularized solution at this iteration is as accurate as those obtained by the hybrid mr - ii and minres for the moderately ill - posed problem blur . therefore , mr - ii has the full regularization for this problem . in contrast , minres has only the partial regularization because its regularized solution at semi - convergence is much less accurate than that obtained by mr - ii . in addition , we observe that the hybrid minres and the hybrid mr - ii simply make the regularized solutions almost stabilize with the minimum error . figure [ fig12 ] ( d ) exhibits the restored image , which is a good approximation to the original image .

we call a regularized solution best possible if it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition ( tsvd ) method . in this paper , we analyze their regularizing effects and establish the following results : ( i ) the filtered svd expression are derived for the regularized solutions by minres ; ( ii ) a hybrid minres that uses explicit regularization within projected problems is needed to compute a best possible regularized solution to a given ill - posed problem ; ( iii ) theth iterate by minres is more accurate than theth iterate by mr - ii until the semi - convergence of minres , but mr - ii has globally better regularizing effects than minres ; ( iv ) bounds are obtained for the 2-norm distance between an underlying-dimensional krylov subspace and the-dimensional dominant eigenspace . they show that mr - ii has better regularizing effects for severely and moderately ill - posed problems than for mildly ill - posed problems , and a hybrid mr - ii is needed to get a best possible regularized solution for mildly ill - posed problems ; ( v ) bounds are derived for the entries generated by the symmetric lanczos process that mr - ii is based on , showing how fast they decay . stronger than our theory , the regularizing effects of mr - ii are experimentally shown to be good enough to obtain best possible regularized solutions for severely and moderately ill - posed problems .

for large scale symmetric discrete ill - posed problems , minres and mr - ii are often used iterative regularization solvers . we call a regularized solution best possible if it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition ( tsvd ) method . in this paper , we analyze their regularizing effects and establish the following results : ( i ) the filtered svd expression are derived for the regularized solutions by minres ; ( ii ) a hybrid minres that uses explicit regularization within projected problems is needed to compute a best possible regularized solution to a given ill - posed problem ; ( iii ) theth iterate by minres is more accurate than theth iterate by mr - ii until the semi - convergence of minres , but mr - ii has globally better regularizing effects than minres ; ( iv ) bounds are obtained for the 2-norm distance between an underlying-dimensional krylov subspace and the-dimensional dominant eigenspace . they show that mr - ii has better regularizing effects for severely and moderately ill - posed problems than for mildly ill - posed problems , and a hybrid mr - ii is needed to get a best possible regularized solution for mildly ill - posed problems ; ( v ) bounds are derived for the entries generated by the symmetric lanczos process that mr - ii is based on , showing how fast they decay . numerical experiments confirm our assertions . stronger than our theory , the regularizing effects of mr - ii are experimentally shown to be good enough to obtain best possible regularized solutions for severely and moderately ill - posed problems . * keywords : * symmetric ill - posed problem , regularization , partial regularization , full regularization , semi - convergence , mr - ii , minres , lsqr , hybrid . * mathematics subject classifications ( 2010 ) * : 65f22 , 65j20 , 15a18 .

cond-mat0204325

i

the understanding of spin dephasing is of paramount interest in all fields of nuclear magnetic resonance ( nmr ) sciences . in nmr spectroscopy it determines the line shape , in nmr imaging it is besides longitudinal relaxation the major mechanism determining the contrast and contains morphological as well as functional information . the processes contributing to spin dephasing are related to the spin environment . in biological tissues , for example , spin dephasing may result from dipole - dipole interaction of water proton spins with paramagnetic ions like fe@xmath0 . another cause is diffusion within inhomogeneous magnetic fields generated by native or contrast agent induced susceptibility differences that are related to tissue composition and/or cellular and sub cellular compartments . in magnetic resonance imaging spin dephasing in external gradient fields is exploited to get information about diffusion within biological systems . these diffusion sensitive imaging techniques are applied to study tissue anisotropy and restrictions of diffusion that are given by membranes of cells and sub cellular structures . essential for dephasing of spins are the field fluctuations that induce the phase modulations . it is important to note that in biological tissues the relevant processes cover almost the whole range of time scales . for example , the dynamics of interactions of water proton spins with paramagnetic macromolecules as ferritin is so fast that it can be considered to be within the motional narrowing regime . on the other hand , dephasing of spins in magnetic field gradients around larger vessels is almost coherent . i.e. it is in the static dephasing regime @xcite . hence , for biological applications it is important to obtain results from theory that are valid over the whole motion regime . however , in most cases this is not possible analytically . therefore , most efforts have focused on limiting cases . the _ motional narrowing limit _ is well investigated and a number of analytical results were obtained for it@xcite . the characteristic of this limit is that the mean phase shift induced by a field realization is much smaller than one , i.e. @xmath1 , where the correlation time @xmath2 gives the mean duration of some field realization , and @xmath3 is the variance of the inhomogeneous field . the relaxation time is then obtained as @xmath4 . in the other limiting case , i.e. the _ static dephasing regime _ ( @xmath5 ) , yablonski@xcite derived analytical expressions for coherent dephasing of spins in inhomogeneous fields around magnetic centers like cylinders or spheres . kiselev @xcite extended yablonski s static dephasing approach by considering diffusion of spins within local linear gradients . however , this approach requires that the diffusion length @xmath6 during dephasing is within the linear approximation of the inhomogeneous fields @xmath7 . note that expansion around the limiting cases by perturbation approaches leads to divergences in the respective other limits . therefore , the intermediate motion regime , i.e. almost everything between the static dephasing and motional narrowing limit , was in most situations accessible by simulations only @xcite . recently , we used a strong collision ( sc ) approach to characterize spin dephasing in a particular situation : an inhomogeneous field around regularly arranged parallel cylinders filled with a paramagnetic substance @xcite , a model reflecting the capillary network of the cardiac muscle . the results agreed well with simulations @xcite over the whole dynamic range , and with experimental data @xcite . the basic idea behind the sc approach is to replace the original generator of the markov process by a simpler one , the sc generator , that conserves particular features of the original process . in particular , by an appropriate choice of its parameter the sc process reproduces the correlation time of the field fluctuations induced by the original markov process . there are several advantages of the sc approximation . first , it is is correct both in the motional narrowing and the static dephasing limits ; thereby also the error in the intermediate regime is reduced considerably , when compared to perturbation approaches . and second , it provides a simple expression for the magnetization decay which may be solved analytically or at least numerically . however , the drawback of the sc approach was up to now that is not part of a systematic approximation to or an expansion of the original generator . therefore , it was unclear how results could be improved beyond the sc approximation . the aim of this paper is to extend the sc approach and provide a framework for a systematic approximation . in the next section we will present a formal description of spin dephasing that will be the basis for our analytical analysis . in sect . iii we will introduce the extended strong collision ( esc ) approximation proper and show how it is used to describe free induction and spin echo decay . in sect . iv we will apply it to two generic cases : spin dephasing induced by an anderson - weiss process @xcite and by restricted diffusion in a linear field gradient . we will close the paper with a summary and a discussion of our results .

rev . lett . 83 ( 1999 ) 4215 ) into a systematic framework . our new approach provides expressions for the free induction and spin echo magnetization decays that may be solved analytically or at least numerically . it is tested for the generic cases of dephasing due to an anderson - weiss process and due to restricted diffusion in a linear field gradient .

for markovian dynamics of field fluctuations we present here an extended strong collision approximation , thereby putting our previous strong collision approach ( phys . rev . lett . 83 ( 1999 ) 4215 ) into a systematic framework . our new approach provides expressions for the free induction and spin echo magnetization decays that may be solved analytically or at least numerically . it is tested for the generic cases of dephasing due to an anderson - weiss process and due to restricted diffusion in a linear field gradient . * * ii 2

cond-mat0107553

c

by considering the gfe for a system , this paper proposed a vpa scheme for the generalized statistical mechanics based on the tsallis entropy . for approximating the gfe , we derived the truncated expressions for @xmath57 up to the third order in the variational perturbation expansion , and the classical harmonic oscillator was considered in detail for an illustration . the model investigation , albeit being a little academic @xcite , illustrates that the approximation up to the first order amounts to a variational method and covers the variational method in ref . @xcite , and the approximations up to the second and third orders improve the variational result and tend to approach the exact result . frankly , the variational perturbation expansion technique is formally similar to the perturbation expansion in ref . @xcite , the work in the present paper is to introduce the variational perturbation idea into the perturbation expansion and use the pms for determining the auxiliary parameter in the vpa scheme . it is these revisions that make the variational perturbational approximation method be non - perturbational , take the variational result as the first - order approximation and systematically improve the variational result . we believe that the investigation in the present paper is useful , at least , for calculating the generalized thermodynamical functions based on the tsallis statistics . finally , we intend to point out that since path - integral formalism has been developed @xcite , it is worth while developing vpa method within the formalism of the generalized thermodynamics . 99 s. abe and y. okamoto , eds . , _ nonextensive statistical mechanics and its applications _ , lectures notes in physics , vol . * 560 * , ( springer , berlin , 2001 ) . a full and timely list of references can be found in the following webpage : http://tsallis.cat.cbpf.br/biblio.htm . c. tsallis , j. stat . phys . * 52 * , 479 ( 1988 ) . r. p. feynman , _ statistical mechanics : a set of lectures _ , ( addison - wesley publishing company , inc . , reading , massachusetts , 1972 ) . a. plastino and c. tsallis , j. phys . a * 26 * , l893 ( 1993 ) . e. k. lenzi , l. c. malacarne and r. s. mendes , phys . lett . * 80 * , 218 ( 1998 ) . r. s. mendes , c. a. lopes , e. k. lenzi and l. c. malacarne , physica a , * 344 * , 562 ( 2004 ) . e. hylleraas , z. phys . * 65 * , 209 ( 1930 ) ; l. cohen , proc . soc . a 68 , 419 ( 1955 ) ; a 68 , 425 ( 1955 ) . h. kleinert and v. schulte - frohlinde , _ critical properties of @xmath247-theories _ , world scientific , singapore , 2001 , chapter 19 ; h. kleinert , _ path integrals in quantum mechanics , statistics , polymer physics , and financial markets _ , 3rd ed . , world scientific , singapore , 2004 . w. f. lu , c. k. kim , j. h. yee , k. nham , phys . d * 64 * , 025006 ( 2001 ) ; w. f. lu , s. k. you , jino bak , c. k. kim and k. nham , j. phys . a * 35 * , 21 ( 2002 ) ; w. f. lu , c. k. kim , jae hyung yee , k. nham , phys . lett . b * 540 * , 309 ( 2002 ) ; w. f. lu , c. k. kim , k. nham , phys . b * 546 * , 177 ( 2002 ) ; w. f. lu , phys . lett . b * 602 * , 261 ( 2004 ) . p. m. stevenson , phys . d * 23 * , 2916 ( 1981 ) ; phys . b * 100 * ( 1981 ) 61 ; phys . rev . d * 24 * , 1622 ( 1981 ) ; s. k. kaufmann and s. m. perez , j. phys . a * 17 * , 2027 ( 1984 ) ; p. m. stevenson , nucl . b * 231 * , 65 ( 1984 ) . e. m. f. curado and c. tsallis , j. phys . a * 24 * , l69 ( 1991 ) ; corrigendum , j. phys . a * 24 * , 3187 ( 1991 ) ; corrigendum , j. phys . a * 25 * , 1019 ( 1992 ) . c. tsallis , r. s. mendes and a. r. plastino , physica a * 261 * , 534 ( 1998 ) . s. martnez , f. nicols , f. pennini and a. r. plastino , physica a * 286 * , 489 ( 2000 ) ; g. l. ferri , s. martnez and a. r. plastino , physica a * 347 * , 205 ( 2005 ) . g. l. ferri , s. martnez and a. r. plastino , j. stat . p04009 ( 2005 ) . p. m. stevenson , phys . d * 30 * , 1712 ( 1984 ) . r. p. feynman , phys . rev . * 56 * , 340 ( 1939 ) ; d. b. lichtenberg , phys . d * 40 * , 4196 ( 1989 ) ; w. namgung , j. kr . . soc . * 32 * , 649 ( 1998 ) . a. duncan and h. f. jones , phys . rev . d 47 , 2560 ( 1993 ) ; c. m. bender , a. duncan , and h. f. jones , phys d * 49 * , 4219 ( 1994 ) ; r. guida , k. konishi , and h. suzuki , ann . ( n.y . ) * 241 * , 152 ( 1995 ) ; r. guida and k. konishi , ann . phys . * 249 * , 109 ( 1996 ) ; e. braaten and e. radescu , phys . 89 * , 271602 ( 2002 ) ; j .- kneur , m. b. pinto and r. o. ramos , phys . * 89 * , 210403 ( 2002 ) . i. s. gradshteyn and i. m. ryzhik , _ table of integrals , series , and products _ ( academic press , new york , 1980 ) .

for the generalized statistical mechanics based on the tsallis entropy , a variational perturbation approximation method with the principle of minimal sensitivity is developed by calculating the generalized free energy up to the third order in variational perturbation expansion . the approximation up to the first order amounts to a variational approach which covers the variational method developed in phys . rev . lett .

for the generalized statistical mechanics based on the tsallis entropy , a variational perturbation approximation method with the principle of minimal sensitivity is developed by calculating the generalized free energy up to the third order in variational perturbation expansion . the approximation up to the first order amounts to a variational approach which covers the variational method developed in phys . rev . lett . 80 , 218 ( 1998 ) by lenzi , and the approximations up to higher orders can systematically improve variational result . as an illustrated example , the generalized free energy for a classical harmonic oscillator ( considered in the lenzi s joint work ) are calculated up to the third order , and the resultant approximations up to the first , second , and third orders are numerically compared with the exact result .

1101.4890

i

in summary , einstein s field equation is extended to include the quantum effect of gravity , from the general assumption that gravity force originates from the coupling and thermal equilibrium between matter and vacuum background . with this new field equation and physical mechanism of gravity force , without any fitting parameter , the accelerating universe of remarkable astronomical observations is quantitatively explained . we stress that the pioneering works on the connection between thermodynamics and gravity force , and the astronomical observations of accelerating universe play key role in the construction of the present theory . in fact , the present work is the result of the direct stimulation of these two works . the present paper is a significant extension of our previous works @xcite , with a lot of physical concepts being clarified and new method added to calculate the dark energy density by including general relativity . the present theory has several significant features . \(1 ) it is a simple extension of einstein s field equation to include quantum effect of gravity . \(2 ) different from newton s gravity law and einstein s general relativity , the present theory gives an interpretation why the gravity force between two classical objects is attractive . at least , the weak equivalence principle also becomes a derivable property . \(3 ) with the same reason to explain the attractive gravity force between two classical objects , the present theory predicts new physical effect , i. e. abnormal quantum gravity effect . \(4 ) the present theory quantitatively agrees with the astronomical observations of accelerating universe without any fitting parameter . \(5 ) if the present theory is verified , it has wide applications , even in condensed matter physics and experimental test of the many - world interpretation and de broglie - bohm theory , etc . \(6 ) last but maybe most important , our theory can be falsified by experiment on earth by measuring gravity acceleration in the interior of a superfluid helium sphere . because of the above features , we believe the present theory should be taken seriously . in particular , these features mean great opportunity to relevant experiments on earth and astronomical observations . one of the main results of the present work is the possible emergence of the quantum effect of gravity at the macroscopic scale . nevertheless , it is an interesting issue to consider whether this macroscopic and abnormal quantum gravity effect could be derived from the microscopic mechanism about the quantum gravity , such as superstring with a positive cosmological constant @xcite , loop quantum gravity and twistor theory @xcite , _ etc_. recently , the relation between gravity and thermodynamics has been studied in the frame of loop quantum gravity @xcite , which provides possible clue to this problem . although the microscopic interaction mechanism of quantum gravity at planck scale is an unsolved problem , the thermodynamics of the macroscopic quantum gravity effect of this work is still meaningful . when statistical mechanics was initiated in 1870 by boltzmann , the concept of atom is still an unrecognized hypothesis , not to mention the collision mechanism between atoms due to electromagnetic interaction . however , this does not influence the power of statistical mechanics in the description of gas dynamics . on the contrary , the theoretical and experimental advances of statistical mechanics greatly promoted further understanding of atoms . once the thermodynamic origin of gravity is verified by future experiments and astronomical observations , the abnormal quantum gravity effect would also greatly promote the understanding of microscopic quantum gravity at planck scale . in this work , the repulsive gravity effect for superfluid helium sphere and dark energy physically originate from the same mechanism that the wave packet of energy is delocalized in the space scale we study . if the de broglie - bohm theory is correct , we would not observe the abnormal gravity effect for superfluid helium . however , for dark energy originating from various vacuum excitations , it is still possible that the energy is carried by various wave packets , rather than particles , because there is no stable particles in the vacuum excitations . hence , even the abnormal gravity effect for superfluid helium is excluded in future experiments , it is still possible that the theory of the present work could be applied to accelerating universe . one may consider the problem that whether there is abnormal quantum gravity effect in the interior of a neutron star . if the temperature ( not the vacuum temperature ) of the neutron star is zero and the wave packets of all neutrons are delocalized in the whole interior of the neutron star , we would expect an abnormal quantum gravity effect . however , the temperature of the neutron state is in fact extremely high . the typical temperature of a neutron star is about @xmath303 kelvins . in addition , the neutron star is not an ideal fermi system without the consideration of the interaction between neutrons . the neutron star comprises very complex structures , such as the out core consisting of neutron - proton fermi liquid and the inner core consisting of possible quark gluon plasma . in quantum statistical mechanics , the approximate model of the neutron star as an ideal fermi system holds because of the validity of the local density approximation . the validity of this ideal fermi system does not mean that the wave packets of all neutrons are delocalized in the whole interior of the neutron star . considering the extremely high temperature and complex structure of the neutron star , it is more reasonable to assume that the delocalized scale of the wave packet of the neutrons is much smaller than the size of the neutron star ( the typical size of a neutron star is about @xmath304 km ) . in this situation , we still expect the normal gravity effect in the interior of the neutron star . we thank the discussions with prof . biao wu , and his great encouragements . we also thank prof . w. vincent liu s great encouragements . this work was supported by national key basic research and development program of china under grant no . 2011cb921503 and nsfc 10875165 .

from the general assumption that gravity originates from the coupling and thermal equilibrium between matter and vacuum , after a derivation of newton s law of gravitation and an interpretation of the attractive gravity force between two classical objects , we consider the macroscopic quantum gravity effect for particles whose wave packets are delocalized at macroscopic scale . we predict an abnormal repulsive gravity effect in this work . for a sphere full of superfluid helium

from the general assumption that gravity originates from the coupling and thermal equilibrium between matter and vacuum , after a derivation of newton s law of gravitation and an interpretation of the attractive gravity force between two classical objects , we consider the macroscopic quantum gravity effect for particles whose wave packets are delocalized at macroscopic scale . we predict an abnormal repulsive gravity effect in this work . for a sphere full of superfluid helium , it is shown that with a gravimeter placed in this sphere , the sensitivities of the gravity acceleration below could be used to test the abnormal quantum gravity effect , which satisfies the present experimental technique of atom interferometer , free - fall absolute gravimeters and superconducting gravimeters . we further propose a self - consistent field equation including the quantum effect of gravity . as an application of this field equation , we give a simple interpretation of the accelerating universe due to dark energy . based on the idea that the dark energy originates from the quantum gravity effect of vacuum excitations due to the coupling between matter and vacuum , without any fitting parameter , the ratio between dark energy density and matter density ( including dark matter ) is calculated as , which agrees quantitatively with the result obtained from various astronomical observations .

astro-ph0612517

i

in the favored cold dark matter cosmology , present - day galaxies were assembled hierarchically from smaller building blocks at earlier cosmic times . since all nearby galaxies with stellar spheroids are observed to host nuclear smbhs @xcite , hierarchical merging leads inevitably to the formation of smbh binaries @xcite . if the binary lifetime exceeds the typical time between mergers , then some galactic nuclei should contain systems of three or more smbhs . these systems are particularly interesting as they often lead to the ejection of one of the bhs at a speed comparable to the galactic escape velocity @xcite . in massive elliptical galaxies the typical speeds are @xmath2 , far greater than attainable through gravitational radiation recoil @xcite . spatially resolved pairs of nuclei have been observed in a few active galaxies . the most famous example is ngc 6240 , an ultraluminous infrared galaxy ( ulirg ) in which two distinct active galactic nuclei ( agn ) are clearly seen in hard x - rays at a projected separation of @xmath01 kpc @xcite . @xcite observed a variable uv source , possibly a second active nucleus , at a projected separation of @xmath3 pc from the primary nucleus in the spiral galaxy ngc 4736 , which shows signs of a recent merger . @xcite have detected what is thought to be an smbh binary at a projected separation of just 7.3 pc in the radio galaxy 0402 + 379 , through multi - frequency radio observations using the very long baseline array ( vlba ) . we begin by discussing the theory of how such systems evolve , and the conditions under which they might acquire a third bh . when two galaxies merge , their dense nuclei sink to the center of the merger product by dynamical friction . as the nuclei spiral in , tidal forces gradually strip the two smbhs of their surrounding stars and dark matter . in mergers between galaxies of comparable mass , the bhs are able to come together and form a bound smbh binary on a timescale of order @xmath4 yrs . the binary continues to harden by dynamical friction until it reaches a separation of order @xmath5 known as the `` hardening radius '' ( e.g. @xcite ) . here @xmath6 is the reduced mass of the two bhs with masses @xmath7 and @xmath8 , @xmath9 is the velocity dispersion of the stars beyond the binary s sphere of gravitational influence , @xmath10 is the binary mass ratio @xmath11 , and @xmath12 is the total mass of the binary . for smaller separations the binary looks like a point mass to the distant stars contributing to dynamical friction , but close stellar encounters preferentially harden the binary and so dominate further energy loss . only stars on nearly radial orbits , with periapsis distances of order the binary separation , can extract energy from ( `` harden '' ) the binary in this stage . these stars undergo strong 3-body interactions with the binary and escape its vicinity with speeds comparable to the black holes orbital speed . in the low - density nuclei of large elliptical galaxies , the total mass in stars on such `` loss cone '' orbits is small compared to the mass of the binary . furthermore the two - body stellar relaxation time is long compared to a hubble time , so once the stars initially on loss cone orbits are cleared out , the loss cone remains empty @xcite . since the binary must eject of order its own mass per @xmath13-folding in its semi - major axis , the system stops hardening around @xmath14 unless some other mechanism causes sufficient mass flux through the binary . if the binary reaches a separation around @xmath15^{1/4 } \hspace{2pt } \cdot \notag \\ & \left ( \frac{\tau_{gw}}{10^{10 } \hspace{2pt } { \rm yrs } } \right)^{1/4 } f^{-1/4}(e ) \hspace{6pt } { \rm pc},\end{aligned}\ ] ] where @xmath13 is the orbital eccentricity of the binary and @xmath16 , then it can coalesce on a timescale @xmath17 through gravitational radiation @xcite . to get from @xmath14 to @xmath18 it must bridge a gap @xmath19^{3/4}\ ] ] by some mechanism other than stellar - dynamical friction or gravitational radiation . the question of whether and how it crosses this gap has become known as the `` final parsec problem '' @xcite . in many galaxies there probably are alternative mechanisms for crossing the gap . when gas - rich galaxies merge , tidal torques channel large amounts of gas into the central @xmath20 pc @xcite . the gas may lose energy through radiation and angular momentum through viscous torques , and is therefore not subject to a loss cone problem . using smoothed particle hydrodynamics simulations @xcite compute a merger time of order @xmath21 yrs in an environment typical of the central regions of ulirgs , which are thought to be gas - rich galaxies caught in the act of merging @xcite . the nuclei of galaxies are also observed to contain numerous massive perturbers ( mps ) such as star clusters , molecular clouds , and possibly intermediate - mass black holes ( imbhs ) . these objects scatter stars into the loss cone much more efficiently than other stellar mass objects , since the relaxation rate scales as the perturber mass for a fixed mass density of perturbers . @xcite extended the fokker - planck loss cone formalism @xcite to accomodate a spectrum of perturber masses and account for relaxation by rare close encounters with mps . they show that the population of known mps in the nucleus of the milky way is sufficient to bring a @xmath22 m@xmath23 bh binary to @xmath18 in @xmath24 yrs , and it is reasonable to expect similar perturber populations in other star - forming spiral galaxies . the final parsec problem is often mentioned as a caveat when predicting the smbh coalescence signal in low - frequency gravitational wave detectors such as the upcoming laser interferometer space anntena ( lisa ) . however the lisa event rate is expected to be dominated by small galaxies at high redshift @xcite , where the gas content and central densities tend to be high and the relaxation times short . for this reason _ the stalling problem is probably not a significant concern for the lisa smbh coalescence signal_. on the other hand bh ejections by gravitational radiation recoil @xcite may play an important role in the high - redshift coalescence rate . the long - term survival of smbh binaries is likewise unlikely in the gas - rich cores of quasars and ulirgs . however none of the gap - crossing mechanisms discussed so far are likely to reduce the coalescence time below a hubble time in mergers between giant , gas - poor elliptical galaxies . @xcite show that a significant fraction of stars on `` centrophilic '' orbits in a triaxial potential can greatly increase the mass flux into the loss cone . some non - axisymmetric potentials can also excite bar instabilities that cause rapid mass flow through the binary and efficient coalescence @xcite . however a central smbh can disrupt box orbits and induce axisymmetry in the inner regions of a triaxial galaxy @xcite , and it is uncertain how often these geometry - specific mechanisms bring the coalescence time below a hubble time . one can naively assess the likelihood of coalescence by considering the `` full '' and `` empty '' loss cone hardening times , @xmath25 and @xmath26 , in the nuclei of various galaxies assuming a spherical and isotropic distribution function . @xmath25 is the hardening time assuming every star kicked out of the loss cone is instantly replaced , while @xmath26 is the time assuming stellar two - body relaxation to be the only replenishing mechanism . in small , dense galaxies @xmath27 yrs and @xmath28 yrs while in the lowest - density cores of giant ellipticals and cd galaxies @xmath29 yrs and @xmath30 yrs @xcite . while the empty loss cone rate is difficult to believe in any galaxy given at least some clustering on scales larger than 1 m@xmath23 , it also seems difficult to approach the full loss cone rate if there is no gas around and no strong radial bias in the stellar distribution . from this point of view the stalling of binaries seems unlikely in small galaxies but probable in low - density , gas - poor ellipticals . if some binaries do survive for around a hubble time , then the hierarchical buildup of galaxies will inevitably place three or more smbhs in some merging systems . an inspiralling satellite affects the evolution of a binary smbh even long before it sinks to the center , by perturbing the large - scale potential and scattering stars into the loss cone . we may estimate the extent of this effect as a function of satellite mass and distance from the center of the host galaxy using a rough but simple argument due to @xcite . the change in velocity necessary to deflect a star at radius @xmath10 into the loss cone is @xmath31 , where @xmath32 is the characteristic specific angular momentum of stars on loss cone orbits @xcite , @xmath33 is the smbhs radius of influence , and @xmath34 is the binary separation . the dynamical time at this radius is @xmath35 , so the acceleration required to scatter a star into the loss cone is roughly @xmath36 . equating this with the tidal acceleration caused by the satellite , @xmath37 , where @xmath38 is the satellite s radius , yields @xmath39 , or with @xmath40 , @xmath41^{1/3 } r.\ ] ] the @xmath38-dependence of @xmath42 reflects the tidal stripping of the satellite as it spirals inward . equation defines a critical radius @xmath10 , outside of which the satellite can deflect stars into ( and out of ) the loss cone in one dynamical time . the mass flux through the binary induced by the satellite is then approximately @xmath43 where @xmath44 is the density of the host galaxy at radius @xmath10 and @xmath45 is the geometrical factor accounting for the fraction of stars on loss cone orbits as a function of radius @xmath38 , assuming an isotropic distribution function @xcite . for a fixed satellite mass and distance , we can then define a `` binary feeding '' timescale by @xmath46 to determine whether the scattering of stars into the loss cone by the satellite is sufficient to harden the binary enough to prevent a close 3-body encounter before the intruder arrives at the galactic center , we must compare @xmath47 with the timescale on which the satellite spirals in by dynamical friction . in the approximation of slow inspiral we may write the dynamical friction timescale as @xmath48},\ ] ] where @xmath49 . @xmath50 in equation is computed from @xmath51 , where @xmath52 is the mass of the host galaxy enclosed within radius @xmath38 . @xmath53 is the satellite mass contained within the tidal truncation radius obtained from a simple point mass approximation , @xmath54^{1/3 } r$ ] ( this slightly underestimates @xmath55 as the satellite approaches the center of the host ) . in fig . 1 we plot @xmath56 and @xmath47 as a function of @xmath38 for a satellite with one third the stellar mass of the host , which contains a binary with @xmath57 m@xmath23 . both host and satellite are modelled as hernquist profiles @xcite , with their masses and effective radii set by observed scaling relations . the details of the galactic model are described further in 2.1 - 2 . , on which an inspiralling satellite scatters mass into the loss cone of an smbh binary , with the dynamical friction timescale , @xmath56 , on which the satellite spirals in . upper ( red dashed ) line : @xmath47 computed from equation ; lower curve : @xmath56 computed from equation . both timescales are plotted as a function of the satellite s distance from the center of the host galaxy . for this plot we chose a binary mass of @xmath58 m@xmath23 and merger mass ratio of 3:1 in the stars . the galactic model is discussed in the text . ] since @xmath47 remains about an order of magnitude above @xmath56 throughout the inspiral , this simple calculation makes it plausible that the binary survives the merger process and undergoes close triple interactions with the infalling smbh . the tidal approximation ( as well as our treatment of dynamical friction ) breaks down as the satellite approaches @xmath59 , so the plot is cut off at a separation of @xmath20 pc , when the satellite still has @xmath04 @xmath13-foldings to go to reach @xmath14 . however this final stage of the inspiral is found to proceed very rapidly in n - body simulations @xcite . the merger s effect on the binary may be dominated by violent relaxation or collective effects such as a bar instability @xcite , in which case our two - body approach does not capture its essence . the evolution of the core distribution function under the influence of a major merger is an intriguing open problem for simulators . after the third bh becomes bound to the binary ( but still before the onset of close 3-body interactions ) another hardening mechanism may become important . if the angle of inclination @xmath60 of the outer binary ( formed by the intruder and the inner binary center - of - mass ( com ) ) exceeds a critical angle @xmath61 , then the quadrupolar perturbation from the intruder induces eccentricity oscillations through a maximum @xcite @xmath62 since the gravitational radiation rate increases sharply toward high eccentricities , these `` kozai oscillations '' can greatly enhance the radiation , possibly causing the binary to coalesce before it can undergo strong 3-body interactions with the intruder @xcite . general relativistic precession can destroy the kozai resonance ( e.g. @xcite ) , but @xcite find that this does not happen for @xmath63^{1/3 } \sqrt{\frac{1-e_{in}^{2}}{1-e_{out}^{2}}},\ ] ] where @xmath64 and @xmath65 are the semi - major axes of the inner and outer binaries , @xmath66 is the outer binary mass ratio , and @xmath67 and @xmath68 are the inner and outer eccentricities . this leaves a window of about a factor of 10 in @xmath69 in which the kozai mechanism can operate before unstable 3-body interactions begin . the actual enhancement of the gravitational radiation rate of course depends on the amount of time spent at high eccentricity , but one may place an upper limit on the importance of kozai oscillations by computing the radiation timescale if the inner binary spends all of its time at @xmath70 . the orbit - averaged power radiated by gravitational radiation is given by @xmath71 @xcite , where @xmath72 is the semi - major axis and @xmath13 is the eccentricity . in fig . 2 we plot contours of the gravitational radiation time @xmath73 in the @xmath72-@xmath60 plane by putting @xmath70 into equation , for an equal - mass @xmath74 m@xmath23 binary . m@xmath23 ) . the cosine of the initial inclination angle is plotted on the horizontal axis , and the inner binary semi - major axis is plotted on the vertical axis . contours are plotted for gravitational radiation timescales of @xmath75 , @xmath76 , @xmath77 , and @xmath78 yrs , if the binary were to stay at maximum eccentricity throughout the whole oscillation cycle . the horizontal lines indicate the hardening radius and the separation such that a circular binary would coalesce on a @xmath79 yr timescale . ] this may seem like a gross overestimate of the gravitational radiation rate , especially since the shape of the kozai oscillations is in fact such that the binary spends more time near @xmath80 than near @xmath70 . however since @xmath17 is so strongly dominated by periapsis passages at @xmath81 , the shift in the contours for a realistic high-@xmath13 duty cycle is only modest . see @xcite for comparison with a detailed study of radiation enhancement by kozai oscillations in binaries with initial @xmath82 yrs . for a binary at @xmath14 , kozai oscillations can induce coalescence within @xmath76 yrs in @xmath83% of cases assuming @xmath84 is uniformly distributed . in the remainder of cases the inner binary may survive until the outer binary shrinks to the point of unstable 3-body interactions . if the intruder comes close enough before it causes sufficient hardening of the ( inner ) binary , then a strong 3-body encounter takes place . strong encounters are characterized by a significant transfer of energy between the binary s internal degrees of freedom and the com motion of the binary and third body . when the intruder is slow relative to the binary s orbital speed @xmath85 , energy typically flows from the inner binary to the outer components , so that the binary is more strongly bound after the encounter . this is one manifestation of the negative specific heat characteristic of gravitationally bound systems . the encounter ends in the escape of one of the three bodies , usually the lightest , from the system at a speed comparable to @xmath85 . when the lightest body @xmath86 escapes , momentum conservation requires that the binary com recoil in the opposite direction with a speed smaller by a factor @xmath87 . it is instructive to compare the expected ejection velocities of the binary and @xmath86 with the typical galactic escape velocity . for a circular binary with @xmath88 , the binding energy at the hardening radius is @xmath89^{3/2}$ ] erg . the binding energy at the radius where @xmath90 yrs is @xmath91^{5/4}$ ] erg . the mean energy @xmath92 harvested from the binary in close encounters with slow intruders is about @xmath93 , though the median @xmath92 is somewhat lower @xcite . energy conservation implies that the escaper leaves the system with kinetic energy @xmath94 $ ] while the binary leaves with @xmath95 $ ] in the system com frame . for an equal mass binary with @xmath96 m@xmath23 , this gives ejection velocities of @xmath97 and @xmath98 for the binary at @xmath14 , and @xmath99 and @xmath100 for the binary at @xmath18 . any nonzero eccentricity of the binary will increase the semi - major axis corresponding to a fixed @xmath17 , lowering the ejection velocities for the binary at @xmath18 . also any deviation from equal masses will result in a smaller fraction of the extracted energy being apportioned to the binary and a smaller binary recoil velocity . the typical escape velocity for galaxies hosting @xmath101 m@xmath23 bhs is around 1500 @xmath102 , accounting for both the stars and the dark matter . from these numbers , it appears that single escapes will be fairly common as repeated encounters harden the binary to @xmath103 . however accounting for realistic mass ratios and eccentricities ( the first 3-body encounter tends to thermalize the eccentricity even if it starts off circular ) , binary escapes should be rare . since the binary must come near the escape velocity to remain outside the nucleus for a significant amount of time , we do not expect triple interactions to empty many nuclei of bhs . we will quantify these statements with our triple - bh simulations . the formation of triple smbh systems through inspiral of a merging satellite leads to a rather specific initial configuration . the three bhs start off as a bound `` hierarchical triple , '' consisting of an inner binary with @xmath104 and a more widely separated outer binary with semi - major axis @xmath65 . for very large @xmath69 we expect hierarchical triples to exhibit very regular behavior ; in this case the third body sees the inner binary as a point mass and the system essentially consists of two independent ( inner and outer ) binaries . however as @xmath69 approaches unity , secular evolution gives way to chaotic 3-body interactions in which the orbits diverge and the system becomes subject to escape of one its components . @xcite derive a criterion for the stability of 3-body systems based on an analogue with the problem of binary tides . the most distant intruder orbit at which unstable interactions can begin is reliably estimated by @xmath105^{2/5},\ ] ] where @xmath106 is the periapsis separation of the outer binary , @xmath64 is the semimajor axis of the inner binary , @xmath107 is the outer binary mass ratio , and @xmath68 is its eccentricity . this criterion has great practical importance due to the high numerical cost of unnecessarily following weak hierarchical systems . it specifies an optimal starting point for our simulations , which aim to study strong interactions in 3-body systems starting off as hierarchical triples . naively one might expect a strong 3-body encounter following a merger with a galaxy hosting a binary , so long as the intruder does not induce coalescence of the binary before it reaches the center . however the stability criterion implies a condition for close interactions much more stringent than this . to undergo a chaotic encounter with the inner binary , the intruder must reach the stability boundary before the outer binary hardens and stalls . a triple system covers somewhat more stellar phase space than a binary of the same size , but not by much for a stable hierarchical system . this means that the merger process can not cause the binary to harden by more than around an @xmath13-folding for a nearly circular , equal - mass system before the intruder arrives at the center . though the order - of - magnitude estimates in the previous section make this plausible , further study is needed to determine the likelihood of unstable triple interactions in realistic merger situations . an eccentric outer binary relaxes the criterion somewhat , but dynamical friction tends to circularize the orbits of satellites with moderate initial eccentricities before they reach the nucleus @xcite . we therefore assume near - circular initial orbits and begin each simulation from a weakly hierarchical configuration . triple smbh systems in galactic nuclei were first considered by @xcite , who computed an extensive series of newtonian 3- and 4-body orbits , and compared the slingshot ejection statistics to the observed structure of extragalactic radio sources . @xcite included a gravitational radiation drag force in the 3-body dynamics . he showed that this perturbing force could in some cases yield much higher ejection velocities than would be possible in newtonian gravity , with associated bursts of gravitational waves . the more complex problem of three or four smbhs coming together in the hierarchical merging process and interacting in a galactic potential was first addressed by @xcite and @xcite , who experimented with a variety of initial bh configurations . @xcite studied binary - binary scattering in galactic nuclei using initial conditions ( ics ) based on extended press schechter theory @xcite . @xcite followed the formation of triple bh systems in halo merger trees tracking the hierarchical buildup of smbhs from @xmath108 m@xmath23 seeds in high-@xmath9 peaks at @xmath109 . using a simple analytic prescription for the ejection velocities , they inferred the presence of a large population of smbhs and imbhs wandering through the halos of galaxies and intergalactic space . @xcite performed the first full n - body simulations of equal - mass triple bh systems embedded in stellar bulges , an important contribution to our understanding of galactic nuclei . because of the large computation time required for each run , they could not statistically sample the highly varied outcomes of the 3-body encounters as the previous authors did . in this paper we study the dynamics of repeated triple - smbh interactions in galactic nuclei . between close encounters we follow the wandering bhs through the galaxy as their orbits decay by dynamical friction . we use physically - motivated initial bh configurations and mass distributions , and updated galactic models characteristic of the low - density , massive elliptical galaxies in which smbh binaries are most likely stall . we include both a stellar and a dark matter component , with the stellar spheroid fixed to lie on the observed @xmath110 and @xmath111 relations @xcite . the close encounters are treated using a ks - regularized bulirsch - stoer integrator provided by sverre aarseth @xcite . the inner density profile is updated throughout the simulations to roughly account for core heating by dynamical friction and stellar mass ejection . gravitational radiation losses are modelled as a drag force determined by the relative coordinates and velocities of each pair . each simulation takes only a few minutes to run , so we can try a variety of distributions of ics and statistically sample the outcomes for each . we use this algorithm to study a variety of consequences of the ongoing encounters , such as the merging efficiency of bh pairs , the time spent wandering at various distances from the galactic center , the distribution of final sizes and eccentricities of the binaries remaining in the galaxy after a steady state has been reached , and the extent of the core scouring caused by the triple smbh systems . aside from the motivating order - of - magnitude calculations in previous sections , this paper does not address the question of _ whether _ close triple smbh systems form in galactic nuclei . we start our simulations from a state that the system must reach shortly before the onset of unstable 3-body interactions _ assuming _ that they occur , and proceed to derive the subsequent evolution . our results may be used to argue for or against the occurrence of triple systems in real galaxies , as observations support or disfavor the signatures that we derive . in 2 we describe our model and code methods . in 3 we present the results of our study , and in 4 we discuss these results and conclude .

galaxies with stellar bulges are generically observed to host supermassive black holes ( smbhs ) . the hierarchical merging of galaxies should therefore lead to the formation of smbh binaries . if the binary lifetime exceeds the typical time between mergers , then triple black hole ( bh ) systems may form . we study the statistics of close triple - smbh encounters in galactic nuclei by computing a series of 3-body orbits with physically - motivated initial conditions appropriate for giant elliptical galaxies . is left wandering through the galactic halo or escapes the galaxy altogether , but escape of all three smbhs is exceedingly rare . [ firstpage ] black hole physics cosmology : theory galaxies : elliptical and lenticular , cd galaxies : interactions galaxies : nuclei methods : numerical

galaxies with stellar bulges are generically observed to host supermassive black holes ( smbhs ) . the hierarchical merging of galaxies should therefore lead to the formation of smbh binaries . merging of old massive galaxies with little gas promotes the formation of low - density nuclei where smbh binaries are expected to survive over long times . if the binary lifetime exceeds the typical time between mergers , then triple black hole ( bh ) systems may form . we study the statistics of close triple - smbh encounters in galactic nuclei by computing a series of 3-body orbits with physically - motivated initial conditions appropriate for giant elliptical galaxies . our simulations include a smooth background potential consisting of a stellar bulge plus a dark matter halo , drag forces due to gravitational radiation and dynamical friction on the stars and dark matter , and a simple model of the time evolution of the inner density profile under heating and mass ejection by the smbhs . we find that the binary pair coalesces as a result of repeated close encounters in% of our runs , and in% of cases a new eccentric binary forms from the third smbh and binary remnant and coalesces during the run time . in about 40% of the runs the lightest bh is left wandering through the galactic halo or escapes the galaxy altogether , but escape of all three smbhs is exceedingly rare . the triple systems typically scour out cores with mass deficits - 2 their total mass , which can help to account for the large cores observed in some massive elliptical galaxies , such as m87 . the high coalescence rate , prevalence of very high - eccentricity orbits , and gravitational radiation `` spikes '' during close encounters in our runs , may provide interesting signals for the future laser interferometer space antenna ( lisa ) . [ firstpage ] black hole physics cosmology : theory galaxies : elliptical and lenticular , cd galaxies : interactions galaxies : nuclei methods : numerical

astro-ph0612517

m

to get a physically motivated distribution of bh mass ratios , we associate the formation of the inner and outer binaries with the last two major mergers in the history of the galactic halo hosting the triple system . we use extended press - schechter theory @xcite to calculate the probability distributions of the halo formation times and progenitor masses , and randomly select the parameters of the previous two mergers from these distributions . we then assign a bh to each progenitor halo using a simple prescription based on the assumption of a flat galactic rotation curve . @xcite derive the instantaneous halo merger rate , @xmath112}{[1-\sigma^{2}(m_{f})/\sigma^{2}(m_{1})]^{3/2}}.\end{gathered}\ ] ] this equation gives the probability , per unit time per unit mass of @xmath113 , of a given halo of mass @xmath114 merging with another halo of mass @xmath113 to form a product of mass @xmath115 @xmath116 @xmath117 at time @xmath118 . here @xmath119 is the present - day variance of the linear density field on mass scale @xmath120 , @xmath121 where @xmath122 is the power spectrum of density fluctuations today , @xmath123 is a tophat window function , and @xmath38 is related to @xmath120 through @xmath124 , the volume times the present - day matter density . @xmath122 is related to the primordial power spectrum through the transfer function @xmath125 , which encapsulates the suppression of perturbations on small scales due to radiation pressure and damping over the history of the universe . for @xmath125 we adopt the standard fitting formulae of @xcite . for the linear growth function @xmath126 we use the approximation @xmath127 \left [ 1+\frac{\omega_{\lambda}(z)}{70 } \right ] } , \ ] ] good to within a few percent for all plausible values of @xmath128 and @xmath129 @xcite . @xmath130 is the growth function for an einstein - de sitter universe , @xmath131 $ ] is the matter density ( normalized to the critical density ) as a function of redshift , and we take @xmath132 assuming the rest of the density is in the form of a cosmological constant . @xmath133 has the weak redshift dependence @xcite @xmath134.\ ] ] we adopt the cosmological parameters obtained from three years of data collection by the wilkinson microwave anisotropy probe ( wmap ) , @xmath135 , @xmath136 , @xmath137 , @xmath138 , and @xmath139 @xcite . since the merger rate diverges as @xmath140 @xmath141 0 , applications of the formula that track individual merging halos must employ a cutoff mass ratio @xmath140 @xmath142 @xmath143 , such that all mergers below @xmath143 are treated as smooth _ accretion _ rather than as discrete mergers ( see @xcite for further discussion ) . the instantaneous rate of accretion onto a halo of mass @xmath120 at redshift @xmath144 is @xmath145 to get the growth history ( `` accretion track '' ) of a halo of mass @xmath146 at time @xmath147 due to accretion since the last merger , one need only solve the differential equation @xmath148 @xmath116 @xmath149 $ ] , subject to the initial condition @xmath150 @xmath116 @xmath146 . we integrate this equation backward in time using a 4th - order runge - kutta method to get the accretion tracks of the halos in our simulations . since we are interested in bh binary formation , we loosely associate @xmath143 with the halo mass ratio such that tidal stripping of the satellite would prevent the eventual merging of the two nuclei . n - body simulations of galaxy mergers place this mass ratio in the range @xmath143 @xmath0 @xmath151 , depending on the density and orbital parameters of the satellite @xcite . hence our canonical choice is @xmath143 @xmath116 0.3 , and we also try values of @xmath152 ( runs d1 ) and 0.5 ( runs d5 ) , the latter being the halo mass that corresponds to a stellar mass ratio of @xmath0 3:1 in our prescription . following @xcite , we write the probability , per unit time , of a halo with mass @xmath115 at time @xmath118 arising from a merger with a smaller halo of mass between @xmath120 and @xmath153 ( the `` capture rate '' ) as @xmath154 \hspace{2pt } \frac{n(m , t)}{n(m_{f},t ) } \hspace{2pt } dm,\end{gathered}\ ] ] the eps merger rate excluding halos below the threshold @xmath143 , and weighted by the number of mass @xmath120 halos per unit halo of mass @xmath115 . the rate at which halos of mass @xmath115 form through all mergers at time @xmath118 is @xmath155 the probability distribution function ( pdf ) of formation times of halos with mass @xmath146 at time @xmath147 is @xmath156 \hspace{2pt } e^{-\int_{t}^{t_{0 } } r_{f}[m(t'),t ' ] dt'}.\ ] ] given a formation time @xmath157 and corresponding mass @xmath158 along the past accretion track of @xmath146 , the mass of the larger progenitor @xmath114 is distributed according to @xmath159 = \frac{2g(m_{1},m)}{\int_{m\delta_{m}/(1+\delta_{m})}^{m/(1+\delta_{m } ) } g(m',m ) dm'},\ ] ] where @xmath160^{-3/2}.\ ] ] by choosing formation times and progenitor masses randomly according to and , we capture the stochasticity of the intervals between mergers above @xmath143 , but treat merging below this threshold only in the mean . see @xcite , @xcite , and @xcite for further details and derivations of and . 3 shows the distribution of formation times , progenitor masses , and accretion tracks for a present - day @xmath161 m@xmath23 halo for @xmath162 0.1 , 0.3 , and 0.5 , and the accretion tracks for 1 , 2.3 , and 5 @xmath163 m@xmath23 halos with @xmath143 fixed at 0.3 . all accretion tracks are normalized to the present - day mass @xmath146 . note the insensitivity of the shape of these tracks to @xmath146 , as expected for masses above the critical mass @xmath164 . halo at @xmath165 . _ upper right : _ pdf of masses of the larger progenitor of the same halo given by equation , normalized to the mass of the merger product , @xmath166 . _ lower left : _ past accretion tracks of a present - day @xmath167 halo back to @xmath168 , normalized to the mass at @xmath165 . _ lower right : _ normalized accretion tracks for three different halo masses . ] our algorithm for generating the bh masses is illustrated schematically in fig . 4 . for each run we begin with a halo of mass @xmath146 @xmath116 @xmath161 @xmath169 at time @xmath147 @xmath142 @xmath170 , choose it s formation time @xmath171 randomly according to equation , and find the mass @xmath172 along its accretion track at that time . the mass @xmath173 is assigned to the dark matter halo hosting the triple bh system , and the physical time for the run to end if other termination conditions are not met first is set to @xmath174 . to explore the dependence of the results on the absolute mass scale , we also try beginning with a @xmath175 @xmath169 halo ( runs h1 ) . we model the halo as a hernquist profile @xcite , which is identical to an nfw profile @xcite in its inner regions if the scale radius @xmath176 is related to the nfw scale radius by @xmath176 @xmath116 @xmath177}$ ] , where @xmath178 is the halo concentration defined by @xmath179 . the hernquist model falls off as @xmath180 instead of @xmath181 far outside @xmath176 @xcite . the virial radius @xmath182 is given by @xmath183^{1/3 } h^{-1 } { \rm kpc},\end{gathered}\ ] ] where @xmath184 @xmath116 @xmath185 - 39[\omega_{m}(z)-1]^{2}$ ] @xcite , and @xmath178 roughly follows the median relation from the @xmath186cdm simulations of @xcite , @xmath187^{0.13}/(1+z)$ ] . the @xmath144 dependence of @xmath188 and @xmath178 nearly cancel to make @xmath176 depend only weakly on redshift , so we simply use the @xmath165 relation between @xmath189 and @xmath176 in our simulations . we choose the mass @xmath114 of the larger progenitor of @xmath173 randomly according to equation , and assign a mass @xmath113 @xmath116 @xmath190 to the smaller progenitor . before the merger the larger progenitor is assumed to have hosted a bh binary , while the smaller one hosted a single bh . repeating the procedure used for @xmath146 , we assign formation times @xmath191 and @xmath192 to @xmath114 and @xmath113 using equation , and choose progenitor masses @xmath193 , @xmath194 , @xmath195 , and @xmath196 according to equation . having constructed a set of progenitor halos , we now need a bh - halo relation @xmath197 to complete our algorithm . we obtain such a relation by equating the halo virial velocity @xmath198 to the circular velocity @xmath199 of the stellar spheroid , and using empirical @xmath200 and @xmath201 correlations to connect @xmath199 to @xmath202 , similar to the approaches in @xcite and @xcite . combining @xmath203^{1/6 } { \rm km \hspace{3pt } s^{-1 } } \end{gathered}\ ] ] @xcite with @xmath199 @xmath204 @xmath205^{0.84}$ ] @xmath102 @xcite and @xmath206 @xmath204 @xmath207^{1/4.02}$ ] @xcite , we arrive at the relation @xmath208 where @xmath209^{-1/2}$ ] . in our canonical runs we set the masses of the inner binary members to @xmath210 and @xmath211 , and that of the intruding bh to @xmath212 . note that in this prescription the intruder is usually lighter than the heavier binary member , so that most of the 3-body interactions result in an exchange . to examine the effect of more interactions without exchange , we try choosing @xmath213 $ ] for the intruder mass in runs ( mx ) . as there is neither a direct causal relationship between @xmath202 and @xmath189 predicted by theory @xcite nor a tight correlation directly observed between these two variables , and we know that identical halos may host galaxies of different morphologies and occupation numbers , @xmath197 should be taken with something of a grain of salt . nevertheless it is a useful way to generate simple but physically - motivated bh mass distibutions when no information other than the halo mass is available . for the halo @xmath146 hosting the bh triple system randomly from equation . given @xmath214 and @xmath215 , we select two progenitor masses @xmath114 and @xmath113 according to equation , assign the binary to the larger one and the third bh to the smaller one . we repeat this process going back one step further in the `` merger tree '' to get the masses of the binary constituents . ] we make one final modification to the set of bh masses used in our simulations . if the outer binary s hardening radius lies outside the stability boundary given by equation with @xmath216 , then the decay of the outer orbit is expected to stall before a strong encounter can begin . to roughly account for this we exclude all ics where @xmath217 @xmath218 @xmath219 . the final distribution of bh mass ratios is shown in fig . 5 for @xmath1620.1 , 0.3 , and 0.5 . . _ lower panel : _ @xmath220 , where @xmath221 is the mass of the lightest bh and @xmath222 is the sum of the masses of the other two bhs . ] in the upper panel we plot the inner binary mass ratios , while the lower panel shows the distribution of @xmath220 , where @xmath221 is the mass of the lightest bh and @xmath222 is the sum of the masses of the other two bhs . this ratio determines the binary recoil speed when the lightest bh is ejected from the system . the total bh mass is typically @xmath24 m@xmath23 in our canonical runs . to complete the galactic model we surround the bh system by a smooth stellar potential superimposed on the dark matter halo . the stars are modelled using the `` @xmath223-models '' of @xcite , with a sharp break to shallower slope @xmath224 added at @xmath225 : @xmath226 our canonical model is the @xmath227 ( hernquist ) profile , and we also try @xmath228 ( runs sc ) to explore the effect of a steeper inner profile and higher central density ( @xmath229 for @xmath228 vs. @xmath230 for the hernquist profile at the bh radius of influence ) . @xmath231 and @xmath232 were initialized to reflect the cusp destruction caused by the inspiralling bhs in reaching their initial configuration , and were updated throughout the simulation to account for the continued core heating and mass ejection . our algorithm for updating the core is described further in 3.5 . the parameters @xmath120 and @xmath72 in the @xmath223-models were set based on the tight correlations observed between smbh mass @xmath202 and stellar bulge mass @xcite and velocity dispersion @xcite . @xcite found the relation @xmath233 between @xmath202 and the virial mass @xmath234 of the stellar bulge , where @xmath235 is the half - light radius and @xmath236 is the effective bulge velocity dispersion . they set @xmath237 ( @xmath238 would be 8/3 for an isothermal sphere ) to get an average ratio of unity between @xmath239 and the dynamically measured masses @xmath240 of galaxies with more direct stellar - dynamical mass determinations @xcite . @xmath236 is typically measured over either a circular aperture of radius @xmath241 @xcite or a linear aperture out to @xmath235 @xcite @xcite discuss the essential agreement between the velocity dispersions measured in these two ways . thus for each model ( @xmath242 ) we compute the projected radius @xmath243 containing half the integrated surface brightness ( assuming a constant mass - to - light ratio ) , and velocity dispersion @xmath244 at radius @xmath241 . here @xmath245 and @xmath246 are constants depending on the density profile ( see @xcite for relevant formulae ) . the parameter @xmath120 is then chosen to satisfy @xmath247 @xmath248 @xmath249 . for the hernquist model with @xmath250 and @xmath251 , @xmath252 . the scale radius is then obtained from @xmath253 , where @xmath254 is the velocity dispersion computed from the @xmath110 relation of @xcite . in each of these relations @xmath202 is set to the total mass of the triple bh system . in a perfectly smooth , spherically symmetric galactic potential , bhs ejected on distant radial orbits return directly to the center to interact strongly with any other nuclear black holes . since real galaxies are clumpy and triaxial , the interaction will more realistically be delayed until the orbit of the ejected bh decays by dynamical friction . to mitigate this problem we flattened the @xmath223-models by adding two low - order spherical harmonic terms to the spherical potential @xcite : @xmath255 where @xmath256 is the potential of the spherical @xmath223-model , @xmath257 , and @xmath258 . since near - sphericity in the inner regions is probably a necessary prerequisite for the survival of the inner binary for of order a hubble time until the next merger @xcite , the parameters @xmath259 were chosen to give a spherical profile near the galactic center , and axis ratios approaching 1.3 and 1.5 for @xmath260 . a similar triaxial modification was applied to the dark matter halo , and the relative orientation of the stellar and halo potentials was chosen randomly . by misaligning their axes we eliminate any artificial stable orbits ( e.g. along the long axis of an ellipsoid ) near which ejected bhs tend to return on a perfectly radial orbit to the center . this triaxial modification had the desired effect of preventing frequent strong encounters at periapsis on distant orbits , but had little influence on the global outcome statistics . we assume that the 3-bh system starts off as a hierarchical triple on the verge of unstable 3-body interactions . in our canonical runs we initialize the inner binary semi - major axis @xmath64 to @xmath14 . to study the effect of varying @xmath64 we also try runs with @xmath261 ( runs ba ) and @xmath262 ( runs sa ) . the outer binary semi - major axis @xmath65 is set by the stability criterion of @xcite , equation the initial eccentricity of the inner ( outer ) binary was chosen uniformly between 0.0 and 0.2 ( 0.3 ) , in accordance with the low eccentricities found in galaxy merger simulations where dynamical friction tends to circularize the orbits of satellites as they spiral inward @xcite . the three euler angles of the intruder s orbital plane were chosen randomly relative to the reference plane of the binary orbit , as was the phase of the initial periapsis of the binary . both orbits were always started at periapsis ; since many orbital periods elapse before unstable interactions begin , the relative phase is effectively randomized in any case . having defined an initial configuration of three bhs embedded in a stellar+dark matter potential , we next describe how we evolve the system forward in time . we treat the close 3-body encounters using sverre aarseth s _ chain _ code , an implementation of the n - body regularization technique of mikkola and aarseth @xcite . the masses are first ordered so that neighbors in the chain are the dominant two - body interactions , then the ks - transformation @xcite is applied to neighboring pairs . this transformation eliminates the singularity at @xmath263 in newtonian gravity and transforms the equations of keplerian motion to the simple harmonic oscillator equation @xcite . external perturbing forces of arbitrary strength depending on the coordinates , velocities , and/or time are simply incorporated into the formulation ( though of course singularities in these perturbing forces need not be eliminated by the change of variables ) . we use this to add a galactic potential ( 2.1 - 2 ) , a gravitational radiation back - reaction force , and a stellar - dynamical friction force on the intruding bh . the regularized equations of motion are integrated using the bulirsch - stoer ( bs ) method @xcite based on romberg extrapolation . for unperturbed sinusoidal motion , the bs integrator requires only two or three timesteps per orbital period ! when the binary and third body are far apart we switch to two - body motion ( of the single bh and binary com ) using a 4th - order runge - kutta ( rk4 ) method . we simultaneously evolve the binary semi - major axis and eccentricity using orbit averaged equations , @xmath264 dt$ ] and @xmath265 dt$ ] , where @xmath266 and @xmath267 are the contributions from stellar interactions and gravitational radiation . the timesteps are adaptively controlled with a simple step - doubling scheme : at each step the 14 numbers @xmath268 are all required to remain the same to within an error @xmath269 @xmath116 @xmath270 under doubling of the step size . to avoid wasting computation time when any of these values approach zero , we accept agreement to @xmath271 decimal places as an alternative criterion for convergence . for the calculations reported in this paper we set @xmath272 . the relative perturbation to the binary from the third body at apoapsis , @xmath273 is used to decide which integration method to use at any given time . here @xmath274 is the apoapsis distance between @xmath275 and @xmath276 , @xmath86 is the intruder mass , and @xmath277 is the distance of the intruder from the binary com . we switch to two - body rk4 integration each time @xmath278 falls below @xmath279 and call the _ chain _ code again when @xmath278 reaches @xmath280 . we choose different @xmath278 thresholds for beginning and ending close encounters to prevent overly frequent toggling between the two methods . when @xmath281 bhs remain in the simulation ( after coalescence of the inner binary or escape of one or more bhs from the galaxy ) , we primarily use the rk4 integrator , but call the _ chain _ code to treat very close two - body encounters . since chain regularization is defined only for three or more bodies , we add a light and distant `` dummy '' particle when using this method for two - body motion . during the two - body motion we declare the single bh or binary ( remnant ) to have escaped if its distance from the galactic center exceeds 500 kpc and its specific energy @xmath282 exceeds @xmath283 , the energy needed to escape from @xmath284 to infinity . we declare the binary to have coalesced during a close encounter when ( i ) @xmath285 , where @xmath286 is the schwarzchild radius of the larger member of the pair , or ( ii ) @xmath287 @xmath288 @xmath289 and @xmath287 @xmath288 50 yrs while @xmath290 , where @xmath291 is the current outer binary dynamical time . during the rk4 integration we require that @xmath287 @xmath288 50 yrs or @xmath292 at periapsis , where @xmath293 are the schwarzchild radii of the two binary members . upon coalescence we replace the pair with a single body of mass @xmath222 and the com position and velocity . a run ends when ( a ) only one smbh remains in the galaxy and it has settled to the center of the potential by dynamical friction ; ( b ) two bhs remain and have formed a hard binary at the galactic center ; ( c ) the physical time exceeds @xmath294 , the current age of the universe minus the halo formation time ; ( d ) all bhs have escaped the galaxy ; or ( e ) the physical time spent in a call to _ chain _ exceeds a maximum allowed time @xmath295 . the last condition is added to avoid spending too much computation time on very long close encounters . during the two - body evolution we apply a dynamical friction force given by chandrasekhar s formula @xcite , @xmath296}{v^{2 } } \hat{v},\ ] ] where @xmath297 , to the single bh and binary com . the factor in square brackets @xmath204 @xmath298 for @xmath299 and @xmath204 @xmath300 for @xmath301 . we take @xmath302 , 1 \right\}\ ] ] for the coulomb logarithm , where @xmath38 is the bh s distance from the galactic center . for @xmath303 in equation we use @xmath304 $ ] , effectively capping the density at its value at the bh radius of influence , @xmath305 , when the bhs pass through the core . the semi - major axis @xmath72 of the binary also evolves under stellar - dynamical friction as it wanders through the galaxy . however chandrasekhar s formula applied separately to the binary constituents does not give a good description of this evolution , since the hard binary loses energy through close 3-body encounters with stars , while equation relies on the assumption that the energy loss is dominated by weak two - body encounters . we approximate the evolution of @xmath72 using a formulation for the decay rate of a hard , massive binary in a uniform and isotropic sea of stars developed in @xcite and @xcite . the formulation was calibrated with an extensive series of 3-body scattering experiments in @xcite and tested against n - body simulations in @xcite . the binary decay rate is given by @xmath306 where the hardening rate @xmath307 can be approximated by the empirical fitting function @xcite @xmath308^{1/2}}.\ ] ] here @xmath309 is the characteristic velocity distinguishing the hard binary regime stars with @xmath310 can not be easily captured into bound orbits and preferentially harden the binary in close encounters . in our simulations the binary com is often speeding through the stellar medium at @xmath311 after an energetic ejection , so the stellar medium looks `` hotter '' in its frame of reference . to account for this we replace @xmath9 in equations and with @xmath312 , a good approximation since @xmath307 is not very sensitive to the shape of the distribution function ( e.g. @xmath313 for a maxwellian vs. @xmath314 for a uniform velocity distribution ) . for @xmath303 in equation we took @xmath304 $ ] as we did for the drag on the com . we ignored the mild eccentricity evolution @xmath315 , which is shown in @xcite to be far weaker than that predicted by chandrasekhar s formula for hard eccentric binaries . when the amplitude of oscillation of one of the two masses falls below @xmath316 , we stop integrating its motion and place it at rest at the galactic center until the second body returns to within a distance of twice the break radius , @xmath317 . if the settled mass is the binary , then we also stop updating its semi - major axis for stellar hardening , assuming that it clears out its loss cone and stalls once it stops moving about the nucleus and encountering new stars . since the total mass in loss cone stars is small compared to the bh mass in the low - density galaxies that we consider , to good approximation the binary stalls as soon as the replenishing mechanism ( motion ) shuts off . during close encounters between the three bhs an orbit - averaged prescription for stellar - dynamical friction is not feasible . however the triples are still marginally stable at the boundary given by equation , so we apply a drag force given by chandrasekhar s formula with @xmath318 to the intruder at the beginning of each run . at the onset of chaotic interactions in the first encounter ( defined loosely by the first time the closest pair is not formed by the original binary members ) this perturbation is shut off , and it remains off in all later close encounters . fortunately the chaotic interactions occur on timescales very short compared to a dynamical friction time , so it is valid to neglect stellar dissipation during close encounters . gravitational radiation is modelled using the @xmath319 $ ] post - newtonian ( 2.5pn ) back - reaction acceleration computed by @xcite , evaluated in the two - body com frame ( e.g. gultekin et al . 2006 ) , @xmath320 } + \vec{v } { \big [ } -\frac{6g(m_{1}+m_{2})}{r}-2v^{2 } { \big ] } { \big \}}.\end{gathered}\ ] ] @xmath321 and @xmath322 are the relative positions and velocities of the two masses . we sum the force linearly over all pairs , a valid approximation provided the perturbations from the third body and other external tidal forces are instantaneously small at periapsis . when averaged over a complete orbit , equation is equivalent to the @xcite equations for the binary semi - major axis and eccentricity , [ pet ] @xmath323 however when @xmath324 comes close to one on hyperbolic orbits , so that @xmath325 , @xmath326 as given by equation becomes _ positive _ , though we know physically that gravitational waves can only carry energy away from the system . to give the correct answer averaged over an orbit , this positive contribution must be cancelled by extra energy loss near periapsis , making the equation potentially sensitive to numerical error . this effect is much less pronounced in the @xcite form than in other expressions derived for the radiation back - reaction acceleration - they derived the formula specifically for practical use on the problem of two point masses ( see appendix of @xcite and references therein ) . for computational ease we neglect the lower - order 1 - 2pn terms ( precession of the periapsis ) in the post - newtonian expansion . though much larger in magnitude than the radiation reaction force , these terms are unimportant in the statistical sense because they conserve the intrinsic properties of the system , such as energy @xcite . we need not concern ourselves with relativistic precession destroying the kozai resonance since the semi - major axis ratio given by equation is much smaller than that of equation . one way to establish the reliability of our integration methods is to test them on problems with known solutions . 6 shows an example on the two - body problem with gravitational radiation . @xmath116 @xmath21 yrs at the beginning of each integration . _ left : _ semi - major axis evolution for initial eccentricities of @xmath327 @xmath116 0.0 , 0.5 , 0.9 , and 0.99 ( bottom to top ) . _ right : _ eccentricity evolution for @xmath327 @xmath116 0.5 , 0.9 , and 0.99 . the curves indistinguishability demonstrates the reliability of all three methods . _ lower panels : _ hyperbolic orbits with impact parameters @xmath328 set to 80% and 120% of the critical value for gravitational radiation capture , computed using the dd acceleration in the _ chain _ code . _ left : _ 0.8@xmath329 ; bh is captured . _ right : _ 1.2@xmath329 ; bh is not captured . the blue asterisks are points along the newtonian trajectory ( without gravitational radiation ) . the deviation from the newtonian trajectory after periapsis can be seen in both plots , even though the energy remains positive in the latter . ] the upper panels show the evolution of the semi - major axis @xmath72 and eccentricity @xmath13 of four decaying elliptical orbits , computed using ( a ) our rk4 integrator and equation , with an error tolerance of @xmath330 , ( b ) the _ chain _ code and equation , with @xmath331 , and ( c ) the @xcite equations . in each case the initial semi - major axis @xmath332 was chosen to give a gravitational radiation timescale of @xmath333 yrs , and the four curves ( from bottom to top ) are for eccentricities of 0.0 , 0.5 , 0.9 , and 0.99 . the agreement of the three computation methods demonstrates the reliability of both the rk4 integrator and our implementation of the _ chain _ code in handling dissipative forces . the lower panels show two hyperbolic orbits with periapsis distances around 30 times the schwarzchild radius @xmath286 of the larger bh , computed using equation in _ chain_. the rk4 integrator was found to fail some tests on very close approaches from hyperbolic orbits with gravitational radiation , so we treat all such approaches using the regularized _ chain _ code in our runs , even during the unperturbed binary evolution . the blue asterisks are points along the newtonian orbits while the red solid lines show the trajectories with gravitational radiation . there is a simple analytic expression for the maximum periapsis distance for gravitational radiation capture from a hyperbolic orbit , @xmath334^{2/7},\ ] ] where @xmath275 and @xmath276 are the masses of the two bodies and @xmath335 is their relative velocity at infinity . the orbit on the lower left begins at 80% of the critical impact parameter and the incoming bh is captured . on the right the intruding bh starts at 120% of the critical impact parameter and is not captured , though the deviation from the newtonian trajectory due to the energy radiated at periapsis can be seen on the way out . we tried iterating over impact parameters close to the critical value and found that the code reproduces equation to within a part in @xmath78 for periapsis distances @xmath336 , and to within a part in @xmath337 for @xmath338 . we also evaluated the performance of the code by repeating our canonical set of 1005 runs with a static inner profile to check the precision of energy conservation . in fig . 7 we histogram the energy errors , computed as @xmath339 - e_{f}}{e_{0}-e_{f } } \right|,\ ] ] where @xmath340 and @xmath341 are the initial and final energies , and the two terms in the sum under the integral are the work done by dynamical friction and gravitational radiation during the current stage of the code . in the upper panel we separately plot the errors for close 3-body encounters , rk4 integration of the unperturbed binary motion ( `` far '' ) , and close two - body encounters computed with _ chain _ during the unperturbed binary evolution . the plot includes all code stages where the energy dissipated was at least @xmath342 in code units , or about a part in @xmath343 of the initial binding energy of the system . the black ( heavy ) histogram shows the errors for close encounters normalized to the _ initial _ energy instead of the dissipated energy in the denominator of equation , since the energy dissipated was very small in many close encounters . in the lower panel we combine the energy errors from the various code stages to get an effective energy error for each entire run , @xmath344 we had to combine the separate errors to obtain @xmath345 since the galactic potential is handled slightly differently during different stages of the code , e.g. the triaxial modification is applied only during the rk4 integration . in a large majority of cases @xmath345 falls between @xmath346 and @xmath347 , and energy is conserved to better than a part in @xmath348 in every run . the excellent energy conservation gives us confidence in the robustness of our integration methods .

merging of old massive galaxies with little gas promotes the formation of low - density nuclei where smbh binaries are expected to survive over long times . our simulations include a smooth background potential consisting of a stellar bulge plus a dark matter halo , drag forces due to gravitational radiation and dynamical friction on the stars and dark matter , and a simple model of the time evolution of the inner density profile under heating and mass ejection by the smbhs . we find that the binary pair coalesces as a result of repeated close encounters in% of our runs , and in% of cases a new eccentric binary forms from the third smbh and binary remnant and coalesces during the run time . in about 40% of the runs the lightest bh the triple systems typically scour out cores with mass deficits - 2 their total mass , which can help to account for the large cores observed in some massive elliptical galaxies , such as m87 . the high coalescence rate , prevalence of very high - eccentricity orbits , and gravitational radiation `` spikes '' during close encounters in our runs , may provide interesting signals for the future laser interferometer space antenna ( lisa ) .

galaxies with stellar bulges are generically observed to host supermassive black holes ( smbhs ) . the hierarchical merging of galaxies should therefore lead to the formation of smbh binaries . merging of old massive galaxies with little gas promotes the formation of low - density nuclei where smbh binaries are expected to survive over long times . if the binary lifetime exceeds the typical time between mergers , then triple black hole ( bh ) systems may form . we study the statistics of close triple - smbh encounters in galactic nuclei by computing a series of 3-body orbits with physically - motivated initial conditions appropriate for giant elliptical galaxies . our simulations include a smooth background potential consisting of a stellar bulge plus a dark matter halo , drag forces due to gravitational radiation and dynamical friction on the stars and dark matter , and a simple model of the time evolution of the inner density profile under heating and mass ejection by the smbhs . we find that the binary pair coalesces as a result of repeated close encounters in% of our runs , and in% of cases a new eccentric binary forms from the third smbh and binary remnant and coalesces during the run time . in about 40% of the runs the lightest bh is left wandering through the galactic halo or escapes the galaxy altogether , but escape of all three smbhs is exceedingly rare . the triple systems typically scour out cores with mass deficits - 2 their total mass , which can help to account for the large cores observed in some massive elliptical galaxies , such as m87 . the high coalescence rate , prevalence of very high - eccentricity orbits , and gravitational radiation `` spikes '' during close encounters in our runs , may provide interesting signals for the future laser interferometer space antenna ( lisa ) . [ firstpage ] black hole physics cosmology : theory galaxies : elliptical and lenticular , cd galaxies : interactions galaxies : nuclei methods : numerical

1608.08462

i

in @xcite , we constructed a perturbative invariant of 3-manifolds with @xmath8 by modifying fukaya s morse homotopy theoretic invariant ( @xcite ) . in this paper , we extend the invariant of @xcite to 3-manifolds with higher first betti numbers and see that it is effectively utilized in the study of finite type invariants of 3-manifolds defined by garoufalidis and levine ( @xcite ) . a theory of finite type invariant of 3-manifolds was first developed by ohtsuki in @xcite for integral homology 3-spheres . ohtsuki s finite type invariant was defined by means of a filtration on the module spanned by diffeomorphism classes of integral homology 3-spheres . ohtsuki s filtration was defined with respect to surgeries on algebraically split framed links in @xmath9 . it has been shown that the graded quotient of his filtration is isomorphic over @xmath5 to certain space of trivalent graphs ( @xcite ) , with the help of the lmo invariant ( @xcite ) . garoufalidis and levine defined in @xcite a filtration for 3-manifolds equipped with some degree 1 map ( @xmath0-homology equivalence ) to a fixed 3-manifold . they gave an upper bound for the graded quotient of their filtration by the space of @xmath10-decorated graphs . to see that this correspondence is an isomorphism , it is important to find a perturbative invariant that is a counterpart to the lmo invariant . some perturbative invariants were found for 3-manifolds with @xmath8 in @xcite . we construct a perturbative invariant for @xmath0-homology equivalences to the 3-torus via morse homotopy theory and by utilizing it we show that when the target manifold is the 3-torus @xmath11 , the space of @xmath10-decorated graph is naturally isomorphic to the graded quotient in garoufalidis levine s filtration . in this paper , we mainly consider the filtration of @xcite defined by surgery on @xmath12-links ( or graph claspers , @xcite ) . we also give a formula relating the value of our first invariant @xmath13 for the connected sum of @xmath11 with an integral homology sphere @xmath14 to the casson invariant of @xmath14 . first , we recall the definitions of @xcite . we fix an oriented closed 3-manifold @xmath1 and let @xmath2 . a degree 1 map @xmath15 from another oriented closed 3-manifold @xmath16 is said to be a _ @xmath0-homology equivalence _ if the induced map @xmath17 is an isomorphism . here , @xmath18 are the @xmath10-coverings of @xmath19 . two @xmath0-homology equivalences @xmath15 and @xmath20 are _ diffeomorphically equivalent _ if there exists a diffeomorphism @xmath21 such that @xmath22 is homotopic to @xmath23 . let @xmath24 be the set of all diffeomorphism equivalence classes of @xmath0-homology equivalences @xmath15 . in the following , we assume for simplicity that @xmath25 . in @xcite , it was shown that when @xmath25 a @xmath0-homology equivalence @xmath15 can be represented uniquely by a nullhomotopic framed link in @xmath1 is determined . ] . moreover , in @xcite , a `` surgery obstruction map '' @xmath26 to certain semi - group @xmath27 was defined by assigning a matrix of @xmath0-valued linking numbers ( or @xmath10-equivariant linking number ) of the framed link . . an element of @xmath28 is given by a _ @xmath10-algebraically split framed link _ , namely , a framed link such that its @xmath0-linking matrix is a diagonal matrix all of whose diagonal entries are @xmath30 and that its components are nullhomotopic . let @xmath31 be the vector space over @xmath5 spanned by the set @xmath28 . in @xcite , it was shown that any element of @xmath28 can be obtained from @xmath1 by surgery on a @xmath12-link . we consider the filtration of @xmath31 given by surgery on @xmath12-links . for the terminology related to @xmath12-links ( or graph - claspers ) , we refer the reader to @xcite . let @xmath32 be a @xmath12-link consisting of disjoint @xmath12-graphs in @xmath16 with only nullhomotopic leaves and let @xmath33=\sum_{g'\subset g}(-1)^{|g'|}m^{g'},\ ] ] where @xmath34 denotes the map @xmath35 obtained by surgery on @xmath36 . namely , by surgery a 4-cobordism @xmath37 between @xmath34 and @xmath16 is obtained and the identity map on @xmath16 extends over it uniquely up to homotopy , and then compose with the @xmath0-homology equivalence @xmath15 . let @xmath38 ( @xmath39 ) be the subspace of @xmath31 spanned by @xmath40 $ ] for @xmath12-link @xmath41 with @xmath42 disjoint @xmath12-graphs . this defines a descending filtration on @xmath31 : @xmath43 . a linear map @xmath44 to a vector space @xmath45 is said to be a _ finite type invariant of type @xmath42 _ if @xmath46 . we call a pair @xmath47 of an abstract , vertex - oriented , edge - oriented trivalent graph @xmath48 and a map @xmath49 a _ @xmath10-decorated graph_. we will denote @xmath47 also by @xmath50 or @xmath51 ( @xmath52 ) . we define the _ degree _ of a @xmath10-decorated graph as the number of vertices . let @xmath53 be the vector space over @xmath5 spanned by @xmath50 of degree @xmath42 quotiented by the relations as , ihx , orientation reversal , linearity and holonomy ( figure [ fig : relations ] ) . when @xmath54 , we call @xmath47 a _ monomial _ @xmath10-decorated graph . a monomial @xmath10-decorated graph @xmath47 determines uniquely a homotopy class @xmath55 of a map @xmath56 . we choose a ribbon structure on @xmath48 that is compatible with the vertex - orientation , choose an embedding of the ribbon graph into @xmath1 that represents @xmath55 and replace the image with a set @xmath57 of @xmath42 disjoint @xmath12-graphs . by assigning @xmath58 $ ] to @xmath50 , a well - defined linear map @xmath59 is obtained . in @xcite , it was shown that this map is surjective$]-coefficients . ] . when @xmath42 is odd , @xmath60 and hence @xmath61 . the main theorem of this paper is the following . [ thm : main ] when @xmath4 , the map @xmath62 is an isomorphism . in the rest of this paper we prove theorem [ thm : main ] . in section [ s : perturbative ] , we define a perturbative invariant @xmath63 for @xmath0-homology equivalence . the way of construction of the invariant is almost the same as @xcite . in section [ s : z - fti ] , we show that @xmath63 is a finite type invariant and compute its values on @xmath64 . the main theorem follows as a corollary of this result and of the surjectivity of @xmath65 of @xcite . the outline of the proof is analogous to that of @xcite . instead of modifying chains , we modify gradients of morse functions to localize the counts of graphs in an alternating sum . in section [ s : example ] , we relate the value of @xmath66 for an integral homology sphere @xmath14 to the casson invariant of @xmath14 , as a straightforward example . throughout this paper , manifolds and maps between them are assumed to be smooth . for a gradient @xmath67 of a morse function , we denote by @xmath68 ( resp . @xmath69 ) the ascending manifold ( resp . descending manifold ) of a critical point @xmath70 of @xmath67 . we denote by @xmath71 ( resp . @xmath72 ) the usual compactification of @xmath68 ( resp . @xmath69 ) into a smooth manifold with corners ( e.g. , @xcite ) . we will often abbreviate @xmath73 etc . as @xmath74 etc . for simplicity . by a ribbon graph , we mean a compact surface equipped with a handle structure with only 0- and 1-handles .

garoufalidis and levine defined a filtration for 3-manifolds equipped with some degree 1 map (-homology equivalence ) to a fixed 3-manifold and showed that there is a natural surjection from a space of-decorated graphs to the graded quotient of the filtration over $ ] . in this paper , we show that in the case of the surjection of garoufalidis levine is actually an isomorphism over . for the proof , we construct a perturbative invariant by applying fukaya s morse homotopy theoretic construction to a local system of the quotient field of . the first invariant is an extension of the casson invariant to-homology equivalences to the 3-torus . the results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with . ( 2016 )

garoufalidis and levine defined a filtration for 3-manifolds equipped with some degree 1 map (-homology equivalence ) to a fixed 3-manifold and showed that there is a natural surjection from a space of-decorated graphs to the graded quotient of the filtration over $ ] . in this paper , we show that in the case of the surjection of garoufalidis levine is actually an isomorphism over . for the proof , we construct a perturbative invariant by applying fukaya s morse homotopy theoretic construction to a local system of the quotient field of . the first invariant is an extension of the casson invariant to-homology equivalences to the 3-torus . the results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with . ( 2016 )

q-bio0702003

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genetic switch systems with mutual repression of two transcription factors , have been studied using a combination of deterministic and stochastic methods . these system exhibit bistability , namely two stable states such that spontaneous transitions between them are rare . induced transitions take place as a result of an external stimulus . we have studied several variants of the genetic switch , which exhibit cooperative binding , exclusive binding , protein - protein interactions and degradation of bound repressors . for each variant we examined the range of parameters in which bistability takes place . numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems . we have shown that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding . the average time @xmath71 between spontaneous transitions was evaluated as a function of the biological parameters . hill - function models are simplifications of rate - law equations . when derived directly from rate laws , @xmath3 is expected to take only integer values , which represents the number of transcription factors required to perform the regulation . however , when these models are used for fitting empirical data , @xmath3 is a fitting parameter which may take non - integer values . extended circuits that include the mrna level have also been studied . it was found that there is a perfect agreement between the results of the res for the simplified and the extended circuits , and excellent agreement with only a slight difference in peak shapes in the me . results of @xmath71 calculated in this procedure , agree with results obtained from averaging a large number of monte carlo simulations , up to a numerical factor of 2 . this factor rises from the fact the in the procedure we described the system can also return from the @xmath2 dominated peak to the @xmath1 dominated peak .

genetic switch systems with mutual repression of two transcription factors are studied using deterministic methods ( rate equations ) and stochastic methods ( the master equation and monte carlo simulations ) . these systems exhibit bistability , namely two stable states such that spontaneous transitions between them are rare . induced transitions may take place as a result of an external stimulus . we study several variants of the genetic switch and examine the effects of cooperative binding , exclusive binding , protein - protein interactions and degradation of bound repressors . numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems . we show that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding . the average time between spontaneous transitions is evaluated as a function of the biological parameters .

genetic switch systems with mutual repression of two transcription factors are studied using deterministic methods ( rate equations ) and stochastic methods ( the master equation and monte carlo simulations ) . these systems exhibit bistability , namely two stable states such that spontaneous transitions between them are rare . induced transitions may take place as a result of an external stimulus . we study several variants of the genetic switch and examine the effects of cooperative binding , exclusive binding , protein - protein interactions and degradation of bound repressors . we identify the range of parameters in which bistability takes place , enabling the system to function as a switch . numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems . we show that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding . the average time between spontaneous transitions is evaluated as a function of the biological parameters .

1408.1653

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hexagonal boron nitride ( hbn ) , a wide - bandgap insulator , has recently emerged as a sort of `` magic '' substrate for exfoliated graphene sheets . early on , it was demonstrated @xcite that hbn is a much better substrate than @xmath2the ordinary substrate @xcite for much of the early work in graphene physics and devices because its surface is much flatter and because it presents a much smaller number of charged impurities . exfoliated graphene sheets deposited on hbn ( g / hbn ) or graphene sheets that are encapsulated in hbn ( hbn / g / hbn ) have therefore much larger mobilities @xcite than those that are deposited on @xmath2 . subsequently , vertical stacks @xcite comprising graphene and hbn have been used for proof - of - concept devices such as field - effect tunneling transistors @xcite and fundamental studies of electron - electron interactions @xcite . more recent experimental work @xcite has demonstrated that hbn substantially alters the electronic spectrum of the massless dirac fermion ( mdf ) @xcite carriers hosted in a nearby graphene sheet . indeed , when graphene is deposited on hbn , it displays a moir pattern @xcite , a modified tunneling density of states @xcite , and self - similar transport characteristics in a magnetic field @xcite . this spectral reconstruction of the mdf energy - momentum dispersion relation is , however , only relevant in the case of long - wavelength moir superlattices ( superlattice period @xmath3 ) , which occur when the twist angle between the graphene and hbn crystals is small . short - wavelength superlattices yield changes of the mdf spectrum at dopings that are not achievable by electrostatic gating . finally , we would like to mention that the authors of ref . have demonstrated that hbn / g / hbn samples , in which the role of contact resistance is minimized by using a suitable geometry , can display very large mobilities , which are solely limited by scattering of electrons against graphene s acoustic phonons . according to boltzmann - transport theory @xcite , this scattering mechanism yields @xcite a mobility @xmath4 that decreases like @xmath5 with increasing carrier density @xmath6in good agreement with experimental data @xcite and , therefore , a drude dc transport scattering time @xmath7 that decreases like @xmath8 in the same limit . we remind the reader that , in the same theoretical framework and by virtue of screening , scattering against charged impurities yields a mobility that increases with increasing carrier density @xcite . high - quality vertical heterostructures comprising graphene and hbn crystals may have a large impact on the success of graphene plasmonics @xcite , an emerging field of research that has recently attracted a great deal of attention . the goal of graphene plasmonics is to exploit the interaction of infrared light with dirac plasmons " ( dps)the self - sustained density oscillations of the mdf liquid in a doped graphene sheet @xcite for a variety of applications such as infrared @xcite and terahertz @xcite photodetectors , strong light - matter interactions @xcite , enhanced light absorption @xcite and bio - sensing @xcite . interest in graphene plasmonics considerably increased after two experimental groups @xcite showed that the dp wavelength is much smaller than the illumination wavelength , allowing an extreme concentration of electromagnetic energy , and that it is easily gate tunable . these early experiments , based on scattering - type near - field optical spectroscopy ( s - snom ) , were not optimized to minimize dp losses and therefore maximize the plasmon inverse damping ratio . microscopic calculations targeting the role of electron - electron interaction effects @xcite beyond the random phase approximation ( rpa ) and charged impurity scattering @xcite indicate that losses can be strongly reduced by using hbn rather than @xmath2 as a substrate . indeed hbn has both a larger static dielectric constant , thus suppressing the strength of electron - electron interactions , and a much lower impurity concentration than @xmath9 . the impact of electron - phonon scattering on the lifetime of hpp modes was recently addressed in ref . , where the authors showed that the inverse damping ratio of mid - infrared hpp modes is strongly limited by scattering against optical phonons of both the sio@xmath10 substrate and the graphene sheet . the hpp mode damping rate was estimated by introducing a self - energy correction in the _ local _ conductivity @xmath11 , which took into account electron - impurity , electron - phonon , and edge scattering . the contribution of electron - phonon interactions to the hpp lifetime was assumed to be independent of momentum . the dependence of the damping rate on momentum stemmed from the contribution due to the scattering against the edges of the sample . in this article we present a theoretical study of the performance of hbn / g / hbn stacks for applications in the field of graphene plasmonics . more precisely , we present a microscopic theory of the damping rate of hpp modes in a graphene sheet encapsulated between two hbn semi - infinite slabs . we focus on scattering of hpp modes in a graphene sheet against i ) graphene s acoustic phonons and ii ) hbn optical phonons , two sources of scattering that are expected to play a key role in limiting the lifetime of collective density oscillations in hbn / g / hbn stacks . besides achieving good quantitative agreement with recent experimental work @xcite , we confirm the important fact that the plasmon lifetime is not necessarily correlated with the mobility @xcite ( i.e. with the transport lifetime that controls the uniform dc conductivity ) . more accurately , the plasmon lifetime is controlled by the non - local conductivity @xmath12 , where @xmath13 is the wavevector and @xmath14 is the plasmon frequency . in the present case , retaining both the wavevector- and frequency - dependence of the non - local conductivity , we are able to show that the plasmon lifetime is in fact _ anti - correlated _ with the transport mobility . indeed , while the mobility decreases with increasing carrier density @xcite , the plasmon lifetime shows exactly the opposite behavior . this article is organized as follows . in sect . [ sect : qualityfactor ] we introduce the hpp mode inverse damping ratio @xmath15 and relate it to the microscopic _ non - local _ dynamical conductivity @xmath16 of a 2d electron liquid embedded in a medium with a generic frequency - dependent dielectric function @xmath17 . in sect . [ sect : theory ] we present our microscopic theory of the hpp mode dispersion relation and losses in hbn / g / hbn stacks due to electron - phonon scattering . in sects . [ sect : intrinsic_acoustic ] and [ sect : optical_substrate ] we describe the details of the electron - phonon interactions we have considered , i.e. scattering of 2d mdfs against graphene s acoustic phonons and hbn optical phonons , respectively . finally , in sect . [ sect : summary ] we report a summary of our main results and conclusions . we have presented a number of relevant technical details in four appendices , with the aim of making our article as self - contained as possible .

we focus on scattering of these modes against graphene s acoustic phonons and hbn optical phonons , two sources of scattering that are expected to play a key role in hbn / g / hbn stacks . we confirm that the plasmon lifetime is not directly correlated with the mobility : in fact , it can be anti - correlated .

graphene sheets encapsulated between hexagonal boron nitride ( hbn ) slabs display superb electronic properties due to very limited scattering from extrinsic disorder sources such as coulomb impurities and corrugations . such samples are therefore expected to be ideal platforms for highly - tunable low - loss plasmonics in a wide spectral range . in this article we present a theory of collective electron density oscillations in a graphene sheet encapsulated between two hbn semi - infinite slabs ( hbn / g / hbn ) . graphene plasmons hybridize with hbn optical phonons forming hybrid plasmon - phonon ( hpp ) modes . we focus on scattering of these modes against graphene s acoustic phonons and hbn optical phonons , two sources of scattering that are expected to play a key role in hbn / g / hbn stacks . we find that at room temperature the scattering against graphene s acoustic phonons is the dominant limiting factor for hbn / g / hbn stacks , yielding theoretical inverse damping ratios of hybrid plasmon - phonon modes of the order of- , with a weak dependence on carrier density and a strong dependence on illumination frequency . we confirm that the plasmon lifetime is not directly correlated with the mobility : in fact , it can be anti - correlated .

1408.1653

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we have presented a theory of hybrid plasmon - phonon modes in a graphene sheet encapsulated between two hexagonal boron nitride ( hbn ) semi - infinite slabs ( hbn / g / hbn ) . by using linear response theory and the random phase approximation , we have calculated the dispersion relation of hybrid plasmon - phonon modes that stem from the hybridization between graphene dirac plasmons and hbn optical phonons . the uniaxial hyperbolic nature of hbn yields three plasmon - phonon branches separated by two reststrahlen bands . we have carried out a detailed study of the inverse damping ratio of these plasmon - phonon modes . we have considered two possible sources of scattering limiting their lifetime : scattering against graphene s acoustic phonons and hbn optical phonons . we have discovered that scattering against intrinsic acoustic phonons is the dominant limiting factor in hbn / g / hbn stacks and that it yields theoretical inverse damping ratios of hybrid plasmon - phonon modes that lie in the range @xmath0-@xmath121 at room temperature , with a weak dependence on carrier density and a strong dependence on illumination wavelength . while the current work focuses on room temperature for its relevance to applications , the inverse damping ratio is expected to increase when temperature is lowered and the scattering of electrons with intrinsic acoustic phonons is suppressed . numerical calculations on the temperature dependence of the inverse damping ratio at a fixed carrier density and illumination frequency will be shown elsewhere @xcite . our theoretical predictions indicate that hbn / g / hbn stacks can be very fruitfully used as a low - loss and gate tunable platform for plasmonics in the mid - infrared spectral range . we will show somewhere else @xcite that our results are in excellent quantitative agreement with recent s - snom measurements . it is a great pleasure to thank andrea tomadin and leonid levitov for many useful discussions . this work was supported by the doe grant de - fg02 - 05er46203 ( a.p . and g.v . ) , a research board grant at the university of missouri ( a.p . and g.v . ) , the e.u . through the graphene flagship ( contract no cnect - ict-604391 ) program ( m.c . , m.l . , f.h.l.k . , and m.p . ) , and the italian ministry of education , university , and research ( miur ) through the programs firb - futuro in ricerca 2010 " - project plasmograph ( grant no . rbfr10m5bt ) and progetti premiali 2012 " - project abnanotech ( m.p . ) . m.c . acknowledges also the support of miur - firb2012 - project hybridnanodev ( grant no . rbfr1236vv ) .

graphene sheets encapsulated between hexagonal boron nitride ( hbn ) slabs display superb electronic properties due to very limited scattering from extrinsic disorder sources such as coulomb impurities and corrugations . such samples are therefore expected to be ideal platforms for highly - tunable low - loss plasmonics in a wide spectral range . in this article we present a theory of collective electron density oscillations in a graphene sheet encapsulated between two hbn semi - infinite slabs ( hbn / g / hbn ) . graphene plasmons hybridize with hbn optical phonons forming hybrid plasmon - phonon ( hpp ) modes . we find that at room temperature the scattering against graphene s acoustic phonons is the dominant limiting factor for hbn / g / hbn stacks , yielding theoretical inverse damping ratios of hybrid plasmon - phonon modes of the order of- , with a weak dependence on carrier density and a strong dependence on illumination frequency .

graphene sheets encapsulated between hexagonal boron nitride ( hbn ) slabs display superb electronic properties due to very limited scattering from extrinsic disorder sources such as coulomb impurities and corrugations . such samples are therefore expected to be ideal platforms for highly - tunable low - loss plasmonics in a wide spectral range . in this article we present a theory of collective electron density oscillations in a graphene sheet encapsulated between two hbn semi - infinite slabs ( hbn / g / hbn ) . graphene plasmons hybridize with hbn optical phonons forming hybrid plasmon - phonon ( hpp ) modes . we focus on scattering of these modes against graphene s acoustic phonons and hbn optical phonons , two sources of scattering that are expected to play a key role in hbn / g / hbn stacks . we find that at room temperature the scattering against graphene s acoustic phonons is the dominant limiting factor for hbn / g / hbn stacks , yielding theoretical inverse damping ratios of hybrid plasmon - phonon modes of the order of- , with a weak dependence on carrier density and a strong dependence on illumination frequency . we confirm that the plasmon lifetime is not directly correlated with the mobility : in fact , it can be anti - correlated .

astro-ph0612256

c

the extreme interest in grb050904 has motivated several groups to analyze the burst data and suggest interpretations . these works fall into two categories : ( 1 ) comparing the properties of grb050904 with other bursts ; and ( 2 ) making fits to the grb050904 data either in the framework of internal shock or external shock models . we give a brief description below of the work of other groups and mention key differences between their work and ours . in category ( 1 ) , @xcite made a composite @xmath15-band light curve starting from @xmath230 days to @xmath231 days . after applying extinction correction , they shifted grb050904 as well as lower - redshift bursts to @xmath232 , and made a comparison of afterglow lightcurves . they found that grb050904 is much brighter than other grbs at early times , but of roughly equal brightness at late times . thus they conclude that grb050904 most likely is still a normal grb . the other analyses are all in category ( 2 ) . @xcite argued that grb050904 is a burst with extremely long central engine activity . they put all the observed data within the framework of the internal shock model . by contrast , we only treat the first several hundred seconds ( bat ) as internal shock activity in our model ( corresponding to the stages ( a ) and ( b ) for model ( a ) , and stage ( a ) for model ( b ) in fig . 1 ) . the late - time flares between @xmath233 and @xmath234 seconds may be due to internal shock , but we have interpreted portions of this emission as being due to the forward - shock afterglow . @xcite argued that the @xmath1s flare is from internal shocks on the basis of its fast decay ( the temporal index is @xmath235 relative to the trigger time ) and also because the optical - to- emission of the flare can not be described by a synchrotron radiation model @xcite . they made fits to all the available @xmath15-band data . they argue that the slow - decay portion of the lightcurve is due to energy injection . in our model , besides fitting over all the additional bands ( and radio ) , we propose a new mechanism for the flattening , namely that it is caused by the suppression of the synchrotron radiation by the interaction between the flare photons and afterglow electrons . separately , in our model ( b ) , we introduced a new reference time @xmath65 which flattens the decay index , and allows us to interpret the @xmath236s optical/ flare as arising in the reverse shock . @xcite made broadband model fits including the , nir / optical , and radio data . the difference with our work is partly that we have used the larger data set that later became available . we have included two data points as early as @xmath237 seconds , and we also included the @xmath238 band data observed @xmath239 days after the burst @xcite . since most of the other data are concentrated around @xmath240 seconds , the introduction of these additional and nir data have some impact on the final fitting result . the other major difference is that we have freed all the possible parameters and applied the mcmc method for the global fitting , making an efficient exploration of the full parameter space , and providing the posterior distributions ( including confidence intervals ) for each parameter . @xcite argued that the power - law - like decay right after the flare @xmath1s should be interpreted as forward shock emission . because the extrapolation of the late time flux to early times is lower than the observed value , they found that a wind - type environment was favored by the closure relationship for this early - time segment ( it should also be noted that their spectral index is smaller than @xcite ) . they propose a density - jump model for the afterglow evolution : before a certain radius , the density goes as @xmath241 and after that , the density is a constant ism model . at the transition point a termination shock is formed which lies around @xmath242pc from the central engine . in our model , we consider the same segment of data , but interpret it differently . we argue that the flux between 600 and 800 seconds arises from the combination of three sources : high - latitude emission of the prompt emission , flux from the flare , and forward shock emission . reviewing the data closely , we see that actually there are two other small flares between 800 and 2000 seconds , so the data around @xmath243 seconds has the contribution from the flares and the forward shock ( see fig . once we subtract the flare contribution from the observed data , we argue , the forward shock contribution is what remains . and in fact , we find that an extrapolation of the late - time flux to early times is consistent with this flare - subtracted early - time flux . there remain some substantial differences in parameter values between the two models that we have presented . for example , the most likely value for @xmath244 in model ( a ) is roughly @xmath245 , but the most likely value for @xmath244 in model ( b ) is around @xmath246 . it turns out that the main difference lies in the optical flux at early times . we take as an example the @xmath15-band light curve for the best fitting parameter set in model ( a ) . the flux slowly decays with an index of @xmath247 before @xmath248 and then follows a faster decay with a temporal index of @xmath249 after @xmath250 . the break at @xmath251 seconds is caused by the crossing of the electron s typical frequency @xmath79 through the optical observing frequency . since we have @xmath252 , then the smaller @xmath244 , the smaller the typical frequency , therefore the earlier the crossing time . the fit in model ( b ) requires an earlier break than in model ( a ) , so a smaller value of @xmath244 is expected in model ( b ) . since the observed flux from the forward shock is the same in both models , we expect a higher kinetic energy in model ( b ) . similarly , at the deceleration time , the reverse shock flux is larger than that from the forward shock , so a larger @xmath253 is expected in the reverse shock . we notice that in both our models we find a small value for @xmath88 , e.g. , the best fitting @xmath254 for model ( a ) and @xmath255 for model ( b ) . considering the radiative correction factor @xmath256 ( where @xmath62 is the deceleration time , and we set @xmath257 seconds for model ( a ) and @xmath258 seconds for model ( b ) ) , that is , the amount by which the kinetic energy at time @xmath42 is reduced by comparison to the kinetic energy at time @xmath62 , we find @xmath259 for model ( a ) and @xmath260 for model ( b ) at @xmath261 seconds after the burst . therefore , the radiative losses are mild in either case , consistent with our assumption that the afterglow evolves in an adiabatic fashion . our fitting results also show that the host galaxy dust extinction is quite small , @xmath262 mag or even smaller , consistent with the results from @xcite . they applied several different dust models ( mw , lmc and smc ) over the composite @xmath15-band , and all the models suggested zero or negligible extinction . in the context of grb990123 , it was suggested that the negligible extinction to the burst was the result of dust destruction by the strong burst and early - afterglow emission @xcite .

we suggest that the period of accelerated decay in the nir may be due to suppression of synchrotron radiation by inverse compton interaction of flare photons with electrons in the forward shock ; a subsequent interval of slow decay would then be due to a progressive decline in this suppression .

grb050904 at redshift , discovered and observed by _ swift_and with spectroscopic redshift from the subaru telescope , is the first gamma - ray burst to be identified from beyond the epoch of reionization . since the progenitors of long gamma - ray bursts have been identified as massive stars , this event offers a unique opportunity to investigate star formation environments at this epoch . apart from its record redshift , the burst is remarkable in two respects : first , it exhibits fast - evolving and optical flares that peak simultaneously ats in the observer frame , and may thus originate in the same emission region ; and second , its afterglow exhibits an accelerated decay in the near - infrared ( nir ) froms tos after the burst , coincident with repeated and energetic flaring activity . we make a complete analysis of available , nir , and radio observations , utilizing afterglow models that incorporate a range of physical effects not previously considered for this or any other grb afterglow , and quantifying our model uncertainties in detail via markov chain monte carlo analysis . in the process , we explore the possibility that the early optical and flare is due to synchrotron and inverse compton emission from the reverse shock regions of the outflow . we suggest that the period of accelerated decay in the nir may be due to suppression of synchrotron radiation by inverse compton interaction of flare photons with electrons in the forward shock ; a subsequent interval of slow decay would then be due to a progressive decline in this suppression . the range of acceptable models demonstrates that the kinetic energy and circumburst density of grb050904 are well above the typical values found for low - redshift grbs .

astro-ph0612256

c

in this paper we have performed the most extensive multiband analysis so far of the grb050904 afterglow . we have considered two scenarios : ( a ) only forward shock emission is considered , and the flares peaking at @xmath263 after the burst are assumed to be due to internal shocks ( or are otherwise independent of the afterglow ) ; and ( b ) the nir and flares at @xmath1s are ascribed to emission from the reverse shock when the ejecta has swept up enough material and starts to decelerate , the synchrotron radiation in the reverse shock produces the optical flare , and the self - compton scattering of synchrotron photons generates the flare observed in the xrt and bat energy bands . combining the early afterglow data with late - time observations in the , optical and radio , and using a markov chain monte carlo method , we present a full characterization of the posterior distributions ( including confidence intervals ) for the various parameters of our model fits . our best - fit parameter values for model ( a ) are @xmath203 , @xmath264 , @xmath201 , @xmath265 , @xmath266 , @xmath267 , and @xmath268mag , with a reduced chi - squared value of @xmath269 . our best - fit parameter values for model ( b ) are @xmath209 , @xmath270 , @xmath271 , @xmath272 , @xmath273 , @xmath274 , @xmath275mag , @xmath276 , and @xmath277 , with a reduced chi - squared value of @xmath278 . note that the subscripts @xmath90 and @xmath89 refer to the reverse shock and forward shock respectively , @xmath279 , and we have assumed @xmath280 . we have compared the density , the geometrically - corrected kinetic energy and the luminosity at @xmath192 hours derived here for our two models of grb050904 against those values for other bursts , as derived from afterglow modeling . the results for both models show that although the luminosity of grb050904 falls within the range for low - redshift grbs , the density and geometrically - corrected kinetic energy are both above the typical values for low - redshift grbs , which suggests that grb050904 may be a member of a distinct population of high - redshift , higher kinetic - energy bursts , whose properties differ from those of low - redshift grbs . a clear preference between our ( a ) and ( b ) models is hard to establish at present , since there is only one high - redshift grb known . one would like access to several more high - redshift grb observational datasets before attempting to discriminate between the two models . it is estimated that @xmath281 grbs are located at @xmath282 @xcite , and detection rate simulations by @xcite indicate that _ swift _ could detect grbs out to redshift @xmath283 , if they are present . @xcite also predict that @xmath284 of the _ swift _ grbs originate at @xmath282 . it appears that one can realistically expect a handful ( 5 to 10 ) of additional high - redshift grb detections with rapid follow - up in the next few years of the _ swift _ mission . in this case , the consistent application of mcmc methods , as used here , will lead efficiently to a set of statistically well - quantified , posterior parameter distributions and confidence intervals . this would enable a statistically meaningful comparison of high - redshift and low - redshift grb parameters , which might well lead us to a definite understanding of the physics and environments of grbs as a function of redshift , up to the highest redshifts detected . this would also have a substantial impact on the study of the large scale structure and star formation processes throughout the universe , and the properties of the cosmic reionization at @xmath285 . l. j. gou thanks g. cusumano for providing the data for this burst , as well as j. cummings , b. zhang , s. kobayashi , zh . wang , and j.f . wang for the helpful discussion . l. j. gou also thanks the support of sigma - xi fellowship . this research has been supported in part through nasa nag5 - 13286 . this research has used the resources of high performance computing ( hpc ) group at the penn state university .

apart from its record redshift , the burst is remarkable in two respects : first , it exhibits fast - evolving and optical flares that peak simultaneously ats in the observer frame , and may thus originate in the same emission region ; and second , its afterglow exhibits an accelerated decay in the near - infrared ( nir ) froms tos after the burst , coincident with repeated and energetic flaring activity . we make a complete analysis of available , nir , and radio observations , utilizing afterglow models that incorporate a range of physical effects not previously considered for this or any other grb afterglow , and quantifying our model uncertainties in detail via markov chain monte carlo analysis . in the process the range of acceptable models demonstrates that the kinetic energy and circumburst density of grb050904 are well above the typical values found for low - redshift grbs .

grb050904 at redshift , discovered and observed by _ swift_and with spectroscopic redshift from the subaru telescope , is the first gamma - ray burst to be identified from beyond the epoch of reionization . since the progenitors of long gamma - ray bursts have been identified as massive stars , this event offers a unique opportunity to investigate star formation environments at this epoch . apart from its record redshift , the burst is remarkable in two respects : first , it exhibits fast - evolving and optical flares that peak simultaneously ats in the observer frame , and may thus originate in the same emission region ; and second , its afterglow exhibits an accelerated decay in the near - infrared ( nir ) froms tos after the burst , coincident with repeated and energetic flaring activity . we make a complete analysis of available , nir , and radio observations , utilizing afterglow models that incorporate a range of physical effects not previously considered for this or any other grb afterglow , and quantifying our model uncertainties in detail via markov chain monte carlo analysis . in the process , we explore the possibility that the early optical and flare is due to synchrotron and inverse compton emission from the reverse shock regions of the outflow . we suggest that the period of accelerated decay in the nir may be due to suppression of synchrotron radiation by inverse compton interaction of flare photons with electrons in the forward shock ; a subsequent interval of slow decay would then be due to a progressive decline in this suppression . the range of acceptable models demonstrates that the kinetic energy and circumburst density of grb050904 are well above the typical values found for low - redshift grbs .

1606.05507

i

all graphs in this paper are finite and simple . a graph @xmath15 is a of a graph @xmath16 if @xmath15 can be obtained from a subgraph of @xmath16 by contracting edges . we write @xmath17 if @xmath15 is a minor of @xmath16 . in those circumstances we also say that @xmath16 has an @xmath15 . our work is motivated by the following hadwiger s conjecture @xcite , which is perhaps the most famous conjecture in graph theory , as pointed out by paul seymour in his recent survey @xcite . [ hc ] for every integer @xmath0 , every graph with no @xmath18 minor is @xmath1-colorable . hadwiger s conjecture is trivially true for @xmath19 , and reasonably easy for @xmath20 , as shown by dirac @xcite . however , for @xmath21 , hadwiger s conjecture implies the four color theorem . wagner @xcite proved that the case @xmath22 of hadwiger s conjecture is , in fact , equivalent to the four color theorem , and the same was shown for @xmath23 by robertson , seymour and thomas @xcite . conjecture remains open for @xmath24 . as pointed out by paul seymour @xcite in his recent survey on hadwiger s conjecture , proving that graphs with no @xmath3 minor are @xmath4-colourable is thus the first case of hadwiger s conjecture that is still open . it is not even known yet whether every graph with no @xmath3 minor is @xmath5-colorable . kawarabayashi and toft @xcite proved that every graph with no @xmath3 or @xmath25 minor is @xmath4-colorable . jakobsen @xcite proved that every graph with no @xmath26 minor is @xmath4-colorable and every graph with no @xmath27 minor is @xmath5-colorable , where for any integer @xmath28 , @xmath29 ( resp . @xmath30 ) denotes the graph obtained from @xmath13 by removing one edge ( resp . two edges ) . for more information on hadwiger s conjecture , the readers are referred to an earlier survey by toft @xcite and a very recent informative survey due to seymour @xcite . albar and gonalves @xcite recently proved the following : [ k7k8 ] ( albar and gonalves @xcite ) every graph with no @xmath3 minor is @xmath12-colorable , and every graph with no @xmath31 minor is @xmath32-colorable the proof of theorem [ k7k8 ] is computer - assisted and not simple . in this paper , we apply a kempe - chain argument ( see lemma [ wonderful ] below ) along with the fact that an induced path on three vertices is dominating in a graph with independence number two in order to give a much shorter and computer - free proof of theorem [ k7k8 ] . in addition , we generalize it to the next step by proving the following . [ main ] every graph with no @xmath6 minor is @xmath7-colorable , where @xmath33 . we want to point out that our proof of theorem [ main ] does not rely on mader s deep result on the connectivity of contraction - critical graphs ( see theorem [ 7con ] below ) . theorem [ main ] states that ( i ) every graph with no @xmath3 minor is @xmath12-colorable ; ( ii ) every graph with no @xmath31 minor is @xmath32-colorable ; and ( iii ) every graph with no @xmath34 minor is @xmath35-colorable . we prove theorem [ main ] in section [ kt ] . applying the method we developed in the proof of theorem [ main ] and mader s deep result ( theorem [ 7con ] ) , we then prove two new results theorem [ maink8- ] and theorem [ maink8= ] . [ maink8- ] every graph with no @xmath9 minor is @xmath10-colorable . [ maink8= ] every graph with no @xmath11 minor is @xmath12-colorable . our proofs of theorem [ maink8- ] and theorem [ maink8= ] are both short and computer - free and will be presented in section [ k8- ] and section [ k8= ] , respectively . to end this paper , we first propose a conjecture in section [ remarks ] . we then apply lemma [ wonderful ] to prove that if conjecture [ conj3 ] ( see section [ remarks ] ) is true , then every graph with no @xmath13 minor is @xmath7-colorable for all @xmath14 . our proof of the last result does not rely on the connectivity of contraction - critical graphs and the new idea we introduced yields a different / short proof of theorem [ main ] . to prove our results , we need to investigate the basic properties of contraction - critical graphs . for a positive integer @xmath1 , a graph @xmath16 is if @xmath36 and any proper minor of @xmath16 is @xmath37-colorable . lemma [ dirac ] below is a folklore result which is an extension of dirac s initial work @xcite on contraction - critical graphs . a proof of lemma [ dirac ] can be easily obtained from the definition of @xmath38-contraction - critical graphs . [ dirac](dirac @xcite ) every @xmath38-contraction - critical graph @xmath16 satisfies the following : for any @xmath39 , @xmath40 , where @xmath41 denotes the independence nunber of the subgraph of @xmath16 induced by @xmath42 ; no minimal separating set of @xmath16 is a clique . lemma [ wonderful ] below on contraction - critical graphs turns out to be very powerful , as the existence of pairwise vertex - disjoint paths is guaranteed without using the connectivity of such graphs . if two vertices @xmath43 in a graph @xmath16 are not adjacent , we say that @xmath44 is a of @xmath16 . one possible application of lemma [ wonderful ] is depicted in figure [ wonderfullemma ] . [ wonderful ] let @xmath16 be any @xmath38-contraction - critical graph . let @xmath45 be a vertex of degree @xmath46 with @xmath47 and let @xmath48 with @xmath49 be any independent set , where @xmath50 and @xmath51 are integers . if @xmath52 is not a clique , then for any @xmath53 , where @xmath54 , @xmath55 , the vertices @xmath56 are all distinct , and for any @xmath57 , the set @xmath58 consists of @xmath59 missing edges of @xmath52 with @xmath60 as a common end , then for each @xmath57 there exist paths @xmath61 in @xmath16 such that each @xmath62 has ends @xmath63 and all its internal vertices in @xmath64 $ ] for all @xmath65 . moreover , for any @xmath66 , the paths @xmath61 are vertex - disjoint from the paths @xmath67 . to @xmath68 with @xmath69 and @xmath70.,width=220 ] let @xmath16 , @xmath68 , @xmath71 and @xmath72 be as given in the statement . let @xmath15 be obtained from @xmath16 by contracting @xmath73 into a single vertex , say @xmath74 . then @xmath15 is @xmath75-colorable . let @xmath76 be a proper @xmath75-coloring of @xmath15 . we may assume that @xmath77 . then each of the colors @xmath78 must appear in @xmath52 , else we could assign @xmath68 the missing color and all vertices in @xmath71 the color @xmath79 to obtain a proper @xmath75-coloring of @xmath16 , a contradiction . since @xmath80 , we have @xmath81 for any two distinct vertices @xmath82 in @xmath52 . we next claim that for each @xmath83 and each @xmath84 there must exist a path between @xmath60 and @xmath85 with its internal vertices in @xmath64 $ ] . suppose not . let @xmath83 and @xmath86 be such that there is no such path between @xmath60 and @xmath85 . let @xmath87 be the subgraph of @xmath15 induced by the vertices colored @xmath88 or @xmath89 under the coloring @xmath90 . then @xmath91 . notice that @xmath60 and @xmath85 must belong to different components of @xmath87 as there is no path between @xmath60 and @xmath85 with its internal vertices in @xmath64 $ ] . by switching the colors on the component of @xmath87 containing @xmath60 , we obtain a @xmath75-coloring of @xmath15 with the color @xmath88 missing on @xmath52 , a contradiction . this proves that there must exist a path @xmath62 in @xmath87 with ends @xmath63 and all its internal vertices in @xmath92 $ ] for each @xmath83 and each @xmath84 . clearly , for any @xmath66 , the paths @xmath61 are vertex - disjoint from the paths @xmath67 , because no two vertices of @xmath93 are colored the same under the coloring @xmath90 . width6pt depth2pt + * remark . * if @xmath94 in the statement of lemma [ wonderful ] , we simply write @xmath95 , and so @xmath72 is a matching of missing edges of @xmath52 . in this case , the paths @xmath96 are pairwise vertex - disjoint if @xmath97 . if @xmath98 in the statement of lemma [ wonderful ] , we simply write @xmath99 . in this case , the paths @xmath100 have @xmath101 as a common end and are not necessarily pairwise internally vertex - disjoint if @xmath102 . theorem [ 7con ] below is a deep result of mader @xcite and will be used in the proofs of theorem [ maink8- ] and theorem [ maink8= ] . it seems very difficult to improve theorem [ 7con ] for small values of @xmath38 . for larger values of @xmath38 , some better results can be found . kawarabayashi @xcite has shown that any minimal non - complete @xmath38-contraction - critical graph with no @xmath103 minor is @xmath104-connected , while kawarabayashi and yu @xcite have shown that any minimal such graph is @xmath105-connected . chen , hu and song @xcite recently improved the bound further by showing that any minimal such graph is @xmath106-connected . [ 7con ] ( mader @xcite ) for @xmath107 , every @xmath38-contraction - critical graph is @xmath5-connected . we also need the following lemma in the proofs of theorem [ maink8- ] and theorem [ maink8= ] . [ lem:7conn2k6 ] for any @xmath5-connected graph @xmath16 , if @xmath16 contains two different @xmath108-subgraphs , then @xmath109 . let @xmath110 be two different @xmath108-subgraphs of @xmath16 with @xmath111 and @xmath112 . let @xmath113 . then @xmath114 . we may assume that @xmath115 for all @xmath116 if @xmath117 . assume that @xmath118 . then @xmath119 has a @xmath120-subgraph of @xmath16 . since @xmath16 is @xmath5-connected , it is easy to see that @xmath109 by contracting a component of @xmath121 into a single vertex . so we may assume that @xmath122 . then there exist @xmath123 pairwise disjoint paths @xmath124 between @xmath125 and @xmath126 in @xmath127 . we may assume that @xmath128 has ends @xmath129 for all @xmath130 . then @xmath131 is connected , so there must exist a path @xmath132 with one end , say @xmath133 , in @xmath134 , the other end , say @xmath135 , in @xmath136 , and no internal vertices in any of @xmath137 . we may assume that @xmath133 lies on the path @xmath138 . let @xmath139 be the subpath of @xmath140 with ends @xmath141 , and @xmath142 be the subpath of @xmath143 with ends @xmath144 . now contracting @xmath139 onto @xmath145 , @xmath146 onto @xmath147 , @xmath142 and @xmath148 onto @xmath149 , @xmath150 onto @xmath151 , and each of @xmath152 to a single vertex , together with @xmath153 if @xmath117 , yields a @xmath9 minor in @xmath16 , as desired . depth2pt + we need to introduce more notation . for a graph @xmath16 we use @xmath154 and @xmath155 to denote the number of vertices and minimum degree of @xmath16 , respectively . for a subset @xmath71 of @xmath156 , the subgraph induced by @xmath71 is denoted by @xmath157 $ ] and @xmath158 $ ] . the degree and neighborhood of a vertex @xmath135 in @xmath16 are denoted by @xmath159 and @xmath42 , respectively . by abusing notation we will also denote by @xmath42 the graph induced by the set @xmath42 . we define @xmath160=n(v)\cup \{v\}$ ] , and similarly will use the same symbol for the graph induced by that set . for @xmath161 , if @xmath157 $ ] is connected , then we denote by @xmath162 the graph obtained from @xmath16 by contracting @xmath157 $ ] into a single vertex and deleting all resulting parallel edges and loops . for @xmath163 , we say that @xmath164 is to @xmath165 if each vertex in @xmath164 is adjacent to all vertices in @xmath165 , and @xmath164 is to @xmath165 if no vertex in @xmath164 is adjacent to any vertex in @xmath165 . if @xmath166 , we simply say @xmath167 is complete to @xmath165 or @xmath167 is anti - complete to @xmath165 . the @xmath168 ( resp @xmath169 ) of two vertex disjoint graphs @xmath16 and @xmath15 is the graph having vertex set @xmath170 and edge set @xmath171 ( resp . @xmath172 ) .

hadwiger s conjecture from 1943 states that for every integer , every graph either can be-colored or has a subgraph that can be contracted to the complete graph on vertices . as pointed out by paul seymour in his recent survey on hadwiger s conjecture , proving that graphs with no minor are-colorable is the first case of hadwiger s conjecture that is still open . it is not known yet whether graphs with no minor are-colorable . using a kempe - chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two , we first give a very short and computer - free proof of a recent result of albar and gonalves and generalize it to the next step by showing that every graph with no minor is-colorable , where . we then prove that graphs with no minor are-colorable and graphs with no minor are-colorable . finally we prove that if mader s bound for the extremal function for minors is true , then every graph with no minor is-colorable for all . this implies our first result .

hadwiger s conjecture from 1943 states that for every integer , every graph either can be-colored or has a subgraph that can be contracted to the complete graph on vertices . as pointed out by paul seymour in his recent survey on hadwiger s conjecture , proving that graphs with no minor are-colorable is the first case of hadwiger s conjecture that is still open . it is not known yet whether graphs with no minor are-colorable . using a kempe - chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two , we first give a very short and computer - free proof of a recent result of albar and gonalves and generalize it to the next step by showing that every graph with no minor is-colorable , where . we then prove that graphs with no minor are-colorable and graphs with no minor are-colorable . finally we prove that if mader s bound for the extremal function for minors is true , then every graph with no minor is-colorable for all . this implies our first result . we believe that the kempe - chain method we have developed in this paper is of independent interest .

astro-ph0009157

c

as discussed in sections[ls5130_dis][ls5127_dis ] , all four stars have relatively normal oxygen abundances . if the oxygen abundance gradients derived in the solar neighbourhood and outer disk are extrapolated inwards ( smartt & rolleston 1997 , rsdr ; @xmath10.07 dexkpc@xmath7 ) their o abundances would be 0.3@xmath10.4dex above the solar neighbourhood value . ls5130 and ls4419 show no evidence at all of oxygen - rich photospheres , and both sample the oii line spectra extremely well , with 43 and 28 features measured respectively . if the statistical errors are normally distributed around the mean , then the random error in our mean differential abundance ( @xmath39 ; where @xmath40 is the standard deviation quoted in tables[ls5130_abundances ] and [ ls4419_abundances ] ) may be as small as @xmath410.04 for both these stars . furthermore , a systematic error in the atmospheric parameters significantly greater than that quoted in table[typical_errors ] would be required to make these stars o - rich which would have major consequences for the other elements and for the continuum fitting of the @xmath0 indices . the large number of oii lines well observed in these stars and the possible error sources in table[typical_errors ] make this a robust result . the other two stars ls4784 and ls5127 show marginal evidence for oxygen abundances higher than their comparison stars , with standard errors of the mean @xmath410.1dex . however again these results appear to be incompatible with an o abundance that is 0.3@xmath10.4dex above solar . a number of previous authors have speculated that the oxygen abundance gradient ( in particular ) may steepen towards the centre ( vlchez & esteban @xcite , shaver et al . @xcite ) . however , we can find no evidence that this is the case . figure[osimg_gradients ] shows the oxygen abundances of our stars together with those of the large galactic b - star sample of rsdr . the low metallicity stars in the outer galaxy ( @xmath42kpc ) are clearly evident , but between 6@xmath19kpc there is little evidence of o abundance variations . the four new stellar points toward the centre accentuate this relation , and it would appear that the o abundance gradient in the galaxy flattens off at around 9kpc and does not significantly increase at smaller galactocentric distances . afflerbach et al . ( @xcite ) have carried out an extensive study of the n , s and o abundances in hii regions with @xmath43kpc . in fig.[on_hii_comp ] , their o abundances are plotted together with those for our stars and the stellar sample of rsdr with @xmath43kpc . the results from the hii regions and the rsdr data - set in the outer galaxy are in excellent agreement ( see rsdr and smartt @xcite for further details ) . however differences appear at smaller galactocentric distances . in particular while the hii region data is consistent with a linear gradient @xmath10.06dexkpc@xmath7 , no significant variations are apparant for the b - stars . one possible explanation is that the zero - points for the two studies is somewhat different . for example , if the afflerbach et al . data were shifted upwards by @xmath40.2dex ( the difference in the mean results within @xmath44kpc ) , then good agreement would be found within the solar neighbourhood , but the b - stars towards the centre would not reflect the hii region results . the cause of this discrepancy is unclear and appears to be limited to oxygen ( see sect . [ discussion_ii ] and [ discussion_iii ] ) . we note that afflerbach et al . is the only hii region study of the inner galaxy to quote oxygen abundances . rudolph et al . ( 1997 ) were unable to independently determine s and o , while shaver et al . ( 1983 ) point out that their two inner galaxy objects ( @xmath11 4kpc ) are somewhat peculiar , and that their radio and optical sources may not be the same . afflerbach et al . have recalculated n abundances from the data of simpson et al . ( @xcite ) and find differences of a factor of @xmath42 in some of the highest metallicity regions . additionally , on average there seems to be a 50% discrepancy in the sulphur abundances . therefore it appears that the nebular abundances in the inner galaxy require further study to constrain their absolute values , while a more extensive sample of b - stars is required . smartt et al . ( @xcite ) have discussed the problem of determining reliable absolute nebular abundances in regions of solar abundance and above in m31 , and find significant discrepancies across different analysis methods . in terms of effective radius ( where @xmath45kpc , from de vaucouleurs & pence @xcite ) , the break in the observed o abundance gradient occurs at @xmath46 . the spiral galaxy ngc2403 has been studied by garnett et al . ( @xcite ) who estimated hii region abundances at 12 points across the disk . in fig[mw_ngc2403_grads ] the oxygen abundance is plotted as a function of effective radius for both these galaxies . a similar gradient change may be present in ngc2403 , although the sampling is relatively sparse . this would imply that a two component model for the radial variation of oxygen is not exclusive to our own galaxy and suggests indirect evidence that it is a real effect . certainly the data suggests that for both galaxies , at @xmath47 , there is little evidence for increasing oxygen abundances . however , the well studied galaxy m33 ( garnett et al . @xcite , henry & howard @xcite , monteverde et al . @xcite ) which is similar in size , mass and metallicity to ngc2403 does not appear to show such a break . additionally , m101 and ngc4303 are other examples of extensively studied galaxies ( kennicutt & garnett @xcite , martin & roy @xcite ) , with 30@xmath180 hii region data points sampled in each . neither show evidence for anything other than a linear relation across the whole disk . the magnesium and silicon abundances for the galactic centre stars and the corresponding rsdr dataset are plotted in fig[osimg_gradients ] . for oxygen and silicon we note that the absolute abundances derived in our two solar neighbourhood standard stars are similar to the mean of the rsdr data set within 1kpc of the sun . hence the absolute results of the galactic centre stars are on a consistent scale to the rest of the data . however with mg , the abundance derived in @xmath48 peg and @xmath49 her is 7.2dex , whereas in rsdr the mean local value is 7.85 dex ( again estimated from the mean of stellar values within 1kpc of the sun ) . the stars analysed in this paper have effective temperatures in the range @xmath50k , while the rsdr sample have a mean temperature of 25275k and it is possible that the modelling of this line is temperature dependent . it is important to use homogeneous datasets when looking for abundance trends and that is indeed what was done in rsdr . they showed that although the zero - point of any particular abundance may be in the error , any gradient biasing is removed by using suitable sub - samples of stars . hence to put the four galactic centre stars onto the same scale as rsdr , we have added 7.85dex to their _ differential _ mg abundances . this assumes that @xmath48 peg , @xmath49 her and the stars within 1kpc of the sun in rsdr have similar abundances and that any offset is due to temperature dependent modelling inaccuracies . it appears that the galactic centre stars are richer in mg than their local counterparts ( see also sect[ls5127_mgii_line ] ) . from table[typical_errors ] an underestimate of the microturbulence ( @xmath51 ) in each star could reduce this mg overabundance as it is based on a relatively strong line . however an increase in @xmath51 by 5 km s@xmath7 or more in each star would be required . while observational uncertainties might cause a particular determination of @xmath51 to have such an error , there is no evidence that we have _ systematically _ underestimated the microturbulence in each star . the mg abundance gradient in fig . [ osimg_gradients ] hence appears to steadily increase towards the centre . a value of @xmath52dexkpc@xmath7is derived by fitting all points , which is slightly larger the rsdr result , although they agree within the uncertainties . the silicon abundances of the galactic centre stars are also plotted in fig.[osimg_gradients ] , and again appear compatible with a steadily increasing abundance gradient towards the centre . a fit through all the si points gives a gradient of @xmath53dexkpc@xmath7 , again in good agreement with the rsdr value for the outer galaxy only . the differential abundances derived for each star are between 0.3@xmath10.6dex higher than their respective standards , strongly supporting the idea that these stars are richer in si than solar neighbourhood material . in fig.[fig_sal_grads ] the al abundances of the galactic centre stars are plotted again with the data from rsdr . aluminium appears to increase steadily toward the centre , and a gradient of @xmath54 dexkpc@xmath7 is derived from the whole data set , similar to that produced by rsdr for the anti - centre direction . sulphur abundances for our stars are also available , although we have no similar data at greater radii from rsdr . in fig[fig_sal_grads ] our stellar abundances are compared with with those from afflerbach et al . reasonable agreement is found with these stars appearing to be s - rich and having abundances similar to those found in nebular studies towards the centre ( although the caveat regarding the nebular absolute values discussed in the previous section may be important ) . the correlation of of o , mg , and si abundances found in the outer galaxy ( see rsdr ) and in the low metallicity magellanic clouds ( rolleston et al . @xcite ) is theoretically expected as these elements are all produced and returned to the ism through supernovae typeii . one would expect that this trend should continue towards the inner galaxy , and it is surprising that our o abundances show no sign of increase , while mg and si appear significantly higher than their comparison stars . the s abundances in these stars are also higher than normal , although the dataset is somewhat less robust given the weakness of the absorption features of this element . prantzos et al . ( @xcite ) have investigated the effect of metallicity dependent yields ( taken from the model calculations of maeder @xcite ) on the galactic chemical evolution of c and o and concluded that oxygen yields from massive stars tend to decrease towards higher metallicities . this is due to mass loss in low metallicity stars , resulting in most of the star s mass being retained up until the final explosion . however , at high metallicities significantly more mass is ejected in the stellar wind ( which is rich in helium and carbon ) , and leaves a smaller mass fraction which can be converted into oxygen . the relatively high carbon and low oxygen abundances derived here are in qualitative agreement with the situation predicted by these models . a full study of the evolution of elements heavier than oxygen has not been undertaken using metallicity dependent yields . however maeder ( 1992 ) has discussed how the combined yield of `` heavy elements '' ( defined as ne , mg , si , s , ca & fe ) from massive stars varies with original metal composition . interestingly , he finds that the `` heavy element '' contributions of the stellar models at high metallicity ( @xmath55 ) are not as low as is the case for oxygen . the situation is further complicated by the fact that chemical yields are dependent on the conditions for black hole formation since the final yield of a pre - supernova massive star ( pictured in the conventional onion - skin model ) depends on what fraction of the stellar layers are retained during core collapse . the observational findings presented here should provide a stimulus to investigations of the evolution of the galaxy at high metallicities , given the probability that stellar yields are highly variable with metallicity , and dependent on the limits of black hole formation . furthermore , we should attempt to determine the true quantitative variation in the ratios of the @xmath14-elements in the inner galaxy to investigate if this trend is real . the recent chemical evolution models of portinari & chiosi ( @xcite , @xcite ) do indicate that the o abundance gradient may indeed flatten off in the region @xmath56kpc , which is encouraging . however their sn typeii yields would suggest that the other @xmath14-processed elements ( mg and si ) should follow a similar trend , which we do not observe . boissier & prantzos ( @xcite ) also suggest that their evolutionary models for the milky way predict a flattening of the metallicity gradient with time , and that saturation is reached in the inner galaxy . this is in reasonable quantitative agreement with what we find for oxygen abundances , but does not explain the behaviour of the other elements . each of the four stars do appear to have nitrogen abundances significantly higher than their solar neighbourhood counterparts . the result for ls5130 is more marginal than the others , but with 19 features measured an error in the mean ( @xmath57 ) of 0.03dex would suggest that the enrichment is real although small . figure4(b ) shows the nitrogen abundances of the galactic centre stars plotted along with a sample of stars from rsdr , and the hii regions from afflerbach et al . the gradient found in rsdr appears to continue towards the centre ; certainly the abundance estimates do not appear to flatten as for oxygen . reasonably good agreement with the absolute values of the hii region analyses is found , and we calculate quite a significant gradient for @xmath58n / o . the full sample of rsdr yielded a gradient in n / o of @xmath59dexkpc@xmath7 , and this is increased slightly by the inclusion of the four galactic centre stars . the latter effect is clearly due to the stars showing enhanced nitrogen in their atmospheres , but normal oxygen . we have previously discussed the possibility of n - rich core gas contaminating the surface of young massive stars , and hence polluting the natal photospheric material ( rsdr ) . however due to the lack of a strong anti - correlation of c and n , rsdr found no evidence for such contamination and the same argument holds for the four stars analysed here . an explanation of the n / o gradient was discussed in rsdr , based on the original ideas of vila - costas & edmunds ( @xcite ) . at metallicities higher than @xmath60 o / h @xmath48.3 , @xmath58 n / o would be proportional to @xmath58 o / h , because n enrichment would be dominated by secondary production , i.e. from c and o seed nuclei in the natal interstellar material through the cno - cycle . primary production would play a minimal role in contributing to the ism abundance of n , and simple closed - box models in which the production of n is dominated by secondary production do predict a linear trend . at low metallicities the secondary component becomes less important ( due to its inherent dependence on metallicity ) , and the dominant mechanism for the production of nitrogen is primary . the metallicity at which this happens is around 8.3dex , close to the value found in the outer most regions of the galactic disk . these ideas stem from the fact that the n / o ratios in low metallicity dwarf galaxies show a very large scatter , with little discernable trend . as primary nitrogen orginates in low - intermediate mass stars there is a time delay between is enrichment and that of primary oxygen ( from short - lived massive stars ) , hence the scatter at low metallicitities is often interpreted as time dependent delays between oxygen and nitrogen enrichments . however at the high metallicities we are dealing with , it is likely that secondary production ( in massive stars is dominant ) , and there would thus be no delay between n and o production . our n / o ratios toward the galactic centre tend to support the conclusions of rsdr that the n / o gradient in the galaxy is real , and due to secondary n production dominating across the disk . in fig.[fig_co_grads ] carbon abundances for the four galactic centre stars and from the rsdr compilation are plotted . as three of our new points are c - rich , it is not surprising that the abundances appears to steadily increase toward the centre . as discussed for nitrogen , because the stars are not rich in oxygen the c / o ratio is greater than normal , hence producing a larger ( and statistically more significant ) gradient than that given in rsdr . as discussed by maeder ( 1992 ) this would be consistent with c being produced at a greater rate than o in metal - rich stars ( section[discussion_ii ] ) , due to the increased mass - loss in the post he - burning phase .

the stars appear to be rich in all the well sampled chemical elements ( c , n , si , mg , s , al ) , _ except for oxygen_. oxygen abundances derived in the atmospheres of these four stars are very similar to that in the solar neighbourhood . if the photospheric composition of these young stars is reflective of the gaseous ism in the inner galaxy , then the values derived for the enhanced metals are in excellent agreement with the extrapolation of the galactic abundance gradients previously derived by rolleston et al . however , the data for oxygen suggests that the inner galaxy may not be richer than normal in this element , and the physical reasons for such a scenario are unclear .

high - resolution , high signal - to - noise spectral data are presented for four young b - type stars lying towards the galactic centre . determination of their atmospheric parameters from their absorption line profiles , and photometric measurement of the continua indicate that they are massive objects lying slightly out of the plane , and were probably born in the disk between 2.5kpc from the centre . we have carried out a detailed absolute and differential line - by - line abundance analyses of the four stars compared to two stars with very similar atmospheric parameters in the solar neighbourhood . the stars appear to be rich in all the well sampled chemical elements ( c , n , si , mg , s , al ) , _ except for oxygen_. oxygen abundances derived in the atmospheres of these four stars are very similar to that in the solar neighbourhood . if the photospheric composition of these young stars is reflective of the gaseous ism in the inner galaxy , then the values derived for the enhanced metals are in excellent agreement with the extrapolation of the galactic abundance gradients previously derived by rolleston et al . ( 2000 ) and others . however , the data for oxygen suggests that the inner galaxy may not be richer than normal in this element , and the physical reasons for such a scenario are unclear .

astro-ph0009157

c

the analysis of high - quality data for four blue stars lying towards the galactic centre , and a comparison with young stars throughout the galactic disk , leads to the following conclusions 1 . the four stars appear to be normal , young , massive , early b - type stars . they lie slightly out of the galactic plane and their calculated distances indicate that they are at distances from the disk of @xmath61kpc . a kinematic analysis indicates that they were probably born in the disk within @xmath62kpc of the galactic centre and then subsequently ejected . comparison with galactic structure models suggests they may have originated in the stellar or molecular ring at @xmath63kpc . 2 . a differential line - by - line abundance analysis of the four stars with two bright stars near the sun indicates a surprising abundance pattern : * two of the stars ( ls5130 and ls4419 ) have oxygen abundances very similar to stars in the solar neighbourhood . the other two ( ls4784 and ls5130 ) show only marginal evidence for enhanced oxygen . in all cases any enhancement is significantly below the @xmath64dex that would be expected given the galactic position of the stars and the gradient from rsdr . oxygen is a very reliably determined abundance in these particular types of stars . * the relatively well measured abundances of si , al , mg and s however do indicate that these stars have enhanced abundances in most other elements . however , the physical cause of the natal gas having a `` normal '' oxygen content and enriched @xmath14-processed elements is unclear . the si and mg abundances are consistent with the continuation of the linear gradient derived by rsdr for stars in the solar neighbourhood and the outer galaxy . * the four stars also appear rich in both c and n , again in agreement with a continuation of the rsdr gradients towards the centre . * we discuss the possible errors in our analysis , and conclude that it is unlikely that these stars have higher oxygen abundances than those typically found in the solar neighbourhood . it is also very likely they are rich in all the other elements studied ( c , n , mg , si , s , al ) although the physical pocesses that produces such a pattern is unclear . * appendix a : table9 and 10 are the lists of equivalent widths of all metal lines measured in the spectra of the galactic centre stars and the standards @xmath49her , and @xmath48peg . and is available electronically * afflerbach a. , churchwell e. , werner m.w . , 1997 478 , 190 anders e. , grevesse n. , 1989 , geochim . acta 53 , 197 bahcall j.n . , 1986 , ara&a 24 , 577 bahcall j.n . , soneira r.m . , 1980 , apjs 44 , 73 baldwin j.a . , ferland g.a . , martin p.g . , et al . 1991 , apj 374 , 580 bertelli g. , bressan a. , chiosi c. , ng y.k . , ortolani s. , 1995 , a&a 301 , 381 blommaert j.a.d.l . , langevelde van h.j . , michiels w. , 1994 , a&a 287 , 479 brown p.j.f . , 1987 , phd thesis , queens university of belfast boissier s. , prantzos n. , 1999 , mnras 307 , 857 cardelli j.a . , meyer d.m . , jura m. , savage b.d . , 1996 , apj 467 , 334 carr j.s . , sellgren k. , balachandran s.c . , 2000 , apj 530 , 307 carraro g. , ng y.k . , portinari l. , 1998 , mnras 296 , 1054 clemens d.p . , sanders d.b . , scoville n.z . , 1998 , apj 327 , 139 cunha k. , lambert d. , 1994 , apj 426 , 170 davis l.e . , 1994 , a reference guide to the iraf / daophot package , noao laboratory deutschman w.a . , davis r.j . , schild r.e . , 1976 , apjs 30 , 97 de vaucouleurs g. , pence w. d. , 1978 , aj 83 , 1163 edvardsson b. , andersen j. , gustafsson b. , et al . 1993 , a&a 75 , 101 fich m. , silkey m. , 1991 , apj 366 , 107 fitzsimmons a. , dufton p.l . , rolleston w.r.j . , 1992 , mnras 259 , 489 fuhrmann k. , 1998 , a&a 338 , 161 garnett d.r . , shields g.a . skillman e.d . , sagan s.p . , dufour r.j . , 1997 , apj 489 , 63 gies d.r . , lambert d.l . , 1992 , apj 387 , 673 grevesse n. , noels a. , 1993 in origin and evolution of the elements , ed . n. prantzos , e. vangioni - flam , & m. casse ( cambridge : cambridge univ . press ) , 14 gummersbach c.a . , kaufer a. , shfer d.r . , szeifert t. , wolf b. , 1998 , a&a 338 , 881 henry r. b. c. , howard j. w. , 1995 , apj 438 , 170 heger a. , langer n. , 2000 , apj in press , ( astro - ph/0005110 ) . house f. , kilkenny d. , 1980 , a&a 81 , 251 howarth i.d . , murray j. , mills d. , berry d.s . , 1998 starlink user notice , no.50.21 israelian g. , garcia lopez r.j . , rebolo r. , 1998 , apj 507 , 805 jeffery c.s . , 1991 , in : newsletter on analysis of astronomical spectra no . 16 , pg . 17 kennicutt r.c . , garnett d.r . , 1996 , apj 456 , 504 kerr f.j . , lynden - bell d. , 1986 , mnras 221 , 1023 kilian j. , 1994 , a&a 282 , 867 kilkenny d. , balona l.a . , carter d.b . , ellis d.t . , woodhouse g.f.w . , 1998 , mnassa 47 , 69 kurucz r.l . , 1979 , apjs 40 , 1 kurucz r.l . , 1991 , in precision photometry : astrophysics of the galaxy , ed . davis - philip , a.r . upgren , & p.l . janes ( schenectady : davis ) leonard p. j. t. , 1993 , in luminous high - latitude stars , ed . d. sasselov ( san francisco : asp ) lester j.b . , gray r.o . , kurucz r.l . , 1986 , apjs 61 , 509 maciel w. j. , quireza 1999 , a&a 345 , 629 maciel w.j . , kppen j. , 1994 , a&a 282 , 43 maeder a. , 1992 , a&a 264 , 105 maeder a. , & meynet g. , 2000 , astro - ph/0006405 martin p. , roy j .- , 1992 , apj 397 , 63 massey p. , 1992 , a user s guide to ccd reductions with iraf , noao laboratory matteucci f. , francois p. , 1989 , mnras 239 , 885 matteucci f. & chiappini c. , 1999 , in proc . eso conf . on chemical evolution from zero to high redshift , springer conference series , eds . j. walsh & m. rosa meyer d. m. , jura m. , cardelli j. a. , 1998 , apj 493 , 222 monteverde i. , herrero a.h . , lennon d.j . , 2000 , apj , submitted mooney c.j . , rolleston w.r.j . , keenan f.p . , et al . 2000 , a&a 357 , 553 najarro f. , krabbe a. , genzel r. , et al . 1997 , a&a 325 , 700 napiwotzki r. , schonberner d. , wenske v. , 1993 , a&a 268 , 653 ortiz r. , lpine j.r.d . , 1993 , a&a 279 , 90 paczynski b. , stanek k.z . , udalski a. , et al . 1994 , aj 107 , 2060 pagel b.e.j . , tautvaiin g. , 1995 , mnras 276 , 505 peters g.j . , 1976 , apjs 30 , 551 pilyugin l.s . , edmunds m.g . , 1996 a&a 313 , 792 portinari l. , chiosi c. , 1999 , a&a 350 , 827 portinari l. , chiosi c. , 2000 , a&a 355 , 929 prantzos n. , vangioni - flam e. , chauveau s. , 1994 , a&a 285 , 132 prantzos n. , aubert o. , 1995 , a&a 302 , 69 reed b.c . , beatty a.e . , 1995 , apjs 97 , 189 reed b.c . , 1993 , apjs , 87 , 367 rolleston w.r.j . , smartt s.j . , dufton p.l . , ryans r.s.i . , 2000 , a&a , in press rolleston w.r.j . , hambly n.c . , keenan f.p . , dufton p.l . , saffer r.a . , 1999 , a&a 347 , 69 rolleston w.r.j . , hambly n.h , dufton p.l . , et al . 1997 , mnras 290 , 422 rolleston w.r.j . , brown p.j.f . , dufton p.l . , howarth i.d . , 1996 , a&a 315 , 95 ryans r.s.i , hambly n.c . , dufton p.l . , keenan f.p . , 1996 , mnras 278 , 132 rudolph a.l . , simpson j.p . , haas m.r . , erickson e.f . , fich m. , 1997 , apj 489 , 94 seaton m.j . , 1979 , mnras 187 , 73 serabyn e. , shupe d. , figer d. , 1998 , nature 394 , 448 schaller g. , schaerer d. , meynet g. , maeder a. , 1992 , a&as 96 , 269 shaver p.a . , mcgee r.x . , newton m.p . , danks a.c . , pottasch s.r . , 1983 , mnras 204 , 53 shortridge k. , meyerdierks h. , currie m. , clayton m. , lockley j. , 1997 , starlink user note 86.14 simpson j. p. , colgan s. w. j. , rubin r. h. , erickson e. f. , haas m. r. , 1995 , apj 444 , 721 smartt s.j . , rolleston w.r.j . , 1997 , apj 481 , l47 smartt s.j . , dufton p.l . , lennon d.j . , 1997 , a&a 326 , 763 smartt s.j . , dufton p.l . , rolleston w.r.j . 1996 , a&a 305 , 164 smartt s.j . , 2000 , in the chemical evolution of the milky way : stars vs. clusters , rome - trieste workshop , es f. matteucci , f. giovanelli smartt s.j . , crowther p.a . , dufton p.l . , et al . , 2000 mnras , to be submitted snow t.p . , witt a.n . , 1996 , apj 468 , l65 stephenson c.b . , sanduleak n. , 1971 , publ . warner & swasey obs . , 1 , 1 venn k.a , smartt s.j . , lennon d.j . , dufton p.l . , 1998 , a&a 334 , 987 vlchez j. m. , esteban c. , 1996 , mnras 280 , 720 vila - costas m. b. , edmunds m. g. , 1993 , mnras 265 , 199 rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + oii 3911.96 & 57a & 96c & 76b & 50a + + cii 3918.98 & 138a & 130a & 261a@xmath65 & + nii 3919.01 & & & & + oii 3919.28 & & & & + + cii 3920.69 & 127a & 113a & & + + oii 3945.04 & 30b & 63b & 58c & + + oii 3954.37 & 51a & 63b & 74b & 48a + + hei 3964.73 & 177a & 187a & 188a & 155a + + oii 3982.72 & 29c & 72b@xmath66 & 60a@xmath66 & 31a + + siii 3983.77 & 28c & & & 14a + + nii 3995.00 & 102a & 104a & 93a & 78a + + nii 4035.08 & 35b & 46a & 42c & 25a + + nii 4041.31 & 44b & 51a & 60b & 37a + + oii 4069.62 & 95a & 115a & 97b & 94 + oii 4069.89 & & & & + + oii 4072.16 & 62a & 75a & 43c & 65a + + cii 4074.48 & 48b & 147a@xmath67 & 141b@xmath67 & 42a + cii 4074.85 & & & & + + cii 4075.85 & & & & + oii 4075.86 & 124a & & & + cii 4076.53 & & & & + nii 4076.91 & & & & + + oii 4078.84 & 24c & 19c & & 26a + + oii 4085.11 & 32c & 39b & 34b & 26a + + oii 4087.15 & 22c & & & 12a + + siiv 4088.85 & 37c & 44b & 61a & + oii 4089.28 & & & & + + siiv 4116.10 & 20c & & & + + oii 4119.22 & 62a & & & + + oii 4120.28 & 291a & & & + hei 4120.81 & & & & + hei 4120.99 & & & & + oii 4121.46 & & & & + + siii 4128.07 & 53a & 38c & 65b & 33a + + siii 4130.89 & 54a & 86c@xmath68 & 70c@xmath68 & 34a + + arii 4131.71 & 38b & & & 17a + + oii 4132.80 & 33b & & 54c@xmath68 & 30a + + aliii 4149.50 & & 56b & 64a & + aliii 4150.14 & & & & + + sii 4153.10 & 60a & 125b & 105a & + oii 4153.30 & & & & + + sii 4162.70 & 47b & & & + + feiii 4164.79 & 45c & 32b & & + + feiii 4166.86 & 26c & & & 13a + + hei 4168.97 & 99a & 99a & 104a & + oii 4169.22 & & & & + + sii 4174.24 & 40b & 28c & & + + nii 4179.67 & 14c & 21c & & + + oii 4185.45 & 36b & 43b & & 30a + + oii 4189.79 & 46b & 54b & 65b & + + piii 4222.15 & 21c & 20c & & 21a + + nii 4227.74 & 30b & 55b & 61b & 18a + + nii 4236.86 & 51a & 55a & 66c & 28a + nii 4236.98 & & & & + + nii 4241.78 & 49a & 86a & 105b & + + rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + siii 4253.59 & & 118a & & + oii 4253.90 & & & & + oii 4254.13 & & & & + + cii 4267.02 & 262a & 245a & 305a & + cii 4267.27 & & & & + oii 4267.71 & & & & + + oii 4275.56 & & 97b & 123a@xmath69 & + + oii 4277.40 & & 72b & & 16a + + siii 4284.99 & & & 47c & 28a + + sii 4294.43 & 35b & 50b & 37c & + oii 4294.79 & & & & + + oii 4303.84 & 30c & 62a & 54c & 26a + + cii 4313.30 & 17c & & & + + oii 4317.14 & 58a & 76a & 64b & 56a + cii 4317.26 & & & & + + oii 4319.63 & 48a & 78a & 63b & 52a + + oii 4325.76 & 40b & 23b & 29c & 20a + + oii 4345.56 & 30c & & & 24a + + oii 4347.42 & 26c & & & 21a + + arii 4348.11 & 32c & & & + + oii 4349.43 & 66b & & & 63a + + oii 4351.26 & 31c & & & 41a + + oii 4366.89 & 60a & 61a & 32c & 48a + + oii 4369.27 & & & & 14a + + cii 4372.49 & 52a & 91a & 67c@xmath70 & + + cii 4374.27 & 32b & & & + + rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + feiii 4395.78 & 40b & 60a & 58b & + oii 4395.94 & & & & + + cii 4411.20 & 38b & 63b & & + cii 4411.52 & & & & + + oii 4414.90 & 84a & 95a & 52c & 75a + + oii 4416.97 & 64a & 76a & 28c & 61a + + feiii 4419.59 & 44b & 32c & 35c & 34a + + feiii 4430.95 & 33c & & 39c & 24a + + nii 4432.74 & 38c & & 51c & 20a + + hei 4437.55 & 126a & 103a & 135a & 118a + + nii 4442.02 & & & 38c & 9a + + nii 4447.03 & 60a & & 72b & 15a + oii 4448.19 & & & & + + oii 4452.37 & 31c & & 29c & 25a + + aliii 4479.89 & 55a & & & 38a + aliii 4479.89 & & & & + + mgii 4481.13 & 202a & 196a & 265a & 160a + mgii 4481.33 & & & & + + aliii 4512.54 & 57a & 65a & 63a & 39a + + aliii 4528.91 & 73a & 144a@xmath71 & 135a@xmath71 & 66a + aliii 4529.20 & & & & + + nii 4530.40 & 29c & & & 25a + + nii 4552.53 & 174a & 199a & 187a & 114a + siiii 4552.62 & & & & + + siiii 4567.82 & 150a & 152a & 151a & 102a + + siiii 4574.76 & 81a & 97a & 90a & 69a + + oii 4590.97 & 52a & 87a & 67a & + + oii 4596.18 & 59a & 74a & 52a & 47a + + nii 4601.48 & 62b & 107a & 90a & 37a + oii 4602.13 & & & & 15a + + rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + nii 4607.16 & 56a & 64a & 68a & 35a + + oii 4609.44 & 41b & 63a & 50a & 30a + + nii 4613.87 & 55a & 79a & 56a & 27a + + cii 4618.40 & 137c & 82a & 213a@xmath72 & + cii 4619.23 & & & & + + nii 4621.29 & 41b & 73a & & 26a + + nii 4630.54 & 92a & 122a & 123a & + + oii 4638.86 & 61b & 81a & 71a & 54a + + oii 4641.82 & 74b & 171a@xmath73 & 172a@xmath73 & 76a + + nii 4643.09 & 58b & & & 42a + + ciii 4647.42 & & 230a @xmath74 & 192a@xmath74 & + + oii 4649.14 & 90a & & & 92a + + oii 4650.84 & 64b & & & 52a + + oii 4661.63 & 67a & 94a & 75a & 54a + + oii 4673.74 & 26c & & & 19a + + oii 4676.24 & 44c & 45c & & 46a + + nii 4678.10 & 24c & & & + + nii 4694.70 & 15c & & & 9a + + oii 4696.35 & 15c & & & 12a + + oii 4699.00 & 42a & 69c & 37a & + oii 4699.22 & & & & + oii 4703.16 & & 66a@xmath74 & 53a@xmath74 & + + oii 4705.35 & 47b & & & + + oii 4710.01 & 30b & 40c & & 20a + + hei 4713.14 & 289a & 283a & 250a & 271a + hei 4713.37 & & & & + + rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + siiii 4716.65 & 41b & 52b & & 12a + + nii 4779.67 & 18c & 36c & 44b & 10a + + cii 4802.70 & 28c & & & 15a + + nii 4803.29 & 37c & 61a & 74a & 18a + + arii 4806.07 & 29c & 21c & & 18a + + siiii 4813.30 & 31b & 41a & 75b@xmath75 & 21a + + sii 4815.52 & 54b & 27b & & 27a + + siiii 4819.72 & 55b & 55a & 50a & 40a + + sii 4824.07 & 12c & & 18c & + + siiii 4828.96 & 44b & 64a & 59a & + + arii 4879.14 & 28b & & 19c & + + oii 4890.93 & 21c & 22b & & 11a + + oii 4906.83 & 38b & & 29b & + + oii 4941.07 & 25b & 38b & 14c & 15a + + oii 4943.00 & 39b & 52b & 45c & 28a + + sii 4991.94 & 16c & & 24c & 22a + + nii 4994.36 & 31b & 55a & 59b & + + sii 5009.62 & 32b & & & + + nii 5010.62 & 40b & 40c & & + + sii 5014.03 & 52b & & & + + hei 5015.68 & 322a & 319a & 354a & + + sii 5032.41 & 66b & 56a & 91a & + + siii 5041.03 & 24c & & 30c & 15a + + rcccc species and line & ls5130 & ls4419 & ls4784 & @xmath48peg + nii 5045.09 & 54b & 59a & 47c & + + hei 5047.74 & 207a & 163a & 179b & + + siii 5055.98 & 35c & & 56b & 22a + siii 5056.31 & & & & + + sii 5103.30 & & & 20c & + + cii 5132.96 & 97a & 74a & 99a & + cii 5133.29 & & & & + + oii 5159.94 & 23a & 38a & 37a & + notes : + 1 . blend of oii 3982.72 and siii 3983.77 . + 2 . includes the oii , nii and cii lines at 40754076aa . + 3 . blended with arii 4131.71 . + 4 . blended with nii 4643.09 . blended with oii 4649.14 & 4650.84 . + 6 . blended with oii 4705.35 . + 7 . also includes cii 3920.69 . blended with oii 4277.40 . + 9 . blended with cii 4374.27 . blended with nii 4530.40 . + 11 . blended with nii 4621.29 . + 12 . blended with sii 4815.52 . + rcc species and line & ls5127 & @xmath49her + oii 3911.96 & 50c & 12a + + cii 3918.98 & 340b & 72a + nii 3919.01 & & + oii 3919.28 & & 8a + cii 3920.69 & & 88a + + oii 3954.37 & 44c & 20a + + hei 3964.73 & 210c & 132a + + nii 3995.00 & 108a & 40a + + nii 4041.31 & 57b & 19a + + nii 4043.53 & 59b & 9a + + oii 4069.62 & 182a & 16a + oii 4069.89 & & 19a + oii 4072.16 & & 27a + + cii 4074.48 & 185a & 15a + cii 4074.85 & & 9a + oii 4075.86 & & 32a + cii 4076.53 & & + nii 4076.91 & & + + oii 4119.22 & 324a & + oii 4120.28 & & + hei 4120.81 & & + hei 4120.99 & & + oii 4121.46 & & + feiii 4122.06 & & + feiii 4122.98 & & + + hei 4168.97 & 70b & 81a + + nii 4241.78 & 67a & 16a + + siii 4253.59 & 69a & 13a + oii 4253.90 & & 4a + oii 4254.13 & & 2a + + cii 4267.02 & 338a & 184a + cii 4267.27 + oii 4267.71 & & + siiii 4267.80 & & 26a + rcc species and line & ls5127 & @xmath49her + sii 4294.43 & 43a & 24a + oii 4294.79 & & 3a + + oii 4303.84 & 39a & 9a + + oii 4325.76 & 55c & 6a + cii 4325.83 & & + cii 4326.16 & & + + siii 4361.53 & 39c & 4a + + oii 4366.89 & 42c & 16a + + cii 4372.49 & 48c & + + feiii 4395.78 & 43b & 15a + + oii 4414.90 & 101b & 28a + oii 4416.97 & & 23a + + feiii 4419.59 & 53a & 23a + + hei 4437.55 & 101a & 102a + + nii 4447.03 & 65b & 18a + + sii 4463.58 & 36b & 14a + + mgii 4481.13 & 423a & 205a + mgii 4481.33 & & + + aliii 4512.54 & 60a & 20a + + sii 4524.95 & 47b & 22a + + aliii 4528.91 & 100b & 6a + aliii 4529.20 & & 28a + nii 4530.40 & & 7a + + rcc species and line & ls5127 & @xmath49her + feii 4549.47 & 49b & + + sii 4552.38 & 191a & 71a + siiii 4552.62 & & + + siiii 4567.82 & 156a & 59a + + siiii 4574.76 & 93a & 27a + + feii 4583.83 & 39b & 12a + + oii 4590.97 & 39c & 17a + + oii 4596.18 & 32c & 12a + + nii 4601.48 & 49c & 15a + + nii 4607.16 & 47c & 14a + + cii 4618.40 & 114c & + cii 4619.23 & & + nii 4621.29 & & + siii 4621.42 & & + siii 4621.72 & & + + feii 4629.34 & 107a & + nii 4630.54 & & + + oii 4638.86 & 43a & 21a + + oii 4641.82 & 108a & 24a + nii 4643.09 & & 17a + + sii 4648.17 & 170a & 5a + oii 4649.14 & & 34a + oii 4650.84 & & 17a + + sii 4656.74 & 26b & 10a + + oii 4661.63 & 65b & 15a + aliii 4663.05 & & 13a + + hei 4713.14 & 306a & 239a + hei 4713.37 & & + + siiii 4813.30 & 81b & 8a + sii 4815.52 & & 40a + + siiii 4819.72 & 47c & + + sii 5014.03 & 522a & 38a + hei 5015.68 & & 233a + + sii 5032.41 & 128a & 47a + + siii 5041.03 & 85a & 42a + + nii 5045.09 & 190a & + sii 5047.28 & & + hei 5047.74 & & + + siii 5055.98 & 182a & 82a + siii 5056.31 & & + +

high - resolution , high signal - to - noise spectral data are presented for four young b - type stars lying towards the galactic centre . determination of their atmospheric parameters from their absorption line profiles , and photometric measurement of the continua indicate that they are massive objects lying slightly out of the plane , and were probably born in the disk between 2.5kpc from the centre . we have carried out a detailed absolute and differential line - by - line abundance analyses of the four stars compared to two stars with very similar atmospheric parameters in the solar neighbourhood .

high - resolution , high signal - to - noise spectral data are presented for four young b - type stars lying towards the galactic centre . determination of their atmospheric parameters from their absorption line profiles , and photometric measurement of the continua indicate that they are massive objects lying slightly out of the plane , and were probably born in the disk between 2.5kpc from the centre . we have carried out a detailed absolute and differential line - by - line abundance analyses of the four stars compared to two stars with very similar atmospheric parameters in the solar neighbourhood . the stars appear to be rich in all the well sampled chemical elements ( c , n , si , mg , s , al ) , _ except for oxygen_. oxygen abundances derived in the atmospheres of these four stars are very similar to that in the solar neighbourhood . if the photospheric composition of these young stars is reflective of the gaseous ism in the inner galaxy , then the values derived for the enhanced metals are in excellent agreement with the extrapolation of the galactic abundance gradients previously derived by rolleston et al . ( 2000 ) and others . however , the data for oxygen suggests that the inner galaxy may not be richer than normal in this element , and the physical reasons for such a scenario are unclear .

astro-ph0409268

i

classical t tauri stars ( cttss ) appear to be roughly solar mass pre - main sequence stars still surrounded by disks of material which actively accrete onto the central star . their properties have been recently reviewed by mnard and bertout ( 1999 ) . it is within the disks surrounding cttss that solar systems form . understanding the processes through which young stars interact with and eventually disperse their disks is critical for understanding the formation of planets and the rotational evolution of stars . over the past several years , strong stellar magnetic fields have come to play a central role in current theories describing the interaction of a ctts with its circumstellar accretion disks . current theories posit that the stellar magnetic field truncates the disk at or inside the corotation radius , directing the flow of accreting disk material toward the polar regions of the central star ( e.g. camenzind 1990 , knigl 1991 , cameron & campbell 1993 , shu et al . 1994 , paatz & camenzind 1996 ) . this magnetospheric accretion model is driven by the need to explain how cttss can accrete substantial amounts of material ( and presumably angular momentum ) , yet remain relatively slow rotators ( hartmann & stauffer 1989 ) . early observations showed that cttss later than k7 had rotation periods of 7 - 10 days while diskless t tauri stars ( the naked ttss : nttss ) rotated with a wide range of periods ( edwards et al . 1993 ) . while this dichotomy in the rotation rates between cttss and nttss has recently been called into question ( stassun et al . 1999 ) for stars in the orion nebula cluster , it is likely that some process must be regulating the rotation of cttss since many of these stars do not appear to be spun up by disk accretion . the magnetospheric accretion model provides a means of regulating the stellar rotation , and it has been used to successfully describe some of the observed characteristics of high spectral resolution line profiles observed in cttss ( hartmann , hewett , & calvet 1994 ; muzerolle , calvet , & hartmann 1998 ) . the most successful approach for measuring fields on late type stars in general has been to measure zeeman broadening of spectral lines in unpolarized light ( e.g. , robinson 1980 ; saar 1988 ; valenti , marcy , & basri 1995 ; johns krull & valenti 1996 ) . for any given zeeman @xmath5 component , the splitting resulting from the magnetic field is @xmath6 where @xmath7 is the land-@xmath7 factor of the transition , @xmath8 is the strength of the magnetic field ( given in kg ) , and @xmath9 is the wavelength of the transition ( specified in angstroms ) . the first direct detection of a magnetic field on a tts was reported by basri , marcy , and valenti ( 1992 ) . these authors used the fact that marginally saturated magnetically sensitive lines will have an increase in their equivalent width in the presence of a magnetic field as the line components split to either side , increasing line opacity in the wings . using this equivalent width technique , basri et al . ( 1992 ) measured the field of the ntts tap 35 , obtaining a value of the magnetic field strength , @xmath8 , times the filling factor , @xmath10 , of field regions of @xmath11 kg . as the importance of magnetic fields on ttss has become recognized , more effort has recently been put into their measurement . guenther et al . ( 1999 ) used the same basic technique as basri et al . ( 1992 ) to analyze spectra from 5 ttss , claiming significant field detections on 2 stars . from the perspective of cool star research , the high apparent field strengths on ttss is a little surprising . spruit & zweibel ( 1979 ) computed flux tube equilibrium models , showing that magnetic field strength is expected to scale with gas pressure in the surrounding non - magnetic photosphere . field strengths set by pressure equipartition appears to be observed in g and k dwarfs ( e.g. saar 1990 , 1994 , 1996 ) and possibly in m dwarfs ( e.g. johns krull & valenti 1996 ) . ttss have relatively low surface gravities and hence low photospheric gas pressures , so that equipartition flux tubes would have relatively low magnetic field strengths compared to cool dwarfs . indeed , safier ( 1999 ) examined in some detail the ability of tts photospheres to confine magnetic flux tubes via pressure balance with the surrounding quiet photosphere , concluding that the maximum field strength allowable on ttss is substantially below the current detections . safier does point out a number of potential problems in the data analysis which might affect the results of basri et al . ( 1992 ) and guenther et al . ( 1999 ) and lead to an overestimate of @xmath12 on these stars . on the other hand , it is doubtful that pressure equipartition arguments should hold when the magnetic field filling factor approaches unity on the stellar surface . saar ( 1996 ) finds that the magnetic field strength does rise above the pressure equipartition value for very active k and m dwarfs once the filling factor reaches a value near one . as a result , super - equipartition fields on ttss , if confirmed , would be an extention of the behavior of very active main sequence stars . johns krull , valenti , and koresko ( 1999b - hereafter paper i ) have examined three ( knigl 1991 , cameron & campbell 1993 , shu et al . 1994 ) of the magnetospheric accretion flow theories which provide detailed analytic expressions for the surface magnetic field strength on cttss . in the context of these theories , the surface field strength depends on the stellar mass , radius , rotation rate , and the mass accretion rate onto the star from the disk . taking values of these quantities from the literature , paper i shows that the predicted magnetic field strengths vary over a large range of values , including strengths as high as 11 kg . the mass accretion rate estimates used in paper i are preferentially those of hartigan , edwards , and ghandour ( 1995 ) . gullbring et al . ( 1998 ) argue that these accretion rates are an order of magnitude too high . paper i showed that the predicted field strengths depend on the square root of the accretion rate . adopting the lower accretion rates favored by gullbring et al . ( 1998 ) suggests that the fields on cttss should range up to @xmath13 kg , still quite a measurable value . paper i made use of the wavelength dependence of the zeeman effect ( see eq . [ 1 ] ) to detect actual zeeman broadening of a line at 2.2 @xmath14 m on the ctts bp tau , measuring an average field strength of @xmath15 kg . the detection of zeeman broadening yields magnetic field strengths largely immune to the uncertainties pointed out by safier ( 1999 ) in equivalent width analyses ; however , paper i points out that the observations could be reproduced by an accretion disk with unusual properties . paper i further suggests that observations of additional ir lines ( particularly magnetically insensitive co lines ) are needed to validate the magnetic field detection . paper i also points out that the mean field detected on bp tau is larger than that which can be confined by pressure balance with the quiet photosphere as discussed by safier ( 1999 ) ; however , this is not a problem if the magnetic filling factor is unity in the visible photosphere ( see also solanki 1994 , saar 1996 ) . indeed , the filling factor is unity in the dipole field configurations favored in current theoretical models ( knigl 1991 , cameron & campbell 1993 , shu et al . 1994 ) . another method for detecting magnetic fields is to look for net circular polarization in zeeman sensitive lines . observed along a line parallel to the magnetic field , the zeeman @xmath5 components are circularly polarized , with the components split to either side of the nominal line wavelength having opposite helicity . if there is a net longitudinal component of the magnetic field on the stellar surface , net circular polarization results , and is usually easier to detect than zeeman broadening of optical lines in unpolarized light . if the magnetic topology on the star is complex and small scale ( as on the sun ) , nearly equal amounts of both field polarities will generally be present producing negligible net polarization . krull et al . ( 1999a ) detect net polarization in the @xmath95876 _ emission _ line on the ctts bp tau , indicating a net longitudinal magnetic field of @xmath16 kg in the line formation region . this emission line is believed to form in the post - shock region where disk material accretes onto the stellar surface ( hartmann , hewett , & calvet 1994 ; edwards et al . if the accretion shock forms at or above the stellar surface , this observation provides additional evidence that super - equipartition fields do exist on at least one ctts . in an effort to further test the reality of strong magnetic fields on ttss , we present here an analysis of the ntts hubble 4 we detect zeeman broadening in four lines in the k band . since nttss do not have close circumstellar accretion disks , there can be no confusing the zeeman broadening signature in magnetically sensitive ir lines with kinematic broadening due to formation in a disk . to further verify the magnetic nature of the detected broadening , several magnetically insensitive co lines are also observed in the ir which show no broadening beyond that expected from stellar rotation . hubble 4 is a k7 ntts which shows strong , circularly polarized radio continuum emission which suggests the presence of large scale , ordered magnetic fields in this star ( skinner 1993 ) . in the recent compilation of briceo et al . ( 2002 ) , they adopt = 4060 k and @xmath17 for hubble 4 , implying @xmath18 . using the pre - main sequence evolutionary tracks of palla and stahler ( 1999 ) , these values of the effective temperature and luminosity imply @xmath19 . combining this mass with the above radius gives an expected gravity of log@xmath20 . we present below optical and ir high resolution spectroscopy that implies strong magnetic fields are indeed present on the surface of the ntts hubble 4 . the observations and data reduction are described in 2 . in 3 the analysis of the data is presented : 3.1 examines the optical data used to determine the basic atmospheric parameters of hubble 4 ; while 3.2 presents the analysis of the ir data to measure the magnetic properties of this star . a discussion of the results is given in

high resolution optical and infrared ( ir ) echelle spectra of the naked ( diskless ) t tauri star hubble 4 are presented . the k band ir spectra include 4 zeeman sensitive lines along with several magnetically insensitive co lines . these stellar parameters are used to synthesize k band spectra to compare with the observations .

high resolution optical and infrared ( ir ) echelle spectra of the naked ( diskless ) t tauri star hubble 4 are presented . the k band ir spectra include 4 zeeman sensitive lines along with several magnetically insensitive co lines . detailed spectrum synthesis combined with modern atmospheric models is used to fit the optical spectra of hubble 4 in order to determine its key stellar parameters : = k ; = ; [ m / h ] = ; = . these stellar parameters are used to synthesize k band spectra to compare with the observations . the magnetically sensitive lines are all significantly broadened relative to the lines produced in the non - magnetic model , while the magnetically insensitive co lines are well matched by the basic non - magnetic model . models with magnetic fields are synthesized and fit to the lines . the best fit models indicate a distribution of magnetic field strengths on the stellar surface characterized by a mean magnetic field strength of kg . the mean field is a factor of 2.0 greater than the maximum field strength predicted by pressure equipartition arguments . to confirm the reality of such strong fields , we attempt to refit the observed profiles using a two component magnetic model in which the field strength is confined to the equipartition value representing plage - like regions in one component , and the field is allowed to vary in a cooler component representing spots . it is shown that such a model is inconsistent with the optical spectrum of the tio bandhead at 7055 .

0811.3149

c

in this paper we have studied conformal perturbation theory beyond the leading order . we have shown that , at least up to quadratic order , the combined bulk boundary perturbation problem is renormalisable , using the minimal subtraction scheme . we also discussed the more commonly used ` wilsonian ' ope scheme , and found it to have some shortcomings at higher order in perturbation theory . we identified systematically the universal ( scheme - independent ) quantities , and gave explicit formulae for them at third order in terms of integrals of conformal 4-point functions . finally , we explained how essentially the same analysis works for the pure bulk and pure boundary case . it seems plausible that similar techniques should allow one to prove renormalisability at arbitrary order in perturbation theory , but we have not attempted to do so . our work was originally motivated by the question of how the dependence of the conformal dimension of a boundary changing field upon a bulk modulus can be understood from the world - sheet perspective . our considerations demonstrate that this effect is captured by a certain universal quadratic rg coefficient , for which we gave an explicit formula . this result should also have interesting applications in other contexts ; in particular , it provides a world - sheet method to study the stability of brane setups under arbitrary bulk deformations .

we identify systematically the universal quantities in the beta function equations , and we give explicit formulae for the universal coefficients at next - to - leading order in terms of integrated correlation functions . as an example , we analyse the radius - dependence of the conformal dimension of some boundary operators for the case of a single neumann brane on a circle , and for an intersecting brane configuration on a torus , reproducing in both cases the expected geometrical answer .

higher order conformal perturbation theory is studied for theories with and without boundaries . we identify systematically the universal quantities in the beta function equations , and we give explicit formulae for the universal coefficients at next - to - leading order in terms of integrated correlation functions . as an example , we analyse the radius - dependence of the conformal dimension of some boundary operators for the case of a single neumann brane on a circle , and for an intersecting brane configuration on a torus , reproducing in both cases the expected geometrical answer .

astro-ph9907082

c

our main goal here is to use observed properties of the blue compact dwarf galaxy i zw 18 to put constraints on its age : the high ionization state of the gas and the presence of wr stars in the main body , the existence of ionized gas in the c component , and the colour - magnitude diagrams from hst images . we were motivated by the study of izotov & thuan ( 1999 ) who have analyzed the c / o and n / o abundance ratios of a sample of the most metal - deficient bcds known , including i zw 18 . those authors found that these ratios are constant for galaxies with @xmath9 @xmath3 @xmath10/20 with a very small dispersion around the mean . this strongly suggests that intermediate - mass stars ( @xmath11 @xmath3 8 @xmath13 ) have not had time to release their carbon and primary nitrogen production , establishing an age upper limit of @xmath4 100 myr for very metal - deficient bcds . the conclusion that galaxies with @xmath9 @xmath3 @xmath10 / 20 are younger than 100 myr has been supported by photometric and spectroscopic studies of two very metal - deficient bcds , sbs 0335052 ( @xmath10/40 , thuan et al . 1997 , papaderos et al . 1998 ) and sbs 1415 + 437 ( thuan et al . 1999a ) . here we examine the age evidence for i zw 18 . to put constraints on the age of i zw 18 , we have followed 3 independent lines of investigation : colour - magnitude diagram , spectral synthesis and hydrodynamical age constraints . we have arrived at the following main conclusions : \1 . the distance to i zw 18 must be increased by a factor of 2 from the previously adopted value of 10 mpc to 20 mpc . such a distance is required to have stars bright and massive enough in i zw 18 to account for its high state of ionization and the presence of wolf - rayet stars in its nw component . ( @xmath49 ) vs. @xmath1 and ( @xmath43 ) vs. @xmath2 cmd studies with the new distance of 20 mpc give ages derived from the main sequence turn - off of i zw 18 of @xmath4 15 myr and @xmath4 5 myr for the c component and main body respectively . the location of the resolved luminous red stars with @xmath76 @xmath4 6 mag and ( @xmath43 ) @xmath4 0.6 1.0 mag is consistent with an age @xmath3 100 myr for the c component . the star formation in this component is likely to have stopped 15 20 myr ago . as for the main body , cmd analysis implies that star formation started 20 50 myr ago and still continues nowadays . analysis of shell structures seen in h@xmath5 images also suggests that star formation in the main body began @xmath4 20 myr ago in different locations at the nw side and has been propagating mainly in the se direction . the age upper limit of @xmath4 50 myr derived for the main body is a whole order of magnitude smaller that the one derived by aloisi et al . ( 1999 ) from cmd analysis of similar hst data . the difference is mainly due to the increase in distance of i zw 18 by a factor of 2 . in summary , all three lines of investigation ( cmds , the distribution of h@xmath5 shells and spectroscopy ) lead to the same conclusion , that i zw 18 did not start to form stars until @xmath3 100 myr ago . this supports the contention of izotov & thuan ( 1999 ) that all very metal - deficient galaxies ( @xmath9 @xmath3 @xmath10/20 ) are young . and n.g.g . thank the universitts sternwarte of gttingen and y.i.i . thanks the university of virginia for warm hospitality . we are grateful to d. schaerer for sending to us his stellar evolutionary synthesis models in electronic form and a. aloisi for communicating her results . we acknowledge the financial support of volkswagen foundation grant no . i/72919 ( y.i.i . , , p.p . and k.j.f . ) and of national science foundation grants ast-9616863 ( t.x.t . and y.i.i . ) and ast-9803072 ( c.b.f . ) . research by k.j.f . and p.p . has been supported by deutsche agentur fr raumfahrtangelegenheiten ( dara ) gmbh grants 50 or 9407 6 and 50 or 9907 7 .

hubble space telescope ( hst ) colour - magnitude diagrams in , and along with long - slit multiple mirror telescope ( mmt ) spectrophotometric data are used to investigate the evolutionary status of the nearby blue compact dwarf ( bcd ) galaxy i zw 18 . we find that the distance to i zw 18 should be as high as 20 mpc , twice the previously accepted distance , to be consistent with existing observational data on the galaxy : colour - magnitude diagrams , the high ionization state of the gas and presence of wr stars in the main body , and the ionization state of the c component . however , the presence of large - scale shells observed around the main body suggests that star formation began 20 myr at the nw end and propagated in the se direction . our analysis of colour - magnitude diagrams and of the spectral energy distribution of the c component implies that star formation in this component started 100 myr ago at the nw end , propagated to the se and stopped 15 myr ago .

hubble space telescope ( hst ) colour - magnitude diagrams in , and along with long - slit multiple mirror telescope ( mmt ) spectrophotometric data are used to investigate the evolutionary status of the nearby blue compact dwarf ( bcd ) galaxy i zw 18 . we find that the distance to i zw 18 should be as high as 20 mpc , twice the previously accepted distance , to be consistent with existing observational data on the galaxy : colour - magnitude diagrams , the high ionization state of the gas and presence of wr stars in the main body , and the ionization state of the c component . the spectral energy distribution ( sed ) of the main body of i zw 18 is consistent with that of a stellar population with age 5 myr . however , the presence of large - scale shells observed around the main body suggests that star formation began 20 myr at the nw end and propagated in the se direction . our analysis of colour - magnitude diagrams and of the spectral energy distribution of the c component implies that star formation in this component started 100 myr ago at the nw end , propagated to the se and stopped 15 myr ago . thus , i zw 18 is likely to be one of the youngest nearby extragalactic objects .

1303.1906

i

in this paper i have studied the rest - frame uv / optical / near - ir ( @xmath14 $ \sim$}}~}3 \ , \rm \mu m$ ] ) seds of 174 @xmath4-selected galaxies with secure spectroscopic redshifts @xmath2 , analysing explicitly the presence and significance of a hot - dust , power - law component . i modelled the residual light produced after subtracting different possible power - law components , using stellar templates , on 11 broad - bands from the @xmath34-band through @xmath45 . the subtracted power - law components are characterised by three different spectral indices @xmath3 , 2.0 and 3.0 , and different normalisation weights . i found that in 60 out of 174 ( 35% ) cases , the sed fitting with stellar templates without any previous power - law subtraction can be rejected with @xmath5 confidence , i.e. 35% of the @xmath4 galaxies at @xmath0 are characterised by a significant power - law component in their rest near - ir sed . i referred to these galaxies as plcgs . this high percentage of plcgs at @xmath0 appears to be inherent to the ir bright galaxy population . a similar analysis performed on a control sample of 247 irac - selected galaxies with @xmath2 and similar stellar masses as the @xmath4 galaxies , but which are not @xmath4-detected , shows that the incidence of plcgs is of only 18% . the incidence of plcgs among @xmath4 galaxies is much higher at @xmath8 ( 47% ) than at @xmath62 ( 21% ) . the lower incidence at @xmath62 is explained by the fact that the presence of very hot dust ( @xmath9 $ \sim$}}~}1300 \ , \rm k$ ] ) is necessary to have a significant power - law component manifested at observed wavelengths @xmath42 . therefore , only sources with the most extreme dust conditions would be recognised with the plcg criterion at these higher redshifts . the impact of a significant power - law component in the rest near - ir sed on the derived galaxy stellar masses needs to be analysed on a case - by - case basis , but it is small in general . the correction for stellar masses is within a factor of 1.5 in @xmath7 of cases . only in a small percentage ( @xmath135 ) of the plcgs , basically all corresponding to powerful agn , the correction to the stellar mass is so large that a stellar mass calculation based on the original photometry should clearly be avoided . a key issue is understanding whether the plcgs are truly related to the presence of an agn , particularly at @xmath8 , where the plcg incidence is very high . to investigate this , i studied different properties of these sources . firstly , i looked for identifications of the plcgs in the ultra - deep 4ms x - ray catalogue for the cdfs . i found that 77% of the x - ray agn within my @xmath4 sample with @xmath61 are classified as plcgs . this indicates that the plcg criterion provides a high completeness in selecting agn at these redshifts . interestingly , 78% of the x - ray normal galaxies within my @xmath4 sample with @xmath61 are also plcgs . but their plcg classification or x - ray luminosities do not correlate with the @xmath4 luminosities , so the x ray emission or the power - law sed excess is not simply a tracer of high star formation rates . a stacking analysis of the plcgs that are not individually x - ray detected at @xmath8 reveals that they have on average similar properties as x - ray - detected normal galaxies . the stacked x - ray images yield a significant detection in the soft x - ray band ( @xmath136 ) , and no signal in the hard x - ray band ( @xmath137 ) . around two thirds of the stacked plcgs have a best - fitting power law with @xmath71 . the visual inspection of optical _ hst_/acs images has proven very useful to shed light on the nature of the plcgs at @xmath8 . i found a clear trend relating the plcg morphology and the spectral - index @xmath44 of the best - fitting sed power - law component : plcgs with @xmath3 or 2.0 look regular and typically have a nuclear matter concentration . instead , the plcgs with @xmath10 typically show irregular morphologies , indicating a disturbed galaxy dynamics , which in some cases suggest the presence of galaxy mergers . this morphology trend is independent of the plcg x - ray detection and classification . the @xmath10 power - law component appears then to be a signature of quite extreme ism conditions , where the interstellar dust is heated to temperatures @xmath98 $ \sim$}}~}700 \ , \rm k$ ] by the gas shocks produced in the disturbed ism . of course , this does not preclude the @xmath10 plcgs to also host agn . actually , the presence of an agn is confirmed in some of them by the x - ray data . but an agn presence does not appear to be necessary to produce a significant @xmath10 power - law component . for the plcgs with @xmath3 or 2.0 , instead , higher dust temperatures are necessary @xmath98 $ \sim$}}~}900 \ , \rm k$ ] . this property , along with the regular morphologies , take to the direct conclusion that these sources must host an agn . one could speculate that the @xmath10 plcgs may all be on the way to form an agn as well , but this hypothesis can not be tested with the current data . considering that all the plcgs with @xmath3 or 2.0 in my sample contain agn , and adding all the other x - ray classified agn within my @xmath4 sample , i obtain that a total of 30% of the @xmath4 sources at @xmath8 host agn . this should be considered as a lower limit , as some of the @xmath10 plcgs that are not individually x - ray detected could also contain agn . if all of them had agn , that would give an upper limit of @xmath138 on the agn fraction among lirgs and ulirgs at @xmath8 . very recently , different works have concluded on a higher fraction of agn among ir galaxies than previously known at @xmath139 , obtaining percentages of 37 - 50% ( e.g. * ? ? ? * ; * ? ? ? the results of my study are in line with such conclusions . at @xmath62 , a similar criterion as that used at @xmath96 would indicate that @xmath140 of the @xmath4 galaxies contain an agn , although this figure is largely based on the x - ray detections , rather than the plcg incidence , which is relatively low ( for a comparison , pozzi et al . 2012 derived a fraction of 35% based on the mid-/far - ir sed analysis of 24 ulirgs at @xmath47 ) . in this work , i have calibrated the plcg analysis technique making use of galaxies with secure spectroscopic redshifts . a complete analysis of the applicability of this technique with the simultaneous determination of photometric redshifts ( @xmath141 ) will be presented in a future paper . a preliminary run of my sed fitting code leaving the redshift as a free parameter indicates that , in fact , the consideration of a power - law subtraction produces some improvement in the overall @xmath141-@xmath25 comparison , reducing the number of catastrophic outliers . this suggests that the identification of plcgs can be done , and the power - law subtraction is actually recommendable , when attempting a @xmath141 determination . although less straightforward than a simple irac colour - colour selection , the plcg identification criterion introduced here is much more complete to select agn of different luminosities , especially in the redshift range @xmath8 . keeping only those plcgs with spectral index @xmath3 and 2 should result in the best compromise between completeness and reliability , as most x - ray normal galaxies and x - ray non - detections are among the plcgs with @xmath71 . in the absence of very deep x - ray data , the plcg selection with spectral index segregation offers the most efficient method to identify the agn presence in any large galaxy sample . based on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . also based on observations undertaken at the european southern observatory ( eso ) _ very large telescope ( vlt ) _ under different programmes ; the nasa / esa _ hubble space telescope _ , obtained at the space telescope science institute , operated by the association of universities for research in astronomy , inc . ( aura ) , under nasa contract nas 5 - 26555 ; and the _ chandra x - ray observatory _ , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas8 - 03060 . i thank almudena alonso - herrero , maurilio pannella , brigitte rocca - volmerange , and john silverman for very useful discussions , and an anonymous referee for a very constructive report . i am also grateful to the goods team for the public release of their superb data products . alonso - herrero , a. , quillen , a. c. , rieke , g. h. , ivanov , v. d. , efstathiou , a. , 2003 , , 126 , 81 alonso - herrero , a. , prez - gonzlez , p. g. , alexander , d. m. et al . , 2006 , , 640 , 167 alonso - herrero , a. , pereira - santaella , m. , rieke , g. h. , rigopulou , d. , 2012 , , 744 , 2 armus , l. , charmandaris , v. , bernard - salas , j. et al . , 2007 , , 656 , 148 barthel , p. , haas , m. , leipski , c. , wilkes , b. , 2012 , , 757 , l26 barvainis , r. , 1987 , , 320 , 537 bauer , f. e. , alexander , d. m. , brandt , w. n. et al . , 2004 , , 128 , 2048 bavouzet , n. , dole , h. , le floch , e. , caputi , k. i. , lagache , g. , kochanek , c. s. , 2008 , , 479 , 83 bertin , e. , arnouts , s. , 1996 , , 117 , 393 bruzual , g. , charlot , s. , 2003 , , 344 , 1000 bruzual , g. , 2007 , aspc , 374 , 303 calzetti , d. , armus , l. , bohlin , r. c. et al . , 2000 , , 533 , 682 caputi , k. i. , dole , h. , lagache , g. et al . , 2006a , , 454 , 143 caputi , k. i. , dole , h. , lagache , g. et al . , 2006b , , 637 , 727 caputi , k. i. , lagache , g. , yan , l. et al . , 2007 , , 660 , 97 caputi , k. i. , lilly , s. j. , aussel , h. et al . , 2008 , , 680 , 939 desai , v. , soifer , b. t , dey , a. et al . , 2008 , , 679 , 1204 dole , h. , lagache , g. , puget , j .- l . et al . , 2006 , , 451 , 417 donley , j. l. , rieke , g. h. , prez - gonzlez , p. g. , rigby , j. r. , alonso - herrero , a. , 2007 , , 660 , 167 donley , j. l. , koekemoer , a. m. , brusa , m. et al . , 2012 , , 748 , 142 fadda , d. , yan , l. , lagache , g. et al . , 2010 , , 719 , 425 fazio , g. g. , hora , j. l. , allen , l. e. et al . , 2004 , , 154 , 10 fiore , f. , grazian , a. santini , p. et al . , 2008 , , 672 , 94 giavalisco , m. , ferguson , h. c. koekemoer , a. m. et al . , 2004 , , 600 , l93 gruppioni , c. , pozzi , f. , andreani , p. et al . , 2010 , , 518 , l27 hainline , l. j. , blain , a. w. , smail , i. et al . , 2011 , , 740 , 96 hernn - caballero , a. , prez - fournon , i. , hatziminaoglou , e. et al . , 2009 , , 395 , 1695 hanami , h. , ishigaki , t. , fujishiro , n. et al . , 2012 , , 64 , 70 hnig , s. f. , kishimoto , m. , gandhi , p. et al . , 2010 , , 515a , 23 hwang , h. s. , geller , m. j. , kurtz , m. j. , dellantonio , i. p. , fabricant , d. g. , 2012 , , 758 , 25 hopkins , a. m. & beacom , j. f. , 2006 , , 651 , 142 juneau , s. , dickinson , m. , bournaud , f. et al . , 2013 , , 764 , 176 kennicutt , r. c. jr . , 1998 , , 36 , 189 kocevski , d. d , faber , s. m. , mozena , m. et al . , 2012 , , 744 , 148 kurk , j. , cimatti , a. , daddi , e. et al . , 2013 , , 549a , 63 lacy , m. , storrie - lombardi , l. j. , sajina , a. et al . , , 154 , 166 lacy , m. , petric , a. , sajina , a. , et al . , 2007 , , 133 , 186 lagache , g. , dole , h. , puget , j .- l . , 2004 , , 154 , 112 le fvre , o. , vettolani , g. , paltani , s. et al . , 2004 , , 428 , 1043 le floch , e. , papovich , c. , dole , h. et al . , 2005 , , 632 , 169 lehmer , b. d. , brandt , w. n. , alexander , d. m. et al . , 2008 , , 681 , 1163 lu , n. , helou , g. , werner , m. w. et al . , 2003 , , 588 , 199 mainieri , v. , bongiorno , a. , merloni , a. et al . , 2011 , , 535a , 80 mas - hesse , j. m. , ot - floranes , h. , cervio , m. , 2008 , , 483 , 71 matsuta , k. , gandhi , p. , dotani , t. et al . , 2012 , , 753 , 104 merloni , a. , bongiorno , a. , bolzonella , m. et al . , 2010 , , 708 , 137 mineo , s. , gilfanov , m. , sunyaev , r. , 2013 , , submitted , arxiv : 1207.2157 mor , r. , netzer , h. , 2012 , , 420 , 526 mullaney , j. r. , pannella , m. , daddi , e. et al . , 2012 , , 419 , 95 murakami h. , baba , h. , barthel , p. et al . , 2007 , , 59s , 369 nardini , e. , risaliti , g. , salvati , m. et al . , 2008 , , 385 , l130 oke , j. b. , gunn , j. e. , 1983 , , 266 , 713 osterbrock , d. e. & ferland , g. j. , 2006 , _ astrophysics of gaseous nebulae and active galactic nuclei _ , 2nd edition , university science books papovich , c. , dole , h. , egami , e.et al . , 2004 , , 154 , 70 petric , a. o. , armus , l. , howell , j. et al . , 2011 , , 730 , 28 pilbratt , g. l. , 2003 , spie , 4850 , 586 polletta , m. , wilkes , b. c. , siana , b. et al . , 2006 , , 642 , 673 popesso , p. , dickinson , m. , nonino , m. et al . , , 494 , 443 pozzi , f. , vignali , c. , gruppioni , c. et al . , 2012 , , 423 , 1909 puget , j .- l . , abergel , a. , bernard , j .- p . , boulanger , f. , burton , w. b. , dsert , f .- x . , hartmann , d. , 1996 , , 308 , l5 rieke , g. h. , young , e. t. , engelbracht , c. w. et al . , 2004 , , 154 , 25 rigby , j. r. et al . , 2004 , , 154 , 160 rosario , d. j. , mozena , m. , wuyts , s. et al . , 2013 , , 763 , 59 sajina , a. et al . , 2007 , , 664 , 713 salpeter , e. e. , 1955 , , 121 , 161 schartmann , m. , meisenheimer , k. , camenzind , m. , wolf , s. , henning , t. , 2005 , , 437 , 861 silverman , j. d. , lamareille , f. , maier , c. et al . , 2009 , , 696 , 396 stark , a. a. , gammie , c. , wilson r. w. et al . , 1992 , , 79 , 77 stern , d. , eisenhardt , p. , gorjian , v. et al . , 2005 , , 631 , 163 symeonidis , m. , georgakakis , a. , seymour , n. et al . , 2011 , , 417 , 2239 szokoly , g. p. , bergeron , j. , hasinger , g. et al . , 2004 , , 155 , 271 teplitz , h. i. , desai , v. , armus , l. et al . , 2007 , , 659 , 941 vanzella , e. , cristiani , s. , dickinson , m. et al . , 2008 , , 478 , 83 werner , m. w. et al . , 2004 , , 154 , 1 xue , y. q. , luo , b. , brandt , w. n. et al . , 2011 , apjs , 195 , 10

i derive this diagnostic from the analysis of 174 galaxies with and spectroscopic redshifts in the chandra deep field south , for which i study the rest - frame uv / optical / near - ir spectral energy distributions ( seds ) , after subtracting a hot - dust , power - law component with three possible spectral indices , 2.0 and 3.0 . i obtain that 35% of these sources are power - law composite galaxies ( plcgs ) , which i define as those galaxies for which the sed fitting with stellar templates , without any previous power - law subtraction , can be rejected with confidence . subtracting the power - law component from the plcg seds produces stellar - mass correction factors in of cases .

i present a generalized power - law diagnostic that allows to identify the presence of active galactic nuclei ( agn ) in infrared ( ir ) galaxies at , down to flux densities at which the extragalactic ir background is mostly resolved . i derive this diagnostic from the analysis of 174 galaxies with and spectroscopic redshifts in the chandra deep field south , for which i study the rest - frame uv / optical / near - ir spectral energy distributions ( seds ) , after subtracting a hot - dust , power - law component with three possible spectral indices , 2.0 and 3.0 . i obtain that 35% of these sources are power - law composite galaxies ( plcgs ) , which i define as those galaxies for which the sed fitting with stellar templates , without any previous power - law subtraction , can be rejected with confidence . subtracting the power - law component from the plcg seds produces stellar - mass correction factors in of cases . the plcg incidence is especially high ( 47% ) at . to unveil which plcgs host agn , i conduct a combined analysis of 4ms x - ray data , galaxy morphologies , and a greybody modelling of the hot dust . i find that : 1 ) 77% of all the x - ray agn in my sample at are recognised by the plcg criterion ; 2 ) plcgs with or 2.0 have regular morphologies and $ \sim$}}~}1000 \,\rm k$ ] , indicating nuclear activity . instead , plcgs with are characterised by disturbed galaxy dynamics , and a hot interstellar medium can explain their dust temperatures . overall , my results indicate that the fraction of agn among sources is between% and 52% at .

1501.07888

i

the study of the intersection theory on the moduli space of riemann surfaces with boundary ( often viewed , with the boundary removed , as open riemann surfaces ) was recently initiated in @xcite . the authors constructed a descendent theory in genus @xmath0 and obtained a complete description of it . in all genera , they conjectured that the generating series of the descendent integrals satisfies the open kdv equations . this conjecture can be considered as an open analog of the famous witten s conjecture @xcite . a higher genus construction of the open descendent theory will appear in @xcite . a combinatorial formula for the open intersection numbers in all genera was found in @xcite . in this paper , using the combinatorial formula from @xcite , we present a matrix integral for the generating series of the open intersection numbers . then applying some analytical tools to this matrix integral , we prove the main conjecture from @xcite . the introduction is organized as follows . in section [ wittc ] we briefly recall the original conjecture of e. witten from @xcite . in section [ konts ] we recall kontsevich s combinatorial formula and kontsevich s proof ( @xcite ) of witten s conjecture . section [ okdv ] contains a short account of the main constructions and conjectures in the open intersection theory from @xcite . section [ ocomb ] describes the combinatorial formula of @xcite for the open intersection numbers of @xcite . throughout this text @xmath1 $ ] will denote the set @xmath2 a compact riemann surface is a compact connected smooth complex curve . given a fixed genus @xmath3 and a non - negative integer @xmath4 the moduli space of all compact riemann surfaces of genus @xmath3 with @xmath5 marked points is denoted by @xmath6 p. deligne and d. mumford defined a natural compactification of it via stable curves in @xcite in 1969 . given @xmath7 as above , a stable curve is a compact connected complex curve with @xmath5 marked points and finitely many singularities , all of which are simple nodes . the collection of marked points and nodes is the set of special points of the curve . we require that all the special points are distinct and that the automorphism group of the curve is finite . the moduli of stable marked curves of genus @xmath3 with @xmath5 marked points is denoted by @xmath8 and is a compactification of @xmath6 it is known that this space is a non - singular complex orbifold of complex dimension @xmath9 for the basic theory the reader is referred to @xcite . in his seminal paper @xcite , e. witten , motivated by theories of @xmath10-dimensional quantum gravity , initiated new directions in the study of @xmath8 . for each marking index @xmath11 he considered the tautological line bundles @xmath12 whose fiber over a point @xmath13\in { { \overline{\mathcal{m}}}}_{g , l}\ ] ] is the complex cotangent space @xmath14 of @xmath15 at @xmath16 . let @xmath17 denote the first chern class of @xmath18 , and write @xmath19 the integral on the right - hand side of is well - defined , when the stability condition @xmath20 is satisfied , all the @xmath21 s are non - negative integers , and the dimension constraint @xmath22 holds . in all other cases @xmath23 is defined to be zero . the intersection products are often called _ descendent integrals _ or _ intersection numbers_. note that the genus is uniquely determined by the exponents @xmath24 . let @xmath25 ( for @xmath26 ) and @xmath27 be formal variables , and put @xmath28 let @xmath29 be the generating function of the genus @xmath3 descendent integrals . the bracket @xmath30 is defined by the monomial expansion and the multilinearity in the variables @xmath25 . concretely , @xmath31 where the sum is over all sequences of non - negative integers @xmath32 with finitely many non - zero terms . the generating series @xmath33 is called the _ ( closed ) free energy_. the exponent @xmath34 is called the _ ( closed ) partition function_. put @xmath35 . witten s conjecture ( @xcite ) says that the closed partition function @xmath36 becomes a tau - function of the kdv hierarchy after the change of variables @xmath37 . in particular , it implies that the closed free energy @xmath38 satisfies the following system of partial differential equations : @xmath39 these equations are known in mathematical physics as the kdv equations . e. witten ( @xcite ) proved that the intersection numbers satisfy the string equation @xmath40 for @xmath41 . this equation can be rewritten as the following differential equation : @xmath42 e. witten also showed that the kdv equations together with the string equation actually determine the closed free energy @xmath38 completely . there was a later reformulation of witten s conjecture due to r. dijkgraaf , e. verlinde and h. verlinde ( @xcite ) in terms of the virasoro algebra . define differential operators @xmath43 , @xmath44 , by @xmath45 while for @xmath46 @xmath47 the virasoro equations say that the operators @xmath43 , @xmath48 , annihilate the closed partition function @xmath36 : @xmath49 it is easy to see that the virasoro equations completely determine all intersection numbers . r. dijkgraaf , e. verlinde and h. verlinde ( @xcite ) proved that this description is equivalent to the one given by the kdv equations and the string equation . witten s conjecture was proven by m. kontsevich @xcite . see @xcite for other proofs . kontsevich s proof @xcite of witten s conjecture consisted of two parts . the first part was to prove a combinatorial formula for the gravitational descendents . let @xmath50 be the set of isomorphism classes of trivalent ribbon graphs of genus @xmath3 with @xmath51 faces and together with a numbering @xmath52 $ ] . denote by @xmath53 the set of vertices of a graph @xmath54 . let us introduce formal variables @xmath55 , @xmath56 $ ] . for an edge @xmath57 let @xmath58 where @xmath11 and @xmath59 are the numbers of faces adjacent to @xmath60 . then we have @xmath61 the second step of kontsevich s proof was to translate the combinatorial formula into a matrix integral . then , by using non - trivial analytical tools and the theory of tau - functions of the kdv hierarchy , he was able to prove that @xmath36 is a tau - function of the kdv hierarchy and , hence , the free energy @xmath38 satisfies the kdv equations . in @xcite r. pandharipande , j. solomon and r.t . constructed an intersection theory on the moduli space of stable marked disks . let @xmath62 be the moduli space of stable marked disks with @xmath63 boundary marked points and @xmath5 internal marked points . this space carries a natural structure of a compact smooth oriented manifold with corners . one can easily define the tautological line bundles @xmath64 for @xmath65 $ ] , as in the closed case . in order to define gravitational descendents , as in , we must specify boundary conditions . indeed , given a smooth compact connected oriented orbifold with boundary , @xmath66 of dimension @xmath51 , the poincar - lefschetz duality shows that @xmath67 thus , given a vector bundle on a manifold with boundary , only _ relative euler class _ , relative to nowhere vanishing boundary conditions , can be integrated to give a number . the main construction in @xcite is a construction of boundary conditions for @xmath68 in @xcite , vector spaces @xmath69 of _ multisections _ of @xmath70 which satisfy the following requirements , were defined . suppose @xmath71 are non - negative integers with @xmath72 then 1 . [ it : canonic1 ] for any generic choice of multisections @xmath73 for @xmath74 the multisection @xmath75\\1\le j\le a_i } } s_{ij}\ ] ] vanishes nowhere on @xmath76 2 . [ it : canonic2 ] for any two such choices @xmath77 and @xmath78 we have @xmath79 where @xmath80 and @xmath81 is the relative euler class . the multisections @xmath82 , as above , are called _ canonical_. with this construction the open gravitational descendents in genus @xmath0 are defined by @xmath83 where @xmath84 is as above and @xmath77 is canonical . in a forthcoming paper @xcite , j. solomon and r.t . define a generalization for all genera . in @xcite a moduli space @xmath85 which classifies real stable curves with some extra structure is constructed . the moduli space @xmath85 is a smooth oriented compact orbifold with corners , of real dimension @xmath86 the stability condition is @xmath87 note that naively , without adding an extra structure , the moduli of real stable curves of positive genus is non - orientable . on @xmath85 one defines vector spaces @xmath88 , for @xmath65,$ ] for which the genus @xmath3 analogs of requirements [ it : canonic1],[it : canonic2 ] from above hold . write @xmath89 for the corresponding higher genus descendents . introduce one more formal variable @xmath77 . the _ open free energy _ is the generating function @xmath90 where @xmath91 , @xmath92 , and again we use the monomial expansion and the multilinearity in the variables @xmath93 the following initial condition follows easily from the definitions ( @xcite ) : @xmath94 in @xcite the authors conjectured the following equations : @xmath95 they were called the open string and the open dilaton equation correspondingly . put @xmath96 the main conjectures in @xcite are [ conjecture : open kdv ] the following system of equations is satisfied : @xmath97 in @xcite equations were called the open kdv equations . it is easy to see that @xmath98 is fully determined by the open kdv equations , the initial condition and the closed free energy @xmath38 . let @xmath99 be the _ open partition function_. in @xcite the authors introduced the following modified operators : @xmath100 where the operators @xmath43 were defined in section [ subsubsection : closed virasoro ] . [ conjecture : open virasoro ] the operators @xmath101 , @xmath44 , annihilate the open partition function : @xmath102 in @xcite equations were called the open virasoro equations . again it is easy to see that the open free energy @xmath98 is fully determined by the open virasoro equations , the initial condition @xmath103 and the closed free energy @xmath38 . from the closed string equation it immediately follows that the open string equation is equivalent to , for @xmath104 . moreover , from the equation @xmath105 it follows that the open dilaton equation @xmath106 is equivalent to , for @xmath107 . more precisely , in @xcite it was conjectured that there exists a definition of open intersection numbers for @xmath108 , for which the open kdv and open virasoro equations hold . the definition was later given in @xcite . although it was not clear at all that the open kdv and the open virasoro equations are compatible , in @xcite it was proved that they indeed have a common solution . for @xmath109 the conjectures were proved in @xcite . in @xcite the conjectures are proved for @xmath110 and the open string and the open dilaton equations are proved for all @xmath111 the main result of this paper is the following theorem . [ theorem : main ] conjectures [ conjecture : open kdv ] and [ conjecture : open virasoro ] are true . let @xmath112 be a power series in the variables @xmath113 with the coefficients from @xmath114 $ ] . in @xcite the following system of equations was introduced : @xmath115 it was called the half of the burgers - kdv hierarchy . this system is obviously stronger than the system of the open kdv equations . in @xcite it was actually shown that the half of the burgers - kdv hierarchy is equivalent to the open kdv equations together with equation . denote by @xmath116 a unique solution of system - specified by the initial condition @xmath117 in @xcite it was shown that @xmath116 satisfies the open kdv equations , the initial condition and the open virasoro equations . this proved the equivalence of the open analog of witten s conjecture and the open virasoro conjecture . this also shows that theorem [ theorem : main ] immediately implies the following corollary . the open free energy @xmath98 satisfies the half of the burgers - kdv hierarchy . consider more variables @xmath118 and let @xmath119 . let @xmath112 be a power series in the variables @xmath120 with the coefficients from @xmath114 $ ] . let us extend the half of the burgers - kdv hierarchy by the following equations : @xmath121 in @xcite the extended system - was called the ( full ) burgers - kdv hierarchy . let @xmath122 be a unique solution of it specified by the initial condition @xmath123 we obviously have @xmath124 . in @xcite it was proved that the function @xmath125 satisfies the following extended virasoro equations : @xmath126 where @xmath127 here we , by definition , put @xmath128 . in @xcite it was conjectured that , by adding descendents for boundary marked points , one can geometrically define intersection numbers which will be the coefficients of @xmath122 . in @xcite j. solomon an r.t . give a complete proposal for the construction in all genera , and it is proved to produce the correct intersection numbers in genus @xmath129 moreover , with this proposal the extended open free energy @xmath130 is defined , as well as the extended open partition function @xmath131 it is proved in @xcite that the following equations hold @xmath132 in @xcite , section 5.2 , it is shown that @xmath133 satisfies these equations as well . thus , theorem [ theorem : main ] implies that @xmath134 and we immediately obtain the following generalization of theorem [ theorem : main ] . [ theorem : extended main ] the extended open free energy @xmath130 is a solution of the full burgers - kdv hierarchy . from the recent result of a. alexandrov @xcite it also follows that the extended open partition function @xmath135 becomes a tau - function of the kp hierarchy after the change of variables @xmath37 and @xmath136 . in @xcite r.t . proved a combinatorial formula for the geometric models which were defined in @xcite . he showed that all these intersection numbers can be calculated as sums of amplitudes of diagrams which will be described below . in this paper a matrix model is constructed out of this combinatorial formula . using this matrix model we prove our main theorem [ theorem : main ] . a topological @xmath137-surface with boundary @xmath15 , is a topological connected oriented surface with non - empty boundary , @xmath63 boundary marked points @xmath138},$ ] and @xmath5 internal marked points @xmath139},$ ] and genus @xmath111 by genus we mean , as usual in the open theory , the doubled genus , that is , the genus of the doubled surface obtained by gluing two copies of @xmath15 along @xmath140 we require the stability condition @xmath141 let @xmath142 be non - negative integers such that @xmath143 be a finite set and @xmath144\to a$ ] a map . @xmath145 will be implicit in the definition . a _ @xmath137-ribbon graph with boundary _ is an embedding @xmath146 of a connected graph @xmath147 into a @xmath137-surface with boundary @xmath15 such that * @xmath148}\subseteq \iota(v(g))$ ] , where @xmath53 is the set of vertices of @xmath147 . we henceforth consider @xmath149 as vertices . * the degree of any vertex @xmath150 is at least @xmath151 . * @xmath152 . * if @xmath153 then @xmath154 } d_i,\ ] ] where each @xmath155 is a topological open disk , with @xmath156 . we call the disks @xmath155 faces . * if @xmath157 , then @xmath158 . the genus @xmath159 of the graph @xmath147 is the genus of @xmath15 . the number of the boundary components of @xmath147 or @xmath15 is denoted by @xmath160 and @xmath161 stands for the number of the internal vertices . we denote by @xmath162 the set of faces of the graph @xmath163 and we consider @xmath164 as a map @xmath165 by defining for @xmath166 where @xmath16 is the unique internal marked point in @xmath167 the map @xmath164 is called the labeling of @xmath168 denote by @xmath169 the set of boundary marked points @xmath148}.$ ] two ribbon graphs with boundary @xmath170 are isomorphic , if there is an orientation preserving homeomorphism @xmath171 and an isomorphism of graphs @xmath172 , such that 1 . @xmath173 2 . @xmath174 for all @xmath175.$ ] 3 . @xmath176 where @xmath177 are the labelings of @xmath178 respectively and @xmath179 is any face of the graph @xmath168 note that in this definition we do not require the map @xmath180 to preserve the numbering of the internal marked points . a ribbon graph is _ critical _ , if * boundary marked points have degree @xmath10 . * all other vertices have degree @xmath151 . * if @xmath181 then @xmath109 and @xmath182 such a component is called _ a ghost_. a _ nodal ribbon graph with boundary _ is @xmath183 , where * @xmath184 are ribbon graphs with boundary . * @xmath185 is a set of _ ordered _ pairs of boundary marked points @xmath186 of the @xmath187 s which we identify . after the identification of the vertices @xmath188 and @xmath189 the corresponding point in the graph is called a node . the vertex @xmath188 is called the legal side of the node and the vertex @xmath189 is called the illegal side of the node . * we also require the graph @xmath147 to be connected . we require that elements of @xmath190 are disjoint as sets ( without ordering ) . the set of edges @xmath191 is composed of the internal edges of the @xmath187 s and of the boundary edges . the boundary edges are the boundary segments between successive vertices which are not the illegal sides of nodes . for any boundary edge @xmath60 we denote by @xmath192 the number of the illegal sides of nodes lying on it . the boundary marked points of @xmath147 are the boundary marked points of @xmath187 s , which are not nodes . the set of boundary marked points of @xmath147 will be denoted by @xmath169 also in the nodal case . a nodal graph @xmath183 is _ critical _ , if * all of its components @xmath187 are critical . * any boundary component of @xmath187 has an odd number of points that are the boundary marked points or the legal sides of nodes . * ghost components do not contain the illegal sides of nodes . a nodal ribbon graph with boundary is naturally embedded into the nodal surface @xmath193 . the ( doubled ) genus of @xmath15 is called the genus of the graph . the notion of an isomorphism is also as in the non - nodal case . in figure [ nodal ] there is a nodal graph of genus @xmath0 , with @xmath194 boundary marked points , @xmath195 internal marked points , three components , one of them is a ghost , two nodes , where a plus sign is drawn next to the legal side of a node and a minus sign is drawn next to the illegal side . denote by @xmath196 the set of isomorphism classes of critical nodal ribbon graphs with boundary of genus @xmath3 , with @xmath63 boundary marked points , @xmath5 faces and together with a bijective labeling @xmath197,$ ] the combinatorial formula in @xcite is fix @xmath198 such that @xmath199 . let @xmath200 be formal variables . then we have @xmath201 where @xmath202 we thank a. alexandrov , l. chekhov , d. kazhdan , o. g. louidor , a. okounkov , r. pandharipande , p. rossi , s. shadrin , j. p. solomon and d. zvonkine for discussions related to the work presented here . a. b. was supported by grant erc-2012-adg-320368-mcsk in the group of r. pandharipande at eth zurich , the grants rffi 13 - 01 - 00755 and nsh-4850.2012.1 . r.t . was supported by isf grant 1747/13 and erc starting grant 337560 in the group of j. p. solomon at the hebrew university of jerusalem . part of the work was completed during the visit of a.b to the einstein institute of mathematics of the hebrew university of jerusalem in 2014 and during the visits of r.t . to the forschungsinstitut fr mathematik at eth zrich in 2013 and 2014 .

in a recent work , r. pandharipande , j. p. solomon and the second author have initiated a study of the intersection theory on the moduli space of riemann surfaces with boundary . they conjectured that the generating series of the intersection numbers satisfies the open kdv equations . in this paper we prove this conjecture . our proof goes through a matrix model and is based on a kontsevich type combinatorial formula for the intersection numbers that was found by the second author .

in a recent work , r. pandharipande , j. p. solomon and the second author have initiated a study of the intersection theory on the moduli space of riemann surfaces with boundary . they conjectured that the generating series of the intersection numbers satisfies the open kdv equations . in this paper we prove this conjecture . our proof goes through a matrix model and is based on a kontsevich type combinatorial formula for the intersection numbers that was found by the second author .

0907.4392

i

the values of the fundamental constants determine the nature of the physical universe , from the size of mountains on earth to the eventual fate of the universe as a whole . historically we have assumed that these constants are invariant in space and time . speculation on the possibility of a time variation of the constants was first discussed by @xcite , @xcite and @xcite . in very rare cases , such as the oklo mine @xcite , there exists a terrestrial laboratory to test for time varying constants . it has been known for over thirty years @xcite that damped lyman alpha systems ( dlas ) @xcite offer the opportunity to measure the value of the fundamental constant @xmath0 , the proton to electron mass ratio as the proton to electron mass ratio rather than the inverse to be consistent with the other recent astronomical determinations of @xmath0 discussed here ] , at early times in the universe . the opportunity stems from the direct dependence of the rotational energy of molecules on @xmath0 and the square root dependency on @xmath0 of the vibrational energy relative to the electronic energy . each absorption line has a unique shift for a change in @xmath0 that depends on the vibrational and rotational quantum numbers of the upper and lower energy states . at the time of @xcite , however , the observational capabilities of astronomical spectroscopy and the accuracy of molecular hydrogen laboratory spectroscopy allowed only very crude determinations of @xmath0 at relatively modest look back times . the high line density of atomic hydrogen lines in dlas and the rarity of dlas with measurable amounts of molecular hydrogen further complicated progress . @xcite and @xcite made early measurements of @xmath0 at significant look back times and found no change to accuracies of @xmath10 and @xmath11 in the spectrum of pks 0528 - 250 at a redshift of 2.811 . at the same time calculations of the expected shifts were made by @xcite who developed a method of sensitivity constants for each line that will be discussed later in this work . an additional constraint of @xmath12 was obtained on the same object by @xcite . an excellent review of studies relevant to a determination of the time history of @xmath0 and other fundamental constants is given in @xcite . three advances now provide the opportunity to measure @xmath0 at large look back times and at accuracies that are starting to impact other areas of physics such as dark energy and string theory . the first advance is the construction of large telescopes such as the keck telescopes , the very large telescopes ( vlt ) and now the large binocular telescope ( lbt ) . a second advance is the installation of stable , high resolution and sensitive spectrometers such as hires @xcite at keck and uves @xcite at the vlt . the third key advance is the measurement of the wavelengths of the h@xmath13 lyman and werner electronic transitions to accuracies of a few parts in @xmath14 @xcite . in addition @xcite has recalculated the sensitivity constants , taking into account both adiabatic and nonadiabatic perturbations , to provide an invaluable set of wavelengths at the present day value of @xmath0 and wavelength sensitivities to @xmath0 for the evaluation of the astronomical observations . the most recent efforts to measure @xmath0 at high redshifts have centered on the ultraviolet and visible echelle spectrometer ( uves ) on the vlt . the spectra of two quasars observed in january of 2002 ( q0347 - 383 ) and january of 2003 ( q0405 - 443 ) contain h@xmath13 absorption lines at redshfits of 3.0249 and 2.5947 . the first observations of q0347 - 383 were commissioning observations carried out in 1999 and described by @xcite . @xcite used this data along with a uves spectrum of q 1232 + 082 to investigate possible changes in @xmath0 . they found two results , @xmath15 and @xmath16 at the 3@xmath17 level for two different sets of thorium argon wavelength lists . a subsequent analysis by @xcite using just the q0347 - 383 spectra found a result of @xmath18 . a later reanalysis of the q0347 - 382 data by @xcite produced a limit at a confidence level of @xmath19 of @xmath20 . @xcite combined the line lists of @xcite and @xcite and found that @xmath21 at the 2@xmath17 level . the 2002 and 2003 uves vlt observations of q0347 - 383 and q0405 - 443 @xcite had higher signal to noise than the 1999 observations . using new laser determined h@xmath13 wavelengths from @xcite and the uves pipeline reduction of the spectra they found @xmath22 . @xcite subsequently utilized a new set of laser determined h@xmath13 wavelengths and the pipeline data to find a change in @xmath0 of @xmath23 . @xcite details the determination of the h@xmath13 parameters and gives a more complete list of laser determined wavelengths that slightly alters the result to @xmath24 . the @xcite h@xmath13 parameters essentially remove the properties of h@xmath13 from the error budget leaving the data reduction and signal to noise of the observed spectrum as the primary error contributors . the @xcite result for q0347 - 383 was examined by @xcite who concluded that the data were consistent with @xmath25 at the @xmath19 confidence level . @xcite have taken the same data set as @xcite with the addition of spectra of q0528 - 250 and found a value of @xmath26 for the combined data set . a key element in their analysis is an improved wavelength calibration as described in @xcite . at this time there are two analyzes of the same data that lead to two different conclusions . our independent analysis of the same data concludes that there is no evidence for a change in @xmath0 , consistent with the results of @xcite . for completeness we consider radio frequency measurements of @xmath0 that are more precise but at significantly lower redshift . although the wavelength determinations are more precise , transitions in different molecules must be compared to provide information on any change in @xmath0 . recently @xcite have looked for variations in @xmath0 using the radio emission lines of ammonia and carbon monoxide . they take advantage of the high sensitivity of the inversion spectrum of ammonia to changes in @xmath0 with @xmath27 where @xmath28 is the redshift of the inversion lines of ammonia , @xmath29 is the redshift of the rotational lines of co and @xmath30 is the cosmological redshift of the galaxy . for the galaxy b0218 + 357 at a redshift of they find @xmath31 . @xcite have improved this result to @xmath32 . this result is at relatively low redshift and it depends on ammonia and carbon monoxide having identical kinetic velocities in the molecular clouds . this is probably unlikely since , unlike the ubiquitous co molecule , nh@xmath33 is concentrated in the colder denser cores of molecular clouds . the fact that it is a null result , however , adds credence to the result since an offset in kinetic velocity would have to accurately match any change in @xmath0 to produce a null result . the result is also for a relatively low redshift , placing it within the current dark energy dominated epoch of the universe . some dark energy theories predict that the fundamental constants only roll during the matter dominated epoch and freeze out at their present values once dark energy becomes dominant around a redshift of 1 , ( eg . @xcite ) . table [ tab - comp ] provides a summary of the astronomical determinations of @xmath0 . in our own galaxy @xcite have reported variations in @xmath0 based on the same ammonia transition along different lines of sight . in this case the variation is relative to the ccs molecule and is manifested by a general positive velocity offset between the ammonia and ccs emission lines . their result gives @xmath34 . slight errors in the line frequencies could mimic such a result . laboratory experiments have set limits on the present day rate of change of @xmath0 . even though their time base is brief by cosmological standards , their wavelength accuracy is far better than can be achieved in astronomical observations . the current best laboratory limits appear to be the results of @xcite which give a result of @xmath35 . to put this in perspective if the rate of change is constant at @xmath36 then the change at the 11 gigayear look back time of q0347 - 383 would be @xmath37 , similar to the astronomical results given in this work . there is no real expectation that the rate of change would be constant so both the astronomical and laboratory results work in concert to constrain possible physical models that predict changes in time of the values of the fundamental constants . the results of @xcite depend on the schmidt model for the nuclear magnetic moment and therefore may be deemed as model dependent . a laboratory result that is independent of the schmidt model is given by @xcite who give @xmath38 . other limits on the present rate of variation in @xmath0 based on the weak equivalence principle and various theories of particle physics are discussed by @xcite . the remainder of the paper addresses the measurement of @xmath0 in the spectra of q0347 - 383 and q0405 - 443 . the wavelength calibration and data reduction to produce the spectra used in this work will only be summarized since it is discussed in detail in @xcite . that separate publication is intended to give a full description of the data analysis in order to allow the reader to concentrate on the measurement of @xmath0 described here without a lengthy data reduction description at the beginning . in this paper we bring those analysis methods to bear in an effort to discriminate between the positive and null results for a variation in @xmath0 .

in an effort to resolve the discrepancy between two measurements of the fundamental constant , the proton to electron mass ratio , at early times in the universe we reanalyze the same data used in the earlier studies . both of the previous studies and this study are based on the same data but with differing analysis methods . this leads to the conclusion that the fundamental constant is unchanged to an accuracy of over the last of the age of the universe , well into the matter dominated epoch . new instruments , both planned and under construction , will provide opportunities to greatly improve the accuracy of these measurements .

in an effort to resolve the discrepancy between two measurements of the fundamental constant , the proton to electron mass ratio , at early times in the universe we reanalyze the same data used in the earlier studies . our analysis of the molecular hydrogen absorption lines in archival vlt / uves spectra of the damped lyman alpha systems in the qsos q0347 - 383 and q0405 - 443 yields a combined measurement of a value of , consistent with no change in the value of over a time span of 11.5 gigayears . here we define as where is the value of at a redshift of z and is the present day value . our null result is consistent with the recent measurements of , , and inconsistent with the positive detection of a change in by . both of the previous studies and this study are based on the same data but with differing analysis methods . improvements in the wavelength calibration over the uves pipeline calibration is a key element in both of the null results . this leads to the conclusion that the fundamental constant is unchanged to an accuracy of over the last of the age of the universe , well into the matter dominated epoch . this limit provides constraints on models of dark energy that invoke rolling scalar fields and also limits the parameter space of super symmetric or string theory models of physics . new instruments , both planned and under construction , will provide opportunities to greatly improve the accuracy of these measurements .

0907.4392

c

our basic conclusion , based on the combination of data from q0347 - 383 and q0405 - 443 , is that there has been no change in the value of @xmath0 to 1 part in @xmath78 over a time span of 11.5 gigayears . this is approximately @xmath9 of the age of the universe . the accuracy of the limit on @xmath1 is set by both the spectral resolution and the signal to noise ratio of the flux . this conclusion is consistent with the results of @xcite but inconsistent with the results of @xcite . starting with the same raw data , the primary difference in this analysis is the use of improved wavelength calibration techniques that eliminated the systematic variations in the calibration used in the uves pipeline at the time of the @xcite analysis . the line selection is also most likely different from @xcite but without a list of those lines it is difficult to assess the influence of the lines chosen . there is a marginal possibility of the detection of a change in @xmath0 based on the q0347 - 383 data alone . we , however , feel that this result while suggestive is certainly not conclusive . what implications does a limit on @xmath1 of @xmath8 have on theories of dark energy that invoke a rolling scalar field potential as the source of the dark energy ? @xcite have despaired about distinguishing between a universe with a cosmological constant relative to a universe with a quintessence rolling scalar field , however , the former predicts no change in @xmath0 while the latter predicts a change even though the magnitude or even the sign of the change is not presently calculable . detection of a change in @xmath0 or its companion the fine structure constant @xmath47 would be strong evidence for quintessence as opposed to a cosmological constant . where @xmath82 is @xmath83 , @xmath84 is the planck mass and @xmath85 is a parameter of unknown value ( @xcite and references therein ) . determination of the value of @xmath0 at high redshift is therefore a direct way to distinguish between quintessence and a cosmological constant . in grand unified theories ( guts ) the rolling of @xmath0 is typically given by where @xmath87 is the qcd scale , @xmath88 is the higgs vacuum expectation value ( vev ) , r is a model dependent value ( @xcite and references therein ) and @xmath47 is the fine structure constant . in many gut models the value of r is large and negative @xmath89 @xcite . our current results limit the value of @xmath90 in equation [ eq - dm ] to be on the order of @xmath8 or less , but does not tell us the individual values of @xmath91 or @xmath85 . the results do , however , rule out model a of @xcite at about the 4@xmath17 level where the potential is given by which predicts a value of @xmath93 at a redshift of 3 . this means that even at the current level of accuracy significant bounds on the quintessence models are being established . in all fairness to the model it must be pointed out that it was designed to achieve that result to match the findings of @xcite . if the claim of a detected change in the fine structure constant @xmath47 ( @xmath94 @xcite ) is accepted then this implies a value of r of @xmath95 which is significantly different in sign and magnitude that the typical gut value quoted above . this would mean that either the roll of both the qcd scale and the higgs vev is small or that they are equal to each other by less than a factor of 2 . of course if the claim for a change in @xmath47 is not accepted the current limitation on @xmath96 places no limit on r. rit would like to acknowledge interesting and useful conversations with wim ubachs , dimitrios psaltis , feryal ozel and michael murphy on theory and technique . c.m . wishes to acknowledge very useful discussions with paolo molaro . the work of c.m . is funded by a ciencia2007 research grant . cccc pks 0528 - 250 & @xcite & 2.811 & @xmath97 + pks 0528 - 250 & @xcite & 2.811 & @xmath98 + pks 0528 - 250 & @xcite & 2.811 & @xmath97 + q0347 - 383 + q1232 + 082 & @xcite & 3.0249 & @xmath99 + q0347 - 383 & @xcite & 3.0249 & @xmath100 + q0347 - 383 & @xcite & 3.0249 & @xmath101 + q0347 - 383 & @xcite & 3.0249 & @xmath102 + q0347 - 383 + q0405 - 443 & @xcite & 3.0249 , 2.5974 & @xmath103 + q0347 - 383 + q0405 - 443 & @xcite & 3.0249 , 2.5974 & @xmath104 + q0347 - 383 + q0405 - 443 & @xcite & 3.0249 , 2.5974 & @xmath105 + q0347 - 383 + q0405 - 443 & @xcite & 3.0249 , 2.5974 & @xmath106 + q0347 - 383 + q0405 - 443 & this work & 3.0249 , 2.5974 & @xmath2 + q0347 - 383 + q0405 - 443 + pks 0528 - 250 & @xcite & 3.0249 , 2.5974 , 2.811 & @xmath107 + b0218 + 357 & @xcite & 0.6847 & @xmath108 + b0218 + 357 & @xcite & 0.6847 & @xmath109 + milky way & @xcite & 0.0 & @xmath110 + ccc uves_2002_01_08t00:46:05_351_b.fits & 8 jan . 2002 & 4500 + uves_2002_01_08t02:03:41_018_b.fits & 8 jan . 2002 & 4500 + uves_2002_01_08t03:21:18_348_b.fits & 8 jan . 2002 & 4500 + uves_2002_01_09t00:43:43_109_b.fits & 9 jan . 2002 & 4500 + uves_2002_01_09t02:02:11_833_b.fits & 9 jan . 2002 & 4500 + uves_2002_01_09t03:19:58_841_b.fits & 9 jan . 2002 & 4500 + uves_2002_01_10t00:48:56_171_b.fits & 10 jan . 2002 & 4500 + uves_2002_01_10t02:06:28_725_b.fits & 10 jan . 2002 & 4500 + uves_2002_01_10t03:24:33_981_b.fits & 10 jan . 2002 & 4500 + ccc uves_2003_01_04t00:43:06_274_b.fits & 4 jan . 2003 & 4500 + uves_2003_01_04t02:09:06_464_b.fits & 4 jan . 2003 & 4500 + uves_2003_01_04t03:34:08_623_b.fits & 4 jan . 2003 & 4500 + uves_2003_01_05t00:48:35_827_b.fits & 5 jan . 2003 & 4500 + uves_2003_01_05t02:16:14_922_b.fits & 5 jan . 2003 & 4500 + uves_2003_01_05t03:46:36_522_b.fits & 5 jan . 2003 & 4500 + uves_2003_01_06t00:45:18_207_b.fits & 6 jan . 2003 & 4500 + uves_2003_01_06t02:15:26_790_b.fits & 6 jan . 2003 & 4500 + uves_2003_01_06t03:46:49_242_b.fits & 6 jan . 2003 & 4500 + lcccccccc l15p1 & 123 & 0.05147000 & 3782.21819 & 0.0112 & -0.0094 & 3.56 & 939.70672 & 3.02489213 + l14r1 & 123 & 0.04625000 & 3811.49618 & 0.0069 & -0.0089 & 7.84 & 946.98040 & 3.02489448 + w3q1 & 123 & 0.02149000 & 3813.28002 & 0.0110 & -0.0062 & 2.73 & 947.42188 & 3.02490179 + w3q1 & 122 & 0.02149000 & 3813.28420 & 0.0079 & -0.0055 & 3.59 & 947.42188 & 3.02490620 + w3p3 & 122 & 0.02097000 & 3830.37745 & 0.0056 & -0.0049 & 6.23 & 951.67186 & 3.02489304 + l13r1 & 121 & 0.04821000 & 3844.04623 & 0.0046 & -0.0043 & 11.31 & 955.06582 & 3.02490189 + l13p1 & 121 & 0.04772000 & 3846.62792 & 0.0085 & -0.0076 & 6.33 & 955.70827 & 3.02489760 + w2q1 & 120 & 0.01396000 & 3888.44675 & 0.0052 & -0.0058 & 30.09 & 966.09608 & 3.02490687 + w2q2 & 120 & 0.01272000 & 3893.21423 & 0.0097 & -0.0078 & 8.29 & 967.28110 & 3.02490468 + l12r3 & 120 & 0.03682000 & 3894.80256 & 0.0085 & -0.0088 & 1.16 & 967.67695 & 3.02489959 + w2q3 & 120 & 0.01088000 & 3900.33218 & 0.0042 & -0.0040 & 15.21 & 969.04922 & 3.02490617 + w1q1 & 118 & 0.00487000 & 3971.76790 & 0.0037 & -0.0054 & 3.82 & 986.79800 & 3.02490469 + w1q1 & 117 & 0.00487000 & 3971.76771 & 0.0075 & -0.0130 & 2.04 & 986.79800 & 3.02490450 + l9r1 & 117 & 0.03753000 & 3992.75767 & 0.0034 & -0.0038 & 8.86 & 992.01637 & 3.02489091 + l8r0 & 116 & 0.03475000 & 4032.24698 & 0.0066 & -0.0075 & 6.32 & 1001.82387 & 3.02490607 + l8r1 & 116 & 0.03408000 & 4034.76532 & 0.0051 & -0.0041 & 16.47 & 1002.45210 & 3.02489587 + l8r1 & 115 & 0.03408000 & 4034.75032 & 0.0104 & -0.0097 & 10.49 & 1002.45210 & 3.02488091 + w0r2 & 115 & -0.00525000 & 4061.22312 & 0.0083 & -0.0104 & 11.53 & 1009.02492 & 3.02489873 + w0q2 & 115 & -0.00710000 & 4068.92599 & 0.0067 & -0.0070 & 27.08 & 1010.93845 & 3.02489982 + w0q2 & 114 & -0.00710000 & 4068.91220 & 0.0138 & -0.0157 & 4.71 & 1010.93845 & 3.02488618 + l7r1 & 114 & 0.03027000 & 4078.98076 & 0.0063 & -0.0093 & 7.84 & 1013.43701 & 3.02489816 + l7p3 & 114 & 0.02460000 & 4103.38732 & 0.0046 & -0.0053 & 2.64 & 1019.50224 & 3.02489289 + l6p3 & 112 & 0.02033000 & 4150.43809 & 0.0083 & -0.0182 & 2.55 & 1031.19260 & 3.02489126 + l5p1 & 112 & 0.02064000 & 4178.48451 & 0.0086 & -0.0089 & 12.21 & 1038.15713 & 3.02490566 + l5r2 & 112 & 0.01997000 & 4180.62771 & 0.0074 & -0.0064 & 4.82 & 1038.69027 & 3.02490313 + l4p2 & 110 & 0.01346000 & 4239.36224 & 0.0061 & -0.0046 & 2.54 & 1053.28426 & 3.02489850 + l4p3 & 110 & 0.01051000 & 4252.19544 & 0.0070 & -0.0065 & 4.61 & 1056.47144 & 3.02490335 + l3r1 & 109 & 0.01099000 & 4280.32103 & 0.0042 & -0.0042 & 45.60 & 1063.46014 & 3.02490030 + l3p1 & 109 & 0.01001000 & 4284.92877 & 0.0059 & -0.0058 & 8.87 & 1064.60539 & 3.02489862 + l3r2 & 109 & 0.00953000 & 4286.49249 & 0.0073 & -0.0073 & 12.69 & 1064.99481 & 3.02489519 + l3r3 & 109 & 0.00719000 & 4296.48519 & 0.0041 & -0.0042 & 9.06 & 1067.47855 & 3.02489136 + l2p2 & 107 & 0.00184000 & 4351.98530 & 0.0113 & -0.0135 & 8.78 & 1081.26603 & 3.02489783 + l2p3 & 107 & -0.00115000 & 4365.24899 & 0.0138 & -0.0117 & 18.47 & 1084.56034 & 3.02490192 + l1r1 & 106 & -0.00143000 & 4398.14064 & 0.0054 & -0.0052 & 22.80 & 1092.73243 & 3.02490172 + l1p1 & 106 & -0.00259000 & 4403.45255 & 0.0038 & -0.0036 & 22.80 & 1094.05198 & 3.02490250 + lcccccccc l16p1 & 139 & 0.05297000 & 3351.25678 & 0.0040 & -0.0066 & 5.30 & 932.26621 & 2.59474230 + w4p2 & 139 & 0.02569000 & 3352.45726 & 0.0052 & -0.0066 & 2.66 & 932.60468 & 2.59472489 + w4p3 & 139 & 0.02350000 & 3360.32455 & 0.0060 & -0.0072 & 1.27 & 934.79006 & 2.59473715 + l15p3 & 138 & 0.04676000 & 3394.61670 & 0.0083 & -0.0058 & 10.34 & 944.33046 & 2.59473389 + w3r2 & 137 & 0.02287000 & 3404.62607 & 0.0037 & -0.0041 & 13.42 & 947.11169 & 2.59474612 + l14r2 & 137 & 0.04715000 & 3409.50987 & 0.0049 & -0.0043 & 13.11 & 948.47125 & 2.59474246 + w3q3 & 137 & 0.01828000 & 3416.42562 & 0.0029 & -0.0028 & 5.69 & 950.39773 & 2.59473251 + l13p2 & 136 & 0.04577000 & 3442.49784 & 0.0049 & -0.0049 & 14.30 & 957.65223 & 2.59472649 + l12p2 & 134 & 0.04341000 & 3473.51556 & 0.0142 & -0.0071 & 6.53 & 966.27550 & 2.59474659 + w2p3 & 134 & 0.00992000 & 3488.92549 & 0.0077 & -0.0080 & 28.26 & 970.56332 & 2.59474279 + l11p2 & 133 & 0.04092000 & 3506.10669 & 0.0088 & -0.0053 & 3.54 & 975.34576 & 2.59473208 + l9r2 & 131 & 0.03594000 & 3571.54986 & 0.0023 & -0.0020 & 7.52 & 993.55061 & 2.59473370 + l9p2 & 131 & 0.03489000 & 3576.31685 & 0.0031 & -0.0045 & 9.02 & 994.87408 & 2.59474322 + l9p2 & 130 & 0.03489000 & 3576.30117 & 0.0026 & -0.0028 & 18.84 & 994.87408 & 2.59472746 + l9p3 & 130 & 0.03202000 & 3586.92017 & 0.0078 & -0.0057 & 8.06 & 997.82718 & 2.59473087 + l8r2 & 130 & 0.03251000 & 3609.06404 & 0.0076 & -0.0083 & 6.95 & 1003.98545 & 2.59473739 + l8r2 & 129 & 0.03251000 & 3609.06183 & 0.0073 & -0.0052 & 5.13 & 1003.98545 & 2.59473519 + l8p2 & 129 & 0.03137000 & 3614.13175 & 0.0037 & -0.0038 & 12.62 & 1005.39320 & 2.59474457 + w0r3 & 128 & -0.00631000 & 3631.14901 & 0.0049 & -0.0046 & 7.27 & 1010.13025 & 2.59473346 + w0q2 & 128 & -0.00710000 & 3634.05522 & 0.0035 & -0.0041 & 23.82 & 1010.93845 & 2.59473440 + l7p2 & 128 & 0.02750000 & 3653.90105 & 0.0045 & -0.0044 & 15.12 & 1016.46125 & 2.59472734 + l6p2 & 127 & 0.02324000 & 3695.76334 & 0.0040 & -0.0125 & 21.67 & 1028.10609 & 2.59472954 + l6p2 & 126 & 0.02324000 & 3695.77191 & 0.0034 & -0.0079 & 6.73 & 1028.10609 & 2.59473788 + l5p2 & 125 & 0.01857000 & 3739.84184 & 0.0039 & -0.0037 & 11.57 & 1040.36733 & 2.59473210 + l5r3 & 125 & 0.01759000 & 3742.68792 & 0.0040 & -0.0038 & 17.41 & 1041.15892 & 2.59473260 + l5p3 & 125 & 0.01564000 & 3751.12621 & 0.0030 & -0.0043 & 26.05 & 1043.50319 & 2.59474340 + l5p3 & 124 & 0.01564000 & 3751.12050 & 0.0031 & -0.0031 & 10.63 & 1043.50319 & 2.59473793 + l4r2 & 124 & 0.01497000 & 3779.85697 & 0.0048 & -0.0040 & 15.03 & 1051.49857 & 2.59473334 + l4r3 & 123 & 0.01261000 & 3788.77180 & 0.0054 & -0.0038 & 25.20 & 1053.97610 & 2.59474166 + l3p2 & 122 & 0.00790000 & 3835.21789 & 0.0037 & -0.0040 & 17.12 & 1066.90068 & 2.59472813 + l3r3 & 122 & 0.00719000 & 3837.30732 & 0.0033 & -0.0035 & 10.22 & 1067.47855 & 2.59473951 + l3p3 & 122 & 0.00493000 & 3846.87296 & 0.0057 & -0.0044 & 5.16 & 1070.14088 & 2.59473508 + l3p3 & 121 & 0.00493000 & 3846.87940 & 0.0040 & -0.0054 & 16.28 & 1070.14088 & 2.59474110 + l2r2 & 121 & 0.00360000 & 3879.52758 & 0.0031 & -0.0030 & 8.81 & 1079.22542 & 2.59473332 + l2r2 & 120 & 0.00360000 & 3879.53298 & 0.0021 & -0.0022 & 7.05 & 1079.22542 & 2.59473832 + l1p2 & 119 & -0.00475000 & 3941.40916 & 0.0044 & -0.0034 & 20.26 & 1096.43894 & 2.59473657 + l1r3 & 119 & -0.00509000 & 3942.44064 & 0.0034 & -0.0042 & 8.44 & 1096.72534 & 2.59473835 + l1p2 & 118 & -0.00475000 & 3941.41164 & 0.0031 & -0.0043 & 8.44 & 1096.43894 & 2.59473884 + l1p3 & 118 & -0.00775000 & 3953.45055 & 0.0061 & -0.0050 & 8.44 & 1099.78718 & 2.59474144 + l0r0 & 117 & -0.00800000 & 3983.42760 & 0.0040 & -0.0090 & 8.44 & 1108.12733 & 2.59473816 + l0p2 & 117 & -0.01191000 & 3999.12069 & 0.0028 & -0.0028 & 8.44 & 1112.49600 & 2.59472815 + l0r3 & 117 & -0.01202000 & 3999.42503 & 0.0084 & -0.0105 & 8.44 & 1112.58000 & 2.59473029 +

our analysis of the molecular hydrogen absorption lines in archival vlt / uves spectra of the damped lyman alpha systems in the qsos q0347 - 383 and q0405 - 443 yields a combined measurement of a value of , consistent with no change in the value of over a time span of 11.5 gigayears . here improvements in the wavelength calibration over the uves pipeline calibration is a key element in both of the null results . this limit provides constraints on models of dark energy that invoke rolling scalar fields and also limits the parameter space of super symmetric or string theory models of physics .

in an effort to resolve the discrepancy between two measurements of the fundamental constant , the proton to electron mass ratio , at early times in the universe we reanalyze the same data used in the earlier studies . our analysis of the molecular hydrogen absorption lines in archival vlt / uves spectra of the damped lyman alpha systems in the qsos q0347 - 383 and q0405 - 443 yields a combined measurement of a value of , consistent with no change in the value of over a time span of 11.5 gigayears . here we define as where is the value of at a redshift of z and is the present day value . our null result is consistent with the recent measurements of , , and inconsistent with the positive detection of a change in by . both of the previous studies and this study are based on the same data but with differing analysis methods . improvements in the wavelength calibration over the uves pipeline calibration is a key element in both of the null results . this leads to the conclusion that the fundamental constant is unchanged to an accuracy of over the last of the age of the universe , well into the matter dominated epoch . this limit provides constraints on models of dark energy that invoke rolling scalar fields and also limits the parameter space of super symmetric or string theory models of physics . new instruments , both planned and under construction , will provide opportunities to greatly improve the accuracy of these measurements .

1701.05316

i

in fluid dynamical simulations in astrophysics , large deformations are common and surface tracking is sometimes necessary . mesh - free methods , in which particles move following the motion of fluid , are very useful for such simulations . in particular , smoothed particle hydrodynamics ( sph ; @xcite , @xcite ) has been widely used in astrophysics and also in computer - aided engineering . sph is one of lagrangian methods . in sph , we assume that the fluid equation can be expressed by interactions between fluid particles . therefore , sph is not only suitable for simulation of large deformations , but also can satisfy the conservation laws . recently , however , it has become known that sph has several difficulties . for example , it can not handle contact discontinuities ( e.g. @xcite , @xcite ) or fluid surfaces . in our opinion , the standard formation of sph has the following three problems , 1 ) the density distribution must be differentiable ( e.g. @xcite ) . hence , sph can not handle the contact discontinuity properly . 2 ) since the approximation of quantities in sph is of zeroth order in space , sph does not have the consistency to the original partial differential equation ( e.g. @xcite ) . according to the lax equivalence theorem ( @xcite ) , a method , which does not have the consistency , does not converge to the original partial differential equation in the limit of the infinite resolution . 3 ) there is no mathematically sound way to specify boundary conditions in sph , except for the mirror boundary condition . traditionally , fixed particles have been used to express reflecting boundaries such as walls and bottom of a well . they are necessary because sph can not express a sharp cutoff in the density distribution . however , there is no way to let these fixed particles change their physical quantities correctly . thus , smoothed estimate of physical quantities of particles near the boundary contains large errors . to solve problem one , @xcite , @xcite , @xcite and @xcite proposed modified formulation of sph in which the differentiability of the density is not required . their methods can handle the density discontinuity better than standard sph ( hereafter ssph ) does . in previous studies , high - order scheme have been proposed as the solution to problem two . @xcite formulated moving least squares particles hydrodynamics ( mlsph ) based on the moving least square ( mls ) technique . in reproducing kernel particle method ( rkpm ; @xcite ) , the formulation is not mls , but is similar . these methods were applied to inviscid fluid dynamics simulations . however , they have not been applied to large deformations . corrective sph method ( cspm ; @xcite , @xcite , @xcite , @xcite ) is based on the taylor expansion and method of weighted residuals ( mwr ) . they calculated burgers equation , conduction of heat , linear elastodynamics and others with cphm . in addition , @xcite modified this scheme and they called it modified sph ( msph ) . msph was applied to the elastic wave and the diffusion equation . for non - compressible fluid , @xcite introduced high - order formulation into moving particle semi - implicit method ( @xcite ) . in addition , finite particle method ( fpm ; @xcite ) was developed using mwr to handle viscous fluid . they calculated dam break test with fpm . corrected sph ( csph ) in @xcite successfully handled the dam break test of inviscid fluid by re - fitting the density frequently . the frequent re - fitting generates numerical viscosity , and particles move to reduce the number - density deviation . thus , large viscosity occurs in their simulation . in other word , if fluid particles do not move following fluid line to prevent the number - density deviation , large numerical viscosity is induced . there seems to be no high - order scheme without large numerical viscosity for inviscid fluid tested with the calculation of large deformations . @xcite argued that it is difficult to handle large deformations with a high - order scheme . figure [ fig : khi_noise ] shows the result of a simulation of the kelvin - helmholtz instability performed using the high - order mesh - free method presented in section [ sec : form_norm ] . the simulation time is @xmath0 , where @xmath1 is the time scale of the kelvin - helmholtz instability . we can see large deviations of the number - density of particles . these deviations are the result of the fact that each particle move following the fluid motion at its location accurately . small - scale vortex can easily generate highly disordered distribution of particles . this problem is mathematically same as the generation of large density fluctuation in cold keplerian disk ( e.g. @xcite ) . . the right panel is the enlarge image of low - density region of the left.,width=453 ] from the above , it is necessary to rearrange particles , when the large number - density deviation occurs . one potential problem of high - order method is that they do not completely satisfy the conservation law because particles do not have extensive variables ( @xcite , @xcite ) . to construct a high - order mesh - free method which satisfies the conservation laws , we must define fluid particles which have extensive variables . hence , each particle must also have its volume . this volume has to be represented by some physical shape of the particle . if the mass of a particle is constant , its shape of the particle has to change following the fluid deformation . consider an initially spherical particle in velocity field with a uniform shear . it will become elongated very soon , and thus numerical scheme would break down . therefore , no high - order method can satisfy the conservation law completely . however , we found that the error of the conservation law is very small for high - order methods . concerning problem three , we can apply boundary conditions in mathematically meaningful and well - defined way because we approximate the partial differential equation using fitting polynomial for intensive variables in high - order mesh - free method . we call our method consistent particle hydrodynamics in strong form ( cphsf ) . we performed several numerical tests , and results were excellent . in the rest of this paper , we present cphsf ( section [ sec : form ] ) , and report the results of numerical tests ( section [ sec : test ] ) . finally , we discuss and summarize our study ( section [ sec : sumanddis ] ) .

in fluid dynamical simulations in astrophysics , large deformations are common and surface tracking is sometimes necessary . smoothed particle hydrodynamics ( sph ) method has been used in many of such simulations . recently , however , it has been shown that sph can not handle contact discontinuities or free surfaces accurately . there are several reasons for this problem . the second one is that sph does not have the consistency , and thus the accuracy is zeroth order in space . , we propose a novel , high - order scheme for particle - based hydrodynamics of compressible fluid . we have applied our method to many test problems and obtained excellent result . our method is not conservative , since particles do not have mass or energy , but only their densities . however , because of the lagrangian nature of our scheme , the violation of the conservation laws turned out to be small . we name this method consistent particle hydrodynamics in strong form ( cphsf ) .

in fluid dynamical simulations in astrophysics , large deformations are common and surface tracking is sometimes necessary . smoothed particle hydrodynamics ( sph ) method has been used in many of such simulations . recently , however , it has been shown that sph can not handle contact discontinuities or free surfaces accurately . there are several reasons for this problem . the first one is that sph requires that the density is continuous and differentiable . the second one is that sph does not have the consistency , and thus the accuracy is zeroth order in space . in addition , we can not express accurate boundary conditions with sph . in this paper , we propose a novel , high - order scheme for particle - based hydrodynamics of compressible fluid . our method is based on kernel - weighted high - order fitting polynomial for intensive variables . with this approach , we can construct a scheme which solves all of the three problems described above . for shock capturing , we use a tensor form of von - neumann - richtmyer artificial viscosity . we have applied our method to many test problems and obtained excellent result . our method is not conservative , since particles do not have mass or energy , but only their densities . however , because of the lagrangian nature of our scheme , the violation of the conservation laws turned out to be small . we name this method consistent particle hydrodynamics in strong form ( cphsf ) .

1401.1154

i

there are many situations in which it is necessary to distinguish the topological properties of ring - shaped quasi one - dimensional objects . this is for instance the case of polymers @xcite , vortex structures in nematic liquid crystals @xcite , @xmath2 superfluid @xcite and disclination lines in chiral nematic colloids @xcite . in order to ascertain the type of a knot , it is possible to apply the so - called knot invariants . these mathematical quantities , which remain unchanged under ambient isotopy , are usually represented in the form of polynomials , like for example the alexander @xcite or the homfly polynomials @xcite . alternatively , certain knot invariants may be defined in terms of multiple curvilinear integrals , in which the integrations are performed along the spatial trajectory of the knot or elements of it @xcite . particularly important for applications is the case in which knots are constructed by joining together at their ends a set of @xmath3 segments . discrete knots of this kind are in fact the most common concrete realizations of knots in numerical simulations . formally , a discrete knot is @xmath4curve which is piecewise smooth and is characterized by sharp corners at the joints between contiguous segments . while there exist already well established mathematical algorithms in order to compute numerically polynomial knot invariants , see for instance the pioneering work @xcite , there are not many studies concerning the numerical computation of knot invariants given in the form of multiple line integrals for such discrete knots . of course , the calculation of line integrals over discrete data is a textbook subject @xcite . moreover , problems in which knots are discretized using splines , have been investigated for example in @xcite . however , we are faced with a somewhat different problem , which arises due to the fact that knot invariants expressed as multiple curvilinear integrals are not well defined in the case of discrete knots . the reason of this failure is related to the presence of the non - smooth corners at the joints between two contiguous segments . as a consequence , the main goal pursued here is to replace the piecewise smooth curves representing a discrete knot with more regular ones . to obtain a smoothing algorithm that is able to perform this replacement for general discrete knots without destroying their topology , a strategy has been adopted that can be briefly summarized as follows . first , the sharp corners are surrounded with spheres whose radii are chosen in such a way that they do not intersect with themselves and with other elements of the knot . after that , the elements containing the corners inside the spheres are substituted with arcs of smooth curves . this procedure transforms the original trajectories into @xmath5curvescurve is a tangent vector geometrically continuous curve characterized by the fact that the unit tangent vector to the curve is continuous @xcite . ] without altering their topological configurations . as an application , the case of the vassiliev knot invariant of degree 2 @xcite of a knot @xmath0 , denoted here @xmath1 , is worked out . the main advantages of choosing this invariant are its relative simplicity and the fact that its exact values for different knots can be computed analytically . in this way , it is possible to perform a comparison between numerical and analytical results . after the smoothing procedure proposed in this work , it becomes possible to calculate @xmath1 numerically with an arbitrarily high precision . despite its simplicity , the vassiliev invariant of degree 2 requires the evaluation of complicated quadruple and triple line integrals . having in mind concrete applications , in which the knot invariant @xmath1 must be computed millions of times , see for instance ref . @xcite , we have proposed here several strategies to accelerate its calculation . it turns out that monte carlo integration algorithms are faster than traditional integration methods @xcite . for this reason , a monte carlo integration scheme is adopted and explained in details . moreover , several tricks to speed up the computation of @xmath1 , that are specific to particular applications or situations , are presented . since the time for evaluating @xmath1 is sensitive to the number of segments @xmath3 composing the knot , but not on its length , we have provided an algorithm to reduce by a factor three the number of segments without changing the topology of the knot . this algorithm is valid for knots defined on a simple cubic lattice . secondly , it is shown that the number of points of the trajectory @xmath0 to be sampled during the monte carlo integration procedure may be considerably decreased when the knot invariant @xmath1 is used in order to detect topology changes that may potentially occur after a random transformation of an element of the trajectory of the knot . such random transformations , like for instance the pivot moves @xcite , the pull moves @xcite and the bfacf moves @xcite , are extensively exploited in numerical simulations of polymer knots . the material presented in this paper is divided as follows . in the next section , the vassiliev invariant of degree 2 is defined in the case of general smooth curves . in section [ section3 ] we specialize to general discrete knots , which are represented as piecewise smooth curves parametrized by a continuous variable @xmath6 $ ] . in this way , the calculation of @xmath1 is reduced to that of multiple integrals over a set of variables @xmath7 $ ] and can be tackled by standard monte carlo techniques . a numerical version of the so - called framing @xcite procedure is implemented in order to regularize singularities that are possibly arising in some of the terms to be integrated . while it is analytically proven that the sum of all these terms is always finite , the presence of singularities in single terms may spoil the result of the numerical integration . a smoothing procedure is presented in section [ section4 ] in order to transform a general discrete knot into a @xmath5curve . this procedure allows the calculation of @xmath1 by monte carlo integration techniques with an arbitrary precision depending on the number of used samples . in section [ section5 ] a few methods to speed up the calculations are discussed . finally , the conclusions are drawn in section [ section6 ] .

in mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory describing the spatial conformation of the knot . in this work it is addressed the problem of evaluating invariants of this kind in the case in which the knot is discrete , i. e. its trajectory is constructed by joining together a set of segments of constant length . examples are polymers , the vortex lines in fluids and superfluids like helium and other quantum liquids . formally , the trajectory of a discrete knot is a piecewise smooth curve characterized by sharp corners at the joints between contiguous segments . a smoothing procedure is presented , which eliminates the sharp corners and transforms the original path into a curve that is everywhere differentiable . it is shown that , after the smoothing , the values of may be evaluated with an arbitrary precision .

in mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory describing the spatial conformation of the knot . in this work it is addressed the problem of evaluating invariants of this kind in the case in which the knot is discrete , i. e. its trajectory is constructed by joining together a set of segments of constant length . such discrete knots appear almost everywhere in numerical simulations of systems containing one dimensional ring - shaped objects . examples are polymers , the vortex lines in fluids and superfluids like helium and other quantum liquids . formally , the trajectory of a discrete knot is a piecewise smooth curve characterized by sharp corners at the joints between contiguous segments . the presence of these corners spoils the topological invariance of the knot invariants considered here and prevents the correct evaluation of their values . to solve this problem , a smoothing procedure is presented , which eliminates the sharp corners and transforms the original path into a curve that is everywhere differentiable . the procedure is quite general and can be applied to any discrete knot defined off or on lattice . this smoothing algorithm is applied to the computation of the vassiliev knot invariant of degree 2 denoted here with the symbol . this is the simplest knot invariant that admits a definition in terms of multiple line integrals . for a fast derivation of , it is used a monte carlo integration technique . it is shown that , after the smoothing , the values of may be evaluated with an arbitrary precision . several algorithms for the fast computation of the vassiliev knot invariant of degree 2 are provided .

1401.1154

c

in this work an algorithm to compute the vassiliev invariant of degree 2 @xmath1 for any knot @xmath0 has been provided . particular attention has been devoted to discrete knots whose trajectories consist of segments connected together at their ends forming @xmath4curves in the space . this case is relevant in numerical simulations , where knots are forcefully discretized and represented as @xmath4curves . the vassiliev invariant of degree 2 is probably the simplest knot invariant that can be defined in terms of multiple integrals computed along the contour of the knot itself . for this reason , there are chances that in the future this invariant will play for knots the same role played by the gauss linking number in numerical studies of links formed by ring - shaped quasi one - dimensional objects . a suitable parametrization of discrete knots has been introduced , see eqs . ( [ equ11][xsdefadd ] ) and the problem of computing the multiple contour integrals has been tackled using the monte carlo integration scheme summarized by the general equation ( [ mcf ] ) . in principle , using the scale invariance of @xmath1 , it is possible to reduce arbitrarily the length of the knot , because a change of scale does not alter its topology . however , even in the best case in which the lengths of the segments become infinitesimal , so that the integration over them can be easily approximated , standard methods require the computation of a sum over @xmath325 terms in order to obtain the value of @xmath1 , as shown in @xcite and in the previous section . with the monte carlo integration a considerably smaller number of samples is necessary in order to evaluate @xmath1 with a satisfactory approximation . for this reason , the standard integration methods like the simpson s rule are decidedly slower as it is pointed out by the example provided at the beginning of section [ section5 ] of a knot with @xmath326 segments . further refinements of the naive monte carlo integration scheme presented in section [ section3 ] are difficult to be implemented or do not increase substantially the speed of the computations . for instance , the division of the integration domain , which is recommended as one of the strategies to improve the sampling efficiency , did not lead to significant improvements . on the other side , it is not easy to guess which distribution of the values of the integration variables could be suitable in order to enhance the sampling procedure . in the computations of @xmath1 for discrete knots consisting of a set of segments , we have found that the results never coincide with the analytical values . this is expected because of the sharp corners at the points where the segments join together . there is some correlation between the number of these corners and the systematic error in the evaluation of @xmath1 that apparently depends on the type of the knot and the number of its segments @xmath3 . however , to establish a general relation between that systematic error and @xmath3 which could be valid for every knot has not been possible . to solve the problem of the sharp corners , a procedure for the smoothing of discrete knots has been presented in section [ section4 ] . this procedure transforms the discrete knot into a @xmath5curve . after the smoothing , it has been possible to evaluate @xmath1 with an arbitrary precision by gradually increasing the number of samples used in the monte carlo integration of the multiple integrals entering the definition of @xmath1 , see for example table [ table2 ] . despite the advantages of the adopted monte carlo method with respect to the traditional integration techniques , the calculation of @xmath1 becomes challenging when the number of segments composing the knot is large . for certain physical applications of the knot invariant @xmath1 , we show that the time necessary for its evaluation , which approximately scales as the fourth power of @xmath3 when @xmath3 is large , can be reduced in such a way that scales with the third power of @xmath3 , see eq . ( [ 53 ] ) . moreover , we present an algorithm to reduce the number of segments of a knot defined on a cubic lattice by a factor three without changing the topology . after this reduction , the knot is no longer defined on a lattice , but still the general smoothing procedure of section [ section4 ] and the provided monte carlo integration scheme of section [ section3 ] can be applied in order to obtain the value of @xmath1 . to give an idea of the efficiency of the methods for reducing the number of segments explained in section [ section5 ] , in the case of a knot with @xmath327 originally constructed on a simple cubic lattice , the number of segments in the final configuration obtained after the treatment ranges between 255 and 300 depending on the initial shape of the knot . finally , it is important to notice that , in order to distinguish the topology of different knots , it is not necessary to achieve a standard deviation that is lower than the threshold value given in eq . ( [ varcond ] ) . in this case , in fact , the probability that one knot could be confused with a topologically inequivalent one due to statistical errors is very small , of the order of @xmath328 . work is in progress in order to generalize the methods presented in this work to the case of the triple invariant of milnor that describes the links formed by three knots . to compute the nearest point @xmath329 of a segment @xmath330 from the vertex @xmath109 , we pick up on @xmath331 a general point @xmath332 as follows : @xmath333 with @xmath334 $ ] . the distance between this point and @xmath109 is @xmath335 . if @xmath332 is the nearest point to @xmath109 , then it satisfies the condition @xmath336 inserting eq . ( [ d1 ] ) in ( [ d2 ] ) and solving eq . ( [ d2 ] ) with respect to @xmath286 , we obtain that the point of @xmath331 at the minimal distance from @xmath109 corresponds to the following value of @xmath286 : @xmath337 three cases may occur : * if @xmath338 , this means that the nearest point occurs on a line having the same direction of @xmath331 at a distance @xmath338 from the point @xmath230 . this means that the nearest point to @xmath109 on the segment @xmath331 is @xmath339 and its distance from @xmath109 is @xmath340 . * if @xmath341 , the nearest point occurs on a line having the same direction of @xmath331 at a distance @xmath342 from @xmath230 . on the segment @xmath331 , the nearest point is in this case the point @xmath230 . its distance from @xmath109 is @xmath343 . * if @xmath344 , then @xmath329 lies on the segment @xmath331 and @xmath345 . the distance of @xmath329 from @xmath109 is in this case : @xmath346 ^ 2}{({\boldsymbol}x_{k+1}-{\boldsymbol}x_k)^2}}\ ] ] by repeating this procedure for all segments @xmath331 with @xmath347 and @xmath348 , we obtain the location of the point of the knot which is not belonging to @xmath67 and @xmath108 , and it is the nearest one from @xmath109 . buerle , c. , bunkov , yu . m. , fisher , s. n. , godfrin , h. , pickett , g. r. : laboratory simulation of cosmic string formation in the early universe using superfluid @xmath349he . nature 382 , 332 - 334 ( 1996 ) ; ruutu , v. m. h. , eltsov , v. b. , gill , a. j. , kibble , t. w. b. , krusius , m. , makhlin , yu . g. , plaais , b. , volovik , g. e. , xu , w. : vortex formation in neutron - irradiated superfluid @xmath349he as an analogue of cosmological defect formation . nature 382 , 334 - 336 ( 1996 ) araki , t. , tanaka , h. : colloidal aggregation in a nematic liquid crystal : topological arrest of particles by a single - stroke disclination line . 97 , 127801 - 4 ( 2006 ) ; tkalec , u. , ravnik , m. , opar , s. , umer , s. , muevi , i. : reconfigurable knots and links in chiral nematic colloids . science 333 , 62 - 65 ( 2011 ) ; umer , s. : 21st international liquid crystal conference , keystone , colorado , usa , july 27 , 2006 , kinsley & associates , littletown , ( 2006 ) kontsevich , m. : vassiliev s knot invariants , preprint , max - planck - institut fr mathematik , bonn ; bar - natan , d. : on the vassiliev knot invariants , harvard preprint ( 1992 ) ; bar - natan , d. : on the vassiliev knot invariants , topology 34 , 423 - 472 ( 1995 ) lesh , n. , mitzenmacher , m. , whitesides , s. : a complete and effective move set for simplified protein folding . proceedings of the seventh annual international conference on research in computational molecular biology ( recomb03 ) , 188 - 195 ( 2003 ) aragao de carvalho , c. , caracciolo , s. , frhlich , j. : polymers and @xmath350 theory in four dimensions . b 215 , 209 - 248 ( 1983 ) ; berg , b. , foerster , d. : random paths and random surfaces on a digital computer . b 106 , 323 - 326 ( 1981 ) witten , e. : quantum field theory and the jones polynomial . 121 , 351 - 399 ( 1989 ) qu , l. and he , d. : solving numerical integration by particle swarm optimization . work published in the proceedings of icica 2010 , part ii , zhu , r. et al . ( eds ) , berlin , heidelberg , springer verlag , 228 - 235 ( 2010 )

such discrete knots appear almost everywhere in numerical simulations of systems containing one dimensional ring - shaped objects . the procedure is quite general and can be applied to any discrete knot defined off or on lattice . this is the simplest knot invariant that admits a definition in terms of multiple line integrals . for a fast derivation of

in mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory describing the spatial conformation of the knot . in this work it is addressed the problem of evaluating invariants of this kind in the case in which the knot is discrete , i. e. its trajectory is constructed by joining together a set of segments of constant length . such discrete knots appear almost everywhere in numerical simulations of systems containing one dimensional ring - shaped objects . examples are polymers , the vortex lines in fluids and superfluids like helium and other quantum liquids . formally , the trajectory of a discrete knot is a piecewise smooth curve characterized by sharp corners at the joints between contiguous segments . the presence of these corners spoils the topological invariance of the knot invariants considered here and prevents the correct evaluation of their values . to solve this problem , a smoothing procedure is presented , which eliminates the sharp corners and transforms the original path into a curve that is everywhere differentiable . the procedure is quite general and can be applied to any discrete knot defined off or on lattice . this smoothing algorithm is applied to the computation of the vassiliev knot invariant of degree 2 denoted here with the symbol . this is the simplest knot invariant that admits a definition in terms of multiple line integrals . for a fast derivation of , it is used a monte carlo integration technique . it is shown that , after the smoothing , the values of may be evaluated with an arbitrary precision . several algorithms for the fast computation of the vassiliev knot invariant of degree 2 are provided .

1312.0003

i

there are several ways to test the geometry of a cosmological model and its expansion , i.e. , that redshifts of the galaxies are cosmological and not due to an alternative mechanism ( narlikar 1989 ; lpez - corredoira 2003 , section 2.1 ; lpez - corredoira 2006 ) . however , almost all of the cosmological tests are entangled with the evolution of galaxies and/or other effects . hubble diagrams ( laviolette 1986 ; schade et al . 1997 ; marosi 2013 ) , tolman surface brightness tests ( lubin & sandage 2001 ; andrews 2006 ; lerner 2006 ) , angular size tests ( kapahi 1987 ; kellerman 1993 ; lpez - corredoira 2010 ) are all affected by the evolution of galaxies at high redshifts . so far , there are no standard rods in physical objects . the ultra - compact radio sources which were claimed in the past to be free of evolutionary effects are now thought to present some evolution as well ( lpez - corredoira 2010 ; pashchenko & vitrishchak 2011 ) . lpez - corredoira ( 2010 ) thinks that the huge size evolution necessary to fit an angular size test with an expanding universe is not understood , but it still depends on our understanding of the galaxies rather than on pure cosmological approaches . baryonic acoustic oscillations ( baos ) in the cosmic microwave background radiation ( cmbr ) or in the large scale structure are usually thought to be standard rods to measure the cosmological expansion ( e.g. , rassat & refregier 2012 ) ; however their features in the power spectrum are not a direct observable quantity free of assumptions ( one must assume at least that bao peaks represent a faint imprint of the sound waves in the clustering of galaxies and matter today excited by the initial inflationary perturbations , i.e. the assumption that the scenario of standard cosmology is correct ) and these peaks could be generated with other cosmological assumptions different from the standard interpretation of acoustic peaks ( lpez - corredoira & gabrielli 2013 ) . time dilation tests in type ia supernovae ( snia ) look like one of the most successful tests in favor of the expansion of the universe ( goldhaber et al . 2001 ; blondin et al . 2008 ) , but they do not serve to test the geometry of the universe , and there are still some problems in the interpretation . the fact that sne ia light curves are narrower when redder ( nobili & goobar 2008 ) is an inconvenience for a clean test free of selection effects . other selection effects and the possible compatibility of the results with a wider range of cosmological models , including static ones , were also pointed out by lpez - corredoira ( 2003 , section 2 and references therein ) , brynjolfsson ( 2004b ) ; leaning ( 2006 ) ; crawford ( 2009b , 2011 ) ; holushko ( 2012 ) , and laviolette ( 2012 , section 7.8 ) . moreover , neither gamma - ray bursts ( crawford 2009a ) nor qsos ( hawkins 2010 ) present time dilation , which is puzzling . there are , of course , the cmbr anisotropies as a way to test cosmological models ; they are , perhaps , the most important support for the standard model . however , one should find an independent confirmation , because it is possible to generate / modify cmbr anisotropies by mechanisms different than the standard cosmology ( narlikar et al . 2007 ; angus & diaferio 2011 ; lpez - corredoira & gabrielli 2013 ; lpez - corredoira 2013 ) and/or contamination ( lpez - corredoira 2007 ) . the microwave background temperature measured from rotational excitation of some molecules as a function of redshift ( molaro et al . 2002 ; noterdaeme et al . 2011 ) is another possible test and was quite successful in proving the expansion : results from noterdaeme et al . ( 2011 ) with the exact expected dependence of @xmath4 are impressive . nonetheless , there are other results which disagree with that dependence ( krelowski et al . 2012 ; sato et al . . it might be due to a dependence on collisional excitation ( molaro et al . 2002 ) or bias due to unresolved structure ( sato et al . . the temperature of the intergalactic medium as a function of redshift might also be useful to constrain more directly the geometry of the universe . at present , it is typically inferred to be 20,000 k ; there is no evidence of evolution with redshift ( zaldarriaga et al . 2001 ) , which is puzzling in an expanding universe . here , we propose an alcock & paczyski ( 1979 ) cosmological test : an evaluation of the ratio of observed angular size to radial / redshift size ( see detailed explanation in section [ .method ] ) . the main advantage of this test is that it does not depend on the evolution of the galaxies , but only on the geometry of the universe . however , the redshift distortions ( kaiser 1987 ; hamilton 1998 ) do also have an influence . although there already have been many attempts to carry out this test ( see references in section [ .2pcf ] ) , here we adapt the method ( sections [ .method ] and [ .2pcf_calc ] ) and focus on the application as a test for any cosmological model , whereas other authors assume the standard cosmology and just fit some of their parameters . we apply it to three spectroscopic surveys with a total of @xmath5 galaxies up to redshift 0.8 and @xmath6 qsos at higher redshifts ( sections [ .application ] and [ .fitcosmo ] ) . as will be discussed in section [ .conclusions ] , this test will give us information about which cosmological model fits better . hence , this is a cosmological test free of entanglement with evolution , for checking both the geometry and the expansion of the universe . certainly , this exercise is not going to solve the question once and for all , but the method is a possible way to do it with the increasing possibility of future huge spectroscopic surveys .

in order to test the expansion of the universe and its geometry , we carry out an alcock & paczyski cosmological test , that is , an evaluation of the ratio of observed angular size to radial / redshift size . the main advantage of this test is that it does not depend on the evolution of the galaxies , but only on the geometry of the universe . then , we use the data to evaluate which cosmological model fits them .

in order to test the expansion of the universe and its geometry , we carry out an alcock & paczyski cosmological test , that is , an evaluation of the ratio of observed angular size to radial / redshift size . the main advantage of this test is that it does not depend on the evolution of the galaxies , but only on the geometry of the universe . however , the redshift distortions produced by the peculiar velocities of the gravitational infall do also have an influence , which should be separated from the cosmological effect . we derive the anisotropic correlation function of sources in three surveys within the sloan digital sky survey ( sdss ) : galaxies from sdss - iii / baryon oscillation spectroscopy survey data release 10 ( boss - dr10 ) , and qsos from sdss - ii and sdss - iii / boss - dr10 . from these , we are able to disentangle the dynamic and geometric distortions and thus derive the ratio of observed angular size to radial / redshift size at different redshifts . we also add some other values available in the literature . then , we use the data to evaluate which cosmological model fits them . we used six different models : concordancecdm , einstein - de sitter , open friedman cosmology without dark energy , flat quasi - steady state cosmology , a static universe with a linear hubble law , and a static universe with tired light redshift . only two of the six models above fit the data of the alcock & paczyski test : concordancecdm and static universe with tired light redshift ; whereas the rest of them are excluded at a% confidence level . if we assume thatcdm is the correct one , the best fit with a free is produced for .

1312.0003

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the application of the alcock paczyski test using the anisotropic two - point correlation function was shown here to be a useful tool for discriminating among different cosmological models without any a priori assumptions . we have seen that the major caveat is the disentanglement with the redshift distortions produced by the peculiar velocities of gravitational infall , but it is still possible to separate both effects . the error bars are large for the determination of the ratio of observed angular size to radial / redshift size , @xmath16 ; however , with large collections of data from many surveys , it is possible to get some constraints in cosmological models . we have applied our method to three surveys within sdss : galaxies from sdss - iii / boss , and qsos from both sdss - ii and sdss - iii / boss . we have not used the survey of galaxies from sdss - ii because most of their galaxies are at @xmath145 , and the condition of completeness within a radius @xmath136 would reduce enormously the sample of parent galaxies , producing a result much poorer than with boss galaxies . the alternative of exploring smaller distances in the correlation would alleviate this constraint of completeness areas ; however , it would introduce the problem of the `` fingers of god '' , which we have not taken into account here because they are expected to produce a negligible effect at the scales we were using . furthermore , we have used some estimations of @xmath16 given directly or indirectly by the literature , leading to the results in table [ tab : chi2 ] or in figure [ fig : models2 ] . we observe that there are four cosmological models excluded within @xmath1% cl : einstein de sitter , open friedman cosmology without dark energy , flat quasi - steady - state cosmology ( qssc ) , and a static universe with a linear hubble law . two models fit the data with a higher probability : concordance cosmology , and static universe with tired light redshift . for the standard concordance model ( @xmath146 ) , we have set constraints on @xmath2 to produce a better fit to the data ; they are : @xmath3 , which is in agreement with values derived from other independent methods . note , however , that , since we have not used any a priori assumptions , neither about the redshift distortion model nor other constraints including other data , the error bars here are larger than those derived from other methods e.g . , a recent analysis of cmbr data with _ planck _ gives @xmath147 ( planck collaboration 2013 ) . if we also allow a free value for the equation of state of the dark energy , @xmath148 , instead of a fixed value of -1 , we get as a best fit : @xmath149 , @xmath150 . certainly , our method here is not the best one to explore the possible values of the cosmological parameters once the cosmology is established ; this could be much improved with further constraints , as ross et al . ( 2007 ) and okumura et al . ( 2008 ) did , but this was not our goal in this paper . we have focused on a method to test cosmological models without any a priori assumption . the qssc model is excluded at a @xmath1% cl , but the fit is better ( excluded with a probability of 98.0% instead of 99.56% ) with parameters @xmath151 and @xmath152 , which lead to @xmath153 . this is compatible with the previously established value for the @xmath154-field density of @xmath155 ( banerjee & narlikar 1999 , with the angular redshift test ) or @xmath156 ( banerjee et al . 2000 , with sne data ) . the main problem with these fitted values is that they lead to the maximum redshift of an observable galaxy at @xmath157 , and we know there are galaxies beyond that . anyway , a flat qssc appears to have a very low probability to be a valid model according to only the present analysis . all these considerations are for a flat qssc universe ; if we allowed curvature ( @xmath158 ) , a wider range of probabilities could be obtained . the static models are not totally excluded from the present analysis . the one with a linear hubble law is excluded ( assuming , as we did , that the linear regime applies and that the peculiar velocity of the galaxies is irrotational , i.e. , with zero vorticity , as predicted by the usual gravitational growth theories ) , but not the one with tired - light hypothesis to explain the redshift . this tired - light model is also supported by some other tests , such as the angular size test ( lpez - corredoira 2010 ; laviolette 2012 , 7.4 ) or differential galaxy number counts versus magnitude ( laviolette 2012 , 7.7 ) . fitting the hubble diagram as @xmath159 , with @xmath160 being the luminosity distance , is not good with sne data using a `` simple '' tired light assumption ( where @xmath161 ; the factor @xmath162 stems from the fact that the luminosity is proportional to @xmath163 and is inversely proportional to @xmath164 due to the redshift without expansion , i.e. , without time dilation ) . however , the fit is acceptable using a `` plasma '' tired light redshift ( @xmath165 ; lpez - corredoira 2010 ) , and it is quite good with gamma - ray bursts even up to @xmath166 with a tired - light model of type @xmath167 ( marosi 2013 ) . future spectroscopic surveys with significantly more sources , coverage , and depth , will be able to improve the present results . in particular , at high redshift we still have few spectroscopic qsos , so the error bars are large ( see how noisy the correlations are in figs . [ fig : elipses2q ] , [ fig : elipses2q2 ] ) . one exception is the point from da ngela et al . ( 2005b ) at @xmath168 : @xmath169 ( note , however , that , looking at figure 5 of da ngela et al . ( 2005b ) , one could say that @xmath170 is quite likely within 1@xmath75 , and consequently the error bars might be somewhat larger ) . as it can be observed in figure [ fig : models ] , accurate values of @xmath63 at @xmath171 will be able to discern between the two successful cosmological models in this paper : @xmath0cdm ( concordance ) and static model with tired - light redshift . also , very accurate measurements of @xmath63 at low @xmath11 would be able to test which one is the correct , since @xmath23 for the concordance model , whereas @xmath172 for the tired - light one . it is also worth noting that the most important contribution stems from the already available data from previous analyses of the anisotropic correlation function , combining the results of six references . the contribution of the data ( @xmath143 ) analyzed in this paper does not change significantly the confidence levels of exclusion of the different cosmological models because we get higher error bars . note , however , that we can not guarantee that the results of those six references with somewhat lower error bars are correct . in principle , from the experience developing this paper , the conclusion that comes up is that the quasi - degeneracy between the values of @xmath54 and @xmath63 for the fit of @xmath50 is a severe problem which produces huge error bars of @xmath63 and , consequently , the alcock & paczyski method only becomes powerful when we join the results from different surveys . thanks are given to : juan e. betancort - rijo ( iac , tenerife , spain ) for discussions on the methodology of the alcock - paczyski test , donald schneider ( sdss - iii scientific publications coordinator ) for his recommendations to bring the paper into compliance with the sdss - iii publications policy , and to the anonymous referee for helpful comments . funding for the 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/. the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , university of cambridge , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington . funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s . department of energy office of science . the sdss - iii web site is http://www.sdss3.org/. sdss - iii is managed by the astrophysical research consortium for the participating institutions of the sdss - iii collaboration including the university of arizona , the brazilian participation group , brookhaven national laboratory , university of cambridge , carnegie mellon university , university of florida , the french participation group , the german participation group , harvard university , the instituto de astrofsica de canarias , the michigan state / notre dame / jina participation group , johns hopkins university , lawrence berkeley national laboratory , max planck institute for astrophysics , max planck institute for extraterrestrial physics , new mexico state university , new york university , ohio state university , pennsylvania state university , university of portsmouth , princeton university , the spanish participation group , university of tokyo , university of utah , vanderbilt university , university of virginia , university of washington , and yale university . 999 ahn , c. p. , alexandroff , r. , allende prieto , c. , et al . 2012 , apjs , 203 , 21 alcock , c. , & paczyski , b. 1979 , nature , 281 , 358 anderson , l. , aubourg , e. , bailey , s. , et al . 2012 , mnras , 427 , 3435 andrews , t. b. 2006 , in : first crisis in cosmology conference ( aip conf . 822 ) , e. j. lerner , j. b. almeida , eds . , american institute of physics , p. 3 angus , g. w. , & diaferio , a. 2011 , mnras , 417 , 941 ashmore , l. 2011 , intrinsic plasma redshifts now reproduced in the laboratory a discussion in terms of new tired light , http://vixra.org/pdf/1105.0010 avni , y. 1976 , apj , 210 , 642 ballinger , w. e. , peacock , j. a. , & heavens , a. f. 1996 , mnras , 282 , 877 banerjee , s. k. , & narlikar , j. v. 1999 , mnras , 307 , 73 . erratum : banerjee , s. k. , & narlikar , j. v. 1999 , mnras , 310 , 912 banerjee , s. k. , narlikar , j. v. , wickramasinghe , n. c. , hoyle , f. , & burbidge , g. 2000 , aj 119 , 2583 baryshev y. , & teerikorpi p. 2012 , fundamental questions of practical cosmology , springer , dordrecht betancort - rijo , j. 1991 , a&a , 245 , 347 blake , c. , glazebrook , k. , davis , t. m. , et al . 2011 , mnras 418 , 1725 blanchard , a. 2006 , in : current issues in cosmology , j .- c . pecker , and j. v. narlikar , ed . , cambridge university press , cambridge ( u.k . ) , p. 76 blondin , s. , davis , t. m. , krisciunas , k. , et al . 2008 , apj , 682 , 724 bolton , a. s. , schlegel , d. j. , aubourg , . , 2012 , aj , 144 , 144 brynjolfsson , a. 2004a , arxiv : astro - ph/0401420 brynjolfsson , a. 2004b , arxiv : astro - ph/0406437 chuang , c. h. , & wang , y. 2012 , mnras , 426 , 226 crawford , d. 2009a , arxiv:0901.4169 crawford , d. 2009b , arxiv:0901.4172 crawford , d. 2011 , j. cosmology , 13 , 3875 croft , r. a. c. , & dailey , m. 2011 , arxiv:1112.3108 da ngela , j. , outram , p. j. , shanks , t. , boyle , b. j. , croom , s. m. , loaring , n. s. , miller , l. , & smith , r. j. 2005a , mnras , 360 , 1040 da ngela , j. , outram , p. j. , & shanks , t. 2005b , mnras , 361 , 879 dawson , k. s. , schlegel , d. j. , ahn , c. p. , et al . 2013 , aj , 145 , i d . 10 eisenstein , d. j. , weinberg , d. h. , agol , e. , et al . 2011 , aj , 142 , 72 goldhaber , g. , groom , d. e. , kim , a. , et al . 2001 , apj , 558 , 359 gunn , j. e. , carr , m. , rockosi , c. , et al . 1998 , aj , 116 , 3040 gunn , j. e. , siegmund , w. a. , mannery , e. j. , et al . 2006 , aj , 131 , 2332 hamilton , a. j. s. 1992 , apj , 385 , l5 hamilton , a. j. s. 1998 , in : the evolving universe . selected topics on large - scale structure and on the properties of galaxies , d. hamilton , ed . , kluwer , dordrecht , p. 185 hawkins , m. r. s. , 2010 , mnras , 405 , 1940 holushko , h. 2012 , tired light and type ia supernovae observations , http://vixra.org/pdf/1203.0062 hoyle , f. , outram , p. j. , shanks , t. , boyle , b. j. , croom , s. m. , & smith , r. j. 2002 , mnras , 332,311 ivashchenko , g. , zhdanov , v. i. , & tugay , a. v. 2010 , mnras , 409 , 1691 jackson , j. c. 1972 , mnras , 156 , 1 kaiser , n. 1987 , mnras , 227 , 1 kapahi , v. k. 1987 , observational cosmology ( iau symp . 124 ) , a. hewitt , g. burbidge , and l. z. fang ( eds . ) , reidel , dordrecht , 251 kellerman , k. i. 1993 , nature , 361 , 134 kim , y .- r . , & croft , r. a. c. 2007 , mnras , 374 , 535 krelowski , j. , galazutdinov , g. , gnacinski , p. 2012 , an , 333 , 627 landy , s. d. , & szalay , a. s. 1993 , apj , 412 , 64 laviolette , p. a. 1986 , apj 301 , 544 laviolette , p. a. 2012 , subquantum kinetics : the alchemy of creation , 4th ed . , starlane publ . , niskayana ( ny , us ) leaning , s. p. 2006 , in : 1st crisis in cosmology conference ( aip conf . 822(1 ) ) , e. j. lerner , j. b. almeida , eds . , aip , melville , p. 48 lehnert , m. d. , nesvadba , n. p. h. , cuby , j .- , et al . 2010 , nature , 467 , 940 lerner , e. j. 2006 , in : 1st crisis in cosmology conference ( aip conf . 822(1 ) ) , e. j. lerner , j. b. almeida , eds . , aip , melville , p. 60 lpez - corredoira , m. 2003 , in : recent research developments in astronomy and astrophysics i , ed . s. g. pandalai , research signpost , kerala , p. 561 lpez - corredoira , m. 2006 , apeiron , newswire/2006 [ arxiv : astro - ph/0701061 ] lpez - corredoira , m. 2007 , j. astrophys . astron . , 28 , 101 lpez - corredoira , m. 2010 , int . d , 19 , 245 lpez - corredoira , m. 2013 , int . d , 22 , 1350032 lpez - corredoira , m. , & gabrielli , a. 2013 , physica a , 392 , 474 lpez - corredoira , m. , sylos - labini , f. , & betancort - rijo , j. 2010 , a&a , 513 , a3 lubin , l. m. , & sandage , a. 2001 , aj , 122 , 1084 marinoni , c. , & buzzi , a. 2010 , nature , 468 , 539 marosi , l. a. 2013 , advances in astronomy , 2013 , i d . 917104 marulli , f. , bianchi , d. , branchini , e. , guzzo , l. , moscardini , l. , & angulo , r. e. 2012 , mnras , 426 , 2566 matsubara , t. , & suto , y. 1996 , apj , 470 , l1 molaro , p , levshakov , s. a. , dessauges - zavadsky , m. , & dodorico , s. 2002 , a&a , 381 , l64 mountrichas , g. , sawangwit , u. , shanks , t. , croom , s. m. , schneider , d. p. , myers , a. d. , & pimbblet , k. 2009a , mnras , 394 , 2050 mountrichas , g. , sawangwit , u. , & shanks , t. 2009b , mnras , 398 , 971 narlikar , j. v. 1989 , space science reviews , 50 , 523 narlikar , j. v. , vishwakarma , r. g. , & burbidge , g. 2002 , pasp , 114 , 1092 narlikar , j. v. , burbidge , g. , & vishwakarma , r. g. 2007 , j. astrophys . astron . , 28 , 67 noterdaeme , p. , petitjean , p. , srianand , r. , ledoux , c. , & lpez , s. 2011 , a&a , 526 , l7 nusser , a. 2005 , mnras , 364 , 743 okumura , t. , matsubara , t. , eisenstein , d. j. , kayo , i. , hikage , c. , szalay , a. s. , & schneider , d. p. 2008 , apj , 676 , 889 outram , p. j. , shanks , t. , boyle , b. j. , croom , s. m. , hoyle , f. , loaring , n. s. , miller , l. , & smith , r. j. 2004 , mnras , 348 , 745 pris , i. , petitjean , p. , aubourg , . , et al . 2012 , a&a , 548 , a66 pris , i. , petitjean , p. , aubourg , . , et al . 2013 , arxiv:1311.4870 pashchenko , i. n. , & vitrishchak , v. m. 2011 , astronomy reports , 55(4 ) , 293 patiri , s. g. , betancort - rijo , j. , & prada , f. 2012 , a&a , 541 , l4 phillips , s. 1994 , mnras , 269 , 1077 planck collaboration 2013 , arxiv:1303.5076 porciani , c. , & norberg , p. 2006 , mnras , 371 , 1824 rassat , a. , & refregier , a. 2012 , a&a , 540 , a115 reid , b. a. , samushia , l. , white , m. , et al . 2012 , mnras , 426 , 2719 ross , n. p. , da ngela , j. , shanks , t. , et al . 2007 , mnras , 381 , 573 ryden , b. s. 1995 , apj , 452 , 25 sato , m. , reid , m. j. , menten , k. m. , & carilli , c. l. 2013 , apj , 764 , 132 schade , d. , barrientos , l. f. , & lpez - cruz , o. 1997 , apj , 477 , l17 schneider , d. p. , richards , g. t. , hall , p. b. 2010 , aj , 139 , 2360 smee , s. , gunn , j. e. , uomoto , a. , et al . 2013 , aj , 146 , 32 sutter , p. m. , lavaux , g. , wandelt , b. d. , & weinberg , d. h. 2012 , apj , 761 , 187 sylos labini , f. , vasilyev , n. l. , baryshev , y. v. , & lpez - corredoira , m. 2009 , a&a , 505 , 981 tocchini - valentini , d. , barnard , m. , bennett , c. l. , & szalay , a. s. 2012 , apj , 757 , 131 vauclair , s. c. , blanchard , a. , sadat , r. , et al . 2003 , a&a , 412 , l37 vishwakarma , r. g. 2002 , mnras , 331 , 776 white , m. , blanton , m. , bolton , a. , et al . 2011 , apj , 728 , 126 white , m. , myers , a. d. , ross , n. p. , et al . 2012 , mnras , 424 , 933 york , d. g. , adelman , j. , anderson , j. e. , jr . , 2000 , aj , 120 , 1579 zaldarriaga , m. , hui , l. , & tegmark , m. 2001 , apj , 557 , 519 zehavi , i. , weinberg , d. h. , zheng , z. , et al . 2004 , apj , 608 , 16

we derive the anisotropic correlation function of sources in three surveys within the sloan digital sky survey ( sdss ) : galaxies from sdss - iii / baryon oscillation spectroscopy survey data release 10 ( boss - dr10 ) , and qsos from sdss - ii and sdss - iii / boss - dr10 . from these , we are able to disentangle the dynamic and geometric distortions and thus derive the ratio of observed angular size to radial / redshift size at different redshifts . we used six different models : concordancecdm , einstein - de sitter , open friedman cosmology without dark energy , flat quasi - steady state cosmology , a static universe with a linear hubble law , and a static universe with tired light redshift . only two of the six models above fit the data of the alcock & paczyski test : concordancecdm and static universe with tired light redshift ; whereas the rest of them are excluded at a% confidence level . if we assume thatcdm is the correct one , the best fit with a free is produced for .

in order to test the expansion of the universe and its geometry , we carry out an alcock & paczyski cosmological test , that is , an evaluation of the ratio of observed angular size to radial / redshift size . the main advantage of this test is that it does not depend on the evolution of the galaxies , but only on the geometry of the universe . however , the redshift distortions produced by the peculiar velocities of the gravitational infall do also have an influence , which should be separated from the cosmological effect . we derive the anisotropic correlation function of sources in three surveys within the sloan digital sky survey ( sdss ) : galaxies from sdss - iii / baryon oscillation spectroscopy survey data release 10 ( boss - dr10 ) , and qsos from sdss - ii and sdss - iii / boss - dr10 . from these , we are able to disentangle the dynamic and geometric distortions and thus derive the ratio of observed angular size to radial / redshift size at different redshifts . we also add some other values available in the literature . then , we use the data to evaluate which cosmological model fits them . we used six different models : concordancecdm , einstein - de sitter , open friedman cosmology without dark energy , flat quasi - steady state cosmology , a static universe with a linear hubble law , and a static universe with tired light redshift . only two of the six models above fit the data of the alcock & paczyski test : concordancecdm and static universe with tired light redshift ; whereas the rest of them are excluded at a% confidence level . if we assume thatcdm is the correct one , the best fit with a free is produced for .

quant-ph0507010

i

quantum adiabatic algorithms , first conceived in ref . @xcite , have been developed as an alternative to the traditional circuit model of quantum computation . an important aspect of the adiabatic approach is its expected resilience to various kinds of open system effects , which makes it a promising candidate for robust quantum computation . the basic reason for this appealing feature is that the adiabatic quantum computer operates near the instantaneous ground state of a time - dependent hamiltonian and can therefore be expected to be insensitive to relaxation and open system effects among the excited states . the robustness of adiabatic quantum computation has been discussed in the literature for different kinds of perturbations , such as relaxation @xcite , noise @xcite , decoherence @xcite , and unitary control errors @xcite . in this paper , we elaborate on the robustness of the quantum adiabatic search @xcite in the presence of decoherence in the instantaneous eigenbasis of the search hamiltonian . in particular , we extend the analysis of the local search @xcite in ref . @xcite to global adiabatic search @xcite . we further address the dependence of the asymptotic time complexity of the search procedure on the success probability for finding the searched item . to model adiabatic search in the presence of decoherence we consider master equations of the form ( @xmath4 from now on ) @xmath5 -b\bm{[}w,[w,\varrho]\bm{]},\ ] ] where @xmath6 is the density operator , and @xmath7 are constants such that @xmath8 . for convenience , we also assume that @xmath9 . the operator @xmath10 is taken to be the time - dependent hamiltonian of the adiabatic search @xcite , and @xmath11 is a time - dependent hermitian operator such that @xmath12 = 0 $ ] , which gives decoherence in the instantaneous eigenbasis of @xmath10 . decoherence with respect to the energy eigenbasis of a system has been shown to occur in certain regimes of system - environment interaction @xcite . apart from this general regime of weak interaction and dominating self - hamiltonians one can find other settings where this type of decoherence occurs . one example is control errors in terms of fluctuations in the energy levels of the system @xcite . note also that decoherence in the energy eigenbasis does not necessarily imply the particular form of eq . ( [ enkel ] ) . however , this choice provides a sufficiently simple model to be treated analytically , and yet results in non - trivial behavior . moreover , the form of eq . ( [ enkel ] ) quite generally guarantees the solution to be a proper density operator at all times . our results show that for @xmath13 and list length @xmath1 , it is sufficient with a run time of the order @xmath1 in the global case and @xmath14 in the local case , as is the case for the ideal closed evolution . these results suggest that the adiabatic quantum computer is protected against this type of decoherence , which is promising for physical realizations . in the wide - open case where @xmath15 @xcite , with the special choice @xmath16 , the scaling changes to @xmath0 and @xmath1 , for the global and local search , respectively . although the concept of adiabaticity has a well defined meaning for closed systems , there is no obviously unique generalization to the case of open systems . one may consider several different generalizations focusing on various aspects of the standard adiabaticity . for example , in refs . @xcite the concept of adiabaticity is based upon jordan block decompositions of the superoperator that describe the evolution of the open system . a different approach , designed for systems that are weakly open , has been put forward in ref . this latter approach is based upon taking decoupling of the energy subspaces of the ideal hamiltonian as the starting point . here , we use a third method based upon a feature of the adiabatic quantum computer , namely that the evolution takes place near an eigenstate of the ideal hamiltonian . this property is in contrast with , e.g. , implementations of holonomic quantum gates @xcite , where it is essential that the gate can operate on arbitrary superpositions , without too large errors . for the functioning of the adiabatic quantum computer , however , it is sufficient that the state of the system remains close to the correct eigenstate . in the present interpretation of the word `` adiabaticity '' we only require that the probabilities of finding the system in the instantaneous eigenstates of @xmath17 are conserved , but we allow superpositions of the instantaneous eigenstates to decay into mixtures . the structure of the article is as follows . in sec . [ sec : twolev ] we derive an integral equation from the original master equation . section [ sec : search ] gives a brief overview of the adiabatic search algorithm , and we calculate some quantities which are used later . in sec . [ sec : semiopen ] we prove a sufficient condition for the success probability of the adiabatic search to approach unity in the semi - open case . we consider both global and local search . furthermore , a condition on the decoherence term of the master equation , which is used in the proofs in sec . [ sec : semiopen ] , is examined . in sec . [ sec : wopen ] we demonstrate that the wide - open case is essentially different from the semi - open case . we prove sufficient conditions for the success probability to approach unity in both the global , as well as the local case . we further prove that the sufficient conditions are also necessary . in sec . [ sec : success ] , bounds on the run time for fixed success probabilities less that unity , are given . we end with the conclusions .

in phys . rev . a * 71 * , 060312(r ) ( 2005 ) the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search hamiltonian was examined . as in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case , as long as the hamiltonian dynamics is present . in the case of pure decoherence , where the environment monitors the search hamiltonian , we find that the time complexity of the global quantum adiabatic search scales like , where is the list length .

in phys . rev . a * 71 * , 060312(r ) ( 2005 ) the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search hamiltonian was examined . we expand this analysis to include the case of the global adiabatic quantum search . as in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case , as long as the hamiltonian dynamics is present . in the case of pure decoherence , where the environment monitors the search hamiltonian , we find that the time complexity of the global quantum adiabatic search scales like , where is the list length . we moreover extend the analysis to include success probabilities and prove bounds on the run time with the same scaling as in the conditions for the limit . we supplement the analytical results by numerical simulations of the global and local search .

1609.07930

c

in this section we apply the stability criteria and growth rate estimates of the two - fluid model to reasonable models of protoplanetary disks . because of uncertainties in the parameters and deficiencies in the model itself , we do not attempt to be comprehensive or to hold fast to the quantitative results we obtain . rather the aim is to give a sense of the main trends and qualitative behaviour , and the general range of numbers one might find in a real system . to that purpose we first concentrate on the two populations of strongly and marginally coupled particles , using the simple estimates derived previously . we then solve the full dispersion relationship numerically at each radius . the disk model we use is a variant of the minimum mass solar nebula ( see youdin 2011 ) . the background gas disk surface density is given as a function of disk radius by @xmath210 here @xmath211 is a free dimensionless parameter , and @xmath212 is disk radius in units of au . the temperature of the nebula is given by @xmath213 as a consequence , the gas s toomre parameter is @xmath214 thus at 1 au , @xmath28 is roughly 40 and this falls to about 10 as we approach 100 au . the inverse stoke s number can be approximated by @xmath215 where @xmath216 is particle size in units of mm . we have assumed that the particles always lie in the epstein regime . only the largest particles at the smallest radii enter the stokes drag regime , so to make life simple we omit it . as a result , stability can be determined once @xmath2 , @xmath211 , @xmath33 , particle size , and the disk radius are specified . first consider larger particles with a largish stokes number @xmath217 . from figure 1 , in a standard mmsn these might correspond to @xmath218 10 cm particles at 30 au or more or @xmath219 cm particles at 100 au . in an older less massive disk , this class may also include mm particles but only at 100 au . thus most particles do not fall into this regime . secular gi arises when criterion is fulfilled . taking the standard value for the dust to gas ratio @xmath220 , we only need to estimate the ratio of velocity dispersions . assuming that the particles random motions are controlled by the background turbulence , as in section 2 , we have @xmath221 and thus instability occurs when @xmath222 this criterion includes both the classical gi of a dust layer and the secular gi , in which we are more interested . as discussed earlier , the properties of the turbulence are poorly constrained . we thus allow @xmath33 to vary between @xmath223 ( a perhaps unrealistically low level ) and @xmath224 . next , to fix ideas , we set @xmath225 and find that the critical toomre parameter for the gas is @xmath226 with the lower value corresponding to a thick disk of relatively ` hot ' particles ( @xmath227 ) , and the higher value to a thin disk of colder particles ( @xmath228 ) . our standard mmsn with @xmath14 yields @xmath28 that fall directly in this range . if @xmath227 instability is not possible , but if @xmath229 then instability can occur on most radii . less massive disks exhibit larger @xmath28 ; for instance with @xmath16 , we have @xmath230 at all radii , and thus the existence of sgi is very much conditional on the efficiency of the turbulence . if @xmath231 then cm sized particles , or even smaller , could be unstable in the outer regions of low mass disks . what of the growth rates of the unstable modes ? equation gives an upper bound on @xmath66 , in the regime of larger particles . this can be reworked into @xmath232 where the last equality comes by setting @xmath233 and @xmath234 . for relatively strong turbulence @xmath235 , the e - folding time of an unstable mode is @xmath236 orbits , too long to be relevant at 100 au , but possibly significant at smaller radii , for example 10 cm size particles at @xmath237 au . smaller @xmath33 , of course , yield faster growth on relevant timescales . we next consider well coupled particles , a class that covers most solids of interest ( see figure 1 ) . the relevant stability criterion for this dust is given by . the dimensionless diffusion coefficient for tightly coupled particles is @xmath110 . setting @xmath183 and @xmath238 yields the criterion @xmath239 which is more difficult to satisfy than in the weakly coupled case . if we assume that @xmath240 , the condition becomes @xmath241 where the larger value corresponds to inefficient turbulent diffusion ( @xmath228 ) and the lower bound corresponds to efficient diffusion ( @xmath227 ) . once again , this suggests that instability occurs when the turbulence is sufficiently weak . in fact , given @xmath14 , the sgi is suppressed if @xmath242 . the situation worsens when @xmath243 . the conclusion is that the instability may not be widespread in smaller dust . let us next turn to growth rates , in particular expression . to fix some numbers , we generously set @xmath238 and @xmath183 and after some manipulation obtain @xmath244 if @xmath234 , then we have @xmath245 while @xmath246 yields appreciable growth at all radii , @xmath247 does so only for @xmath248 au . for larger @xmath13 growth times this further reinforces the conclusion that the sgi is only relevant to the dynamics of tightly coupled particles when turbulence is very low indeed . in this subsection we solve the full dispersion relation at each radius of our disk model . figure 1 indicates that particle sizes of a mm and above couple to the gas differently at different radii , potentially passing from the well - coupled to the weakly coupled regimes as we go further out radially . at certain radii @xmath249 and our analytic results are no longer strictly valid , meaning that the dispersion relation must be solved numerically . some stability curves are plotted in fig . [ fig5 ] for two different disk models , @xmath14 ( left panels ) and @xmath16 ( right panels ) , for several values of the turbulence parameter @xmath33 , and for two values of the dust to gas density ratio , @xmath183 ( top row ) and @xmath250 ( bottom row ) . parameter regions above a given curve are subject to instability , and thus for a given @xmath33 and particle size there exists a critical radius within which the sgi is completely suppressed . in fact , for a relatively turbulent disk with @xmath251 , @xmath14 and @xmath183 , all particles smaller than @xmath252 cm are stable , and all particles smaller than 1 mm , when @xmath253 . weaker levels of turbulence , of course , permit instability upon smaller particle sizes and for larger swathes of the disk . but one must drive @xmath33 to levels @xmath254 to obtain sgi at radii @xmath255 au for particles larger than a mm . the sgi s struggles worsen when the disk is less massive ( @xmath16 ) , with mm sized particles unstable only for very low values of @xmath33 . but moving to the lower row of plots , it is immediately clear that increasing @xmath2 improves its range . for example , if @xmath14 and @xmath251 then all particles smaller than mm sizes are stable . if @xmath253 then all particles smaller than @xmath256 mm are stable . on the other hand , when @xmath257 the prospects for instability become increasingly bleak . maximum e - folding times for the sgi at different radii in a standard disk model with @xmath14 and dust - to - gas ratio @xmath183 . the solid curves correspond to cm sized particles , and the dashed curves to mm sized particles . the blue colour represents a turbulent mass diffusivity of @xmath253 , red represents @xmath258 , and green @xmath228.,width=340 ] we also compute the minimum e - folding times for the sgi , for @xmath14 and @xmath183 . these are plotted in figure [ fig6 ] . solid curves correspond to cm sized particles and dashed curves correspond to mm sized particles . we omit smaller particles because they are always in the well - coupled regime treated in section 6.2 . the different colours represent different values of @xmath33 . as is clear , a turbulence level of @xmath259 yields a growth time too long to be important for both particle classes , while a value of @xmath258 yields a growth time of a few @xmath260 years for cm sized particles , at a large range of radii @xmath261 au . milimetre sized particles exhibit appreciable growth only for low turbulence levels @xmath262 and at larger radii @xmath263 au . less massive disks ( such as with @xmath16 ) yield even weaker growth . increasing @xmath2 to 0.1 , exacerbates growth by up to an order of magnitude , whereas decreasing @xmath211 to 0.1 , reduces the growth rate by roughly an order of magnitude . these growth times are on the whole consistent with youdin ( 2011 ) at radii @xmath261 au , but closer to the disk two - fluid effects lead to noticeable departures . the growth times diverge at certain critical radii . these occur , of course , when the sgi is stabilised , as indicated by fig . [ fig5 ] . for a given @xmath33 , any given particle size has a critical radius within which the sgi is stable . the takeaway message is that mm sized particles require very low levels of turbulence to be sgi unstable , and then this is localised to the outer parts of the disk . centimetre sized particles do better , and the sgi may play some role in their dynamics across a range of disk radii and disk properties . finally , older , less massive disks struggle to host the sgi in any form , though a larger dust to gas ratio can mitigate this to some degree . unfortunately , like @xmath33 , the parameter @xmath2 is difficult to constrain .

in protoplanetary disks the aerodynamical friction between particles and gas induces a variety of instabilities that facilitate planet formation . of these in contrast to a single - fluid treatment , the sgi is quenched within a critical disk radius , as large as 10 au and 30 au for cm and mm sized particles respectively , although establishing robust estimates is hampered by uncertainties in the parameters ( especially the strength of turbulence ) and deficiencies in the razor - thin disk model we employ .

in protoplanetary disks the aerodynamical friction between particles and gas induces a variety of instabilities that facilitate planet formation . of these we examine the so called ` secular gravitational instability ' ( sgi ) in the two - fluid approximation , deriving analytical expressions for its stability criteria and growth rates . concurrently , we present a physical explanation of the instability that shows how it manifests upon an intermediate range of lengthscales exhibiting geostrophic balance in the gas component . in contrast to a single - fluid treatment , the sgi is quenched within a critical disk radius , as large as 10 au and 30 au for cm and mm sized particles respectively , although establishing robust estimates is hampered by uncertainties in the parameters ( especially the strength of turbulence ) and deficiencies in the razor - thin disk model we employ . it is unlikely , however , that the sgi is relevant for well - coupled dust . we conclude by applying these results to the question of planetesimal formation and the provenance of large - scale dust rings . instabilities protoplanetary discs planets and satellites : formation

1210.8304

i

the constraints of causality for two - body scattering with finite - range interactions were first derived by wigner @xcite . the causality bound can be understood as a lower bound on the time delay between the incoming and outgoing wave packets , @xmath4 . when @xmath4 is negative , the outgoing wave packet departs earlier than for the non - interacting system . however , the incoming wave must first reach the interaction region before the outgoing wave can leave . in low - energy scattering this manifests itself as an upper bound on the effective range parameter . in ref . @xcite , phillips and cohen derived this bound for s - wave scattering with finite - range interactions . some constraints on nucleon nucleon scattering and the chiral two - pion exchange potential were considered in ref . @xcite , and relations between the scattering length and effective range have been explored for one - boson exchange potentials @xcite and van der waals potentials @xcite . in refs . @xcite the causality bounds for finite - range interactions were extended to an arbitrary number of spacetime dimensions and arbitrary angular momentum . the extension to systems with partial wave mixing was first studied in ref . @xcite . in this paper , we consider the causality constraints for the scattering of two charged particles with an arbitrary finite - range interaction . our analysis is the first study of causality bounds that takes into account the long - range coulomb force . the results presented here are relevant to studies of low - energy scattering of nuclei and nucleons using effective field theory ( eft ) . in particular , they are important for the application of effective field theory to nuclear halo systems ( cf . refs . @xcite ) . this so - called halo eft utilizes the separation of energy scales between the internal excitations of the core nucleus and soft halo physics . an effective lagrangian is built order by order from local interactions and used to describe low - energy phenomena such as shallow bound states , charge radii , near - threshold resonances , radiative capture reactions , and soft photodissociation . halo eft has been used to describe neutron alpha scattering @xcite and alpha alpha scattering @xcite . bound single - neutron halo systems such as @xmath5be @xcite and @xmath6li @xcite have been studied as well as various two - neutron halo systems @xcite . there is an important connection between causality bounds and the convergence of effective field theory calculations with increasing order @xcite . for local contact interactions , the range of the effective interaction is controlled by the momentum cutoff scale of the effective theory . in effective theories with non - perturbative renormalization , which typically occur in nuclear physics , exact cutoff - independence can generally not be achieved . there is a `` natural '' value of the cutoff at which all higher - order corrections scale as expected from dimensional analysis . if the cutoff is taken larger , `` new physics '' intervenes , the corrections scale unnaturally , and unitarity violations may occur . this is different from what one encounters in high - energy particle physics where the renormalization is typically perturbative and cutoff momenta can be taken arbitrarily large . for calculations using dimensional regularization , the renormalization scale plays a similar role in regulating ultraviolet physics . the term `` new physics '' , in this context , refers to details left out ( integrated out ) in the effective theory . in the case of halo eft , these details are the finite size of the core nucleus and its internal excitations as well as the exponential tail of the pion - exchange interaction . problems with convergence of the effective theory can occur if the cutoff scale is set higher than the scale of the new physics . however , it is useful to have a more quantitative measure of when problems may appear , and this is where the causality bound provides a useful diagnostic tool . for each scattering channel we use the physical scattering parameters to compute a quantity called the causal range , @xmath7 . it is the minimum range for finite - range interactions consistent with the requirements of causality and unitarity . for any fixed cutoff scale , the causality bound marks a branch cut of the effective theory when viewed as a function of physical scattering parameters @xcite . the coupling constants of the effective theory become complex when scattering parameters violating the causality bound are enforced . these branch cuts do not appear in perturbation theory ; however , a nearby branch point can spoil the absolute convergence of the perturbative expansion . our results can be viewed as a guide for improving the convergence of halo eft calculations . in particular , if the cutoff momentum used in a calculation is too high , then problems with convergence may appear in some observables . consequently , the causal range can be used to estimate the `` natural '' ultraviolet cutoff @xmath8 of the effective theory as @xmath9 . the natural cutoff is optimal in the sense that no known infrared physics is left out of the theory and that all corrections involving the ultraviolet cutoff scale naturally @xcite . increasing the cutoff beyond the natural value will not improve the accuracy of the calculation . the causality bounds also have an impact in the regime of bound states . for two - body halo states , or more generally whenever there is a shallow two - body bound state close to threshold , the same integral identity that yields the causality bound for the effective range can be used to derive a relation between the asymptotic normalization constant ( anc ) of the bound - state wave function , the binding momentum , and the effective range for the scattering of the two halo constituents . this relation can be shown to be equivalent to a result previously derived by sparenberg _ et al . _ its significance lies in the fact that the anc is an important input parameter for the calculation of near - threshold radiative capture and photodissociation reactions . the causality bounds also constrain the range of model potentials that are fitted to scattering data in order to extract ancs . the organization of this paper is as follows . we first review the general theory of scattering for two charged particles with additional short - range interactions . our analysis includes both attractive and repulsive coulomb forces . in the next section we derive the charged - particle causality bounds for arbitrary values of the orbital angular momentum . using the causality bounds , we extract and discuss the causal range for several nuclear scattering processes including proton proton , proton deuteron , proton@xmath10he , proton alpha , and alpha alpha scattering . we then elucidate the relation for the anc and extract the ancs of the excited @xmath0 and @xmath1 states in @xmath2o from @xmath3c scattering data as an application . we conclude with a summary of the main results and provide an outlook . in the appendices , we provide technical details of the derivation as well as numerical examples .

the main result is an upper bound for the coulomb - modified effective range parameter . we discuss the implications of this bound to the effective field theory for nuclear halo systems . for the bound - state regime , we find relations for the asymptotic normalization coefficients ( ancs ) of nuclear halo states . as an application of these relations , we extract the ancs of the excited and states ino fromc scattering data .

in quantum systems with short - range interactions , causality imposes nontrivial constraints on low - energy scattering parameters . we investigate these causality constraints for systems where a long - range coulomb potential is present in addition to a short - range interaction . the main result is an upper bound for the coulomb - modified effective range parameter . we discuss the implications of this bound to the effective field theory for nuclear halo systems . in particular , we consider several examples of proton nucleus and nucleus nucleus scattering . for the bound - state regime , we find relations for the asymptotic normalization coefficients ( ancs ) of nuclear halo states . as an application of these relations , we extract the ancs of the excited and states ino fromc scattering data .

astro-ph0601336

i

the determination of water abundance in space is a long standing problem in astronomy . theoretical models predict that water can be the most abundant species in _ warm _ molecular clouds after h@xmath0 and co @xcite . therefore , the determination of its spatial distribution and abundance contributes to a better knowledge of the chemical and physical processes that take place in the interstellar medium ( ism ) . unfortunately , water is an abundant molecule in our atmosphere making particularly difficult the observation of its rotational lines and vibrational bands from earth . even so , some observations of water lines have been performed from ground based and airborne telescopes : the @xmath21 at 22 ghz @xcite , the @xmath22 at 325 ghz @xcite , the @xmath23 at 321 ghz @xcite and the @xmath5 at 183.31 ghz ( watson et al . 1980 ; cernicharo et al . 1990 , 1994 , 1996 , 1999a ; gonzlez - alfonso et al . , 1998 ) . due to the maser nature of these lines , the analysis and interpretation of the spectra is not obvious . among these lines , only two have been used to map the extended emission of water vapor in orion : the @xmath5 -hereafter the 183 ghz line- @xcite and the @xmath22 at 325 ghz @xcite . however , although we know from iso and swas observations that water is extended ( @xmath325@xmath24@xmath1025@xmath24 ) in sgr b2 ( cernicharo et al . 1997 , neufeld et al . 2003 , goicoechea et al . 2004 ) , little is known about its excitation conditions and its detailed spatial distribution . an alternative to indirectly estimate the water abundance in the galactic center ( gc ) is to use related species such as hdo @xcite or h@xmath25o@xmath26 @xcite . in none of these cases , the determination of @xmath13(h@xmath0o ) is straightforward . the iso mission @xcite , and specially , the _ long wavelength spectrometer _ , lws , @xcite , and the _ short wavelength spectrometer _ , sws , @xcite , have provided a unique opportunity to observe several h@xmath0o lines in a great variety of astronomical environments . nevertheless , the majority of these observations were performed at the low spectral resolution of the grating mode ( @xmath31000 km s@xmath4 ) , which produces a critically strong dilution in the search for molecular features in most ism sources . nevertheless , the sgr b2 cloud has been analyzed and studied in detail with the lws/_fabry perot _ , which provided a velocity resolution of @xmath335 km s@xmath4 ( goicoechea et al . 2004 , hereafter g04 ) . opposite to what is found toward other star forming regions such as orion @xcite , the observations of the @xmath27 line at @xmath3179.5 @xmath28 m in sgr b2 show that the line appears in absorption rather than in emission @xcite . afterward , the launch of the _ submillimeter wave astronomy satellite _ , swas , @xcite , and odin @xcite allowed the observation of the @xmath29 fundamental transition of both h@xmath30o at 557 and h@xmath31o at 548 ghz , first detected by the _ kuiper airborne observatory _ , kao , @xcite . although the velocity resolution is @xmath31 km s@xmath4 , the large beam of swas ( @xmath34@xmath24 ) makes these observations more sensitive to the cold and less dense gas . swas observations have provided a reliable estimate of the water vapor abundance in the low excitation clouds located in the line of sight toward sgr b2 @xcite . however , the fact that only the ground - state absorption line is detected makes difficult a detailed study of water vapor excitation mechanisms in sgr b2 itself . this is the most massive cloud in the galaxy , with @xmath310@xmath32 , @xcite , and a paradigmatic object in the gc region as its geometrical properties , physical conditions and chemical characteristics make it a miniature galactic nucleus with @xmath315@xmath24 extent ( g04 and references therein ) . the main star - forming regions in sgr b2 are located within three dust condensations , labeled as ( n ) , ( m ) and ( s ) . these condensations are embedded in a @xmath310 pc moderate density ( @xmath33 @xmath8 ) cloud @xcite . in addition , these structures are surrounded by lower density components of warm ( t@xmath1@xmath34100 k ) gas , hereafter the sgr b2 warm envelope . these conditions have been derived mainly from absorption observations of nh@xmath25 metastable lines @xcite , oh lines ( goicoechea & cernicharo , 2001 ; 2002 ) , and h@xmath0co lines @xcite in the radio domain . nevertheless , the warm and low density gas is poorly traced by radio observations of other molecular species ( generally excited by collisions in the denser regions and thus observed in emission ) . however , the warm envelope represents the strongest contribution to the absorption features produced by many light hydrides in the far - ir spectrum of sgr b2 @xcite . the origins of the observed rich chemistry and the heating mechanisms in the sgr b2 warm envelope are a subject of intense debate . the matter is complicated due to the different observational signatures to be integrated in the same picture : high temperatures derived from nh@xmath25 absorption lines @xcite , fine structure emission from the photo ionized and photo dissociated gas @xcite , sio and x ray distribution @xcite , etc . in all possible scenarios , water plays a significant role . several mechanisms allow its formation and survival in the warm envelope . dissociative recombination of h@xmath25o@xmath26 leads to the production of h@xmath0o and oh . this processes depend on the specific @xmath35 branching ratio for the h@xmath0o formation channel . unfortunately , the determination of @xmath35 with different experimental procedures has also yielded different values , from @xmath35=0.05 @xcite to @xmath35=0.25 @xcite , while most chemical models have used @xmath35@xmath30.35 . in addition , water could also be produced in the gas phase by the endothermic reaction : @xmath36 however , the gas temperature must exceed @xmath3300 k to overcome the activation barrier @xcite . at these temperatures , the reaction : @xmath37 also contributes to the formation of oh . therefore , h@xmath0o and oh column densities can be used to determine the role of the neutral - neutral reactions in their formation / destruction routes . still , the exact h@xmath0o / oh ratio will be determined by @xmath35 , the temperature and by photodissociation processes if uv radiation is present . as an example , neufeld et al . @xcite studied a diffuse cloud ( @xmath38@xmath31 , @xmath39=100 @xmath8 ) toward w51 and showed that the presence of a warm gas component ( t@xmath1@xmath40400 k ) could explain the observed variations of the h@xmath0o / oh ratio respect to other diffuse clouds . finally , high oxygen depletion onto water ice mantles in dust grains could have taken place during the evolution of the cool gas in sgr b2 . photodesorption and/or evaporation for dust temperatures above @xmath390 k , could release some water back into the gas phase enhancing the h@xmath0o abundance expected from pure gas phase formation @xcite . however , gas and dust are thermally decoupled in the outer layers of sgr b2 , where the dust temperatures are significantly lower , t@xmath41@xmath142030 k @xcite , than gas temperatures , t@xmath1@xmath14300 k @xcite . thus , t@xmath41 seems too low to produce significant evaporation of water ice mantles . therefore , a detailed study of the far ir h@xmath0o lines and of the 183.31 ghz extended emission is needed to constrain the water abundance and the physical characteristics of the absorbing / emitting region . in this work we present and analyze the far - ir observations of several thermal lines of water vapor toward sgr b2(m ) and the first map of the 183.31 ghz maser emission of para h@xmath0o around sgr b2 main condensations . the layout of the paper is as follows : in section [ sct : observ ] we summarize the far ir , submm and mm observations and data reduction . the spectra and maps are presented in sec.[sct : results ] . section [ sct : analysis ] is devoted to the analysis of co observations ( [ ssct : co ] ) and water vapor observations ( [ ssct : h2o ] ) with different radiative transfer methods . the main implications of our work are discussed in sec . [ sct : discussion ] , where photochemistry models for h@xmath0o and oh are also presented . a summary is given in sec . [ sct : summary ] .

in the region of sgr b2 there are several condensations heated externally by nearby hot stars . therefore ho far we have conducted a study combining ho lines in both spectral regions . first , observations of several ho thermal lines seen in absorption toward sgr b2(m ) at a spectral resolution of km s have been analyzed . second , an _ iram_30 m telescope map of the para radiative transfer models indicate that these lines are quite insensitive to temperature and gas density , and that ir photons from the dust play a dominant role in the excitation of the involved ho rotational levels . in order to get the physical conditions of the absorbing gas we have also analyzed the co emission toward sgr b2(m ) . hence , the water vapor abundance in this region ,(ho ) , is(1 - 2) . neutral reactions producing ho and oh . on the other hand , the 183.31 ghz data provide a much better spatial and spectral resolution than the far - ir iso data . the emission is very strong toward the cores . the derived water vapor abundance for this component is a few .

in the region of sgr b2 there are several condensations heated externally by nearby hot stars . therefore ho far ir lines are expected to probe only an external low density and high temperature section of these condensations , whereas millimeter - wave lines can penetrate deeper into them where the density is higher and t lower . we have conducted a study combining ho lines in both spectral regions . first , observations of several ho thermal lines seen in absorption toward sgr b2(m ) at a spectral resolution of km s have been analyzed . second , an _ iram_30 m telescope map of the para ho line at 183.31 ghz , seen in emission , has also been obtained and analyzed . the ho lines seen in absorption are optically thick and are formed in the outermost gas of the condensations in front of the far ir continuum sources . they probe a maximum visual extinction of to 10 mag . radiative transfer models indicate that these lines are quite insensitive to temperature and gas density , and that ir photons from the dust play a dominant role in the excitation of the involved ho rotational levels . in order to get the physical conditions of the absorbing gas we have also analyzed the co emission toward sgr b2(m ) . we conclude , based on the observed co=76 line at 806.65 ghz with the _ caltech submillimeter observatory _ , and the lack of emission from the far ir co lines , that the gas density has to be lower than . using the values obtained for the kinetic temperature and gas density from oh , co , and other molecular species , we derive a water column density of ( 9) in the absorbing gas . hence , the water vapor abundance in this region ,(ho ) , is(1 - 2) . the relatively low ho / oh abundance ratio in the region , - 4 , is a signature of uv photon dominated surface layers traced by far ir observations . as a consequence the temperature of the absorbing gas is high , t - 500 k , which allows very efficient neutral neutral reactions producing ho and oh . on the other hand , the 183.31 ghz data provide a much better spatial and spectral resolution than the far - ir iso data . this maser line allows to trace water deeper into the cloud , i.e. , the inner , denser ( ) and colder ( t k ) gas . the emission is very strong toward the cores . the derived water vapor abundance for this component is a few . there is also moderate extended emission around sgr b2 main condensations , a fact that supports the water vapor abundance derived from far ir ho lines for the outer gas .

astro-ph0601336

c

the warm gas present in the outer layers of sgr b2(m ) might be expected to radiate in high@xmath6 co lines . the co @xmath6=1413 transition at 185.999 @xmath28 m is the one with the lowest energy level ( e@xmath53@xmath14581 k ) within the range of iso / lws detectors . however , we have not detected any emission / absorption from co ( 3@xmath55 limits are @xmath452@xmath1010@xmath56 w @xmath12 ) in the iso / lws spectra toward the sgr b2 region @xcite , with both grating and fp . co spectroscopical data and line flux upper limits are tabulated in table [ tab - co_lines ] . nevertheless , recent studies of the large scale co @xmath6=76 emission toward the gc with the _ antarctic submillimeter telescope and remote observatory _ ( ast / ro ) have shown that the emission is concentrated toward sgr b and sgr a complexes @xcite . [ co_7_6_cso ] shows higher spectral / angular resolution observations of this line taken with the cso telescope toward the sgr b2(m ) position . hence , it seems that at a given @xmath57 level , the rotational co line emission disappears from the sgr b2 spectrum . = 76 line at 806 ghz toward sgr b2(m).,width=302 ] we have performed nonlocal radiative transfer calculations to try to reproduce the lack of high@xmath6 co lines in the far ir spectrum of sgr b2(m ) , and to help constraining the physical parameters needed to model the water vapor absorption . in fig . [ co_models ] we show the predictions of a nonlocal model for several high@xmath6 transitions of co. the nonlocal model used here is an adaptation of the radiative transfer code developed by gonzlez alfonso & cernicharo @xcite with the inclusion of dust in the transfer @xcite . we have implemented it for high@xmath6 levels of co ( see sec . [ sssct : nonlocal ] for further details on the specific model developed for sgr b2(m ) ) . the collisional rates have been taken from flower ( 2001 ) . the dust continuum emission has been modeled following goicoechea and cernicharo ( 2002 ) and g04 ( see section 4.2.1 ) . we have run an array of models for co abundances of 10@xmath58 and 10@xmath15 with t@xmath1 varied from 100 to 500 k and @xmath17 from 5.0@xmath1010@xmath59 to 6.4@xmath1010@xmath60 @xmath8 . from these results it is clear that in order to suppress the far ir co emission , and to match the co @xmath6=76 line emission , low h@xmath0 density is required . in particular , if the co @xmath6=76 line arises from a layer of gas at t@xmath1@xmath3100 k the model with @xmath17=2@xmath1010@xmath7 @xmath8 correctly reproduces the absence of far ir lines . however , if it arises from the same outer layer of warm oh ( t@xmath1@xmath3300 k ) detected in the far ir @xcite , the limit for the density will be @xmath4510@xmath7 @xmath8 . in any case , the lack of high@xmath6 co lines is an evidence of the low density gas located in front of the far ir continuum source . in the following section we will constrain the physical parameters of this layer by studying the available water vapor lines . the main observational result of this work is that the far ir water lines toward sgr b2(m ) appear in absorption , while the 183.31 ghz line is seen in emission around and in the main condensations . the fact that the continuum emission in the far ir is optically thick @xcite indicates that the h@xmath0o absorption lines arise from regions where the excitation temperatures ( t@xmath61 ) are smaller than the dust temperatures inferred from the continuum emission . however , the problem of the h@xmath0o line excitation toward sgr b2 is not straightforward . apart from self absorption and the possible excitation by collisions with molecules ( e.g. h@xmath0 ) , atoms ( e.g. he ) and @xmath62 ( if the ionization fraction is significant ) , the level population can be primarily determined by the thermal emission of dust , the role of which is essential in the excitation of molecules such as h@xmath0o or oh which have many rotational lines in the far and mid ir . this represents a major difference with respect to the excitation treatment of molecules observed in the radio domain where , generally , one can neglect the excitation by dust photons . if the excitation is dominated by ir photons , the two transitions arising from the ground levels of ortho - water , 2@xmath511@xmath52 and 1@xmath631@xmath52 , will determine how the higher energy levels will be populated . even if collisions are important , the presence of an optically thick far ir continuum will strongly affect the t@xmath61 of water and , thus , it must be carefully taken into account in the models . in the case of sgr b2 , this means that the external dust layers of the cloud will absorb the possible water line emission from the inner regions . knowledge of the geometry of the region to be modeled and the relative filling factors of the dust and gas in the beam of the lws instrument , are also important for the models . for this reason , high angular resolution ground - based observations of the 183.31 ghz water line are particularly important to model the water vapor radiative transfer in sgr b2 . in this section we analyze the h@xmath0o observations with a _ multi - molecule large velocity gradient _ ( lvg ) model . for sgr b2(m ) we have adopted a spherical geometry with two components ( see fig . [ fig_h2o_geometry ] ) : a uniform continuum core with a diameter of @xmath323@xmath44 ( @xmath31 pc for a distance of 8.5 kpc ) and a shell of variable thickness and distance to the core . the presence of an external shell of molecular gas ( not resolved by the iso / lws beam ) surrounding a central condensation is indicated by the analysis of the far ir oh @xcite and nh@xmath25 lines @xcite and it is also supported by h@xmath0o 183.31 ghz line observations ( fig . [ mapa_183 ] ) . in particular goicoechea & cernicharo ( 2002 ) found an angular size of @xmath342@xmath44 for the oh envelope . the bulk of the h@xmath0o absorption lines must arise from the oh layers , or inside them . following the oh model geometry and figs . [ mapa_183 ] and [ co_7_6_cso ] , a total size of @xmath330@xmath44 ( @xmath31.2 pc ) for the core@xmath48shell cloud has been adopted . the core is considered as a gray body with an opacity at 80 @xmath28 m of 2.5 , with a dust opacity law given by @xmath64 $ ] , and a dust temperature of 30 k. these values are consistent with the color temperatures and dust emissivities derived from the analysis of the iso / lws continuum observations at the same wavelengths of the detected far ir h@xmath0o lines @xcite . although dust temperatures can be slightly larger ( t@xmath41@xmath36080 k ) inside the cloud ( from the analysis of the millimeter continuum emission ) or slightly lower ( t@xmath41@xmath320 k ) in the outer and colder diffuse layers ( from the analysis of extended iras continuum emission ; gordon et al . 1993 ) , we judged t@xmath41@xmath1430 k as the most representative value for the dust grains coexisting with the molecular species detected in the far ir . the bulk of the continuum emission detected by iso can be hardly fitted with dust temperatures below 30 k. the emission of higher temperature dust arising from the innermost regions of sgr b2 is hidden in the fir by the huge amount of foregroung gas and colder dust . for these models we have considered an ortho h@xmath0o column density of 1.8@xmath1010@xmath11 @xmath12 and a para h@xmath0o column density of 0.6@xmath1010@xmath11 @xmath12 . these are lower limits suggested by the nonlocal radiative transfer models ( see below ) . the lvg model computes the statistical equilibrium population of the rotational levels for ortho h@xmath0o and para h@xmath0o independently . the collisional rates were scaled from those of h@xmath0o he collisions @xcite . several lvg computations for some selected ortho and para h@xmath0o transitions in the thz domain are shown in fig . [ lvg_mod ] . the excitation temperature t@xmath61 of each transition is shown in each panel as a function of t@xmath1 and @xmath17 . the thick contour corresponds to the equivalent temperature of the continuum core . therefore , to the left of this contour , water lines appear in absorption . from the lvg models it is clear that high density and temperature are required to observe the far ir water lines in emission . for the range of densities implied by the co observations ( @xmath4510@xmath7 @xmath8 ) , the h@xmath0o lines observed by the iso / lws are correctly predicted in absorption . the temperature of the absorbing layer is , however , more difficult to estimate because the possible solutions for a given ( low ) density model are not very sensitive to temperature variations , as indicated by the smooth change of t@xmath61 for constant density as the kinetic temperature goes from 500 to 100 k. lvg models predict that , in addition to dust photons , collisions play a role in the excitation of the lowest h@xmath0o rotational levels . the higher energy levels are pumped from the lowest levels by absorption of far ir photons . in the case of the ortho 1@xmath631@xmath52 line observed by _ swas _ ( neufeld et al . 2000 ) , only moderate densities ( @xmath655@xmath1010@xmath7 @xmath8 ) are needed to observe the line in emission . this is the case of the extended emission in the 1@xmath631@xmath52 line observed in the _ 180 pc molecular ring _ around the gc between @xmath66@xmath3@xmath4880 and 120 km s@xmath4 ( neufeld et al . 2003 ) . for the density conditions derived for the sgr b2 velocity range and for the line of sight clouds , the 557 ghz line is correctly predicted in absorption . a similar behavior is expected for the para 1@xmath671@xmath68 line at @xmath3269.3 @xmath28 m ( @xmath31113 ghz ) that will be observed by future heterodyne instruments such as hifi / herschel . among all the moderate excitation para h@xmath0o lines ( e@xmath69@xmath45450 k ) , only the 3@xmath702@xmath71 appears in the mm domain . contrary to other h@xmath0o maser lines accessible from ground based telescopes , relatively low t@xmath1 and density are required for the 3@xmath702@xmath71 line inversion . these conditions allowed the first detection of extended water emission in orion ( cernicharo et al . 1994 ) , and a strong dependence of the emission with t@xmath1 was revealed . hence , the 183.31 ghz line could be an excellent tracer of the warm gas in molecular clouds . we have used the same lvg model to analyze the 183.31 ghz line inversion mechanism that produces extended emission in sgr b2 ( fig . [ mapa_183 ] ) . [ lvg_mod_183 ] shows different excitation models for the 3@xmath702@xmath71 transition . since a fraction of the 183.31 ghz line emission may arise from the warm envelope in front of the far ir continuum source , models with and without a central continuum source ( the same described at the beginning of the section ) have been considered . for a given t@xmath1 ( from 20 to 500 k ) , each panel shows computations of different para h@xmath0o column densities as a function of t@xmath72 ( in k ) and h@xmath0 volume density . for low temperatures ( t@xmath1@xmath4540 k ) and moderate densities ( @xmath4510@xmath60 @xmath8 ) , the line will be observable for large @xmath73(p h@xmath0o ) values only if a continuum source is present . due to the minor role played by collisions in the pumping of the rotational levels at these temperatures , the line intensity is almost independent of the density . nevertheless , as the temperature increases , collisions start to be significant , and the expected intensity of the 183.31 ghz line becomes less sensitive to the models with or without the continuum source . the 183.31 ghz emission is produced by the population inversion of the 3@xmath70 and 2@xmath71 levels . this is due to the different rates ( far from the thermalization densities ) at which both levels can be populated . the 3@xmath70 level is radiatively connected with the 2@xmath71 and 2@xmath74 lower energy levels . the a@xmath75 coefficient for the 3@xmath702@xmath74 at @xmath3138.5 @xmath28 m is @xmath310@xmath7 times larger than the a@xmath75 of the 3@xmath702@xmath71 transition . the @xmath3138.5 @xmath28 m line is seen in strong absorption in the iso / lws spectra ( see fig . [ obs_iso_agua ] ) . the 2@xmath71 level is radiatively connected with the 2@xmath67 and 1@xmath67 lower energy levels . the 2@xmath711@xmath67 at @xmath3100.9 @xmath28 m produces the strongest absorption of para h@xmath0o in the far ir ( see fig . [ obs_iso_agua ] ) . on the other hand , the 3@xmath70 level is radiatively connected with the 3@xmath76 , 4@xmath76 and 4@xmath77 higher energy levels , while the 2@xmath71 level is connected with the 3@xmath78 ( see the @xmath367.1 @xmath28 m line in fig . [ obs_iso_agua ] ) and with the 3@xmath70 higher energy levels . the different radiative pathways and rates ( for @xmath79@xmath80@xmath81 ) at which the 3@xmath70 and 2@xmath71 levels can be populated produce the inversion . lvg models ( see fig . [ lvg_mod_183 ] ) show that this mechanism can be efficient to produce 183.31 ghz line emission in regions of relatively low density . the observed extended emission at 183.31 ghz has a brightness temperature of 10 k in average ( see fig . [ mapa_183 ] ) . assuming that it arises from the low density regions in which the the far ir continuum sources are embedded , with t@xmath1@xmath82300 - 500 k and @xmath17@xmath310@xmath7 @xmath8 , we derive a value for @xmath73(p h@xmath0o ) of @xmath405@xmath1010@xmath83 @xmath12 . this value is higher than the one derived from the far - ir water lines ( see sect . [ sssct : nonlocal ] ) possibly because the 183.31 ghz line penetrates deeper into this dusty environment . toward the main condensation , the bulk of the emission seems to arise from the cold ( t@xmath8440 k ) and dense gas . under these conditions , a brightness temperature of the line of @xmath3100 k would translate into a p h@xmath0o column density of @xmath310@xmath85 @xmath12 . however , an important contribution could come from the embedded high temperature and high density core condensations where the column density should also be much larger . to take into account the radiative coupling between regions with different physical and/or excitation conditions , the radiative transfer has to be treated with nonlocal techniques , more sophisticated than the lvg approximation . we have adapted the radiative transfer used in sec . 5.1 for co , to the ground vibrational states of ortho and para h@xmath0o respectively . the model includes all the water rotational levels with transitions between 40 @xmath28 m and 183.31 ghz . the level population is computed in statistical equilibrium considering collisional excitation and radiative excitation by line and continuum photons . this is computed consistently assuming that the water molecules and the dust grains are coexistent . the geometry , core@xmath48shell dimensions , are the same of that considered in the lvg models . the shell was divided in 41 layers . the central dust condensation has been modeled with identical parameters than in section 4.2.1 . we have considered 14 rotational levels ( e@xmath86 @xmath45 608 k ) of ortho - water for model with t@xmath1 below 100 k , and up to 30 rotational levels ( e@xmath87 @xmath45 1290 k ) for the higher temperature models . we have adopted a turbulence velocity of 8 km s@xmath4 . the continuum radiation field has been treated as in gonzlez - alfonso & cernicharo @xcite by considering a spectral range of 70 km s@xmath4 centered on each water transition . collisional rates have been taken from green et al . obviously all the water transitions in the continuum core are thermalized to the dust temperature due to the large opacity in the far ir . the computed line profiles are a result of convolving the brightness temperature with the angular resolution of the lws detectors ( @xmath380@xmath44 ) . the resulting spectral resolution of the synthetic water lines is 1 km s@xmath4 , as no convolution with the spectral resolution has been performed . to test the sensitivity of the model to the physical parameters , models were computed with @xmath73(h@xmath0o ) and t@xmath1 of 1.8@xmath1010@xmath11 , 9@xmath1010@xmath11 and 4.5@xmath1010@xmath83 cm @xmath88 , and 40 , 100 , 200 , 300 and 500 k respectively , while densities were increased from 5@xmath1010@xmath89 to 6.4@xmath1010@xmath90 cm @xmath91 multiplying by 2 in each step . the different nonlocal radiative transfer models for the first two h@xmath0o column densities are shown in figs . [ mod_1_h2o ] and [ mod_2_h2o ] . models for other column densities have been also ran . however , the observed absorption depth is not reproduced for column densities below 10@xmath11 cm @xmath88 . in the @xmath73(h@xmath0o)=1.8@xmath1010@xmath83 cm @xmath88 case too many lines would be in emission , contrary to the observations . the main problem to interpret the h@xmath0o absorption toward sgr b2 arises from the large opacities of the far ir water lines , @xmath310@xmath59 , and even @xmath310@xmath7 for the ortho 2@xmath511@xmath52 line at @xmath3179.5 @xmath28 m . under these conditions , many weak water lines have to be detected to constrain the physical conditions and the column density . therefore , we strength that it is difficult to obtain physical parameters from the observation of few far ir water rotational lines . in addition , as the radiative excitation dominates the population of the far ir levels , lines with a weak dependence on the dust excitation should be investigated . taking into account the 14 water lines detected in the far ir , the nonlocal results imply that models are not very sensitive to the temperature and that the only indication about the column density has to be searched in weak far ir h@xmath0o lines or in the 183.31 ghz line ( see sec . even so , the far ir absorption arises in the low density external layers of gas , while it is very likely that the 183.31 ghz line could have an important contribution from inner and denser regions . another complication arises from the fact that for models of low @xmath73(h@xmath0o ) values and t@xmath1@xmath92200 k , it is also difficult to distinguish between different h@xmath0 densities ( see fig . [ mod_1_h2o ] ) . limits to t@xmath1 and @xmath79(h@xmath0 ) have to be searched in weak lines below 70 @xmath28 m . the models for high water column densities ( fig . [ mod_2_h2o ] ) predict absorption lines in the lws range at @xmath356.3 , @xmath357.6 , @xmath358.7 , @xmath378.7 , @xmath399.5 , @xmath3125.4 and @xmath3136.5 @xmath28 m . at the spectral resolution and sensitivity of the lws / fp , none of these lines has been detected . this implies that @xmath73(h@xmath0o)@xmath934.5@xmath9410@xmath83 @xmath12 toward the warm envelope of sgr b2 . the @xmath3136.5 @xmath28 m line ( also predicted by the models with large column densities ) is contaminated by the absorption produced by the c@xmath25 @xmath95(8 ) rovibrational line , which is also predicted by the models of tri atomic carbon @xcite . some of these lines are predicted ( even in emission ) by the models with large @xmath73(h@xmath0o ) . models with @xmath73(h@xmath0o)@xmath821.8@xmath1010@xmath11 @xmath12 are consistent with iso detections and upper limits to other iso lines . only the @xmath367.1 @xmath28 m line is weaker than the observations . therefore , models shown in fig . [ mod_1_h2o ] give a lower limit to the water vapor column density in the outer and warm ( 300 - 500 k ) envelope . taking into account the difficulties implied by the h@xmath0o modeling in the far ir , we found that @xmath96 @xmath12 is the best h@xmath0o column density to fit the iso observations ( see figure [ mod_2_h2o ] ) . the co analysis ( sec . 5 ) , and the studies in the far ir of oh @xcite and the ammonia lines @xcite and our co data , also support that the water absorption lines arise in the warm and low density ( @xmath9210@xmath7 @xmath8 ) layer in front of sgr b2(m ) . the h@xmath0o column density derived from iso observations is below the lower limit of @xmath73(h@xmath0o ) estimated for the 183.31 ghz line for this component ( see sec . [ sssct : lvg ] ) as it is likely that an important fraction of the 183.31 ghz emission arises from the inner and denser regions closer to sgr b2 main cores or even from them , as pointed out above . this component , with a mean h@xmath0 density of @xmath6510@xmath19 @xmath8 , has been traced by the nh@xmath25 non metastable emission lines , t@xmath1@xmath14100 k , @xcite and by the emission of hc@xmath25n rotational lines , t@xmath1@xmath142040 k , @xcite . the determination of the temperature from mm emission lines is also complicated in these regions completely obscured to iso observations . therefore , the observed differences in the 3@xmath702@xmath74 line intensity and shape can be a combination of @xmath73(h@xmath0o ) and/or t@xmath1 variations across the region . the results presented in this section have been compared to those obtained from another radiative transfer based on a different approach ( asensio ramos & trujillo bueno 2003 ; asensio ramos & trujillo bueno 2006 ) . this code is a generalization to spherical geometry of the very fast iterative multilevel gauss - seidel ( muga ) and multilevel successive overrelaxation ( musor ) methods developed by trujillo bueno & fabiani bendicho ( 1995 ) for the case of a two - level atom and generalized by fabiani bendicho , trujillo bueno & auer ( 1997 ) to multilevel atoms in cartesian geometries . the code also allows the application of the standard multilevel accelerated @xmath97-iteration ( mali ) ( olson , auer & buchler 1986 ) . the angular information for the calculation of the mean intensity is obtained by solving the radiative transfer equation along its characteristic curves ( straight trajectories with constant impact parameter ) with the aid of the short - characteristics formal solver with parabolic precision ( kunasz & auer 1988 ) . the statistical equilibrium equations are linearized with the aid of the preconditioning scheme developed by rybicki & hummer ( 1991 , 1992 ) with the introduction of an approximate @xmath97 operator that can be efficiently obtained in the framework of the short - characteristics technique . the convergence rate of the muga and musor schemes is equivalent to that obtained with the introduction of a nonlinear @xmath97 operator , with the advantage of not being necessary neither to calculate nor to invert such nonlinear operator . interestingly , the time per iteration is similar to that obtained for the standard @xmath97-iteration or the mali method . the computing time for the muga scheme is reduced by a factor 4 with respect to mali , while the musor scheme leads to an order of magnitude of improvement in the total computing time with respect to mali . the calculations have been performed with the same geometry , the same molecular data ( collisional and radiative transitions ) , and the same physical conditions of the previously described nonlocal code . the emerging line profiles from both codes are very similar , with differences below 2 - 3 % . both codes predict lines in absorption / emission for the same physical conditions , with identical spectral shapes in the cases where re - emission is found in the line wings . this test of consistency allows us to be very confident in the results presented in figs . [ mod_1_h2o ] and [ mod_2_h2o ] since they were obtained with numerical methods based on completely different approaches .

ir lines are expected to probe only an external low density and high temperature section of these condensations , whereas millimeter - wave lines can penetrate deeper into them where the density is higher and t lower . ho line at 183.31 ghz , seen in emission , has also been obtained and analyzed . the ho lines seen in absorption are optically thick and are formed in the outermost gas of the condensations in front of the far ir continuum sources . ir observations . as a consequence the temperature of the absorbing gas is high , t - 500 k , which allows very efficient neutral

in the region of sgr b2 there are several condensations heated externally by nearby hot stars . therefore ho far ir lines are expected to probe only an external low density and high temperature section of these condensations , whereas millimeter - wave lines can penetrate deeper into them where the density is higher and t lower . we have conducted a study combining ho lines in both spectral regions . first , observations of several ho thermal lines seen in absorption toward sgr b2(m ) at a spectral resolution of km s have been analyzed . second , an _ iram_30 m telescope map of the para ho line at 183.31 ghz , seen in emission , has also been obtained and analyzed . the ho lines seen in absorption are optically thick and are formed in the outermost gas of the condensations in front of the far ir continuum sources . they probe a maximum visual extinction of to 10 mag . radiative transfer models indicate that these lines are quite insensitive to temperature and gas density , and that ir photons from the dust play a dominant role in the excitation of the involved ho rotational levels . in order to get the physical conditions of the absorbing gas we have also analyzed the co emission toward sgr b2(m ) . we conclude , based on the observed co=76 line at 806.65 ghz with the _ caltech submillimeter observatory _ , and the lack of emission from the far ir co lines , that the gas density has to be lower than . using the values obtained for the kinetic temperature and gas density from oh , co , and other molecular species , we derive a water column density of ( 9) in the absorbing gas . hence , the water vapor abundance in this region ,(ho ) , is(1 - 2) . the relatively low ho / oh abundance ratio in the region , - 4 , is a signature of uv photon dominated surface layers traced by far ir observations . as a consequence the temperature of the absorbing gas is high , t - 500 k , which allows very efficient neutral neutral reactions producing ho and oh . on the other hand , the 183.31 ghz data provide a much better spatial and spectral resolution than the far - ir iso data . this maser line allows to trace water deeper into the cloud , i.e. , the inner , denser ( ) and colder ( t k ) gas . the emission is very strong toward the cores . the derived water vapor abundance for this component is a few . there is also moderate extended emission around sgr b2 main condensations , a fact that supports the water vapor abundance derived from far ir ho lines for the outer gas .

astro-ph0601336

i

we have carried out far ir observations of several thermal absorption lines of water vapor toward sgr b2(m ) and have mapped the 183.31 ghz water line around the main dust condensations of the complex . the main conclusions of this work are the following : 1 . the detected water absorption lines are very opaque and arise from the warm envelope around sgr b2(m ) . the observation of the co @xmath6=76 line and the lack of far ir co lines at iso s sensitivities ( 3@xmath55 limits below @xmath32@xmath1010@xmath56 w @xmath12 ) imply that the density of such layer is @xmath79(h@xmath0)@xmath310@xmath7 @xmath8 . 3@xmath702@xmath74 line at 183.31 ghz shows @xmath340@xmath44@xmath1040@xmath44 extended emission around sgr b2(m ) and @xmath340@xmath44@xmath1020@xmath44 around sgr b2(n ) . this is the first observation of that line in a gc source and represents further evidence that water vapor is extended in warm molecular clouds . lvg and nonlocal radiative transfer calculations have been carried out to extract the water vapor abundance and to constrain the physical parameters of the absorbing / emitting regions . because of the radiative excitation by dust photons , the far ir water lines are not very sensitive to t@xmath1 . taking into account the analysis of the related species oh @xcite , the water absorption must arise from warm gas at similar temperatures , i.e. , from 300 to 500 k. for this warm envelope , we found @xmath73(h@xmath0o)=@xmath96 @xmath12 . an important fraction of the 183.31 ghz emission arises from the inner , denser and colder gas located closer to the main cores . we estimate a water abundance of a few@xmath1010@xmath20 in the denser regions . photochemistry models show that a component of warm gas , @xmath3300 - 500 k , is needed to activate the neutral - neutral reactions and reproduce the large h@xmath0o and oh column densities observed in the envelope . we show that oh and h@xmath0o far ir observations toward sgr b2 are surface tracers of the cloud ( a maximum a@xmath101 of 5 to 10 mag ) . we found @xmath13(h@xmath0o)@xmath14(1 - 2)@xmath1010@xmath15 in these regions . although irradiated by fuv , and possibly more energetic photons , affecting the h@xmath0o / oh ratio in the outermost layers , a clumpy structure for the pdr is needed . alternatively , low velocity shocks could maintain the gas heating through the envelope . due to the complexity of sgr b2 ( and also of the gc ism as a whole ) a multiple scenario is needed to explain the modest angular resolution far ir observations . the input of chemical models , and higher sensitivity and larger spatial resolution far ir observations will lead to a better understanding of the gc environment . we thank spanish dges and pnie for funding support under grants panaya2000 - 1784 , esp2001 - 4516 , aya2002 - 10113-e , esp2002 - 01627 , aya2003 - 02785-e and aya2004 - 05792 . cso observations were supported by nsf grant ast-9980846 . jrg was also supported by the french _ direction de la recherche _ in the latest stages of the work . we thank j. le bourlot for useful suggestions and for his help with the pdr model . zmuidzinas , j. , blake , g. a. , carlstrom , j. , keene , j. , miller , d. , schilke , p. & ugras , n. g. 1995 , in astronomical society of the pacific , airborne astronomy symposium on the galactic ecosystem : from gas to stars to dust , volume 73 , 33 - 40 . cccccccccc p h@xmath0o & @xmath5 & 1635.439 & 183.3 & 205 & 3.53e@xmath11506 & 1120@xmath960 k km s@xmath4 & & 30@xmath93 + o h@xmath31o & @xmath27 & 181.053 & 1655.8 & 79 & 5.45e@xmath11502 & @xmath115(2.68@xmath90.26)e@xmath11518 & 0.73@xmath90.02 & 52@xmath92 + o h@xmath0o & @xmath116 & 180.488 & 1661.0 & 160 & 2.99e@xmath11502 & @xmath115(3.44@xmath90.46)e@xmath11518 & 0.74@xmath90.01 & 64@xmath93 + o h@xmath0o & @xmath27 & 179.527 & 1669.9 & 80 & 5.47e@xmath11502 & @xmath115(6.76@xmath90.29)e@xmath11514 & 0.06@xmath90.03 & 204@xmath923 + o h@xmath0o & @xmath117 & 174.626 & 1716.7 & 163 & 4.94e@xmath11502 & @xmath115(1.86@xmath90.46)e@xmath11518 & 0.82@xmath90.02 & 44@xmath92 + p h@xmath0o & @xmath118 & 138.527 & 2164.1 & 205 & 1.22e@xmath11501 & @xmath115(1.27@xmath90.03)e@xmath11518 & 0.89@xmath90.02 & 41@xmath92 + o h@xmath0o & @xmath119 & 121.719 & 2462.9 & 516 & 1.20e@xmath11501 & @xmath115(5.40@xmath90.29)e@xmath11518 & 0.96@xmath90.01 & 62@xmath99 + o h@xmath0o & @xmath120 & 108.073 & 2773.9 & 160 & 2.52e@xmath11501 & @xmath115(6.07@xmath90.05)e@xmath11518 & 0.64@xmath90.02 & 53@xmath92 + p h@xmath31o & @xmath121 & 102.008 & 2938.9 & 194 & 2.52e@xmath11501 & @xmath115(9.38@xmath92.62)e@xmath11519 & 0.96@xmath90.01 & 72@xmath98 + p h@xmath0o & @xmath121 & 100.983 & 2968.7 & 196 & 2.55e@xmath11501 & @xmath115(6.02@xmath90.11)e@xmath11518 & 0.66@xmath90.01 & 55@xmath99 + p h@xmath0o & @xmath122 & 89.988 & 3331.4 & 297 & 3.45e@xmath11501 & @xmath115(1.99@xmath90.14)e@xmath11518 & 0.90@xmath90.01 & 67@xmath92 + o h@xmath0o & @xmath123 & 75.380 & 3977.0 & 271 & 3.25e@xmath11501 & @xmath115(4.32@xmath90.07)e@xmath11518 & 0.69@xmath90.02 & 51@xmath92 + p h@xmath0o & @xmath124 & 67.089 & 4468.6 & 410 & 1.20e@xmath4800 & @xmath115(1.36@xmath90.08)e@xmath11518 & 0.93@xmath90.03 & 76@xmath98 + o h@xmath0o & @xmath125 & 66.437 & 4512.4 & 376 & 1.22e@xmath4800 & @xmath115(8.39@xmath90.14)e@xmath11519 & 0.92@xmath90.02 & 46@xmath93 + ccccccccc co & @xmath126 & 185.9 & 1611.8 & 581 & 2.95e@xmath11504 & @xmath452.28e@xmath11518 & @xmath451.007 + co & @xmath127 & 173.6 & 1726.6 & 663 & 3.64e@xmath11504 & @xmath451.02e@xmath11518 & @xmath451.003 + co & @xmath128 & 162.8 & 1841.3 & 752 & 4.42e@xmath11504 & @xmath451.14e@xmath11518 & @xmath451.003 + co & @xmath129 & 153.3 & 1956.0 & 846 & 5.31e@xmath11504 & @xmath451.49e@xmath11518 & @xmath451.003 +

we conclude , based on the observed co=76 line at 806.65 ghz with the _ caltech submillimeter observatory _ , and the lack of emission from the far ir co lines , that the gas density has to be lower than . using the values obtained for the kinetic temperature and gas density from oh , co , and other molecular species

in the region of sgr b2 there are several condensations heated externally by nearby hot stars . therefore ho far ir lines are expected to probe only an external low density and high temperature section of these condensations , whereas millimeter - wave lines can penetrate deeper into them where the density is higher and t lower . we have conducted a study combining ho lines in both spectral regions . first , observations of several ho thermal lines seen in absorption toward sgr b2(m ) at a spectral resolution of km s have been analyzed . second , an _ iram_30 m telescope map of the para ho line at 183.31 ghz , seen in emission , has also been obtained and analyzed . the ho lines seen in absorption are optically thick and are formed in the outermost gas of the condensations in front of the far ir continuum sources . they probe a maximum visual extinction of to 10 mag . radiative transfer models indicate that these lines are quite insensitive to temperature and gas density , and that ir photons from the dust play a dominant role in the excitation of the involved ho rotational levels . in order to get the physical conditions of the absorbing gas we have also analyzed the co emission toward sgr b2(m ) . we conclude , based on the observed co=76 line at 806.65 ghz with the _ caltech submillimeter observatory _ , and the lack of emission from the far ir co lines , that the gas density has to be lower than . using the values obtained for the kinetic temperature and gas density from oh , co , and other molecular species , we derive a water column density of ( 9) in the absorbing gas . hence , the water vapor abundance in this region ,(ho ) , is(1 - 2) . the relatively low ho / oh abundance ratio in the region , - 4 , is a signature of uv photon dominated surface layers traced by far ir observations . as a consequence the temperature of the absorbing gas is high , t - 500 k , which allows very efficient neutral neutral reactions producing ho and oh . on the other hand , the 183.31 ghz data provide a much better spatial and spectral resolution than the far - ir iso data . this maser line allows to trace water deeper into the cloud , i.e. , the inner , denser ( ) and colder ( t k ) gas . the emission is very strong toward the cores . the derived water vapor abundance for this component is a few . there is also moderate extended emission around sgr b2 main condensations , a fact that supports the water vapor abundance derived from far ir ho lines for the outer gas .

1109.1515

c

much as the fwhm of h@xmath70 , strength of ( relative to h@xmath70 ) , and soft x - ray properties can be used to divide low-@xmath34 quasars into extrema in the `` eigenvector 1 '' context @xcite , so can the properties of the emission line at high redshift . in @xcite , we showed that the extrema of large ( small ) eqw and small ( large ) blueshift can be attributed to different components of the belr : a disk and a wind , respectively . here , we consider the uv through x - ray properties of radio - quiet quasars in the eqw - blueshift parameter space and argue that these extrema are likely due to a hard ionizing spectrum in the `` disk''-dominated systems and a soft ionizing spectrum in the `` wind''-dominated systems ( figures [ fig : aox ] , [ fig : daox ] , and [ fig : sed ] ) . indeed these extrema could instead be considered as hard - spectrum quasars ( hsq ) and soft - spectrum ( ssq ) quasars . more work is needed to understand the nature of quasars with weak that appears at the systemic redshift and the relationship between the optical to x - ray flux ratio ( @xmath6 ) and both the soft and hard x - ray spectral indices . we argue that differences in the properties of quasars could be used to better trace the structure of the sed in the two decades of frequency space between uv and x - ray measurements , allowing for more accurate bolometric corrections for individual quasars ( figures [ fig : sed ] and [ fig : bb ] ) . the emission line may further allow a quantification the differences in the `` unseen '' part of the ionizing sed ( e.g. , an excess in the helium continuum above the nominal assumed from @xmath6 ) and may help identify which objects are likely to show a soft x - ray excess feature . support for this work was provided by the national aeronautics and space administration through chandra award number g08 - 9103x issued by the chandra x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of the national aeronautics space administration under contract nas8 - 03060 . gtr acknowledges support from an alfred p. sloan research fellowship and nasa grant 07-adp07 - 0035 . scg thanks the national science and engineering research council of canada and an ontario early research award for support . kml acknowledges support by nsf ast-0707703 . 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 is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , cambridge university , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington . ccccccccc chandra c9 & 10 & 0 & 1 & 0 & 0 & 6 & 1.651.90 & 31.4831.68 + gallagher 05 & 14 & 1 & 2 & 0 & 3 & 10 & 1.692.17 & 31.0031.73 + strateva 05 & 155 & 0 & 10 & 94 & 89 & 23 & 1.552.24 & 30.3531.54 + kelly 07 & 157 & 1 & 9 & 95 & 112 & 6 & 1.572.03 & 30.0730.93 + just 07 & 32 & 3 & 10 & 26 & 21 & 5 & 1.912.27 & 32.0532.23 + gibson 08 & 536 & 52 & 71 & 100 & 210 & 181 & 1.702.31 & 30.3432.11 + green 09 & 281 & 11 & 29 & 146 & 182 & 46 & 1.542.26 & 30.4931.79 + young 09 & 792 & 33 & 78 & 423 & 604 & 91 & 1.552.27 & 30.1031.75 + lcccccrl 005102.42@xmath49010244.4 & 1.889 & 17.366 & 31.571 & 1597 & 20.3 & 3.5 & 9224 + 014812.23 + 000153.2 & 1.708 & 17.386 & 31.411 & 1590 & 15.2 & 10.5 & 9225 + 020845.53 + 002236.0 & 1.898 & 16.723 & 31.854 & 1555 & 23.1 & 3.5 & 9223 + 090007.14 + 321921.9 & 1.851 & 17.095 & 31.650 & 1955 & 21.6 & 11.0 & 9323 + 100401.28 + 423123.0 & 1.666 & 16.733 & 31.725 & 2281 & 16.8 & 8.1 & 9322 + 102907.06 + 651024.6 & 2.171 & 16.730 & 31.968 & 1560 & 17.2 & 10.6 & 9228 + 115351.11 + 113649.2 & 1.681 & 17.262 & 31.463 & 2026 & 22.9 & 11.0 & 9230 + 141949.39 + 060654.0 & 1.649 & 17.119 & 31.385 & 1941 & 23.5 & 9.9 & 9226 + 150313.63 + 575151.6 & 1.721 & 17.075 & 31.593 & 1555 & 15.9 & 10.0 & 9227 + 162622.06 + 295237.4 & 1.902 & 17.017 & 31.700 & 1501 & 20.6 & 10.9 & 9229 005102.42@xmath49010244.4 & 1.889 & 17.366 & 31.571 & 1597.2 & 20.31 & 3.5 & 9224 + 014812.23 + 000153.2 & 1.708 & 17.386 & 31.411 & 1589.7 & 15.16 & 10.5 & 9225 + 020845.53 + 002236.0 & 1.898 & 16.723 & 31.854 & 1554.9 & 23.13 & 3.5 & 9223 + 090007.14 + 321921.9 & 1.851 & 17.095 & 31.650 & 1954.5 & 21.64 & 11.0 & 9323 + 100401.28 + 423123.0 & 1.666 & 16.733 & 31.725 & 2280.5 & 16.83 & 8.1 & 9322 + 102907.06 + 651024.6 & 2.171 & 16.730 & 31.968 & 1560.0 & 17.21 & 10.6 & 9228 + 115351.11 + 113649.2 & 1.681 & 17.262 & 31.463 & 2026.1 & 22.87 & 11.0 & 9230 + 141949.39 + 060654.0 & 1.649 & 17.119 & 31.385 & 1941.2 & 23.54 & 9.9 & 9226 + 150313.63 + 575151.6 & 1.721 & 17.075 & 31.593 & 1554.8 & 15.91 & 10.0 & 9227 + 162622.06 + 295237.4 & 1.902 & 17.017 & 31.700 & 1501.3 & 20.61 & 10.9 & 9229 lcccccccr 005102.42@xmath49010244.4 & 2.75@xmath110 & @xmath111 & @xmath112 & @xmath113 & 31.510 & @xmath114 & @xmath115 + 014812.23 + 000153.2 & 2.13@xmath116 & @xmath117 & @xmath118 & @xmath119 & 31.434 & @xmath120 & @xmath121 + 020845.53 + 002236.0 & 1.62@xmath122 & @xmath123 & @xmath124 & @xmath125 & 31.768 & @xmath126 & @xmath127 + 090007.14 + 321921.9 & 2.20@xmath128 & @xmath129 & @xmath130 & @xmath131 & 31.635 & @xmath132 & @xmath133 + 100401.28 + 423123.0 & 1.65@xmath134 & @xmath135 & @xmath136 & @xmath137 & 31.677 & @xmath138 & @xmath139 + 102907.06 + 651024.6 & 2.10@xmath140 & @xmath141 & @xmath142 & @xmath143 & 31.971 & @xmath144 & @xmath127 + 115351.11 + 113649.2 & 2.10@xmath140 & @xmath145 & @xmath146 & @xmath147 & 31.484 & @xmath148 & @xmath149 + 141949.39 + 060654.0 & 2.65@xmath150 & @xmath151 & @xmath152 & @xmath153 & 31.533 & @xmath154 & @xmath155 + 150313.63 + 575151.6 & 2.30@xmath156 & @xmath157 & @xmath158 & @xmath159 & 31.574 & @xmath160 & @xmath161 + 162622.06 + 295237.4 & 2.60@xmath162 & @xmath163 & @xmath164 & @xmath165 & 31.679 & @xmath120 & @xmath166 .... # obsid , ra , dec , soft , soft_err , hard , hard_err , tot , tot_err , rate_soft , rate_soft_err , rate_hard , rate_hard_err , rate_tot , rate_tot_err , hr_obs 9223 , 32.189583 , 0.376667 , 48.99 , 7.07 , 17.65 , 4.24 , 66.47 , 8.25 , 1.4061e-02 , 2.0296e-03 , 5.0658e-03 , 1.2178e-03 , 1.9079e-02 , 2.3670e-03 , -0.47 9224 , 12.760000 , -1.045667 , 26.38 , 5.20 , 0.00 , 0.00 , 29.25 , 5.48 , 7.3846e-03 , 1.4544e-03 , 0.0000e+00 , 0.0000e+00 , 8.1877e-03 , 1.5331e-03 , -- 9225 , 27.050833 , 0.031444 , 64.36 , 8.12 , 6.70 , 2.65 , 77.71 , 8.89 , 6.1131e-03 , 7.7161e-04 , 6.3632e-04 , 2.5132e-04 , 7.3813e-03 , 8.4420e-04 , -0.81 9226 , 214.955833 , 6.115000 , 93.50 , 9.80 , 10.86 , 3.32 , 106.24 , 10.34 , 9.4342e-03 , 9.8862e-04 , 1.0953e-03 , 3.3466e-04 , 1.0719e-02 , 1.0437e-03 , -0.79 9227 , 225.806667 , 57.864333 , 114.30 , 10.86 , 18.41 , 4.36 , 136.27 , 11.75 , 1.1404e-02 , 1.0838e-03 , 1.8371e-03 , 4.3489e-04 , 1.3596e-02 , 1.1721e-03 , -0.72 9228 , 157.279583 , 65.173500 , 111.99 , 10.77 , 25.43 , 5.10 , 140.03 , 11.92 , 1.0539e-02 , 1.0136e-03 , 2.3927e-03 , 4.7983e-04 , 1.3177e-02 , 1.1214e-03 , -0.63 9229 , 246.592083 , 29.877056 , 78.08 , 8.94 , 9.79 , 3.16 , 89.98 , 9.54 , 6.8972e-03 , 7.9014e-04 , 8.6453e-04 , 2.7935e-04 , 7.9489e-03 , 8.4270e-04 , -0.78 9230 , 178.462917 , 11.613667 , 112.09 , 10.68 , 24.62 , 5.00 , 137.65 , 11.79 , 1.0166e-02 , 9.6836e-04 , 2.2329e-03 , 4.5345e-04 , 1.2484e-02 , 1.0693e-03 , -0.64 9322 , 151.005417 , 42.523056 , 46.77 , 6.93 , 14.53 , 3.87 , 63.41 , 8.06 , 6.1342e-03 , 9.0868e-04 , 1.9058e-03 , 5.0804e-04 , 8.3172e-03 , 1.0574e-03 , -0.53 9323 , 135.029724 , 32.322716 , 90.38 , 9.59 , 17.54 , 4.24 , 108.60 , 10.49 , 8.3068e-03 , 8.8161e-04 , 1.6117e-03 , 3.8992e-04 , 9.9817e-03 , 9.6399e-04 , -0.67 ....

we explore the connection between these emission - line properties and the uv through x - ray spectral energy distribution ( sed ) for radio - quiet ( rq ) quasars . we find that rq quasars with both strong emission and small blueshifts can be classified as `` hard - spectrum '' sources that are ( relatively ) strong in the x - ray as compared to the uv . on the other hand , rq quasars with both weak emission and large blueshifts are instead `` soft - spectrum '' sources that are ( relatively ) weak in the x - ray as compared to the uv . detailed analysis of the emission line may allow for sed - dependent corrections to these quantities .

in the restframe ultra - violet ( uv ) , two of the parameters that best characterize the range of emission - line properties in quasar broad emission - line regions are the equivalent width and the blueshift of the line relative to the quasar rest frame . we explore the connection between these emission - line properties and the uv through x - ray spectral energy distribution ( sed ) for radio - quiet ( rq ) quasars . our sample consists of a heterogeneous compilation of 406 quasars from the sloan digital sky survey ( at ) and palomar - green survey ( at ) that have well - measured emission - line and x - ray properties ( including 164 objects with measured ) . we find that rq quasars with both strong emission and small blueshifts can be classified as `` hard - spectrum '' sources that are ( relatively ) strong in the x - ray as compared to the uv . on the other hand , rq quasars with both weak emission and large blueshifts are instead `` soft - spectrum '' sources that are ( relatively ) weak in the x - ray as compared to the uv . this work helps to further bridge optical / soft x - ray `` eigenvector 1 '' relationships to the uv and hard x - ray . based on these findings , we argue that future work should consider systematic errors in bolometric corrections ( and thus accretion rates ) that are derived from a single mean sed . detailed analysis of the emission line may allow for sed - dependent corrections to these quantities .

1302.5386

i

let us consider a bounded domain @xmath0 in @xmath1 containing the closed unit ball @xmath2 ( see figure 1 ) . let @xmath3 $ ] be a hlder continuous function which is periodic with respect to the orthonomal basis @xmath4 of @xmath1 . more precisely , @xmath5 satisfies @xmath6 with @xmath7 and @xmath5 as given above , we are interested in the limiting behavior of the following problem : @xmath8 here @xmath9 is the outward normal vector at @xmath10 and @xmath11 is a uniformly elliptic , fully nonlinear operator ( see section 2 for the precise definitions and conditions on @xmath11 ) . the set @xmath12 is introduced to avoid discussions of compatibility conditions on the neumann boundary data . the operators under discussion include , for example , the divergence or non - divergence form operator @xmath13 where @xmath14 are hlder continuous and @xmath15 for positive constants @xmath16 and @xmath17 . let us define @xmath18 due to @xcite , there exists a uniformly elliptic operator @xmath19 such that any solution of @xmath20 in @xmath0 with _ fixed _ boundary data on @xmath21 converges uniformly to the solution of @xmath22 in @xmath0 ( see theorem [ original_convergence ] ) . therefore the question under investigation is whether the oscillatory boundary data changes the averaging behavior of @xmath23 . as we will discuss below , several difficulties arise in answering this question due to the nonlinear ( or non - divergence form ) nature of the problem as well as the geometry of the domain . let us state our main result . first we introduce the following cell problem " for given @xmath24 and @xmath25 . @xmath26 [ main ] let @xmath27 and , @xmath28 and @xmath29 be as given above . * for given normal direction @xmath30 and any @xmath25 there exists a unique homogenized neumann data @xmath31 for solutions of @xmath32 . * suppose that @xmath33 only depends on the eigenvalues of @xmath34 . then there exists a continuous function @xmath35 , given as the continuous extension of @xmath31 over @xmath36 , such that the following holds : + suppose @xmath0 is a bounded domain in @xmath1 such that @xmath21 is @xmath37 and does not contain any flat part . let @xmath23 solve @xmath28 . then @xmath23 converges locally uniformly to @xmath38 , which is the unique solution of the homogenized problem @xmath39 * the assumption on @xmath40 in theorem [ main ] ( b ) seems to be necessary condition to achieve the continuity of the homogenized slope @xmath41 , at least from numerical experiments ( see @xcite ) . note that this assumption is equivalent to the rotation and reflection invariance of @xmath40 . for more discussion on the role of the assumption in the analysis , see the discussion in section 4 . + * we point out that the restriction on the geometry of @xmath21 is necessary : in fact , if @xmath21 is locally @xmath42 with @xmath43 and @xmath44 , then the ( normalized ) distribution of boundary data @xmath45 on @xmath46 changes a lot as @xmath47 . consequently in this case there is no unique limit of @xmath23 . @xmath48_discussions on previous results _ our problem is classical for the case of uniformly elliptic , divergence - form equations @xmath49 with the _ co - normal _ boundary data @xmath50 for - , a corresponding result to theorem [ main ] was proved in the classical paper of bensoussan , lions and papanicolau @xcite with explicit integral formula for the limiting operator as well as the limiting boundary data . recently , a corresponding result was shown for systems of divergence type operators with oscillatory dirichlet boundary data in convex domains , by gerard - varet and masmoudi @xcite ( also see @xcite ) . for nonlinear or non - divergence type operators , or even for linear operators with oscillatory neumann boundary data that is not co - normal , most available results concern half - space type domains whose boundary goes through the origin . in @xcite , tanaka considered some model problems in half - space whose boundary is parallel to the axes of the periodicity by purely probabilistic methods . in @xcite arisawa studied special cases of problems in oscillatory domains near half spaces going through the origin , using viscosity solutions as well as stochastic control theory . generalizing the results of @xcite , barles , da lio and souganidis @xcite studied the problem for operators with oscillating coefficients , in half - space type domains whose boundary is parallel to the axes of periodicity , with a series of assumptions which guarantee the existence of approximate corrector . in @xcite the continuity property of the averaged neumann boundary data in half - spaces , with respect to the normal direction , was studied in the case of homogeneous operator @xmath11 . recently , barles and mironescu @xcite showed a corresponding result to @xcite for oscillatory dirichlet boundary data . @xmath48 _ main ingredients and challenges _ the main steps in extending aforementioned results from half - space type domains to general domains are the following . the plan is to use these solutions in strip domains to approximate those in general domains . for the stability of such approximation one requires the distribution of @xmath5 to be invariant on @xmath51 regardless of the choice of @xmath16 , which is the case if @xmath52 is not a multiple of vectors in @xmath53 : we call such vectors _ irrational_. for irrational @xmath52 we show that there is a unique linear profile the solutions converges to , by proving an oscillation lemma as well as using a quantitative version of weyl s equi - distribution theorem . secondly , to establish sufficient stability to address the general domains , some estimate on the convergence mode of @xmath54- solutions in the strip domain , in terms of the variation of @xmath52 , is necessary . establishing this estimate is our second , and most challenging , main step ( theorem [ continuity ] ) . for problems with divergence - form structure , such estimates were obtained in @xcite and @xcite ( also see @xcite ) by means of integral formulas . in our setting we must proceed by maximum principle - type arguments , which requires careful perturbation of the boundary data as well as construction of delicate barriers which describes the averaging behavior of solutions up to the neumann boundary . the proof of theorem 4.1 is based on the observation that , in meso - scopic " scale , the distribution of @xmath5 on two hyperplanes with close irrational normals are similar ( see section 4.1 . for a heuristic discussion of this fact ) . we adopt a multi - scale homogenization argument to address separately the effects on the solution caused by ( a ) microscopic oscillation of @xmath5 near the neumann boundary ( b ) the difference in normal directions and ( c ) the oscillation present in the operator @xmath11 . we mention that a significant difficulty arises due to the presence of the @xmath55-dependence in @xmath11 in addition to the oscillations in @xmath5 . to get around this difficulty , we use localization arguments as well as the existing homogenization results ( e.g. @xcite and @xcite ) to show homogenization occurs away from the neumann boundary . such strategy works since near the neumann boundary the oscillation of the first derivative ( @xmath5 ) dominates the behavior of solutions . @xmath48 _ outline of the paper _ in section 2 we introduce some notations as well as preliminary results which will be used in the rest of the paper . in section 3 we first study the averaging properties of the operator in the strip ( half - space type ) domains , to show that if the hypersurface is normal to an irrational direction then there is a unique homogenized neumann data in the limit @xmath47 . besides the complications that the inhomogeities in the operator @xmath11 cause , the proof of averaging phenomena in this setting is due to the weyl s distribution theorem , whose quantitative version that we need is borrowed from @xcite and is stated in theorem [ lemma - m ] . in section 4 we prove the main estimate on the mode of convergence of @xmath54-solutions in the strip domain as the normal @xmath52 varies around a reference direction . after a heuristic description of the proof , we prove a series of lemmas which describe the behavior of solutions in different regions of the strip domain , divided in terms of the distance to the neumann boundary . based on the lemmas then we are able to construct suitable barriers to obtain the desired estimate ( theorem 4.1 ) . lastly in section 5 we prove theorem [ main ] using the aforementioned estimate , the non - flat geometry of @xmath21 , and the stability of viscosity solutions .

in this article we investigate averaging properties of fully nonlinear pdes in bounded domains with oscillatory neumann boundary data . the oscillation is periodic and is present both in the operator and in the neumann data . our main result states that , when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant , the solutions uniformly converge to the homogenized solution solving a neumann boundary problem . furthermore we show that the homogenized neumann data is continuous with respect to the normal direction of the boundary . our result is the nonlinear version of the classical result in for divergence - form operators with co - normal boundary data . the main ingredients in our analysis are the estimate on the oscillation on the solutions in half - spaces ( theorem [ thm : planar ] ) , and the estimate on the mode of convergence of the solutions as the normal of the half - space varies over _ irrational _ directions ( theorem [ continuity ] ) .

in this article we investigate averaging properties of fully nonlinear pdes in bounded domains with oscillatory neumann boundary data . the oscillation is periodic and is present both in the operator and in the neumann data . our main result states that , when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant , the solutions uniformly converge to the homogenized solution solving a neumann boundary problem . furthermore we show that the homogenized neumann data is continuous with respect to the normal direction of the boundary . our result is the nonlinear version of the classical result in for divergence - form operators with co - normal boundary data . the main ingredients in our analysis are the estimate on the oscillation on the solutions in half - spaces ( theorem [ thm : planar ] ) , and the estimate on the mode of convergence of the solutions as the normal of the half - space varies over _ irrational _ directions ( theorem [ continuity ] ) .

1112.1978

c

in this article we have provided a formal framework for reidentification in general . we have defined @xmath0-confusion as a concept for modelling the anonymity of a database table and we have proved that @xmath0-confusion is a generalization of @xmath1-anonymity . then after a short survey on the different available definitions of @xmath1-anonymity for graphs we provided a new definition for @xmath1-anonymity , which we consider to be the correct definition . it has been explained how this definition can be used in combination with @xmath0-confusion , for the anonymization of data from , for example , social networks . we have provided a description of the @xmath1-anonymous graphs , both for the regular and the non - regular case . however , under some conditions our definition of @xmath1-anonymity is quite strict , so that it is only satisfied by a small number of graphs . in order to avoid this problem , we have introduced the more flexible definition of @xmath2-anonymity . our definition of @xmath2-anonymity for graph is meant to replace the definition in @xcite , which we have proved to have severe weaknesses . we have given two variants of the definition of @xmath2-anonymity , which may serve under different conditions . we have also provided a set of algorithms ; one algorithm that given a graph @xmath101 and a natural number @xmath1 returns a graph @xmath253 based on @xmath101 that is @xmath1-anonymous , two algorithms that given a graph @xmath101 and a natural number @xmath1 calculates the largest @xmath4 such that @xmath101 is @xmath2-anonymous according to our two different definitions of @xmath2-anonymity , and finally , one algorithm that given a graph @xmath101 that satisfies @xmath2-anonymity returns a graph @xmath253 similar to @xmath101 that satisfies @xmath261-anonymity .

in this article we provide a formal framework for reidentification in general . we define-confusion as a concept for modelling the anonymity of a database table and we prove that-confusion is a generalization of-anonymity . after a short survey on the different available definitions of-anonymity for graphs we provide a new definition for-anonymous graph , which we consider to be the correct definition . we provide a description of the-anonymous graphs , both for the regular and the non - regular case . we also introduce the more flexible concept of-anonymous graph . our definition of-anonymous graph is meant to replace a previous definition of-anonymous graph , which we here prove to have severe weaknesses . finally we provide a set of algorithms for-anonymization of graphs .

in this article we provide a formal framework for reidentification in general . we define-confusion as a concept for modelling the anonymity of a database table and we prove that-confusion is a generalization of-anonymity . after a short survey on the different available definitions of-anonymity for graphs we provide a new definition for-anonymous graph , which we consider to be the correct definition . we provide a description of the-anonymous graphs , both for the regular and the non - regular case . we also introduce the more flexible concept of-anonymous graph . our definition of-anonymous graph is meant to replace a previous definition of-anonymous graph , which we here prove to have severe weaknesses . finally we provide a set of algorithms for-anonymization of graphs .

1610.06790

i

the strong numerical approximation of an it stochastic partial differential equation defined in the bounded domain @xmath1 is analyzed . the domain @xmath2 is assumed to be a convex polygon , or has smooth boundary . boundary conditions on the domain @xmath2 are typically neumann , dirichlet or robin conditions . more precisely , we consider in the abstract setting the following stochastic partial differential equation @xmath3dt+b(x(t))dw(t ) , \quad x(0)=x_0 , \quad t\in[0,t],\end{aligned}\ ] ] on @xmath4 , @xmath5 is a final time , @xmath6 and @xmath7 are nonlinear functions , @xmath8 is the initial data which may be random , @xmath9 is a linear operator , unbounded , not necessarily self adjoint , and the generator of an analytic semigroup @xmath10 the noise @xmath11 is a @xmath12 wiener process defined in a filtered probability space @xmath13 . the filtration is assumed to fulfill the usual conditions ( see ( * ? ? ? * definition 2.1.11 ) ) . we assume that the noise can be represented as @xmath14,\end{aligned}\ ] ] where @xmath15 , @xmath16 , @xmath17 are respectively the eigenvalues and the eigenfunctions of the covariance operator @xmath18 , and @xmath19 are independent and identically distributed standard brownian motion . precise assumptions on @xmath6 , @xmath7 , @xmath8 and @xmath9 will be given in the next section to ensure the existence of the unique mild solution @xmath20 of ( [ model ] ) which has the following representation ( see @xcite ) @xmath21.\end{aligned}\ ] ] in few cases , exact solutions are explicitly available , so numerical techniques are the only tools to provide good approximations in more general cases ( see for examples @xcite ) . approximations are done at two levels , spatial approximation and temporal approximation . for the spatial approximation , the finite difference , the finite element method and the galerkin spectral method are usually used @xcite . as for pdes , standard explicit time stepping methods for spdes are usually unstable for stiff problems and therefore severe time step constraint is needed . to overcome that drawback , standard implicit euler methods are usually used @xcite . although standard implicit euler methods are stable , their implementation requires significantly more computational effort , specially full implicit methods , as newton method is usually used to solve nonlinear algebraic equations at each time step . recently , stochastic exponential integrators @xcite were appeared as non standard explicit methods efficient for spde ( [ model ] ) . all stochastic exponential integrators analyzed in the literature for spdes @xcite are bounded on the nonlinear problem as in ( [ model ] ) where the linear part @xmath9 and the nonlinear function @xmath6 are explicitly known a priori . such approach is justified in situations where the nonlinear function @xmath6 is small . indeed when @xmath6 is small the linear operator @xmath9 drives the spde ( [ model ] ) and the good stability of the exponential integrators and semi - implicit method are ensured . in fact , in more realistic applications the nonlinear function @xmath6 can be stronger . typical examples are semilinear advection diffusion reaction equations with stiff reaction term . in such cases , the spde ( [ model ] ) is driven by the nonlinear operator @xmath6 and both exponential integrators @xcite and semi - implicit euler @xcite will behave as explicit euler - maruyama scheme , therefore their good stability properties are lost . to overcome this issue we propose in this work a new scheme called stochastic exponential rosenbrock scheme ( sers ) . coupled with finite element for space discretization , the new scheme is based on a local linearization of the drift term at each time step in the corresponding semi - discrete problem of ( [ model ] ) . the local linearization therefore weaken the nonlinear part of the drift such that the linearized semi - discrete problem is driven by its linear part , which change at each time step . the standard stochastic exponential scheme @xcite is applied at the end to that linearized semi - discrete problem and the corresponding scheme is our new scheme . the challenge here is to deal with the new discrete semigroup which indeed is a semigroup process , called stochastic perturbed semigroup . the key ideal comes from the deterministic exponential rosenbrock method @xcite . note that similar schemes for stochastic differential equation in finite dimensional have been proposed in @xcite . using some deterministic tools from @xcite , we propose a strong convergence proof of the new schemes where the linear operator @xmath9 is not necessarily self adjoint . note that the orders of convergence are the same with stochastic exponential schemes proposed in@xcite . the deterministic part of this scheme is order @xmath22 in time and has been proven to be efficient and robust in comparison to standard schemes in many applications @xcite where the perturbed semigroup and related matrix functions have been computed using the krylov subspace technique @xcite and fast leja points technique @xcite . for new our stochastic scheme , numerical simulations show the good stability behavior of the new scheme compared with a stochastic exponential scheme proposed in @xcite , where the stochastic perturbed semigroup and related matrix functions are computed using krylov subspace technique . the rest of this paper is organized as follows . section [ wellposed ] is devoted to the mathematical setting , the numerical method and the main result . in section [ convergenceproof ] some preparatory results and the proof of the main result are provided . we end the paper in section [ experiment ] with some numerical experiments to sustain our theoretical results .

in this paper , we consider the numerical approximation of a general second order semilinear stochastic partial differential equation ( spde ) driven by multiplicative and additive noise . our main interest is on such spdes where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations . most numerical techniques , including current stochastic exponential integrators lose their good stability properties on such equations . using finite element for space discretization , we propose a new scheme appropriated on such equations , called stochastic exponential rosenbrock scheme ( sers ) based on local linearization at every time step of the semi - discrete equation obtained after space discretization . numerical experiments to sustain theoretical results are provided .

in this paper , we consider the numerical approximation of a general second order semilinear stochastic partial differential equation ( spde ) driven by multiplicative and additive noise . our main interest is on such spdes where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations . most numerical techniques , including current stochastic exponential integrators lose their good stability properties on such equations . using finite element for space discretization , we propose a new scheme appropriated on such equations , called stochastic exponential rosenbrock scheme ( sers ) based on local linearization at every time step of the semi - discrete equation obtained after space discretization . we consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root - mean - square norm . numerical experiments to sustain theoretical results are provided . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

1610.06790

r

before we state the well posedness result , let us define keys functional spaces , norms and notations that we will be use in the rest of the paper . let @xmath23 be a separable hilbert space . for all @xmath24 and for a banach space @xmath25 , we denote by @xmath26 the banach space of all equivalence classes of @xmath27 integrable @xmath25-valued random variables . we denote by @xmath28 the space of bounded linear mappings from @xmath25 to @xmath29 endowed with the usual operator norm @xmath30 . by @xmath31 , we denote the space of hilbert schmidt operators from @xmath25 to @xmath29 . for simplicity we use the notations @xmath32 and @xmath33 . we assume that the covariance operator @xmath34 is positive and self - adjoint . throughout this paper @xmath35 is a @xmath18-wiener process . the space of hilbert schmidt operator from @xmath36 to @xmath29 is denoted by @xmath37 with the corresponding norm @xmath38 defined by @xmath39 where @xmath40 is an orthonormal basis of @xmath29 . this definition is independent of the orthonormal basis of @xmath29 . + in order to ensure the existence and the uniqueness of solution of ( [ model ] ) we make the following assumptions . * [ linear operator @xmath9 ] * [ assumption1 ] @xmath42 is a negative generator of an analytic semigroup @xmath43 . * [ initial value @xmath8 ] * [ assumption2 ] we assume that @xmath45 , @xmath46 . as in the current literature on deterministic exponential rosenbrock - type methods @xcite , we make the following assumption on the nonlinear term . * [ nonlinear term @xmath6 ] * [ assumption3 ] we assume that the nonlinear mapping @xmath48 is lipschitz continuous and frchet derivable with its derivative uniformly bounded , i.e. there exists a constant @xmath49 such that @xmath50 as a consequence , there exists a constant @xmath51 such that @xmath52 for all @xmath53 . following ( * ? ? ? * chapter 7 ) or @xcite we make the following assumption on the diffusion term . * [ diffusion term ] * [ assumption4 ] we assume that the operator @xmath54 satisfies the global lipschitz condition , i.e. there exists a positive constant @xmath55 such that @xmath56 as a consequence , there exists a positive constant @xmath51 such that @xmath57 for all @xmath53 . to establish our @xmath0 strong convergence result , we will also need the following further assumption on the diffusion term when @xmath58 , which was also used in @xcite . [ assumption5 ] we assume that there exist two positive constants @xmath59 , and @xmath60 $ ] very small enough such that @xmath61 and @xmath62 for all @xmath63 , where @xmath64 is the parameter defined in assumption [ assumption2 ] . we equip @xmath65 , @xmath66 with the norm @xmath67 , for all @xmath68 . it is well known that @xmath69 is a banach space @xcite . let us recall in the following proposition some semigroup properties of the operator @xmath70 generated by @xmath9 that will be useful in the rest of the paper . * * [ smoothing properties of the semigroup]**@xcite [ prop1 ] let @xmath71 , @xmath72 and @xmath73 , then there exists a constant @xmath49 such that @xmath74 where @xmath75 , and @xmath76 . + if @xmath77 then @xmath78 . * * [ well posedness result ] * * ( * ? ? ? * theorem 7.4 ) + [ theorem1 ] let assumption [ assumption1 ] , assumption [ assumption3 ] and assumption [ assumption4 ] be satisfied . if @xmath8 is a @xmath79- measurable @xmath29 valued random variable , then there exists a unique mild solution @xmath20 of problem ( [ model ] ) taking the form ( [ mild1 ] ) and satisfying the following @xmath80=1,\end{aligned}\ ] ] and for any @xmath24 there exists a constant @xmath81 such that @xmath82}\mathbb{e}\vert x(t)\vert^p\leq c(1+\mathbb{e}\vert x_0\vert^p).\end{aligned}\ ] ] furthermore from ( * ? ? ? * theorem 1 ) or ( * ? ? ? * theorem 2.6 ) it holds that for all @xmath83 , for all @xmath24 there exists a positive constant @xmath55 such that for all @xmath84 $ ] we have @xmath85 in the rest of this paper , to simplify the presentation , we assume that the linear operator @xmath9 a second order . more precisely , we assume that our spde ( [ model ] ) is a second order semilinear parabolic and take the form @xmath86dt\nonumber\\ & + & b(x , x(t , x))dw(t , x ) , \quad x\in\lambda , \quad t\in[0,t],\end{aligned}\ ] ] where the functions @xmath87 and @xmath88 are continuously differentiable with globally bounded derivatives . in the abstract framework ( [ model ] ) , the linear operator @xmath9 takes the form @xmath89 where @xmath90 , @xmath91 . we assume that there is a positive constant @xmath92 such that @xmath93 the functions @xmath94 and @xmath95 are defined by @xmath96 for all @xmath97 , @xmath68 , @xmath98 , with @xmath4 . for an appropriate eigenfunctions @xmath99 such that @xmath100<\infty$ ] , it is well known ( * ? ? ? * section 4 ) that the nemystskii operator @xmath6 related to @xmath101 and the multiplication operator @xmath7 associated to @xmath102 defined in ( [ nemystskii ] ) satisfy assumption [ assumption3 ] and assumption [ assumption4 ] respectively . as in @xcite , we introduce two spaces @xmath103 and @xmath104 , such that @xmath105 ; the two spaces depend on the boundary conditions of @xmath106 and the domain of the operator @xmath9 . for dirichlet ( or first - type ) boundary conditions we take @xmath107 for robin ( third - type ) boundary condition and neumann ( second - type ) boundary condition , which is a special case of robin boundary condition , we take @xmath108 @xmath109 where @xmath110 is the normal derivative of @xmath111 and @xmath112 is the exterior pointing normal @xmath113 to the boundary of @xmath9 given by @xmath114 using the green s formula and the boundary conditions , the corresponding bilinear form associated to @xmath115 is given by @xmath116 for dirichlet and neumann boundary conditions , and @xmath117 for robin boundary conditions . using the garding s inequality ( @xcite ) we obtain @xmath118 by adding and substracting @xmath119 on the right hand side of ( [ model ] ) , we have a new linear operator that we atill call a corresponding to the new bilinear form that we still call @xmath120 such that the following coercivity property holds @xmath121 note that the expression of the nonlinear term @xmath6 has changed as we included the term @xmath122 in a new nonlinear term that we still denote by @xmath6 . the coercivity property ( [ ellip2 ] ) implies that @xmath9 is sectorial on @xmath123 i.e. there exists @xmath124 such that @xmath125 where @xmath126 ( see @xcite ) . then @xmath9 is the infinitesimal generator of a bounded analytic semigroup @xmath127 on @xmath123 such that @xmath128 where @xmath129 denotes a path that surrounds the spectrum of @xmath130 . the coercivity property ( [ ellip2 ] ) also implies that @xmath115 is a positive operator and its fractional powers are well defined for any @xmath131 by @xmath132 where @xmath133 is the gamma function ( see @xcite ) . let s now turn to the discretization of our problem ( [ model ] ) . we start by spltting the domain @xmath106 in finite triangles . let @xmath134 be the triangulation with maximal length @xmath135 satisfying the usual regularity assumptions , and @xmath136 the space of continuous functions that are piecewise linear over the triangulation @xmath134 . we consider the projection @xmath137 from @xmath4 to @xmath138 defined for every @xmath139 by @xmath140 the discrete operator @xmath141 is defined by @xmath142 like @xmath9 , @xmath143 is also a generator of a semigroup @xmath144 . as any semigroup and its generator , @xmath143 and @xmath145 satisfy the smoothing properties of proposition [ theorem1 ] but with a uniform constant @xmath55 , independent of @xmath135 . following @xcite , we characterize the domain of the operator @xmath146 as follow @xmath147 the semi - discrete version of problem ( [ model ] ) consists to find @xmath148 , @xmath149 $ ] such that @xmath150 and @xmath151dt+p_hb(x^h(t))dw(t ) , \quad t\in(0,t].\end{aligned}\ ] ] we note that @xmath143 and @xmath152 satisfy the same assumptions as @xmath9 and @xmath6 respectively . we also note that @xmath153 satisfies assumption [ assumption4 ] . therefore , theorem [ theorem1 ] ensures the existence of the unique mild solution @xmath154 of ( [ semi1 ] ) such that @xmath155.\end{aligned}\ ] ] this mild solution of ( [ semi1 ] ) is given by @xmath156 the following lemma will be useful in our convergence analysis . [ lemma1 ] the following inequality holds @xmath157 from the equivalence of norms ( see ( * ? ? ? * ( 3.12 ) ) ) we have @xmath158 note that @xmath159 where @xmath160 stands for the weak derivative . let @xmath161 be the set of functions @xmath162 with compact support in @xmath106 . let @xmath163 , for all @xmath164 , we have @xmath165 where @xmath166 is a duality pairing between @xmath167 and @xmath161 , and @xmath168 is the derivative of @xmath169 in the classical sense . from ( * remark 2.1 ) we have @xmath170 since @xmath171 is a linear operator . so , the equality ( [ weak2 ] ) yields @xmath172 since ( [ weak3 ] ) holds for all @xmath164 , it follows that @xmath173 in the weak sense . inserting this latter relation in ( [ weak1 ] ) , using the fact that the projection @xmath137 is bounded with respect to the norm @xmath174 and again the equivalence of norm ( * * ( 3.12 ) ) yields @xmath175 we therefore have @xmath176 note that ( [ inter1 ] ) remains true if we replace @xmath177 by @xmath178 . by interpolation theory we have @xmath179 let us recall the following well known lemma . * * [ it isometry ] * * ( * ? ? ? * proposition 2.3.5 ) + for any @xmath180 $ ] and for any @xmath181-valued predictable process @xmath182 , @xmath183 $ ] the following equality holds @xmath184= \mathbb{e}\left[\int_0^t\vert\phi(s ) \vert^2_{l^0_2}ds\right].\end{aligned}\ ] ] the following two lemmas provide space and time regularity results of the mild solution of the semi - discrete problem ( [ semi1 ] ) . these lemmas play an important role in our convergence analysis . more results on the regularity of the mild solution of problem ( [ model ] ) can be found in @xcite . * [ space regularity of the mild solution @xmath154 ] * + [ lemma3 ] let assumption [ assumption1 ] , assumption [ assumption2 ] , assumption [ assumption3 ] and assumption [ assumption4 ] be fulfilled with @xmath186 , then for all @xmath187 $ ] , @xmath188 . moreover , there exists a positive constant @xmath55 independent of @xmath135 such that @xmath189 the proof follows the sames lines as that of ( * ? ? ? * lemma 2.6 ) or ( * ? ? ? * theorem 1 ) or ( * ? ? * theorem 3.1 ) by making use of lemma [ lemma1 ] . * [ time regularity of the mild solution @xmath154 ] * [ lemma4 ] let @xmath190 be the mild solution of ( [ semi1 ] ) . if assumption [ assumption1 ] , assumption [ assumption2 ] , assumption [ assumption3 ] and assumption [ assumption4 ] are fulfilled with the corresponding @xmath191 . for @xmath192 there exists a positive constant @xmath55 independent of @xmath135 such that for @xmath193 $ ] , @xmath194 , we have @xmath195 moreover , if assumption [ assumption5 ] is fulfilled with @xmath196 , then there exists a positive constant @xmath55 such that @xmath197 the proof follows the same lines as that of ( * ? ? ? * lemma 2.7 ) or ( * ? ? ? * theorem 1 ) or ( * ? ? ? * theorem 4.1 ) by making use of lemma [ lemma1 ] . for the time discretization , we consider the one - step method which provides the numerical approximated solution @xmath198 of @xmath199 at discrete time @xmath200 , @xmath201 . the method is based on the continuous linearization of ( [ semi1 ] ) . more precisely we linearize ( [ semi1 ] ) at each time step as @xmath202dt+p_hb(x^h(t))dw(t ) , \end{aligned}\ ] ] for all @xmath203 , where @xmath204 is the frchet derivative of @xmath152 at @xmath198 and @xmath205 is the remainder at @xmath198 . both @xmath206 and @xmath205 are random functions and are defined for all @xmath207 by @xmath208 before build the new numerical scheme , let us recall the following important lemma . [ lemma5 ] for all @xmath209 and all @xmath210 , the random linear operator @xmath211 is the generator of a strongly continuous semigroup @xmath212 , uniformly bounded on @xmath213 $ ] called random ( or stochastic ) perturbed semigroup . furthermore the following estimate holds @xmath214 where @xmath55 is a positive constant independent of @xmath135 , @xmath215 , @xmath216 , @xmath217 and the sample @xmath218 . the proof follows the same lines as ( * ? ? ? * lemma 3.6 ) or ( * ? ? ? * lemma 4 ) . here our jacobian depends on @xmath207 , the constant @xmath55 is independent of the sample @xmath218 since there exists @xmath51 such that @xmath219 according to assumption [ assumption3 ] . given the solution @xmath199 and the numerical solution @xmath198 at @xmath220 , we obtain from ( [ semi ] ) the following mild representation form of @xmath221 @xmath222 we note that ( [ semi2 ] ) is the exact solution of ( [ semi1 ] ) at @xmath223 . to establish our numerical method we use the following approximations @xmath224 therefore the deterministic integral part of ( [ semi2 ] ) can be approximated as follows @xmath225 inserting ( [ constr1 ] ) in ( [ semi2 ] ) and using the approximation @xmath226 give the following approximation @xmath227 of @xmath221 , called stochastic exponential rosenbrock scheme ( sers ) @xmath228 with @xmath229 . the numerical scheme ( [ erem ] ) can be rewritten in the following equivalent form , which is efficient for implementation @xmath230,\end{aligned}\ ] ] where @xmath231 , @xmath232 are independent , standard , normally distributed random variables with mean @xmath178 and variance @xmath233 , and @xmath234 we note that the operator @xmath235 is uniformly bounded , i.e independently of @xmath135 , @xmath215 and @xmath218 ( see e.g ( * ? ? ? * lemma 2.4 ) ) . note that the corresponding standard stochastic exponential scheme presented in @xcite is given by @xmath236.\end{aligned}\ ] ] this scheme will be called setd1 and will be used in our numerical simulations for comparison with sers scheme . if the deterministic part is also approximated as the diffusion part ( [ napp ] ) , we will obtain the following new scheme @xmath237,\end{aligned}\ ] ] our main result is also valid for scheme ( [ schme0 ] ) and the extension of our proof to that scheme is done as in @xcite without any issue . having the numerical method in hand , our goal is to analyze its strong convergence toward the exact solution in the root - mean - square @xmath0 sense . in the following subsection we state our strong convergence result , which is in fact our main result . throughout this paper we take @xmath238 $ ] , where @xmath239 for @xmath240 , @xmath241 , @xmath242 is fixed , @xmath55 is a generic constant that may change from one place to another . the main result of this paper is formulated in the following theorem . [ mainresult1 ] let @xmath243 and @xmath198 be respectively the mild solution ( [ mild1 ] ) and the numerical approximation given by ( [ erem ] ) at @xmath200 . let assumption [ assumption1 ] , assumption [ assumption2 ] , assumption [ assumption3 ] and assumption [ assumption4 ] be fulfilled . for @xmath244 the following error estimate holds @xmath245 moreover , under a strong regularity of the initial data , that is assumption [ assumption2 ] and assumption [ assumption5 ] are fulfilled with @xmath246 , the following error estimate holds @xmath247 as in ( * ? ? ? * remark 2.9 ) , strong assumptions on the nonlinear functions @xmath6 and @xmath7 can allow to achieve a spatial error of order @xmath248 [ remark2 ] for additive noise , smooth noise with further assumptions on the nonlinear term @xmath6 should improve the time accuracy as in @xcite . [ remark3 ] note that the semi - discrete problem ( [ semi1 ] ) can be replaced by the following semi - discrete problem where the noise is truncated @xmath249dt+p_hb(x^h(t))p_hdw(t ) , t\in[0,t].\end{aligned}\ ] ] it was shown in @xcite that in the case of additive noise with smooth covariance operator kernel , this truncation can be done severely without loosing the spatial accuracy of the finite element method . applying our stochastic exponential rosenbrock scheme to ( [ semi3 ] ) yields @xmath250 we note that theorem [ mainresult1 ] also holds for the numerical scheme ( [ serem2 ] ) . parts of @xcite can be used .

we consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root - mean - square norm . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

in this paper , we consider the numerical approximation of a general second order semilinear stochastic partial differential equation ( spde ) driven by multiplicative and additive noise . our main interest is on such spdes where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations . most numerical techniques , including current stochastic exponential integrators lose their good stability properties on such equations . using finite element for space discretization , we propose a new scheme appropriated on such equations , called stochastic exponential rosenbrock scheme ( sers ) based on local linearization at every time step of the semi - discrete equation obtained after space discretization . we consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root - mean - square norm . numerical experiments to sustain theoretical results are provided . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

1702.02694

i

the _ nariai spacetime _ was discovered by nariai in 1950 ( see the reprints of the original works in @xcite ) . it is the solution of einstein s vacuum equations for spacetime metrics and the sign convention for curvature tensors in @xcite . in this convention the de - sitter spacetime is a solution of eq . with @xmath0 . ] @xmath1 where either as _ any _ positive constant or set @xmath2 . ] @xmath0 is the cosmological constant and @xmath3 is the einstein tensor associated with the metric @xmath4 , with spatial topology @xmath5 and @xmath6 here , @xmath7 is the standard coordinate along the spatial @xmath8-factor and @xmath9 is the metric of the standard round unit two - sphere . the time coordinate is @xmath10 . in the last years , the nariai spacetime has become an object of special interest since ginsparg and perry @xcite found that it can be interpreted as a de - sitter universe containing a black hole of `` maximal size '' . thanks to its geometrical properties , the nariai spacetime has been used to model several situations . one of the most remarkable applications was carried out by bousso and hawking @xcite who used this spacetime to study the quantum pair creation of black holes during inflation . these cosmological models , at the borderline between inflation and gravitational collapse , restricted to spherically symmetric perturbations of the nariai spacetime . it was found that under certain conditions those models asymptotically approach the de sitter universe in agreement with the _ cosmic no - hair conjecture _ @xcite . it states that this behavior is _ generic _ for inhomogeneous and anisotropic expanding solutions of eq . . although there is some mathematical evidence @xcite that supports the validity of this conjecture , the general case still remains unclear . a particular property of the nariai spacetime itself is its peculiar time dependence which is _ not _ consistent with this . while the spatial @xmath8-factor expands exponentially for large @xmath11 , the geometry of the spatial @xmath12-factor remains constant . thus , the expansion of this solution is very anisotropic , and , in fact , is _ inconsistent _ with the cosmic no - hair paradigm . in more geometric terms , this corresponds to the fact , which was proven for the first time in @xcite ( an alternative proof was given in @xcite ) , that the nariai spacetime does not possess even a piece of a smooth conformal boundary . if the cosmic no - hair conjecture really holds , the nariai spacetime must be therefore `` very special '' , and hence in particular be unstable under `` generic '' perturbations . in @xcite one of us has initiated the study of _ homogeneous _ ( but fully nonlinear ) perturbations of the nariai spacetime . the nariai solution is a member of the class of _ kantowski - sachs _ spatially homogeneous ( but anisotropic ) solutions @xcite of einstein s vacuum equation with a positive cosmological constant . the spatial topology of all of these models is @xmath5 . it was found there that the nariai solution is _ critical _ in this family in the following sense . for all kantowski - sachs models , except for the nariai solution , we can choose the time orientation such that the solution either collapses to the future ( a _ big crunch _ characterized by the formation of a curvature singularity and all future inextendible causal curves are incomplete ) or expands eternally to the future in consistency with the cosmic no - hair picture ( existence of a smooth future conformal boundary in consistency with the future asymptotics of de - sitter space so that all future inextendible causal curves are complete ) . the nariai solution is exactly at the borderline between these two extremes as the curvature is bounded everywhere and all inextendible causal curves are both future and past complete , but it nevertheless does not agree with cosmic no - hair . the first rigorous work in @xcite on this topic has recently been extended in @xcite . the numerical studies in @xcite of gowdy - symmetric @xcite ( see section [ sec : formulationeqs ] for more details on gowdy symmetry ) inhomogeneous fully nonlinear perturbations of the nariai solution have revealed evidence that the analogous critical phenomenon also exists in much larger classes of spacetimes . in particular , it was found that all solutions , which are obtained from initial data not too far away from the nariai solutions , always either globally collapse or expand in the same manner as in the spatially homogeneous case with the exception of _ critical solutions _ which are exactly at the borderline between these two cases . in particular , it was found that in contrast to the spherically symmetric models considered by bousso et al . above @xcite , gowdy symmetric models never _ locally _ collapse or expand , and the formation of cosmological black holes in this class was therefore ruled out . because the perturbations considered in @xcite were small in some sense , the question remained open whether this may be different for larger gowdy symmetric perturbations . one of the finding in our paper here now suggests that gowdy symmetric models indeed never form cosmological black holes . for future work , it will be interesting to pose this question again within more general classes of spacetimes , for example @xmath13-symmetric spacetimes , and study whether cosmological black holes may be created by perturbations of the nariai spacetime . before we continue , let us remind the reader about the heuristic idea of the study of the criticality of the cosmological models in @xcite . there we worked with gowdy symmetric initial data ( which satisfy the constraint equation implied by eq . ) given by two real parameters @xmath14 and @xmath15 whose precise definition is irrelevant now ( cf . @xcite for the details ) . the special choice @xmath16 corresponds to nariai initial data while @xmath17 and @xmath18 yields a class of spatially homogeneous models . the larger the value of @xmath19 is , however , the `` more spatially inhomogeneous '' the initial data are . the idea was to fix some non - zero value of the `` inhomogeneity parameter '' @xmath15 and then to study a sequence of ( fully nonlinear ) cosmological models given by a sequence of values of @xmath14 . on the one hand , it was found that if @xmath14 is sufficiently large , the corresponding model expands globally to the future ; in fact , the solution develops a smooth conformal boundary to the future in this case and is hence fully consistent with the cosmic no - hair conjecture . if @xmath14 is sufficiently small on the other hand , the model collapses globally to the future and eventually forms a curvature singularity . at the borderline between these two regimes corresponding to a critical value for @xmath14 , the corresponding model _ neither _ collapses nor expands to the future . however , no further information about such critical models was extracted in @xcite . the purpose of our present paper is manifold . again , we restrict to the class of gowdy - symmetric models with a positive cosmological constant and we revisit the same situation , but tackle it with a more advanced approach . to this end , we use a different class of initial data which now depends on _ three _ parameters @xmath20 ( which has a similar meaning as the `` inhomogeneity parameter '' @xmath15 above ) , @xmath21 ( which has a similar meaning as the parameter @xmath14 above ) and an additional parameter @xmath22 which essentially controls the wave number of the inhomogeneous perturbation ( the initial data in @xcite restricted to the case @xmath23 ) . the details are discussed in section [ sec : familyidn ] . on the one hand , we confirm and strengthen the findings in @xcite by performing a similar numerical analysis . on the other hand , however , we shall focus in great detail on the _ critical solutions _ here and thereby reveal an interesting new oscillatory phenomenon which could potentially be interpreted as gravitational waves . the main finding of our paper are now summarized as three main results . the purpose of this paper is to provide the details and give justifications . in all of what follows we shall assume . the case @xmath24 is a special borderline case which turns out to be not well described by our analytic method discussed in section [ sec : heuristicmode ] . we therefore completely disregard the case @xmath24 in this paper . ] notice that the well - understood @xcite homogeneous case of our models corresponds to @xmath26 . the nariai solution is determined by @xmath27 . one can easily check that if @xmath28 or @xmath29 , the corresponding solution of eq . is not isometric to the nariai solution by comparing the kretschmann scalar with the particular globally constant value for the nariai solution . our first main finding is summarized as follows . [ conj1 ] pick any real value @xmath20 and integer @xmath25 . then there is a constant @xmath30 such that the solution of eq . , determined by initial data given by the parameter @xmath20 , @xmath22 and any real value @xmath21 as in section [ sec : familyidn ] , globally collapses and forms a curvature singularity if @xmath31 and globally expands in consistency with the cosmic no - hair conjecture if @xmath32 . for small values of @xmath20 , this result is in full consistency with the findings in @xcite . here we claim now that this also holds for large values of @xmath20 . as mentioned earlier , this rules out in particular the possibility of `` local '' collapse and hence there are generically indeed only two kinds cosmological models in this class . in this paper here we provide some more refined numerical evidence complemented by a heuristic mode analysis ( see section [ sec : heuristicmode ] ) . we call any of our models _ critical _ if @xmath33 for any given @xmath20 and @xmath22 , and _ almost critical _ or _ close - to - critical _ if @xmath34 , but @xmath35 . the second main finding of our work which significantly goes beyond the results in @xcite is summarized as follows . [ conj2 ] for any non - zero value of @xmath20 , the critical and close - to - critical solutions asserted in result [ conj1 ] are oscillatory . based on the before - mentioned heuristic analysis in section [ sec : heuristicmode ] , we are able to derive formulas for oscillation frequencies , amplitudes and phases and how these depend on the initial data parameters . the only non - oscillatory solutions correspond to the spatially homogeneous case @xmath26 in which the critical solution is known to coincide with the nariai solution ( see @xcite ) and therefore has the peculiar asymptotics discussed above . our numerical work here suggests that _ all _ the critical models , also the inhomogeneous ones , are _ nariai - like _ in the following sense . [ conj3 ] the critical solutions behave asymptotically as follows . while the spatial @xmath8-factor expands exponentially , the spatial @xmath12-factor geometry oscillates around the round unit @xmath36-sphere geometry and is therefore in particular bounded . all these models therefore violate the cosmic no - hair picture by these highly anisotropic asymptotics . we emphasize that our work here is _ not _ actually concerned with the _ instability of the nariai solution _ ; this issue is addressed elsewhere @xcite . the point of our work here is now to identify and describe inhomogeneous critical models and their nariai - like asymptotics _ all _ of which violate the cosmic no - hair paradigm . we remark that all our numerical studies were conducted with slight generalizations of the numerical code presented in @xcite . more details and references regarding our numerical infrastructure are given in section [ sec : evolutionn ] . the paper is organized as follows . in section [ sec : setup ] , we discuss the general setup , i.e. , the formulation of einstein s equation in the presence of symmetries via _ geroch s symmetry reduction _ , our extraction of evolution equations with a well - posed initial value problem and of constraint equations from this , our numerical implementation and our particular family of initial data . section [ sec : criticalnariai ] is devoted to our analytical and numerical studies . first we discuss our heuristic mode analysis which is the basis for all of what follows . then we provide numerical evidence for all of the main results above .

in this paper , we construct and study solutions of einstein s equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the nariai cosmological model . similar to this nariai spacetime , our solutions are at the borderline between gravitational collapse and de - sitter - like exponential expansion . our studies focus in particular on the intriguing oscillatory dynamics which we discover . our investigations are carried out both analytically ( using heuristic mode analysis arguments ) and numerically ( using the numerical infrastructure recently introduced by us ) .

in this paper , we construct and study solutions of einstein s equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the nariai cosmological model . similar to this nariai spacetime , our solutions are at the borderline between gravitational collapse and de - sitter - like exponential expansion . our studies focus in particular on the intriguing oscillatory dynamics which we discover . our investigations are carried out both analytically ( using heuristic mode analysis arguments ) and numerically ( using the numerical infrastructure recently introduced by us ) .

1301.5298

i

optimization appears in many areas of scientific computing , since the solution of a problem can often be described as the minimum of an optimization problem . local methods such as gradient descent are often employed to handle global minimization problems . they can be very efficient to compute a local minimum , but the output depends on the initial guess and they give no guarantee of a global solution . in the case where the function @xmath0 to minimize is a polynomial , it is possible to develop methods which ensure the computation of a global solution . reformulating the problem as the computation of a ( minimal ) critical value of the polynomial @xmath0 , different polynomial system solvers can be used to tackle it ( see e.g. @xcite , @xcite ) . but in this case , the complex solutions of the underlying algebraic system come into play and additional computation efforts should be spent to remove these extraneous solutions . semi - algebraic techniques such as cylindrical algebraic decomposition or extensions @xcite may also be considered here but suffer from similar issues . though the global minimization problem is known to be np - hard ( see e.g. @xcite ) , a practical challenge is to device methods which take into account `` only '' the real solutions of the problem or which can approximate them efficiently . * previous works . * about a decade ago , a relaxation approach has been proposed in @xcite ( see also @xcite , @xcite ) to solve this difficult problem . instead of searching points where the polynomial @xmath0 reaches its minimum @xmath1 , a probability measure which minimizes the function @xmath0 is searched . this problem is relaxed into a hierarchy of finite dimensional convex minimization problems , that can be solved by semi - definite programming ( sdp ) techniques , and which converges to the minimum @xmath1 @xcite . this hierarchy of sdp problems can be formulated in terms of linear matrix inequalities on moment matrices associated to the set of monomials of degree @xmath2 for increasing values of @xmath3 . the dual hierarchy can be described as a sequence of maximization problems over the cone of polynomials which are sums of squares ( sos ) . a feasibility condition is needed to prove that this dual hierarchy of maximization problems also converges to the minimum @xmath1 , ie . that there is no duality gap . from a computational point of view , this approach suffers from two drawbacks : 1 . the hierarchy of optimization problems may not be _ exact _ , ie . it may not always convergence in a finite number of steps ; 2 . the size of the sdp problems to be solved _ grows exponentially _ with @xmath3 . to address the first issue , the following strategy has been considered : add polynomial inequalities or equalities satisfied by the points where the function @xmath0 is minimum . inequality constraints can for instance be added to restrict the optimization problem to a compact subset of @xmath4 and to make the hierarchies exact @xcite , @xcite . natural constraints which do not require apriori bounds on the solutions are for instance the vanishing of the partial derivatives of @xmath0 . a result in @xcite shows that if the gradient ideal ( generated by the first differentials of @xmath0 ) is zero - dimensional , then the hierarchy extended with constraints from the gradient ideal is exact . it is also proved that there is no duality gap if `` good '' generators of the gradient ideal are used . in @xcite , it is proved that the extended hierarchy is exact when the gradient ideal is radical and in @xcite the extended hierarchy is proved to be exact for any polynomial @xmath0 when the minimum is reached in @xmath4 ( see also @xcite ) . in @xcite , the relaxation techniques are analyzed for functions for which the minimum is not reached and which satisfies some special properties `` at infinity '' . from an algorithmic point of view , this is not ending the investigations since a criteria is needed to determine at which step the minimum is reached . it is also important to known if all the minimizers can be recovered at that step . methods based on moment matrices have been proposed to compute generators of the ideal characterizing the real solutions of a polynomial system . in @xcite , the computation of generators of the ( real ) radical of an ideal is based on moment matrices which involve all monomials of a given degree and a stopping criterion related to curto - fialkow flat extension condition @xcite is used . this method is improved in @xcite . combining the border basis algorithm of @xcite and a weaker flat extension condition @xcite , a new algorithm which involves sdp problems of significantly smaller size is proposed to compute the ( real ) radical of an ideal , when this ( real ) radical ideal is zero - dimensional . in @xcite , an algorithm is also proposed to compute the global minimum of the polynomial function @xmath0 based on techniques from @xcite . it terminates when the gradient ideal is radical zero - dimensional . an interesting feature of these hierarchies of sdp problems is that , at any step they provide a lower bound of @xmath1 and the sos hierarchy gives certificates for these lower bounds ( see e.g. @xcite and reference therein ) . in @xcite it is also shown how to obtain `` good '' upper bounds by perturbation techniques , which can directly be generalized to the approach we propose in this paper . * contributions . * we show that if the minimum @xmath1 is reached in @xmath4 , a generalized hierarchy of relaxation problems which involve the gradient ideal @xmath5 is exact and yields the generators of the minimizer ideal @xmath6 from a sufficiently high degree . in the case where the minimizer ideal is zero dimensional , we give a criterion for deciding when the minimum is reached , based on the flat extension condition in @xcite . this criterion is used in a new algorithm which computes a border basis of the minimizer ideal of a polynomial function , when this minimizer ideal is zero - dimensional . the algorithm is an extension of the real radical algorithm described in @xcite . the rows and columns of the matrices involved in the semi - definite programming problem are associated with the family of monomials candidates for being a basis of the quotient space , i.e , a subset of monomials of size much smaller than the number of monomials of the same degree . thus , the size of the sdp problems involved in this computation is significantly smaller than the one in @xcite , @xcite or @xcite . we show that by solving this sequence of sdp problems , we obtain in a finite number of steps the minimum @xmath1 of @xmath0 , with no duality gap . when this minimum is reached , the kernel of the hankel matrix associated to the solution of the sdp problem yields generators of the minimizer ideal which are not in the gradient ideal . assuming that the minimizer ideal @xmath6 is zero dimensional , computing the border basis of this kernel yields a representation of the quotient algebra by @xmath6 and thus a way to compute effectively the minimizer points , using eigenvector solvers . an implementation of this method has been developed , which integrates a border basis implementation and a numerical sdp solver . it is demonstrated on typical examples . * content . * the paper is organized as follows . section 2 recalls the concepts of algebraic tools as ideals , varieties , dual space , quotient algebra , the definitions and theorems about border basis , and the hankel operators involved in the computation of ( real ) radical ideals and in the computation of our minimizer ideal . in section 3 , we describe the main results on the exactness of the hierarchy of sdp problems on truncated hankel operators and show that the minimizer ideal can be computed from the kernel of these hankel . in section 4 , we analyze more precisely the case where the minimizer ideal is zero - dimensional . in section 5 , we describe our algorithm and we prove its correctness . finally in section 6 , we illustrate the algorithm on some classical examples .

in this paper , we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function , assuming that the minimizer ideal is zero - dimensional . the proposed algorithm combines border basis , moment matrices and semidefinite programming . in the case where the minimum is reached at a finite number of points , it provides a border basis of the minimizer ideal

in this paper , we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function , assuming that the minimizer ideal is zero - dimensional . our method is a generalization of lasserre relaxation method and stops in a finite number of steps . the proposed algorithm combines border basis , moment matrices and semidefinite programming . in the case where the minimum is reached at a finite number of points , it provides a border basis of the minimizer ideal

1310.1272

i

the topological he - mckellar - wilkens ( hmw ) phase introduced in 1993 by x.g . he and b.h.j . mckellar @xcite and in 1994 by m. wilkens @xcite was never tested since its theoretical discovery . we have recently published such a test @xcite using our lithium atom interferometer . in a companion paper @xcite quoted here as hmwi , we have recalled the theory of this topological phase and its relations with the aharonov - bohm @xcite and aharonov - casher phases @xcite . we have also discussed the effects of phase dispersion on interferometer signals and we have considered in detail the phase shifts induced by electric and magnetic fields , namely the dynamical phase shifts due to the stark and zeeman hamiltonian and the topological phase shift due to the aharonov - casher effect . the present paper is devoted to a detailed presentation of the experiment , of its results and of the analysis of the stray effects which have complicated the test of the hmw phase . in the following sections , we first describe the experiment , the data recording procedure and the interferometer signals ( section [ expset ] ) . then , we discuss the effects of the electric fields ( section [ elec ] ) and of the magnetic fields ( section [ mag ] ) . section [ data ] presents the data set for the hmw phase measurement and the raw results . the model describing the stray effects due to phase shift dispersion , introduced in hmwi and developed in the appendix ( section [ app ] ) , is tested thanks to numerous and sensitive measurements of the fringe phase and visibility ( section [ test ] ) . thanks to this model , we have been able to reject most of the stray effects and to measure the hmw phase , as detailed in section [ hmwm ] . a conclusion ( section [ conc ] ) summarizes what we have learnt from this experiment , recalls the open questions ( in particular a phase shift presently not understood ) and discusses how to improve this experiment .

we start by describing our setup and we characterize the effects of the electric and magnetic fields needed to observe the hmw effect . then , we develop a model of our interferometer signals including all the defects we have identified . after various tests of this model , we use it to suppress the largest part of the stray phase shifts . * keywords : * atom interferometry ; high phase sensitivity ; stark effect ; zeeman effect ; paschen - back effect ; fringe visibility ; fringe phase shift ; interferometer defects .

in this paper , we describe an experimental test of the he - mckellar - wilkens ( hmw ) topological phase with our lithium atom interferometer . the expected value of the hmw phase shift in our experiment is small and its measurement was difficult because of stray phase shifts due to small experimental defects . we start by describing our setup and we characterize the effects of the electric and magnetic fields needed to observe the hmw effect . then , we develop a model of our interferometer signals including all the defects we have identified . after various tests of this model , we use it to suppress the largest part of the stray phase shifts . we thus obtain a series of measurements of the hmw phase : the results are% larger than expected and this discrepancy is probably due to some limitations of our model . * keywords : * atom interferometry ; high phase sensitivity ; stark effect ; zeeman effect ; paschen - back effect ; fringe visibility ; fringe phase shift ; interferometer defects .

1310.1272

c

we have described a measurement of the he - mckellar - wilkens topological phase by atom interferometry . this experiment was feasible with our atom interferometer because the interferometer arms are well separated in space and the interferometer signal is intense , with a large fringe visibility , near @xmath300% . the arm separation is needed in order to insert a septum between the two interferometer arms without any degradation of the signal . the signal intensity and the large value of the fringe visibility both contribute to enhance the phase sensitivity : its practically achieved value near @xmath301 mrad/@xmath302 is needed for the present measurement . the hmw phase shift is rather small , at most @xmath303 mrad under our experimental conditions , and appears as the combination of four phase measurements for which @xmath304 s of data recording were needed to reduce the uncertainty near @xmath87 mrad . the analysis of the experiment revealed more complex than expected , because of stray phases . the complexity of the signal is due to several factors : * the signal is the sum of the contributions of 8 sublevels which are not exactly in phase because of the zeeman phase shifts due to the slightly different values of the magnetic field on the two interferometer arms ; * we have built a compensator coil to produce an opposite gradient of the magnetic field at another place in the interferometer . the use of this compensator has been very fruitful as it has enabled us to apply substantially higher fields with a limited loss of fringe visibility . however the compensator produces a low field , so that it can correct only the part of the phase shifts due to linear zeeman effect ; * the weights of the various @xmath305 sublevels are functions of the laser frequency and power density in the standing waves used for atom diffraction . we had to control these parameters rather tightly in order to keep these weights almost equal and constant ; * the main phase shifts ( diffraction phase shift , stark and zeeman phase shifts ) present a dispersion with the atomic trajectory described in our calculations by the @xmath51-coordinate . in the presence of several dispersed phase shifts , the visibility of the contribution of a given sublevel to the total fringe signal is better or worse , depending if the dispersions of the different phase shifts subtract or add their effects . we have developed a model taking into account all these effects and this model has been very successful in explaining the variations of the observed phase shifts and visibility with the capacitor voltage @xmath70 and the hmw coil current @xmath8 . however , an extra - phase has been observed and characterized : this phase is odd with the capacitor voltage @xmath70 ; it behaves roughly like @xmath249 ; it appears only when the magnetic field is applied but its value is independent of the magnetic field magnitude in a wide range . we continue our investigations to understand the effect which produces this phase . by combining measured phase shifts with opposite values of the current @xmath8 , we have eliminated this phase and we have obtained a first measurement of the hmw phase shift . the observed effect is larger than its expected value by @xmath290% if we use all the collected data points and only by @xmath289% if we consider only the data points with @xmath306 a. finally , there is a small stray contribution of the aharonov - casher phase to the measured phase shift , and using our model , it was possible to evaluate this contribution and to correct the measured values accordingly . the discrepancy between our corrected measurements and the expected hmw phase shift is then reduced to @xmath0% . it is necessary to improve this experiment in order to reduce the uncertainty on the hmw phase - shift . here are the main possibilities : * reduction of stray effects by a better construction of the hmw interaction region . the present construction has two main defects : the difference of the capacitor thicknesses varies with the @xmath51-coordinate and the septum does not coincide with the symmetry plane of the hmw coils . the construction of capacitors with a better controlled geometry is probably possible but quite difficult , because of the need of using a stretched septum . a better centering of the septum with respect to the hmw coils is probably rather easy and this would reduce substantially the zeeman phase shifts which are the largest source of complication . * reduction of stray effects by optical pumping of the atomic beam . if all the atoms are in one @xmath120 sublevel only , the signal is no more an average on the hyperfine sublevels populations . moreover , the trajectory - averaged zeeman phase shift can be exactly canceled by the compensator if the pumping is done in the @xmath307 , @xmath308 ( or @xmath309 ) sublevel for which zeeman effect is exactly linear . as a consequence , this arrangement , which should reduce most of the stray phase shifts , is feasible with minor modifications of our setup and experiments are in progress . * reduction of stray effects by using another atom : this requires the development of a completely new atom interferometer with separated arms . most of the difficulties are due to the paramagnetic character of lithium and an atom with a @xmath310s@xmath311 non - degenerate ground state ( i.e. with a zero nuclear spin ) would be ideal because there would be no zeeman phase shift and no aharonov - casher phase shift . we may consider either a thermal beam of a light atom or a laser - cooled atomic source . in the case of a thermal beam , the most obvious choice is ground state helium , with which a very nice interferometer using diffraction by material gratings was developed by j.p . toennies and co - workers @xcite . because helium electric polarizability is small ( @xmath312 ) , larger electric and/or magnetic fields will be needed . among atoms which have been laser - cooled , magnesium , calcium , strontium or ytterbium all have a @xmath310s@xmath311 ground state and at least one stable isotope with a nuclear spin equal to @xmath5 . we thank the laboratory technical staff for their help , a. cronin for fruitful discussions , g. trnec , a. miffre and m. jacquey for all the work done on our atom interferometer . we are greatly indebted toward cnrs inp , anr ( grants anr-05-blan-0094 and anr-11-bs04 - 016 - 01 hipati ) and rgion midi - pyrnes for supporting our research .

in this paper , we describe an experimental test of the he - mckellar - wilkens ( hmw ) topological phase with our lithium atom interferometer . we thus obtain a series of measurements of the hmw phase : the results are% larger than expected and this discrepancy is probably due to some limitations of our model .

in this paper , we describe an experimental test of the he - mckellar - wilkens ( hmw ) topological phase with our lithium atom interferometer . the expected value of the hmw phase shift in our experiment is small and its measurement was difficult because of stray phase shifts due to small experimental defects . we start by describing our setup and we characterize the effects of the electric and magnetic fields needed to observe the hmw effect . then , we develop a model of our interferometer signals including all the defects we have identified . after various tests of this model , we use it to suppress the largest part of the stray phase shifts . we thus obtain a series of measurements of the hmw phase : the results are% larger than expected and this discrepancy is probably due to some limitations of our model . * keywords : * atom interferometry ; high phase sensitivity ; stark effect ; zeeman effect ; paschen - back effect ; fringe visibility ; fringe phase shift ; interferometer defects .

astro-ph0211159

i

the one - point probability distribution function ( pdf ) of a random mass density field @xmath3 , or the counts - in - cells ( cic ) statistics of a discrete random distribution such as galaxies , is probably the first statistics used to reveal the clustering feature of galaxy distribution . in the famous work by edwin hubble ( 1934 ) , he showed that the frequency distribution of galaxy count @xmath4 in angular cells is not gaussian , but lognormal . this result indicates that the pdf of galaxy distribution is fundamental in characterizing the cosmic mass and velocity fields . although current samples applied for large - scale structure study have much deeper redshift and much wider angular size than available in hubble s time , the one - point statistics is still frequently applied . this is because the one - point distribution and its moments contain complete information of the field , which might not be easily detected by other conventional methods . unlike the fourier amplitude , cic is not subject to the central limit theorem . it can detect the non - gaussianity of a field consisting of randomly distributed non - gaussian clumps , while the pdf of the fourier amplitudes is still gaussian due to the central limit theorem ( fan and bardeen 1995 ) . even the 2nd moment of one - point distribution is different from the 2nd moment of the fourier decomposition power spectrum . the former contains perturbations on scales larger than the size of the observed sample , while the latter does not . therefore , one - point statistics is applied on various samples of large - scale structures , including galaxy surveys ( e.g. hamilton 1985 ; alimi et al 1990 ; gaztaaga 1992 ; szapudi et al 1996 ; kim & strauss 1998 ) , transmitted flux of quasars ly@xmath5 absorption spectrum ( e.g. meiksin & bouchet 1995 ; gaztanaga & croft 1999 ; zhan & fang 2002 ) , and n - body simulation samples ( e.g. coles & jones 1991 ; taylor & watts 2000 ) . one - point statistics of a density fluctuation field @xmath6 is given by the distribution of the one - point variable @xmath7 , which is a sampling of the field by a window function @xmath8 around position @xmath9 and on scale @xmath10 , i.e. @xmath11 the variable @xmath7 is actually a mean value of the field at position @xmath9 and on scale @xmath10 . the distribution of @xmath7 gives a pdf description of the field . eq.(1 ) is a space(@xmath9)-scale(@xmath10 ) decomposition of the field @xmath6 with bases @xmath12 . most popular windows @xmath8 are gaussian and top - hat filters . the gaussian windows generally are not orthogonal , i.e. the function @xmath8 does not satisfy @xmath13 , if @xmath14 . thus , the variables @xmath7 are either incomplete or redundant . in turn , these may lead to 1 ) loss of information of the field @xmath6 ; and 2 ) false correlation . it is possible to construct an orthogonal and complete set of bases from top - hat windows . however , they are not localized in the fourier space . the index @xmath10 does not refer to a well defined scale @xmath15 . as a consequence , the one - point statistics with conventional windows are not suitable for problems with a scale - dependence . for instance , the redshift distortion on the one - point statistics can be properly estimated only if we know how the redshift distortion depends on the scale and shape of the window function . therefore , a better algorithm for the one - point statistics is needed , for example , to recover the real - space rms density fluctuation @xmath16 from redshift distortion . in this paper , we show that these problems can be solved with a multiresolutional decomposition via the discrete wavelet transform ( dwt ) . the bases of the dwt decomposition are complete , orthogonal , and localized in both physical and fourier spaces . it was shown in the last few years that the dwt can be employed as an alternative representation in most conventional statistics of cosmic mass and velocity fields , including power spectrum ( pando & fang 1998 ; fang & feng 2000 ; yang et al 2001a , 2002 ) , high order correlations ( pando & fang 1998 ; pando , feng & fang 2001 ; feng , pando & fang 2001 ) , bulk and pair - wise velocity ( zhan & fang 2002 ; yang et al . 2001b ) , and identification of halos and clusters ( xu , fang & wu 2000 ) . the dwt representation is also able to reveal statistical features , which might not be easily detected without a set of decomposition bases localized in both physical and scale spaces . with the dwt analysis , the intermittency of the fluctuations of quasars ly@xmath5 transmitted flux has been detected ( jamkhedkar , zhan & fang 2000 ; pando , feng & fang 2000 ; zhan , jamkhedkar & fang 2001 ; pando et al . this property can not be simply detected by the popular @xmath8 or fourier decompositions . we also show that the dwt decomposition is very useful for one - point statistics . we establish the algorithm of the one - point variable and its moments in considering various corrections , such as poisson sampling and selection function . to demonstrate the advantage of the dwt one - point statistics , we show that one can map between the one - point statistics in real and redshift spaces scale - by - scale , taking account the distortion due to bulk velocity , velocity dispersion and selection function . with these results one can construct @xmath1 ( redshift distortion parameter ) estimators with moments of the dwt one - point variables . unlike conventional estimators , the dwt estimators do not need extra - information or _ ad hoc _ assumption when the non - linear redshift distortion is not negligible . this paper is organized as follows . 2 presents the algorithm for the pdf of one - point variables with the dwt space - scale decomposition . 3 discusses the second moment of the one - point variables in considering the effects of poisson sampling and selection function . in 4 we develop the theory of redshift distortion of the dwt one - point variables . 5 tests the theoretical results in 4 on n - body simulation samples . the emphases are the redshift - to - real - space mapping of the diagonal and off - diagonal second moments . 6 shows the application in estimating the redshift distortion parameter . finally , the conclusions and discussions are given in 7 . the appendix provides relevant formulae for quantities defined in 4 . we release the codes for calculating these quantities via anonymous ftp at samuri.la.asu.edu/pub/zhan/dwtcic.tgz .

we also establish the mapping between the dwt one - point statistics in redshift space and real space , i.e. the algorithm for recovering the dwt one - point statistics from the redshift distortion of bulk velocity , velocity dispersion , and selection function . the dwt estimators , however , do not need such extra information .

in this paper , we develop a method of performing the one - point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform ( dwt ) . we establish the algorithm of the one - point variable and its moments in considering the effects of poisson sampling and selection function . we also establish the mapping between the dwt one - point statistics in redshift space and real space , i.e. the algorithm for recovering the dwt one - point statistics from the redshift distortion of bulk velocity , velocity dispersion , and selection function . numerical tests on n - body simulation samples show that this algorithm works well on scales from a few hundreds to a few h mpc for four popular cold dark matter models . taking the advantage that the dwt one - point variable is dependent on both the scale and the shape ( configuration ) of decomposition modes , one can design estimators of the redshift distortion parameter ( ) from combinations of dwt modes . comparing with conventional estimators , such as quadrupole - to - monopole ratio , the dwt estimators are scale - decomposed . it is useful to consider scale - dependent effects . when the non - linear redshift distortion is not negligible , the quadrupole - to - monopole ratio is a function of scale . this estimator would not work without adding information about the scale - dependence , such as the power - spectrum index or the real - space correlation function of the random field . the dwt estimators , however , do not need such extra information . moreover , the scale - decomposed estimators would also be able to reveal the scale - dependence of the bias parameter of galaxies . numerical tests show that the proposed dwt estimators are able to determine robustly with less than 15% uncertainty in the redshift range .

astro-ph0211159

c

we have developed the one - point statistics of a perturbed density field with the multiresolutional decomposition based on discrete wavelet transform . since the scale and shape of the dwt bases are well defined , this frame work is very effective to deal with problems of how the one - point distribution and its moments depend on the scale and shape of the window function . with this property , we have established the algorithm of one - point variable and its moments in considering the effects of poisson sampling and selection function . we have also established the algorithm for recovering the dwt one - point statistics from the redshift distortion due to bulk velocity , velocity dispersion and selection function . because the recovery of the real - space dwt one - point variable and its moments can be realized scale - by - scale , one can design @xmath1 estimators which are sensitive to the scale - dependence of @xmath1 , for instance , caused by the scale - dependence of bias parameter of galaxies . these @xmath1 estimators are effective in avoiding the difficulty caused by the scale - dependence of the non - linear redshift distortion . compared with conventional @xmath1 estimators ( peacock et al 2001 ) , the dwt @xmath1 estimators do not need to assume that the velocity dispersion is scale - independent , or to add extra information , such as the power - spectrum index or the real - space correlation function of the field . numerical tests by n - body simulation samples show that the proposed estimators can yield the correct value of @xmath1 with about 15% uncertainty for all popular cdm models in the redshift range @xmath335 . since dwt decomposition contains two sets of bases , the scaling function @xmath28 and wavelet @xmath29 , a dwt decomposition of a density field @xmath18 actually yields variables for one - point statistics @xmath336 as well as the variables for calculating the power spectrum @xmath337 ( fang & feng 2000 ) . in this sense , one can say that the dwt decomposition unifies the algorithm of cic and power spectrum , which are only two aspects of the statistics with the dwt variables sfcs and wfcs respectively . for a finite size sample , these two aspects of statistics can play different roles : the former contains information of the perturbations on scales larger than the size , while the latter does not . however , the latter generally has diagonalized covariance , while the former does not . therefore , the @xmath1 estimators with wfcs are better than sfcs , while sfcs are useful to estimate the effect of perturbations on large scales . actually , the dwt decomposition can provide more types of statistics variables . for 2-d or 3-d samples , we have variables defined as @xmath338 @xmath339 obviously , these variables are sampled partially by the scaling function @xmath28 , and partially by the wavelet @xmath29 . statistics with these variables are not typical cic , or power spectrum . they are , however , useful to study the one - point statistics , or power spectrum of 2-d and 3-d samples . with the method developed in this paper , it is not difficult to calculate various corrections ( poisson sampling , selection function , redshift distortion ) on the one - point statistics with variables ( 86 ) or ( 87 ) . hz is grateful to daniel eisenstein for extensive discussions on this paper and facilitating the lcdm2 simulation . hz would also like to thank david burstein for hosting the utility codes in this paper .

in this paper , we develop a method of performing the one - point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform ( dwt ) . we establish the algorithm of the one - point variable and its moments in considering the effects of poisson sampling and selection function . numerical tests on n - body simulation samples show that this algorithm works well on scales from a few hundreds to a few h mpc for four popular cold dark matter models . taking the advantage that the dwt one - point variable is dependent on both the scale and the shape ( configuration ) of decomposition modes , one can design estimators of the redshift distortion parameter ( ) from combinations of dwt modes . comparing with conventional estimators , such as quadrupole - to - monopole ratio , the dwt estimators are scale - decomposed . this estimator would not work without adding information about the scale - dependence , such as the power - spectrum index or the real - space correlation function of the random field . numerical tests show that the proposed dwt estimators are able to determine robustly with less than 15% uncertainty in the redshift range .

in this paper , we develop a method of performing the one - point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform ( dwt ) . we establish the algorithm of the one - point variable and its moments in considering the effects of poisson sampling and selection function . we also establish the mapping between the dwt one - point statistics in redshift space and real space , i.e. the algorithm for recovering the dwt one - point statistics from the redshift distortion of bulk velocity , velocity dispersion , and selection function . numerical tests on n - body simulation samples show that this algorithm works well on scales from a few hundreds to a few h mpc for four popular cold dark matter models . taking the advantage that the dwt one - point variable is dependent on both the scale and the shape ( configuration ) of decomposition modes , one can design estimators of the redshift distortion parameter ( ) from combinations of dwt modes . comparing with conventional estimators , such as quadrupole - to - monopole ratio , the dwt estimators are scale - decomposed . it is useful to consider scale - dependent effects . when the non - linear redshift distortion is not negligible , the quadrupole - to - monopole ratio is a function of scale . this estimator would not work without adding information about the scale - dependence , such as the power - spectrum index or the real - space correlation function of the random field . the dwt estimators , however , do not need such extra information . moreover , the scale - decomposed estimators would also be able to reveal the scale - dependence of the bias parameter of galaxies . numerical tests show that the proposed dwt estimators are able to determine robustly with less than 15% uncertainty in the redshift range .

1208.6419

i

star clusters are considered as important tracers for understanding the formation and evolution of their host galaxies @xcite . star cluster systems have been traditionally separated into two populations globular clusters and open clusters ( gcs and ocs)on their ages , masses , metallicities , and positions . however , more recent studies have discovered that the distinction between gcs and ocs becomes increasingly blurred ( see * ? ? ? * for details ) . @xcite listed photometric colors and magnitudes for star clusters in magellanic clouds ( mcs ) and the fornax dwarf system and divided them into two groups . they found that star clusters in blue group have central condensation properties similar to those of the red group , which were considered as gcs , however they could not be identified with the galactic ocs . @xcite termed 23 clusters in the large magellanic cloud ( lmc)differing from gcs in their relative youth and ocs in their richness and shape as `` young populous clusters '' , which were called `` young massive clusters '' ( ymcs ) or `` blue luminous compact clusters '' ( blccs ) by @xcite . actually , the blue integrated colors for a cluster may be influenced by several factors , such as poor metallicity ( the luminosity of the horizontal branch ) , young age ( the position of the main - sequence turnoff stars ) , and some exotic stellar populations ( e.g. , blue stragglers , wolf - rayet stars ) . however , several studies ( e.g. , * ? ? ? * ; * ? ? ? * ) have reached similar conclusions that the exceedingly blue colors of blccs are a direct consequence of their young ages ( see * ? ? ? * for details ) . m31 is the largest galaxy in the local group , and has a large number of star clusters , including young clusters having been studied by many authors . @xcite listed 11 objects in m31 classified as blue clusters using the uv colors , most of which have been proved to be young clusters @xcite , except for b133 and b145 , which were stated as a star and an old gc @xcite , respectively . @xcite derived ages and masses for a large sample of young clusters , and found that these star clusters are less than 2 gyr old , and most of them have ages between @xmath10 and @xmath11 yr and masses ranging from @xmath12 to @xmath13 . these authors also stated that the young star clusters in m31 show a range of structures , most of which have low concentrations typical of ocs in the milky way ( mw ) , however , there are a few with high concentrations similar to the mw gcs . @xcite carried out a survey of compact star clusters in the southwest part of m31 , and suggested a rich intermediate - mass star cluster population in m31 , with a typical age range of 30 myr @xmath14 3 gyr , peaking at @xmath6 70 myr . in order to ascertain the properties of the blccs , @xcite performed an image survey for 20 blccs lying in the disk of m31 using the wide field and planetary camera-2 ( wfpc2 ) on the _ hubble space telescope _ ( _ hst _ ) . in addition , another key aim of this _ hst _ survey was to determine the fraction of contamination of blccs by asterisms , since @xcite suggested that a large fraction of the putative blccs may in fact be just asterisms . @xcite presented the resulting @xmath15 images of six very young or young star clusters in m31 observed with the keck laser guide star adaptive optics system , and indicated that the four youngest out of these six objects are asterisms . however , @xcite presented a conclusion that these four objects are true clusters based on spectra . the _ hst _ images @xcite showed that nineteen of the twenty surveyed candidates are real star clusters , and one ( nb67 ) is a bright star . @xcite measured surface brightness profiles for 23 bright young star clusters using images from the wfpc2 , including the sample clusters of @xcite , and derived the structural properties by fitting the surface brightness profiles to several structural models . the authors stated that the sample young clusters are expected to dissolve within a few gyr and will not survive to become old gcs , and that young star clusters in m31 and mcs follow the same fundamental plane relations as old gcs of m31 , mcs , the mw and ngc 5128 , regardless of their host galaxy environments . @xcite presented a m31 stellar cluster catalog utilizing the panchromatic hubble andromeda treasury survey data , which will cover @xmath16 of m31 disk with multiple filters and allow the identification of thousands of star clusters . the large population of young star clusters reflect a high efficiency of cluster formation , possibly triggered by a current interaction event between m31 and its satellite galaxy @xcite , suggesting that young star clusters should be associated with the star - forming ( sf ) regions of m31 . @xcite found that young clusters ( @xmath17 2 gyr ) are spatially coincident with m31 s disk , including the 10 kpc ring and the outer ring @xcite . although these authors also found the young star clusters in the halo of m31 , all of the clusters outside of the optical disk of m31 are old , globular clusters ( see * ? ? ? * for details ) . @xcite stated that most of young star clusters kinematics have the thin , rotating disk component ( see also * ? ? ? * ) . the young star clusters distribution has a distinct peak around @xmath18 kpc from the center in m31 disk , and some young star clusters show concentration around the 10 kpc ring splitting regions near m32 and most of them have systematically younger ages ( @xmath19 myr ) . @xcite also stated that the young star clusters show a spatial distribution similar to ob stars , uv sf regions , and dust , all of which are important tracers of disk structures . several criteria were developed for selecting young clusters from the integrated spectrum and colors . @xcite comprehensively studied the properties of 67 very blue and likely ymcs in m31 selected according to their color @xmath20 $ ] and/or the strength of @xmath21 spectral index ( @xmath22 ) . @xcite presented a catalog of m31 gcs based on images from the sdss and the wide field camera on the united kingdom infrared telescope and selected a population of young clusters with a definition of @xmath23 $ ] . @xcite published a catalog of m31 young clusters ( @xmath24 gyr ) and supported the selection criteria @xmath25 $ ] and @xmath26 $ ] @xcite . these criterions may play important roles in distinguishing young from old clusters for those whose ages can not be derived accurately . the formation and disruption of young star clusters represent a latter - day example of the hierarchical formation of galaxies @xcite . motivated by that , we decided to describe some basic properties of young star clusters in m31 , such as positions , distributions of ages and masses , correlations of the ages and masses with structure parameters , which may provide important information about the processes involved in their formation and disruption . in this paper , we will provide photometry of a set of young star clusters in m31 using images obtained with the beijing arizona taiwan connecticut ( batc ) multicolor sky survey telescope . by comparing the observed seds with the galev simple stellar population ( ssp ) models , we derive their ages and masses . this paper is organized as follows . in section 2 we present the batc observations of the sample clusters , the relevant data - processing steps , and the _ galex _ ( fuv and nuv ) , optical broad - band , sdss @xmath1 and 2mass nir data that are subsequently used in our analysis . in section 3 we derive ages and masses of the sample clusters . a discussion on the sample young clusters ( @xmath17 2 gyr ) will be given in section 4 . finally , we will summarize our results in section 5 .

in this paper , we present photometry for young star clusters in m31 , which are selected from caldwell et al . these star clusters have been observed as part of the beijing arizona taiwan connecticut ( batc ) multicolor sky survey from 1995 february to 2008 march .

in this paper , we present photometry for young star clusters in m31 , which are selected from caldwell et al . these star clusters have been observed as part of the beijing arizona taiwan connecticut ( batc ) multicolor sky survey from 1995 february to 2008 march . the batc images including these star clusters are taken with 15 intermediate - band filters covering 300010000 . combined with photometry in the _ galex _ far- and near - ultraviolet , broad - band , sdss , and infrared of two micron all sky survey , we obtain their accurate spectral energy distributions ( seds ) from . we derive these star clusters ages and masses by comparing their seds with stellar population synthesis models . our results are in good agreement with previous determinations . the mean value of age and mass of young clusters ( gyr ) is about 385 myr and , respectively . there are two distinct peaks in the age distribution , a highest peak at age 60 myr and a secondary peak around 250 myr , while the mass distribution shows a single peak around . a few young star clusters have two - body relaxation times greater than their ages , indicating that those clusters have not been well dynamically relaxed and therefore have not established the thermal equilibrium . there are several regions showing aggregations of young star clusters around the 10 kpc ring and the outer ring , indicating that the distribution of the young star clusters is well correlated with m31 s star - forming regions . the young massive star clusters ( age myr and mass ) show apparent concentration around the ring splitting region , suggesting a recent passage of a satellite galaxy ( m32 ) through m31 disk .